Patent Publication Number: US-2022222567-A1

Title: Photonic Quantum Networking for Large Superconducting Qubit Modules

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application No. 63/090,966 filed on Oct. 13, 2020, and entitled “Photonic Quantum Networking for Large Superconducting Qubit Modules,” and U.S. Provisional Application No. 63/128,536 filed on Dec. 21, 2020, and entitled “Photonic Quantum Networking for Large Superconducting Qubit Modules.” The above-referenced priority applications are hereby incorporated by reference. 
    
    
     GOVERNMENT SUPPORT 
     This invention was made with government support under Contract No. FA8750-20-P-1716 awarded by the Air Force Research Laboratory. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     The following description relates to quantum processing units with logical qubit hardware modules. 
     Quantum computers can perform computational tasks by storing and processing information within quantum states of quantum systems. For example, qubits (i.e., quantum bits) can be stored in, and represented by, an effective two-level sub-manifold of a quantum coherent physical system. A variety of physical systems have been proposed for quantum computing applications. Examples include superconducting circuits, trapped ions, spin systems, and others. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of an example computing environment. 
         FIG. 2  is a schematic diagram of an example quantum computing network. 
         FIG. 3  is a schematic diagram showing aspects of an example quantum processing unit. 
         FIG. 4  is a table showing parameters of an example transducer device. 
         FIG. 5  is a schematic diagram showing aspects of an example quantum computing network. 
         FIG. 6  is a flow chart showing aspects of an example process. 
         FIG. 7A  is a table showing quantum entanglement generation rates generated by operation of two example types of transducer devices. 
         FIG. 7B  is a plot of simulated transmission in decibels (dB) as a function of detuning in Giga Hertz (GHz) of an example pump filter. 
         FIG. 8  is a table showing a raw link fidelity of a generated quantum entanglement using the example quantum computing network  500  shown in  FIG. 5 . 
         FIG. 9A  is a schematic diagram showing an example quantum logic circuit for quantum entanglement distillation. 
         FIG. 9B  is a schematic diagram showing an example quantum logic circuit for quantum entanglement distillation. 
         FIG. 10A  is a block diagram showing aspects of an example quantum computing network. 
         FIGS. 10B-10C  are schematic diagrams showing aspects of example quantum logic circuits for performing a ZZZZ parity measurement and a XXXX parity measurement with a remote Bell pair of entangled qubits. 
         FIG. 11  is a schematic diagram showing aspects of an example quantum logic circuit for quantum entanglement distillation. 
         FIG. 12  is a log-log plot  1200  of a total number of links (e.g., a multiplexing factor m) as a function of raw entangling probability (p s ) of a single link at various numbers (k) of raw Bell pairs of entangled qubits. 
     
    
    
     DETAILED DESCRIPTION 
     In some aspects of what is described here, quantum computing systems based on superconducting quantum processing units are networked across a photonic communication system (e.g., an optical intranet) to form a quantum computing network for distributed quantum computation and long-distance quantum communication. In some implementations, quantum computing systems, for example in quantum data centers, are communicably coupled through an optical intranet in an end-to-end architecture. Each of the quantum computing systems includes one or more quantum processing units. In some instances, each of the quantum processing units is a modular quantum processing unit including multiple quantum processor modules. In this case, quantum processor modules in different quantum computing systems can be communicably coupled via the optical intranet. 
     In some implementations, each of the quantum processing units includes multiple qubit devices and possibly other quantum circuit devices. By networking multiple qubit devices, for example, in a quantum processor module, a logical qubit can be defined. When the multiple qubit devices are networked in an error correcting pattern, the logical qubit defined by the multiple qubit devices can be sustained by an error correction mechanism. In some implementations, the qubit devices include superconducting qubit devices and other solid-state qubit devices, e.g., spin qubit devices. In some implementations, the qubit devices can be used for performing quantum operations, for performing quantum entanglement distillation, for storing quantum entanglement, (e.g., by logical qubits), or other operations. 
     In some implementations, each of the quantum processing units includes a direct electro-optic transducer device based on a thin-film lithium niobate resonator device. During the creation of remote quantum entanglement between two qubit devices in two respective quantum processing units, two transducer devices in the two respective quantum processing units receive respective optical excitations, and generate respective microwave modes and optical modes. The microwave modes are transmitted within the respective quantum processing units to, and captured by, the two respective qubit devices. In response to a successful detection of the optical modes that are transmitted out of the respective quantum processing units and within the photonic communication system, quantum entanglement transmitted by the microwave modes to the two respective qubit devices, e.g., a remote Bell pair of entangled states, can be identified. In some implementations, a remote Bell pair of entangled states is stored and distilled in local error corrected logical qubits for subsequent processing. 
     In some implementations, the systems and techniques described here can provide technical advantages and improvements. For example, the transduction scheme is mechanically and thermally stable (e.g. does not rely on freestanding structures), broadband (for strong electro-optic coefficients), scalable, and tunable (e.g. using bias voltages). The methods and systems presented here can improve entanglement generation rates and enable operations at increased bandwidth and reduced noise. For example, the methods and systems presented can link superconducting qubit devices operating at a microwave frequency regime (e.g., a few GHz frequencies) in superconducting quantum processing units with a photonic quantum computing network operating at an optical frequency regime (e.g., at hundreds of THz frequencies) without introducing noise. The link created by the methods and systems presented here is quantum-coherent. For another example, the methods and systems presented here can generate remote pairs of entangled states with high fidelity. The methods and systems presented allow for large-scale superconducting quantum processing units to communicate over a photonic communication network without perfect memories. In some cases, a combination of these and potentially other advantages and improvements may be obtained. 
       FIG. 1  is a block diagram of an example computing environment  100 . The example computing environment  100  shown in  FIG. 1  includes a computing system  101  and user devices  110 A,  110 B,  110 C. A computing environment may include additional or different features, and the components of a computing environment may operate as described with respect to  FIG. 1  or in another manner. 
     The example computing system  101  includes classical and quantum computing resources and exposes their functionality to the user devices  110 A,  110 B,  110 C (referred to collectively as “user devices  110 ”). The computing system  101  shown in  FIG. 1  includes one or more servers  108 , quantum computing systems  103 A,  103 B, a local network  109 , and other resources  107 . The computing system  101  may also include one or more user devices (e.g., the user device  110 A) as well as other features and components. A computing system may include additional or different features, and the components of a computing system may operate as described with respect to  FIG. 1  or in another manner. 
     The example computing system  101  can provide services to the user devices  110 , for example, as a cloud-based or remote-accessed computer system, as a distributed computing resource, as a supercomputer or another type of high-performance computing resource, or in another manner. The computing system  101  or the user devices  110  may also have access to one or more other quantum computing systems (e.g., quantum computing resources that are accessible through the wide area network  115 , the local network  109 , or otherwise). 
     The user devices  110  shown in  FIG. 1  may include one or more classical processors, memory, user interfaces, communication interfaces, and other components. For instance, the user devices  110  may be implemented as laptop computers, desktop computers, smartphones, tablets, or other types of computer devices. In the example shown in  FIG. 1 , to access computing resources of the computing system  101 , the user devices  110  send information (e.g., programs, instructions, commands, requests, input data, etc.) to the servers  108 ; and in response, the user devices  110  receive information (e.g., application data, output data, prompts, alerts, notifications, results, etc.) from the servers  108 . The user devices  110  may access services of the computing system  101  in another manner, and the computing system  101  may expose computing resources in another manner. 
     In the example shown in  FIG. 1 , the local user device  110 A operates in a local environment with the servers  108  and other elements of the computing system  101 . For instance, the user device  110 A may be co-located with (e.g., located within 0.5 to 1 km of) the servers  108  and possibly other elements of the computing system  101 . As shown in  FIG. 1 , the user device  110 A communicates with the servers  108  through a local data connection. 
     The local data connection in  FIG. 1  is provided by the local network  109 . For example, some or all of the servers  108 , the user device  110 A, the quantum computing systems  103 A,  103 B, and the other resources  107  may communicate with each other through the local network  109 . In some implementations, the local network  109  operates as a communication channel that provides one or more low-latency communication pathways from the server  108  to the quantum computing systems  103 A,  103 B (or to one or more of the elements of the quantum computing systems  103 A,  103 B). The local network  109  can be implemented, for instance, as a wired or wireless Local Area Network, an Ethernet connection, or another type of wired or wireless connection. The local network  109  may include one or more wired or wireless routers, wireless access points (WAPs), wireless mesh nodes, switches, high-speed cables, or a combination of these and other types of local network hardware elements. In some cases, the local network  109  includes a software-defined network that provides communication among virtual resources, for example, among an array of virtual machines operating on the server  108  and possibly elsewhere. 
     In the example shown in  FIG. 1 , the remote user devices  110 B,  110 C operate remote from the servers  108  and other elements of the computing system  101 . For instance, the user devices  110 B,  110 C may be located at a remote distance (e.g., more than 1 km, 10 km, 100 km, 1,000 km, 10,000 km, or farther) from the servers  108  and possibly other elements of the computing system  101 . As shown in  FIG. 1 , each of the user devices  110 B,  110 C communicates with the servers  108  through a remote data connection. 
     The remote data connection in  FIG. 1  is provided by a wide area network  115 , which may include, for example, the Internet or another type of wide area communication network. In some cases, remote user devices use another type of remote data connection (e.g., satellite-based connections, a cellular network, a virtual private network, etc.) to access the servers  108 . The wide area network  115  may include one or more internet servers, firewalls, service hubs, base stations, or a combination of these and other types of remote networking elements. Generally, the computing environment  100  can be accessible to any number of remote user devices. 
     The example servers  108  shown in  FIG. 1  can manage interaction with the user devices  110  and utilization of the quantum and classical computing resources in the computing system  101 . For example, based on information from the user devices  110 , the servers  108  may delegate computational tasks to the quantum computing systems  103 A,  103 B and the other resources  107 ; the servers  108  can then send information to the user devices  110  based on output data from the computational tasks performed by the quantum computing systems  103 A,  103 B, and the other resources  107 . 
     As shown in  FIG. 1 , the servers  108  are classical computing resources that include classical processors  111  and memory  112 . The servers  108  may also include one or more communication interfaces that allow the servers to communicate via the local network  109 , the wide area network  115 , and possibly other channels. In some implementations, the servers  108  may include a host server, an application server, a virtual server, or a combination of these and other types of servers. The servers  108  may include additional or different features, and may operate as described with respect to  FIG. 1  or in another manner. 
     The classical processors  111  can include various kinds of apparatus, devices, and machines for processing data, including, by way of example, a microprocessor, a central processing unit (CPU), a graphics processing unit (GPU), an FPGA (field programmable gate array), an ASIC (application specific integrated circuit), or combinations of these. The memory  112  can include, for example, a random-access memory (RAM), a storage device (e.g., a writable read-only memory (ROM) or others), a hard disk, or another type of storage medium. The memory  112  can include various forms of volatile or non-volatile memory, media, and memory devices, etc. 
     Each of the example quantum computing systems  103 A,  103 B operates as a quantum computing resource in the computing system  101 . The other resources  107  may include additional quantum computing resources (e.g., quantum computing systems, quantum simulators, or both) as well as classical (non-quantum) computing resources such as, for example, digital microprocessors, specialized co-processor units (e.g., graphics processing units (GPUs), cryptographic co-processors, etc.), special purpose logic circuitry (e.g., field programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), etc.), systems-on-chips (SoCs), etc., or combinations of these and other types of computing modules. 
     In some implementations, the servers  108  generate programs, identify appropriate computing resources (e.g., a QPU or QVM) in the computing system  101  to execute the programs, and send the programs to the identified resources for execution. For example, the servers  108  may send programs to the quantum computing system  103 A, the quantum computing system  103 B, or any of the other resources  107 . The programs may include classical programs, quantum programs, hybrid classical/quantum programs, and may include any type of function, code, data, instruction set, etc. 
     In some instances, programs can be formatted as source code that can be rendered in human-readable form (e.g., as text) and can be compiled, for example, by a compiler running on the servers  108 , on the quantum computing systems  103 , or elsewhere. In some instances, programs can be formatted as compiled code, such as, for example, binary code (e.g., machine-level instructions) that can be executed directly by a computing resource. Each program may include instructions corresponding to computational tasks that, when performed by an appropriate computing resource, generate output data based on input data. For example, a program can include instructions formatted for a quantum computer system, a simulator, a digital microprocessor, co-processor or other classical data processing apparatus, or another type of computing resource. 
     In some cases, a program may be expressed in a hardware-independent format. For example, quantum machine instructions may be provided in a quantum instruction language such as Quil, described in the publication “A Practical Quantum Instruction Set Architecture,” arXiv:1608.03355v2, dated Feb. 17, 2017, or another quantum instruction language. For instance, the quantum machine instructions may be written in a format that can be executed by a broad range of quantum processing units or simulators. In some cases, a program may be expressed in high-level terms of quantum logic gates or quantum algorithms, in lower-level terms of fundamental qubit rotations and controlled rotations, or in another form. In some cases, a program may be expressed in terms of control signals (e.g., pulse sequences, delays, etc.) and parameters for the control signals (e.g., frequencies, phases, durations, channels, etc.). In some cases, a program may be expressed in another form or format. In some cases, a program may utilize Quil-T, described in the publication “Gain deeper control of Rigetti quantum processing units with Quil-T,” available at https://medium.com/rigetti/gain-deeper-control-of-rigetti-quantum-processors-with-quil-t-ea8945061e5b dated Dec. 10, 2020, which is hereby incorporated by reference in the present disclosure. 
     In some implementations, the servers  108  include one or more compilers that convert programs between formats. For example, the servers  108  may include a compiler that converts hardware-independent instructions to binary programs for execution by the quantum computing systems  103 A,  103 B. In some cases, a compiler can compile a program to a format that targets a specific quantum resource in the computer system  101 . For example, a compiler may generate a different binary program (e.g., from the same source code) depending on whether the program is to be executed by the quantum computing system  103 A or the quantum computing system  103 B. 
     In some cases, a compiler generates a partial binary program that can be updated, for example, based on specific parameters. For instance, if a quantum program is to be executed iteratively on a quantum computing system with varying parameters on each iteration, the compiler may generate the binary program in a format that can be updated with specific parameter values at runtime (e.g., based on feedback from a prior iteration, or otherwise); the parametric update can be performed without further compilation. In some cases, a compiler generates a full binary program that does not need to be updated or otherwise modified for execution. 
     In some implementations, the servers  108  generate a schedule for executing programs, allocate computing resources in the computing system  101  according to the schedule, and delegate the programs to the allocated computing resources. The servers  108  can receive, from each computing resource, output data from the execution of each program. Based on the output data, the servers  108  may generate additional programs that are then added to the schedule, output data that is provided back to a user device  110 , or perform another type of action. 
     In some implementations, all or part of the computing environment operates as a cloud-based quantum computing (QC) environment, and the servers  108  operate as a host system for the cloud-based QC environment. The cloud-based QC environment may include software elements that operate on both the user devices  110  and the computer system  101  and interact with each other over the wide area network  115 . For example, the cloud-based QC environment may provide a remote user interface, for example, through a browser or another type of application on the user devices  110 . The remote user interface may include, for example, a graphical user interface or another type of user interface that obtains input provided by a user of the cloud-based QC environment. In some cases the remote user interface includes, or has access to, one or more application programming interfaces (APIs), command line interfaces, graphical user interfaces, or other elements that expose the services of the computer system  101  to the user devices  110 . 
     In some cases, the cloud-based QC environment may be deployed in a “serverless” computing architecture. For instance, the cloud-based QC environment may provide on-demand access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, services, quantum computing resources, classical computing resources, etc.) that can be provisioned for requests from user devices  110 . Moreover, the cloud-based computing systems  101  may include or utilize other types of computing resources, such as, for example, edge computing, fog computing, etc. 
     In an example implementation of a cloud-based QC environment, the servers  108  may operate as a cloud provider that dynamically manages the allocation and provisioning of physical computing resources (e.g., GPUs, CPUs, QPUs, etc.). Accordingly, the servers  108  may provide services by defining virtualized resources for each user account. For instance, the virtualized resources may be formatted as virtual machine images, virtual machines, containers, or virtualized resources that can be provisioned for a user account and configured by a user. In some cases, servers  108  include a container management and execution system that is implemented, for example, using KUBERNETES® or another software platform for container management. In some cases, the cloud-based QC environment is implemented using a resource such as, for example, OPENSTACK®. OPENSTACK® is an example of a software platform for cloud-based computing, which can be used to provide virtual servers and other virtual computing resources for users. 
     In some cases, the server  108  stores quantum machine images (QMI) for each user account. A quantum machine image may operate as a virtual computing resource for users of the cloud-based QC environment. For example, a QMI can provide a virtualized development and execution environment to develop and run programs (e.g., quantum programs or hybrid classical/quantum programs). When a QMI operates on the server  108 , the QMI may engage either of the quantum processing units  102 A,  102 B, and interact with a remote user device ( 110 B or  110 C) to provide a user programming environment. The QMI may operate in close physical proximity to, and have a low-latency communication link with, the quantum computing systems  103 A,  103 B. In some implementations, remote user devices connect with QMIs operating on the servers  108  through secure shell (SSH) or other protocols over the wide area network  115 . 
     In some implementations, all or part of the computing system  101  operates as a hybrid computing environment. For example, quantum programs can be formatted as hybrid classical/quantum programs that include instructions for execution by one or more quantum computing resources and instructions for execution by one or more classical resources. The servers  108  can allocate quantum and classical computing resources in the hybrid computing environment, and delegate programs to the allocated computing resources for execution. The quantum computing resources in the hybrid environment may include, for example, one or more quantum processing units (QPUs), one or more quantum virtual machines (QVMs), one or more quantum simulators, or possibly other types of quantum resources. The classical computing resources in the hybrid environment may include, for example, one or more digital microprocessors, one or more specialized co-processor units (e.g., graphics processing units (GPUs), cryptographic co-processors, etc.), special purpose logic circuitry (e.g., field programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), etc.), systems-on-chips (SoCs), or other types of computing modules. 
     In some cases, the servers  108  can select the type of computing resource (e.g., quantum or classical) to execute an individual program, or part of a program, in the computing system  101 . For example, the servers  108  may select a particular quantum processing unit (QPU) or other computing resource based on availability of the resource, speed of the resource, information or state capacity of the resource, a performance metric (e.g., process fidelity) of the resource, or based on a combination of these and other factors. In some cases, the servers  108  can perform load balancing, resource testing and calibration, and other types of operations to improve or optimize computing performance. 
     Each of the example quantum computing systems  103 A,  103 B shown in  FIG. 1  can perform quantum computational tasks by executing quantum machine instructions (e.g., a binary program compiled for the quantum computing system). In some implementations, a quantum computing system can perform quantum computation by storing and manipulating information within quantum states of a composite quantum system. For example, qubits (i.e., quantum bits) can be stored in, and represented by, an effective two-level sub-manifold of a quantum coherent physical system. In some instances, quantum logic can be executed in a manner that allows large-scale entanglement within the quantum system. Control signals can manipulate the quantum states of individual qubits and the joint states of multiple qubits. In some instances, information can be read out from the composite quantum system by measuring the quantum states of the qubits. In some implementations, the quantum states of the qubits are read out by measuring the transmitted or reflected signal from auxiliary quantum devices that are coupled to individual qubits. 
     In some implementations, a quantum computing system can operate using gate-based models for quantum computing. For example, the qubits can be initialized in an initial state, and a quantum logic circuit comprised of a series of quantum logic gates can be applied to transform the qubits and extract measurements representing the output of the quantum computation. Individual qubits may be controlled by single-qubit quantum logic gates, and pairs of qubits may be controlled by two-qubit quantum logic gates (e.g., entangling gates that are capable of generating entanglement between the pair of qubits). In some implementations, a quantum computing system can operate using adiabatic or annealing models for quantum computing. For instance, the qubits can be initialized in an initial state, and the controlling Hamiltonian can be transformed adiabatically by adjusting control parameters to another state that can be measured to obtain an output of the quantum computation. 
     In some models, fault-tolerance can be achieved by applying a set of high-fidelity control and measurement operations to the qubits. For example, quantum error correcting schemes can be deployed to achieve fault-tolerant quantum computation. Other computational regimes may be used; for example, quantum computing systems may operate in non-fault-tolerant regimes. In some implementations, a quantum computing system is constructed and operated according to a scalable quantum computing architecture. For example, in some cases, the architecture can be scaled to a large number of qubits to achieve large-scale general purpose coherent quantum computing. Other architectures may be used; for example, quantum computing systems may operate in small-scale or non-scalable architectures. 
     The example quantum computing system  103 A shown in  FIG. 1  includes a quantum processing unit  102 A and a control system  105 A, which controls the operation of the quantum processing unit  102 A. Similarly, the example quantum computing system  103 B includes a quantum processing unit  102 B and a control system  105 B, which controls the operation of a quantum processing unit  102 B. A quantum computing system may include additional or different features, and the components of a quantum computing system may operate as described with respect to  FIG. 1  or in another manner. 
     In some instances, all or part of the quantum processing unit  102 A functions as a quantum processing unit, a quantum memory, or another type of subsystem. In some examples, the quantum processing unit  102 A includes a quantum circuit system. The quantum circuit system may include qubit devices, readout devices, and possibly other devices that are used to store and process quantum information. In some cases, the quantum processing unit  102 A includes a superconducting circuit, and the superconducting circuit includes qubit devices operatively coupled to each other by coupler devices. In certain examples, the qubit devices and the coupler devices are implemented as superconducting quantum circuit devices that include Josephson junctions, for example, in Superconducting QUantum Interference Device (SQUID) loops or other arrangements, and are controlled by radio-frequency signals, microwave signals, and bias signals delivered to the quantum processing unit  102 A. 
     In some implementations, quantum processing units  102  in distributed quantum computing systems  103  are communicably coupled together using a photonic communication network to form a quantum computing network. In this case, remote quantum entanglement can be created between qubit devices from the different quantum processing units  102  in the distributed quantum computing systems  103 . In some implementations, each of the quantum processing units  102  includes a transducer device and a superconducting quantum circuit (e.g., qubit devices and other quantum circuit devices). In some implementations, each of the transducer devices is configured to receive an optical excitation from an external source (e.g., the pump laser system  504  of the global controller system  508  in  FIG. 8 ) via the photonic communication network (e.g., the optical fibers that carry the first and second optical excitations  534 A,  534 B) and to generate optical and microwave modes that contains entangled optical-microwave photon pairs. In some implementations, the optical modes are transmitted out of the respective quantum processing units  102 , e.g., from the transducer devices, for example to the global controller system  508  of  FIG. 5 , via the photonic communication network; and the microwave modes are transmitted within the respective quantum processing units  102  from the transducer devices, e.g., to the qubit devices in the superconducting quantum circuits. In this case, a remote pair of entangled microwave states can be created, transmitted by the microwave modes to the qubit devices, and applied on qubits defined by the qubit devices for further processing (e.g., quantum entanglement distillation or other types of quantum operations). In some instances, a quantum processing unit  102  may be implemented as the quantum processing unit  210 ,  300 ,  502  in  FIGS. 2, 3, and 5 , or in another manner. 
     The quantum processing unit  102 A may include, or may be deployed within, a controlled environment. The controlled environment can be provided, for example, by shielding equipment, cryogenic equipment, and other types of environmental control systems. In some examples, the components in the quantum processing unit  102 A operate in a cryogenic temperature regime and are subject to very low electromagnetic and thermal noise. For example, magnetic shielding can be used to shield the system components from stray magnetic fields, optical shielding can be used to shield the system components from optical noise, thermal shielding and cryogenic equipment can be used to maintain the system components at controlled temperature, etc. 
     In some implementations, the example quantum processing unit  102 A can process quantum information by applying control signals to the qubits in the quantum processing unit  102 A. The control signals can be configured to encode information in the qubits, to process the information by performing quantum logic gates or other types of operations, or to extract information from the qubits. In some examples, the operations can be expressed as single-qubit quantum logic gates, two-qubit quantum logic gates, or other types of quantum logic gates that operate on one or more qubits. A quantum logic circuit, which includes a sequence of quantum logic operations, can be applied to the qubits to perform a quantum algorithm. The quantum algorithm may correspond to a computational task, a hardware test, a quantum error correction procedure, a quantum state distillation procedure, or a combination of these and other types of operations. 
     The example control system  105 A includes controllers  106 A and signal hardware  104 A. Similarly, control system  105 B includes controllers  106 B and signal hardware  104 B. All or part of the control systems  105 A,  105 B can operate in a room-temperature environment or another type of environment, which may be located near the respective quantum processing units  102 A,  102 B. In some cases, the control systems  105 A,  105 B include classical computers, signaling equipment (microwave, radio, optical, bias, etc.), electronic systems, vacuum control systems, refrigerant control systems, or other types of control systems that support operation of the quantum processing units  102 A,  102 B. 
     The control systems  105 A,  105 B may be implemented as distinct systems that operate independent of each other. In some cases, the control systems  105 A,  105 B may include one or more shared elements; for example, the control systems  105 A,  105 B may operate as a single control system that operates both quantum processing units  102 A,  102 B. Moreover, a single quantum computing system may include multiple quantum processing units, which may operate in the same controlled (e.g., cryogenic) environment or in separate environments. 
     The example signal hardware  104 A includes components that communicate with the quantum processing unit  102 A. The signal hardware  104 A may include, for example, waveform generators, amplifiers, digitizers, high-frequency sources, DC sources, AC sources, etc. The signal hardware may include additional or different features and components. In the example shown, components of the signal hardware  104 A are adapted to interact with the quantum processing unit  102 A. For example, the signal hardware  104 A can be configured to operate in a particular frequency range, configured to generate and process signals in a particular format, or the hardware may be adapted in another manner. 
     In some instances, one or more components of the signal hardware  104 A generate control signals, for example, based on control information from the controllers  106 A. The control signals can be delivered to the quantum processing unit  102 A during operation of the quantum computing system  103 A. For instance, the signal hardware  104 A may generate signals to implement quantum logic operations, readout operations, or other types of operations. As an example, the signal hardware  104 A may include arbitrary waveform generators (AWGs) that generate electromagnetic waveforms (e.g., microwave or radio-frequency) or laser systems that generate optical waveforms. The waveforms or other types of signals generated by the signal hardware  104 A can be delivered to devices in the quantum processing unit  102 A to operate qubit devices, readout devices, bias devices, coupler devices, or other types of components in the quantum processing unit  102 A. 
     In some instances, the signal hardware  104 A receives and processes signals from the quantum processing unit  102 A. The received signals can be generated by the execution of a quantum program on the quantum computing system  103 A. For instance, the signal hardware  104 A may receive signals from the devices in the quantum processing unit  102 A in response to readout or other operations performed by the quantum processing unit  102 A. Signals received from the quantum processing unit  102 A can be mixed, digitized, filtered, or otherwise processed by the signal hardware  104 A to extract information, and the information extracted can be provided to the controllers  106 A or handled in another manner. In some examples, the signal hardware  104 A may include a digitizer that digitizes electromagnetic waveforms (e.g., microwave or radio-frequency) or optical signals, and a digitized waveform can be delivered to the controllers  106 A or to other signal hardware components. In some instances, the controllers  106 A process the information from the signal hardware  104 A and provide feedback to the signal hardware  104 A; based on the feedback, the signal hardware  104 A can in turn generate new control signals that are delivered to the quantum processing unit  102 A. 
     In some implementations, the signal hardware  104 A includes signal delivery hardware that interfaces with the quantum processing unit  102 A. For example, the signal hardware  104 A may include filters, attenuators, directional couplers, multiplexers, diplexers, bias components, signal channels, isolators, amplifiers, power dividers, and other types of components. In some instances, the signal delivery hardware performs preprocessing, signal conditioning, or other operations to the control signals to be delivered to the quantum processing unit  102 A. In some instances, signal delivery hardware performs preprocessing, signal conditioning, or other operations on readout signals received from the quantum processing unit  102 A. 
     The example controllers  106 A communicate with the signal hardware  104 A to control operation of the quantum computing system  103 A. The controllers  106 A may include classical computing hardware that directly interface with components of the signal hardware  104 A. The example controllers  106 A may include classical processors, memory, clocks, digital circuitry, analog circuitry, and other types of systems or subsystems. The classical processors may include one or more single- or multi-core microprocessors, digital electronic controllers, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit), or other types of data processing apparatus. The memory may include any type of volatile or non-volatile memory or another type of computer storage medium. The controllers  106 A may also include one or more communication interfaces that allow the controllers  106 A to communicate via the local network  109  and possibly other channels. The controllers  106 A may include additional or different features and components. 
     In some implementations, the controllers  106 A include memory or other components that store quantum state information, for example, based on qubit readout operations performed by the quantum computing system  103 A. For instance, the states of one or more qubits in the quantum processing unit  102 A can be measured by qubit readout operations, and the measured state information can be stored in a cache or other type of memory system in one or more of the controllers  106 A. In some cases, the measured state information is subsequently used in the execution of a quantum program, a quantum error correction procedure, a quantum processing unit (QPU) calibration or testing procedure, or another type of quantum process. 
     In some implementations, the controllers  106 A include memory or other components that store a quantum program containing quantum machine instructions for execution by the quantum computing system  103 A. In some instances, the controllers  106 A can interpret the quantum machine instructions and perform hardware-specific control operations according to the quantum machine instructions. For example, the controllers  106 A may cause the signal hardware  104 A to generate control signals that are delivered to the quantum processing unit  102 A to execute the quantum machine instructions. 
     In some instances, the controllers  106 A extract qubit state information from qubit readout signals, for example, to identify the quantum states of qubits in the quantum processing unit  102 A or for other purposes. For example, the controllers may receive the qubit readout signals (e.g., in the form of analog waveforms) from the signal hardware  104 A, digitize the qubit readout signals, and extract qubit state information from the digitized signals. In some cases, the controllers  106 A compute measurement statistics based on qubit state information from multiple shots of a quantum program. For example, each shot may produce a bitstring representing qubit state measurements for a single execution of the quantum program, and a collection of bitstrings from multiple shots may be analyzed to compute quantum state probabilities. 
     In some implementations, the controllers  106 A include one or more clocks that control the timing of operations. For example, operations performed by the controllers  106 A may be scheduled for execution over a series of clock cycles, and clock signals from one or more clocks can be used to control the relative timing of each operation or groups of operations. In some implementations, the controllers  106 A may include classical computer resources that perform some or all of the operations of the servers  108  described above. For example, the controllers  106 A may operate a compiler to generate binary programs (e.g., full or partial binary programs) from source code; the controllers  106 A may include an optimizer that performs classical computational tasks of a hybrid classical/quantum program; the controllers  106 A may update binary programs (e.g., at runtime) to include new parameters based on an output of the optimizer, etc. 
     The other quantum computing system  103 B and its components (e.g., the quantum processing unit  102 B, the signal hardware  104 B, and controllers  106 B) can be implemented as described above with respect to the quantum computing system  103 A; in some cases, the quantum computing system  103 B and its components may be implemented or may operate in another manner. 
     In some implementations, the quantum computing systems  103 A,  103 B are disparate systems that provide distinct modalities of quantum computation. For example, the computer system  101  may include both an adiabatic quantum computing system and a gate-based quantum computer system. As another example, the computer system  101  may include a superconducting circuit-based quantum computing system and an ion trap-based quantum computer system. In such cases, the computer system  101  may utilize each quantum computing system according to the type of quantum program that is being executed, according to availability or capacity, or based on other considerations. 
       FIG. 2  is a block diagram showing aspects of an example quantum computing network  200 . The example quantum computing network  200  for distributed quantum computing is a photonic quantum computing network, which uses optical elements for creating quantum entanglement between remote quantum computing systems  202 . In some instances, the quantum entanglement created among qubit devices in remote quantum computing systems is heralded by “which-path” entanglement. The quantum computing network  200  based on “which-path” entanglement can be implemented as the example quantum computing network  500  described in  FIG. 5 . As shown in  FIG. 2 , the example quantum computing network  200  includes a main input/output (I/O) interface  204 , a global controller system  206 , and multiple quantum computing systems  202 . In some implementations, the example quantum computing network  200  may include additional and different features or components and components of the example quantum computing network  200  may be implemented in another manner. 
     In some implementations, the main I/O interface  204  is configured to arrange the distributed quantum computing. In some instances, the main I/O interface  204  can be used as a portal for receiving instructions, e.g., from the user device  110 , to be computed within the quantum computing network  200  and returning results, e.g., to the user device  110 . In some implementations, classical data flows between the main I/O interface  204  and the global controller system  206 . 
     As shown in  FIG. 2 , the global controller system  206  is communicably coupled with the quantum computing systems  202  forming the quantum computing network  200 . In some implementations, the global controller system  206  is configured to perform synchronization and orchestration within the quantum computing network  200 . In some implementations, the global controller system  206  includes photonic sources, switches, photodetectors, and other optical elements to generate quantum entanglement between qubit devices in remote quantum computing systems  202 . In some instances, the global controller system  206  can link local supporting computations, such as error correction and runtime algorithmic execution, for instance conventional algorithms, quantum algorithms, or hybrid quantum-classical algorithms. 
     As shown in  FIG. 2 , each of the quantum computing systems  202  includes a quantum processing unit  210 , local controllers  214 , and local I/O interface  216 . A quantum processing unit  210  is configured to process quantum information requested by the local controllers  214 , including executing quantum algorithms, quantum error correction, and performing quantum entanglement distillation, or other types of quantum operations. In some instances, the quantum processing unit  210  in a quantum computing system  202  can be configured to perform hybrid quantum-classical algorithms or other types of quantum algorithms. In some instances, a quantum computing system  202  may include multiple quantum processing units  210 . 
     In some implementations, the quantum processing unit  210  of a quantum computing system  202  includes a quantum circuit  218 . In some implementations, the quantum processing unit  210  is a superconducting quantum processing unit and the quantum circuit  218  includes superconducting quantum circuit devices connected by superconducting circuitry. In some instances, the quantum circuit  218  of a quantum processing unit  210  includes qubit devices (e.g., fixed-frequency qubit devices and tunable-frequency qubit devices), or other superconducting quantum circuit devices. In some examples, each of the qubit devices in the quantum circuit  218  of a quantum processing unit  210  can be encoded with a single bit of quantum information. 
     In some implementations, each of the qubit devices in the quantum circuit  218  of a quantum processing unit  210  has two eigenstates that are used as computational basis states (e.g., |0  and |1 ), and each qubit device can transition between its computational basis states or exist in an arbitrary superposition of its computational basis states. In some examples, the two lowest energy levels (e.g., the ground state and first excited state) of each qubit device are defined as a qubit and used as computational basis states for quantum computation. In some examples, higher energy levels (e.g., a second excited state or a third excited state) can be used to define a qubit, a qutrit, or a multi-level quantum computational device in some instances. Quantum states (e.g., qubits) defined by respective qubit devices in a single quantum processing unit  210  can be manipulated by control signals, or read by readout signals, generated by the local controllers  214 . In some instances, the control signals are transmitted on control signal lines as part of the quantum circuit  218  of the quantum processing unit  210 . The qubit devices in a quantum processing unit  210  can be controlled individually, for example, by delivering control signals from the local controllers  214  to the respective qubit devices in the quantum processing unit  210 . In some cases, readout devices can detect the states of the qubit devices, for example, by interacting directly with the respective qubit devices. 
     In some implementations, the quantum processing unit  210  is a modular quantum processing unit which may include multiple quantum processor modules. In certain instances, each of the multiple quantum processor modules includes a subset of the qubit devices in the quantum processing unit  210 , connections among the qubit devices, and potentially other hardware features that define an appropriate lattice for one or more quantum error correction codes to be applied. 
     In some instances, quantum processor modules in a quantum processing unit  210  can be interconnected to each other. Couplings between quantum processor modules can be used to apply multi-qubit quantum logic gates or other types of operations to qubits in distinct quantum processor modules within the same quantum processing unit  210 . Quantum logic gates can be used to provide quantum entanglement between qubits in distinct quantum processor modules of the same quantum processing unit  210 . The quantum processor modules can be arranged in the quantum processing unit  210  as an array in a two-dimensional or three-dimensional lattice structure. In some implementations, the quantum processing unit  210  may include additional and different features or components, and components of the example quantum processing unit  210  may be implemented in another manner. In certain instances, the connections between quantum processor modules can be configured by the global controller system  206  and may vary during the execution of a quantum algorithm. 
     In some implementations, to operate planar surface codes across the quantum processing unit  210 , only boundary qubit devices on quantum processor modules in the same quantum computing system  202  or remote quantum computing systems  202  need to be communicably coupled to one another (e.g., in the example quantum computing network  1000  shown in  FIG. 10A ). Higher-dimensional codes may require more connectivity between the quantum processing units  210  in a quantum computing system  202 . 
     Although each individual qubit device defines a single qubit, a lattice of qubit devices in the quantum processing unit  210  or qubit devices in a quantum processor module may operate collectively as a single logical qubit. For example, a stabilizer code or another type of quantum error correction scheme can be applied to the lattice of qubit devices. In some cases, one of the qubit devices operates as a data qubit device, other qubit devices in the lattice operate as ancilla qubit devices, and a quantum error correction scheme is applied to the lattice. The ancilla qubit devices may be used to detect an error syndrome, which can be used to correct errors on the data qubit device. Examples of stabilizer codes include surface codes, color codes, and other types of quantum error correcting codes. 
     In some instances, the quantum circuit  218  may include flux bias control lines which can provide magnetic flux locally to tunable-frequency qubit devices to tune their frequencies. The quantum circuit  218  may include tunable coupler devices, microwave feedlines, and resonator devices which are capacitively coupled to qubit devices to readout qubits. In some examples, the quantum circuit  218  may include microwave feedlines which are coupled to one or several of the resonator devices to allow microwave excitation of the resonator devices used to readout qubits. In this case, the quantum circuit  218  may include microwave drive lines which are capacitively coupled to qubit devices to drive qubits. The quantum circuit  218  may further include filters, isolators, circulators, amplifiers, or other circuit elements. 
     As shown in  FIG. 2 , each of the quantum computing systems  202  includes a local I/O interface  216  which supports interfacing instructions to be computed within the quantum processing unit  210 . In some implementations, the local controllers  214  are configured to perform orchestration of the execution of instructions on the associated hardware/software within the quantum computing system  202  in coordination with the global controller system  206 , for example, coherent transmission of quantum optical states between qubit devices in remote quantum computing systems  202 , e.g., between the local controllers  214  of the remote quantum computing systems  202 , through the global controller system  206  of the quantum computing network  200 . 
     As shown in  FIG. 2 , a quantum processing unit  210  further includes transducer devices  212 . In some implementations, each of the transducer devices  212  is configured to receive an optical excitation from the global controller system  206  through the local controllers  214 , and to generate optical and microwave modes that contains entangled optical-microwave photon pairs. In some implementations, the microwave mode is transmitted within the respective quantum processing unit  210 , e.g., from the transducer devices  212  to a qubit device in the quantum circuit  218 ; and the optical mode is transmitted out of the quantum processing unit  210 , e.g., from the transducer devices  212  to the global controller system  206  via the local controllers  214 . In some instances, a photon can be generated by a qubit device in the quantum circuit  218  and transmitted back to the transducer device  212  to generate heralded entanglement. One example process is described in the publication entitled “Optically-Heralded Entanglement of Superconducting Systems in Quantum Networks” by S. Krastanov, et al. (arXiv:2012.13408v3 [quantu-ph] Dec. 24, 2020). 
     In some implementations, a transducer device  212  in a quantum processing unit  210  includes an electro-optic transducer device. In some instances, the transducer device  212  may include an electro-optomechanical transducer device, an opto-magnetic transducer device, a transducer device based on trapped atoms, or other types of transducer devices. In some implementations, each of the transducer devices  212  is a direct transducer device which directly converts a signal in one form of energy to a signal in another. In some implementations, a transducer device  212  is configured to convert a pump laser signal, e.g., an optical excitation, generated by a pump laser system of the global controller system  206 , to a microwave mode and an optical mode. The microwave mode includes photons in a microwave frequency range, e.g., microwave photons; and the optical mode includes photons in an optical frequency range, e.g., optical photons, which is distinct from the frequency of optical photons in the pump laser signal. 
     In some implementations, a transducer device  212  in a quantum processing unit  210  includes a photonic circuit with optical circuit elements and a superconducting microwave circuit with microwave circuit elements. For example, a transducer device  212  may include optical waveguides, optical ring resonators, optical couplers, modulators, switches, filters, and other optical circuit elements. A transducer device  212  may include microwave transmission lines, microwave resonators, filters, and other microwave circuit elements. In some implementations, a transducer device  212  is implemented as the transducer devices  304 ,  522 A,  522 B in  FIGS. 3 and 5 , or in another manner. In some implementations, the transducer device  212  includes a lithium niobate thin film material. One example transducer device is described in the publication entitled “Cavity Electro-optics in Thin-film Lithium Niobate for Efficient Microwave-to-optical Transduction” by J. Holzgrafe, et al. (arXiv: 2005.00939V2 [quantu-ph] May 2, 2020). 
     In some aspects of what is described here, the transducer devices  212  and the quantum circuit  218  of the quantum processing unit  210  are communicably coupled to the local controllers  214 . In some instances, the local controllers  214  deliver control signals to the transducer devices  212  and the quantum circuit devices of the quantum circuit  218 . In certain instances, the local controllers  214  can also receive readout signals from the qubit devices of the quantum circuit  218 . In certain instances, the local controllers  214  include connector hardware elements which include signal lines, signal processing hardware, filters, feedthrough devices (e.g., light-tight feedthroughs, etc.), and other types of components. In some implementations, the connector hardware elements of the local controllers  214  can span multiple different temperature and noise regimes. For example, the connector hardware elements can include a series of thermal stages operating at different temperatures, e.g., 60 Kelvin (K), 3 K, 800 milli Kelvin (mK), 150 mK, that decrease between a higher temperature regime of the global controller system  206  and local controllers  214  and a lower temperature regime of the quantum processing unit  210 . In some instances, components of the local controllers  214  can operate in a room temperature regime, an intermediate temperature regime, or both. For example, the local controllers  214  can be configured to operate at much higher temperatures and be subject to much higher levels of noise than are present in the environment of the quantum processing unit  210 . 
     In some implementations, the quantum processing unit  210  including the transducer devices  212  and the quantum circuit  218 , and part of the local controllers  214  can be maintained in a controlled cryogenic environment, (e.g., cooled using liquid helium). One or more electrically conductive layers (or at least a portion) in the quantum circuit  218  can operate as a superconducting layer at that temperature. The environment can be provided, for example, by shielding equipment, cryogenic equipment, and other types of environmental control systems. In some examples, the components in the quantum processing unit  210  operate in a cryogenic temperature regime and are subject to very low electromagnetic and thermal noise. For example, magnetic shielding can be used to shield the system components from stray magnetic fields, optical shielding can be used to shield the system components from optical noise, and thermal shielding and cryogenic equipment can be used to maintain the system components at controlled temperatures, etc. 
     In some instances, information is encoded in the qubit devices of the quantum circuit  218 , and the information can be processed by operation of the qubit devices. For instance, input information can be encoded in the computational states or computational subspaces defined by some or all of the qubit devices of the quantum circuit  218 . The input information can be processed, for example, by applying a quantum algorithm or other operations. 
     In some aspects of operation, the local controllers  214  send control signals to the qubit devices of the quantum circuit  218 . The control signals can be configured to manipulate the qubits defined by the qubit devices. In some implementations, a control signal can be a direct current (DC) signal communicated from the local controller  214  to the individual qubit device. In some implementations, a control signal can be an alternating current (AC) signal communicated from the local controllers  214  to the individual qubit device. In some cases, the AC signal may be superposed with a direct current (DC) signal. Other types of control signals may be used. In some instances, the local controller  214  identifies a quantum logic gate to be applied to qubit devices and possibly other quantum circuit devices in respective quantum processing unit  210 . The local controller  214  can perform the quantum logic gate operations by communicating the control signals to a control line that is coupled to the qubit device in a quantum processing unit  210 . In certain instances, the local controllers  214  shown in  FIG. 2  may include, for example, a signal generator system, a program interface, a signal processing system, and possibly other system components. 
     In some aspects of operation, the local controllers  214  can also send control signals to the transducer devices  212 , optical circuit elements, microwave circuit elements on the quantum processing unit  210 . For example, the local controllers  214  can transmit control signals to the transducer device  212  to tune resonant frequencies of optical ring resonators (e.g., the optical resonators  326 A,  326 B of the transducer device  304  in  FIG. 3 ). For another example, the local controllers  214  may apply control signals to the pump filters for filtering the pump laser signal, e.g., the pump filter  306  in  FIG. 3  and pump filter  712  in  FIG. 7B . 
       FIG. 3  is a schematic diagram showing aspects of an example quantum processing unit  300 . The example quantum processing unit  300  includes a superconducting quantum circuit  320  and a transducer device  304 . The superconducting quantum circuit  320  includes qubit devices  302 , and can further include other quantum circuit devices, for example, coupler devices (e.g., capacitive coupler device, tunable coupler device, or others), readout devices, or other types of quantum circuit devices that are used for quantum information processing in the quantum processing unit  300 . The superconducting quantum circuit  320  may include one or more Josephson junctions, capacitors, inductors, and other types of circuit elements. As shown in  FIG. 3 , the superconducting quantum circuit  320  further includes microwave transmission lines  312 , a Purcell filter  314 , and an intermediate resonator  316 . The example quantum processing unit  300  further includes a pump filter  306 . The quantum processing unit  300  can be implemented as the quantum computing systems  210 ,  502 A,  502 B in the quantum computing networks  200  and  500  as shown in  FIGS. 2 and 5 , or in another manner. In some implementations, the example quantum processing unit  300  may include additional and different features or components and components of the example quantum processing unit  300  may be implemented in another manner. 
     As shown in  FIG. 3 , the transducer device  304  receives an optical excitation. For example, the optical excitation can be an incident pump laser signal from a pump laser source, e.g., in the global controller system  206  in  FIG. 2 . The optical excitation includes optical photons in a first optical frequency range. In some instances, the optical excitation is transmitted to the transducer device  304  via optical fibers. In some instances, an optical coupler may be used to enable communication between the optical fibers that carry the optical excitation and the transducer device  304 . An optical coupler can be further optically coupled to a transducer device  304  through optical waveguides. In certain instances, an optical coupler may include various structures with high coupling efficiency and low reflection losses. For example, an optical coupler may include multiple gratings, and can be fabricated in one or more dielectric layers on the same substrate as the transducer device  304 . 
     In some implementations, the transducer device  304  is a direct electro-optical transducer device, which is configured to receive the optical excitation and convert the received optical excitation to a microwave mode and an optical mode. In other words, an optical photon in the first optical frequency range can be converted, by operation of the transducer device  304 , to a pair of correlated and entangled microwave and optical photons. The generated optical photons in the optical mode are in a second optical frequency range that is different from the first optical frequency range of the optical excitation. 
     As shown in  FIG. 3 , the transducer device  304  includes first and second optical ring resonators  326 A,  326 B which are evanescently coupled; and the transducer device  304  further includes a superconducting microwave resonator  332  which includes an inductor element and a capacitor element. Each of the optical ring resonators  326 A,  326 B includes a lithium niobate triplet resonator device. The superconducting microwave resonator  332  is configured to modulate the received optical excitation. Parametrically coupled optical ring resonators  326 A,  326 B and microwave resonator  332  are used to generate photon pairs. Examples of photon pair generation in parametrically coupled resonators are described in the publication entitled “Non-classical Correlations between Single Photons and Phonons from a Mechanical Oscillator” by Riedinger, et al. (arXiv:1512.05360v2 [quant-ph], Dec. 16, 2015), and in the publication entitled “Cryogenic microwave-to-optical conversion using a triply resonant lithium-niobate-on-sapphire transducer” by T. P. McKenna et al. (arXiv:2005.00897v1 [quant-ph] May 2, 2020) 
     In some implementations, the microwave mode is transmitted within the quantum processing unit  300 . Particularly, the microwave mode, generated by operation of the transducer device  304 , is transmitted from the superconducting microwave resonator  332  to a qubit device  302  of the superconducting quantum circuit  320  through a first output port  308 A. In some implementations, the qubit device  302  may be implemented as a transmon device, a fluxonium device, or another type of superconducting qubit device. In some examples, the qubit frequency of a qubit device  302  is not tunable by application of an offset field and is independent of magnetic flux experienced by the qubit device. For instance, a fixed-frequency qubit device may have a fixed qubit frequency that is defined by an electronic circuit of the qubit device. As an example, a superconducting fixed-frequency qubit device (e.g., a fixed-frequency transmon qubit device) may be implemented without a SQUID (Superconducting Quantum Interface Device) loop. 
     In some examples, the qubit frequency of a qubit device  302  is tunable, for example, by application of an offset field. For instance, a superconducting tunable-frequency qubit device may include a superconducting loop (e.g., a SQUID loop), which can receive a magnetic flux that tunes the qubit frequency of a tunable-frequency qubit device. The transition frequency is also known as “resonant frequency” or “fundamental frequency”, which is defined by the energy difference between the first and second excited states of the qubit divided by Planck&#39;s constant (e.g., according to ω=E/h). The transition frequency also defines the operating frequency of the example tunable-frequency qubit device. A tunable-frequency qubit device may be implemented as a tunable-frequency transmon qubit device or another type of tunable-frequency qubit device. For example, a tunable-frequency qubit device may include two Josephson junctions connected in parallel with each other to form a SQUID loop, which resides adjacent to a control signal line (e.g., a flux-bias control line). The tunable-frequency qubit device further includes a shunt capacitor connected with the two Josephson junctions in parallel. 
     In this case, the superconducting quantum circuit  320  may include flux-bias control lines for tuning the magnetic flux through the SQUID loop of the qubit device  302 . Manipulating the magnetic flux through the SQUID loop, can increase or decrease the operating frequency of the example tunable-frequency qubit device. In some instances, the operating frequency may be tuned in another manner, for instance, by another type of control signal. In some implementations, the flux modulation signal can be applied to a flux bias element to obtain a modulated magnetic flux applied to the SQUID loop. The modulated magnetic flux applied to the SQUID loop can cause a modulation to the transition frequency of the tunable-frequency qubit device. 
     In this case, a tunable-frequency qubit device enables parametric control to coherently absorb microwave radiation from the transmission line plus the resonator circuit with tunable coupling strengths, allowing for pulse shaping the absorption process, for instance with a time-reversed profile relative to the incoming signal from the transducer device  314 . In other words, tuning the coupling strength between elements or the frequencies of resonant modes can be used to achieve suitable shaping of the capture process. One example parametric control on a tunable-frequency qubit device is described in the publication entitled “Parametrically Activated Entangling Gates using Transmon Qubits” by S. Caldwell et al., (arXiv:1706.06562v2 [quantu-ph] Dec. 8, 2017). One example process for catching shaped microwave photons is escribed in the publication entitled “Catching Shaped Microwave Photons with 99.4% Absorption Efficiency”, by Wenner et al. (arXiv:1311.1180v2 [quant-ph] Nov. 16, 2013). 
     As shown in  FIG. 3 , the microwave photons in the microwave mode are transmitted to the qubit devices  302  of the superconducting quantum circuit  320  via a microwave transmission line  312 . In some instances, a microwave transmission line  312  may be implemented as, for example coplanar waveguides, substrate integrated waveguides, or another type of planar transmission line. As shown in  FIG. 3 , the microwave photons are further transmitted through the Purcell filter  314  and the intermediate resonator  316 . In some implementations, the Purcell filter  314  includes one or more coupled linear resonators and is configured to filter microwave photons in a microwave mode generated by the transducer device  304 . This filter allows for rapid collection of the incoming microwave photons during an active state without damping the capture circuit during a storage state. One example Purcell filter is described in the publication entitled “Computational modeling of decay and hybridization in superconducting circuits” by M. G. Scheer and M. B. Block (arXiv:1810.11510v3 [quantu-ph] Mar. 21, 2019). In some examples, the intermediate resonator  316  can be made to have the same frequency as the operating frequencies of the qubit device  302  and the superconducting microwave resonator  332  of the transducer device  304 . In some instances, frequency tuning can be achieved by tuning fabrication parameters during the manufacturing process or by applying control signals on the qubit device (e.g., flux-bias control signals on a tunable-frequency qubit devices). One example intermediate resonator is described in the publication entitled “Microwave Quantum Link between Superconducting Circuits Housed in Spatially Separated Cryogenic Systems” by P. Magnard et al. (arXiv:2008.01642v1 [quantu-ph] Aug. 4, 2020). In some implementations, the intermediate resonator  316  includes one or more Josephson junctions and is configured to maintain coherence of the microwave photons that are transmitted to the qubit device  302 . As shown in  FIG. 3 , the qubit device  302  is capacitively coupled to the intermediate resonator  316  through a first capacitor  326 . The intermediate resonator  316  is further capacitively coupled to the Purcell filter  314  through a second capacitor  328 . The Purcell filter  314  includes one or more optional ports  318  which are configured to allow for control of the qubit device  302 . For example, the optional port  318  can be configured to perform quantum logic gate operations, readout operations, or other types of quantum operations. 
     In some implementations, an optical output including the optical mode together with unconverted optical excitation is transmitted out of the example quantum processing unit  300  through a second output port  308 B. In some instances, to remove the unconverted optical excitation, the optical output can be filtered by the pump filter  306 . As a result, the example quantum processing unit  300  outputs the optical mode generated by the transducer device  304 . In some instances, the optical mode from the quantum processing unit  300  can be transmitted back to the global controller system  206  as shown in  FIG. 2  or other components. In some instances, the pump filter  306  includes a sequence of serial-coupled or parallel-coupled optical ring resonators. As shown in  FIG. 3 , the pump filter  306  includes a pair of serial-coupled optical ring resonators. In certain instances, the pump filters  316  may include a sequence of optical ring resonators configured in different manner (e.g., the example pump filter  712  shown in  FIG. 7B ), or other components. 
     In some instances, the quantum processing unit  300  includes multiple superconducting quantum circuits  320 , each of which includes a qubit device  302  and corresponding control circuitry, e.g., a Purcell filter  314 , an intermediate resonator  316 , and other quantum circuit devices. In this case, each of the qubit devices  302  with the corresponding control circuitry can be, communicably coupled to one or more transducer devices  304  through distinct microwave transmission lines  312  using frequency multiplexing or spatial multiplexing. For example, the quantum processing unit  300  includes a demultiplexer which is configured to select a particular qubit device  302  in a particular superconducting quantum circuit  320  on the quantum processing unit  300  for receiving the microwave mode from a particular transducer device  304 . 
     In this case, multiple qubit devices  302  in the quantum processing unit  300  are arranged in a rectilinear (e.g., rectangular, or square) array that extends in two spatial dimensions (e.g., in the plane of the page), or in another type of ordered array. In some instances, the rectilinear array also extends in a third spatial dimension (e.g., in/out of the page), for example, to form a cubic array or another type of three-dimensional array. In some instances, the example quantum processing unit  300  is a modular quantum processing unit that includes multiple quantum processor modules. Each of the quantum processor modules includes a subset of the qubit devices in the quantum processing unit  300 . In certain examples, qubit devices and corresponding control circuitry in a quantum processor module are communicably coupled to a transducer device  304 . 
     In some instances, the superconducting quantum circuit  320  can be supported by a first substrate; and the transducer device  304 , the pump filter  306 , and other optical circuit elements (e.g., optical coupler, waveguides, etc.) can be supported by a second substrate. In this case, the superconducting quantum circuit  320  on the first substrate and the transducer device  304  on the second substrate are galvanically or capacitively coupled, for example via superconducting bonding bumps or capacitive electrodes. In some instances, the superconducting quantum circuit  320 , the transducer device  304 , the pump filter  306 , and the other circuit elements of the quantum processing unit  300  are supported on a common substrate. 
     In some implementations, the first and second substrates may include a dielectric substrate (e.g., silicon, sapphire, etc.). In certain examples, the first and second substrates may include an elemental semiconductor material such as, for example, silicon (Si), germanium (Ge), selenium (Se), tellurium (Te), or another elemental semiconductor. In some instances, the first and second substrates may also include a compound semiconductor such as silicon germanium (SiGe), silicon carbide (SiC), silicon germanium carbide (SiGeC), aluminum oxide (sapphire), gallium arsenide (GaAs), indium arsenide (InAs), indium phosphide (InP), gallium arsenic phosphide (GaAsP), or gallium indium phosphide (GaInP). In some instances, the first and second substrates may also include a superlattice with elemental or compound semiconductor layers. In some instances, the first and second substrates include an epitaxial layer. In some examples, the first and second substrates may have an epitaxial layer overlying a bulk semiconductor or may include a semiconductor-on-insulator (SOI) structure. 
     In some implementations, the quantum processing unit  300 , e.g., the superconducting quantum circuit  320  and the superconducting microwave resonator  332  of the transducer device  304 , include superconducting materials. In some implementations, the superconducting materials may be superconducting metals, such as aluminum (Al), niobium (Nb), tantalum (Ta), vanadium (V), tungsten (W), indium (In), titanium (Ti), Lanthanum (La), lead (Pb), tin (Sn), and/or zirconium (Zr), that are superconducting at an operating temperature of the example quantum processing unit  300 , or another superconducting metal. In some implementations, the superconducting materials may include superconducting metal alloys, such as molybdenum-rhenium (Mo/Re), niobium-tin (Nb/Sn), or another superconducting metal alloy. In some implementations, the superconducting materials may include superconducting compound materials, including superconducting metal nitrides and superconducting metal oxides, such as titanium-nitride (TiN), niobium-nitride (NbN), zirconium-nitride (ZrN), hafnium-nitride (HfN), vanadium-nitride (VN), tantalum-nitride (TaN), molybdenum-nitride (MoN), yttrium barium copper oxide (Y—Ba—Cu—O), or another superconducting compound material. In some instances, the superconducting materials may include multilayer superconductor-insulator heterostructures. 
     In some implementations, the superconducting circuit elements and the optical circuit elements in the quantum processing unit  300  can be formed on surfaces of the first or the second substrates and patterned using a microfabrication process or in another manner. For example, the superconducting circuit elements and the optical circuit elements in the quantum processing unit  300  may be formed by performing at least some of the following fabrication processes: using chemical vapor deposition (CVD), physical vapor deposition (PVD), atomic layer deposition (ALD), and/or other suitable techniques to deposit respective superconducting layers on the substrates; and performing one or more patterning processes (e.g., a lithography process, a dry/wet etching process, a soft/hard baking process, a cleaning process, etc.) to form openings in the respective superconducting layers. 
     In some implementations, the example quantum processing unit  300  resides in a cryogenic environment. For example, the qubit device  302  and the other superconducting circuit elements in the superconducting quantum circuit  320  can be supported on a thermal stage with a low temperature (e.g., 10 milli Kelvin (mK)) and the transducer device  304  and the other optical circuit elements can be supported on the same thermal stage at the same temperature. In this case, the microwave modes generated by the transducer devices  304  can be thermalized to similar temperature limits. In some examples, the transducer device  304  and the other optical circuit elements can be supported on a different thermal stage with a slightly higher temperature in the cryogenic environment (e.g., 100 mK or another temperature). 
       FIG. 4  is a table  400  showing parameters of an example transducer device. The example transducer device may be implemented as the transducer device  212 ,  304  in  FIGS. 2 and 3  for creating pairs of entangled microwave-optical photons under an external optical excitation. For example, the example transducer device can be used to create quantum entanglement between remote quantum computing systems. In some implementations, the transducer device is a triply resonant, direct electro-optic transducer device operated based on a pulse spontaneous parametric down-conversion scheme. In some implementations, the electro-optic transducer device includes a lithium niobate thin film resonator device. 
     In some implementations, such example transducer device can provide technical advantages. For example, the electro-optic transducer device does not require narrow-bandwidth intermediate states; can provide a fast decay time of the microwave resonator and, thus a high maximum repetition rate. For another example, a lithium niobate resonator device can provide high nonlinearity and low loss, which enables the transducer device to operate on low pump power at low temperature. In some implementations, a direct electro-optic transducer device based on a lithium niobate thin film resonator device can also support tunable optical modes and can be integrated with high efficiency pump filters. In some cases, a combination of these and potentially other advantages and improvements may be obtained. 
     A generation rate of entangled microwave-optical photon pairs by the example transducer device can be determined according to the parameters of the example transducer device shown in  FIG. 4 . For example, based on demonstrated optical quality factors of ≥10 7  in a lithium niobate thin film, an optical linewidth of 100 MHz with a coupling ratio of 3 can be estimated. Assuming a strongly over-coupled microwave resonator with a coupling ratio of 10, a microwave linewidth of 10 MHz can be estimated. A repetition rate for a quantum entanglement generation protocol can be limited by the rate at which a superconducting microwave resonator in a transducer device (e.g., the superconducting microwave resonator  332  of the transducer device  304  in  FIG. 3 ) returns to the ground state, which is approximately one tenth of a decay rate of the superconducting microwave resonator. A total optical loss between a transducer device and the photodetector devices is approximately 6 dB, including 2 dB of fiber-to-chip coupling loss and 4 dB of loss due to other sources including filtering and detector efficiency. The probability (p s ) to successfully generate entangled microwave-optical photon pairs is given by 
         p   s =4 g   0   2   N/κ   0   2   (1)
 
     where g 0  is the single photon coupling rate, e.g., g 0 /2π=2 kHz, N is the number of pump photons in a pulse of the incident pump laser signal, and κ 0 /2π is the optical linewidth. For 2 μW of average input power of the incident pump laser signal, the probability for successfully generating a pair of entangled microwave-optical photons in a single attempt at remote entanglement generation on a single link is 2%. Consider the total optical loss of 6 dB, the probability to herald remote quantum entanglement is 0.5%, corresponding to a quantum entanglement generation rate of p s =4 kHz. 
       FIG. 5  is a schematic diagram showing aspects of an example quantum computing network  500 . As shown in  FIG. 5 , the quantum computing network  500  includes a global controller system which includes a first portion  508 A and a second portion  508 B, and two quantum processing units  502 A,  502 B. The two quantum processing units  502 A,  502 B reside in remote quantum computing systems on distant sites, which are communicably coupled to the global controller system  508 , for example through optical fibers in an optical intranet. In some implementations, the example quantum computing network  500  may include additional and different features or components and components of the example quantum computing network  500  may be implemented in another manner. 
     As shown in  FIG. 5 , the first portion  508 A of the global controller system includes a pump laser system  504  and a first beam splitter device  506 A; and the second portion  508 B of the global controller system includes an interferometer device  540  which includes a phase shifter  510  and a second beam splitter device  506 B. The second beam splitter device  506 B includes two output channels  542 A,  542 B, which are equipped with respective photodetector devices  512 A,  512 B. The first portion  508 A of the global controller system provides optical input signals to the quantum processing units  502 A,  502 B; and the second portion  508 B of the global controller system receives optical output signals from the quantum processing units  502 A,  502 B. In some implementations, when optical output signals are detected on the output channels  542 A/ 542 B of the interferometer device  540  by the photodetector devices  512 A/ 512 B (e.g., a “click” on a photodetector device), quantum entanglement transferred to qubit devices on the two quantum processing units  502 A,  502 B can be identified. 
     In some implementations, the pump laser system  504  is configured to generate an incident pump laser signal  532 . In some instances, the incident pump laser signal  532  includes classical laser pulses. In some instances, the wavelength of the incident pump laser signal  532  from the pump laser system  504  can be tunable. In some instances, the pump laser system  504  can generate an incident pump laser signal in a mm-wave, terahertz, or optical regime, e.g., about 10 11 -10 16  Hz. In some examples, the pump laser system  504  may include other optical components, for example, lenses, mirrors, diffusers, filters, polarization modifier, amplifier, phase modulator, Bragg gratings, attenuators, photonic crystals, and multiplexer. In certain instances, the pump laser system  504  may provide phase modulation, frequency modulation, and amplitude modulation to the incident pump laser signal  532 . 
     In some instances, the first beam splitter  506 A, which is partially reflective and partially transmissive, is used to split the incident pump laser signal  532  into a first optical excitation  534 A and a second optical excitation  534 B, each along a separate path (e.g., a transmitted path and a reflected path). As shown in  FIG. 5 , the first quantum processing unit  502 A is placed in a path (e.g., the reflected path) of the first beam splitter device  506 A to receive a first optical excitation  534 A and the second quantum processing unit  502 B is placed in the other path (e.g., the transmitted path) of the first beam splitter device  506 A to receive a second optical excitation  534 B. 
     In some implementations, optical fibers are used to guide the incident pump laser signal  532  from the pump laser system  504  to the first beam splitter device  506 A. Similarly, optical fibers can be also used to guide the first and second optical excitation  534 A,  534 B to the respective first and second quantum processing units  502 A,  502 B. In some implementations, the optical fibers may include single-mode optical fibers to improve the quality of the laser beam or multi-mode optical fibers to maintain the intensity of the laser beam. In certain instances, the optical fibers may be implemented as polarization-maintaining optical fibers, photonic-crystal fibers, or another type of optical fiber. 
     As shown in  FIG. 5 , each of the quantum processing units  502 A,  502 B includes a transducer device  522 , a pump filter  524 , and a superconducting quantum circuit  526 . Particularly, the first quantum processing unit  502 A includes a first transducer device  522 A, a first pump filter  524 A, and a first superconducting quantum circuit  526 A; and the second quantum processing unit  502 B includes a second transducer device  522 B, a second pump filter  524 B, and a second superconducting quantum circuit  526 B. The first and second transducer devices  522 A,  522 B are illuminated by the respective first and second optical excitations  534 A,  534 B from the first beam splitter device  506 A. Each of the first and second optical excitations  534 A,  534 B includes part of the incident pump laser signal  532 . 
     In some implementations, each of the transducer devices  522 A,  522 B is implemented as the transducer device  212 ,  304  in the quantum processing unit  210 ,  300  in  FIGS. 2-3 . The transducer devices  522 A,  522 B of the respective first and second quantum processing units  502 A,  502 B receive the respective first and second optical excitations  534 A,  534 B and generate microwave modes and optical modes at a certain quantum entanglement generation rate. In some implementations, the superconducting quantum circuits  526 A,  526 B including qubit devices and other quantum circuit elements can be implemented as the superconducting quantum circuit  320  shown in  FIG. 3  or in another manner. In some implementations, the superconducting quantum circuits  526 A,  526 B define respective first paths for the microwave photons in the microwave modes within the first and second quantum processing unit  502 A,  502 B. In some implementations, the microwave photons may be directly used to perform quantum processing tasks by operating qubit devices, readout devices, tunable coupler devices, or other types of quantum circuit elements in the superconducting quantum circuit  526 . 
     As shown in  FIG. 5 , each of the first and second quantum processing units  502 A,  502 B includes a series of optical circuit elements that define respective second paths for the optical photons in the optical modes  536 A,  536 B generated by the respective transducer devices  522 A,  522 B. The pump filters  524 A,  524 B, which are configured to remove unconverted first and second optical excitations  534 A,  534 B, may be implemented as the pump filters  306 ,  712  in  FIGS. 3 and 7B , or in another manner. In some implementations, the optical modes from the first and second quantum processing unit  502 A,  502 B are output to the interferometer device  540 , for example on optical fibers. The interferometer device  540  includes a phase shifter  510  and a second beam splitter  506 B. In some implementations, the phase shifter  510  is configured to introduce a constant phase difference between the two optical modes  536 A,  536 B from the first and second quantum processing unit  502 A,  502 B. In some instances, the second beam splitter  506 B, which is partially reflective and partially transmissive, is used to split each of the optical modes  536 A,  536 B received from the first and second quantum processing units  502 A,  502 B along two separate output channels  542 A,  542 B. 
     As shown in  FIG. 5 , a first photodetector device  512 A is placed in a first output channel  542 A of the second beam splitter device  506 B and a second photodetector device  512 B is placed in a second output channel  542 B of the second beam splitter device  506 B. In some implementations, the photodetector devices  512 A,  512 B are used to convert the optical output signals on the output channels  542 A,  542 B to electrical signals. In some implementations, the photodetector devices  512 A,  512 B include photodiodes, which may be used to detect the optical output signals on the output channels of the interferometer device  540 . The output signals on the output channels  542 A,  542 B may be caused by one of the following: the first optical mode  536 A generated by the first quantum processing unit  502 A, the second optical mode  536 B generated by the second quantum processing unit  502 B, or a combination of both. In some implementations, each of the first and second photodetector devices  512 A,  512 B is a single-photon detector device. In some instances, the photodetector device  512 A,  512 B may operate in the few-photon limit. When the optical output signals on the output channels  542 A,  542 B are detected by the photodetector devices  512 A,  512 B, a successful transmission of quantum entanglements between the qubit devices of the first and second quantum processing units  502 A,  502 B can be identified. 
     In some instances, in one round, a repumping laser pulse signal with a frequency different from the incident pump laser signal can be applied to the transducer devices  522 A,  522 B to set the state back to the ground state. After the transducer devices are set back to the ground state, the same incident pump laser signal is applied to the transducer devices  522 A,  522 B again. Forward scattered Stokes light from the first and second quantum processing units  502 A,  502 B is combined at the second beam splitter device  506 B of the interferometer device  540 . This process can be repeated until the optical output signals on the output channels  542 A,  542 B are detected by the photodetector devices  512 A,  512 B. The probability for the photodetector devices  512 A,  512 B to detect the output optical signals is given by the probability (p s ) of successfully generating entangled microwave-optical photon pairs defined in Equation (1) for each round, and thus, the process has to be repeated about 1/p s  times for a successful quantum entanglement preparation. 
     During operation, the pump laser system  504  generates an incident pump laser signal which is split. A portion of the incident pump laser signal (e.g., the first optical excitation  534 A) enters the first quantum processing unit  502 A; and the other portion of the incident pump laser signal (e.g., the second optical excitation  534 B) enters the second quantum processing unit  502 B. Nonlinear interactions in the transducer devices  522 A,  522 B allow for spontaneous parametric down-conversion of the first and second optical excitations, which generates microwave modes and optical modes containing correlated and entangled microwave-optical photon pairs. The optical photons in the respective optical modes are coupled out of the respective quantum processing units  502 A,  502 B after filtering the unconverted first and second optical excitations. The first output optical signal  536 A from the first quantum processing unit  502 A is modulated by the phase shifter  510  in the interferometer device  540  to control the optical phase. The optical output signals  536 A,  536 B produced by the first and second quantum processing units  502 A,  502 B are interfered and combined on the second beam splitter device  506 B to erase “which-path” information. Detection of the optical photons by the photodetector devices  512 A,  512 B heralds the creation of an entangled microwave-optical photon pair and a successful transmission of the quantum entanglement to the qubit devices in the first and second quantum processing units  502 A,  502 B by the microwave modes. The entangled Bell state |Ψ ±    is defined by: 
     
       
         
           
             
               
                 
                   
                     
                        
                       
                         Ψ 
                         ± 
                       
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                     = 
                     
                       
                         1 
                         
                           2 
                         
                       
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                                   0 
                                   
                                     M 
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                               A 
                             
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                               B 
                             
                           
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                               e 
                               
                                 i 
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     where Δφ is the constant phase shift introduced to the first output optical signal  536 A, |0 MW     A  and |1 MW     A  are microwave states corresponding to |0  and |1  photons in a microwave resonator of a first transducer device (e.g., the transducer device  522 A in the first quantum computing system  502 A in  FIG. 5 ), and |0 MW     B  and |1 MW     B  are microwave states corresponding to |0  and |1  photons in a microwave resonator of a second transducer device (e.g., the transducer device  522 B in the second quantum computing system  502 B in  FIG. 5 ). 
     In some implementations, the quantum entanglement in a form of a Bell pair of entangled microwave states can be applied to qubits defined by the respective qubit devices in the first and second quantum processing units  502 A,  502 B for further processing. Collecting the entangled microwave state in the qubit devices of the superconducting quantum circuit  526 A,  526 B in the first and second quantum processing units  502 A,  502 B by stimulated two-photon Raman absorption, parametrically activated resonant exchange interactions generated by flux modulation, or other techniques, allows the quantum entanglement to be used by the first and second quantum processing units  502 A,  5020 B. Examples of two-photon Raman absorption are described in the publication entitled “Deterministic remote entanglement of superconducting circuits through microwave two-photon transitions” by P. Campagne-Ibarcq et al. (arXiv:1712.05854v1 [quant-ph], Dec. 15, 2017). Examples of flux modulation are described in the publication entitled “Parametrically activated quantum logic gates” by E. Sete et al. (U.S. Pat. No. 10,483,980). 
     In some implementations, the scheme for generating quantum entanglement relies on the indistinguishability of optical photons in the output channels  542 A,  542 B arriving from the two transducer devices  522 A,  522 B of the two quantum processing units  502 A,  502 B and the complete extinction of pump photons in the first and second optical excitations  534 A,  534 B by the pump filters  524 A,  524 B. One example remote quantum entanglement heralded by “which-path” entanglement is described in the publication entitled “Long-distance Quantum Communication with Atomic Ensembles and Linear Optics” by L. Duan et al. (arXiv:quant-ph/0105105v1 May 22, 2001). 
       FIG. 6  is a flow chart showing aspects of an example process  600 . The example process  600  can be performed, for example, by a quantum computing network. For example, the example process  600  can be performed by operation of the quantum computing network  500  as shown in  FIG. 5 , or another type of quantum computing network. In some implementations, the example process  600  can be performed to create quantum entanglement between two qubit devices from two remote quantum computing systems. The example process  600  may include additional or different operations, including operations performed by additional or different components, and the operations may be performed in the order shown or in another order. 
     At  602 , first and second optical excitations are generated. In some implementations, the first and second optical excitations are generated for the two qubit devices as part of two superconducting quantum circuits of two quantum processing units. In some instances, the first and second optical excitations can be generated by a pump laser system and a beam splitter device (e.g., the pump laser system  504  and the first beam splitter device  506 A in  FIG. 5 ). For example, an incident pump laser signal can be generated by the pump laser system, guided by optical fibers from the pump laser system to the beam splitter device, and split by the beam splitter device into the first and second optical excitation along separate paths. Each of the first and second optical excitations contains a portion of the incident pump laser signal. In some implementations, the first and second optical excitations are guided by optical fibers to the respective remote quantum computing systems. 
     At  604 , microwave modes and optical modes are generated. In some implementations, the first and second optical excitations are received by respective transducer devices of respective quantum processing units in the remote quantum computing systems (e.g., the first transducer device  522 A of the first quantum processing unit  502 A and the second transducer device  522 B of the second quantum processing unit  502 B in  FIG. 5 ). In some implementations, nonlinear interactions in the transducer devices allow for spontaneous parametric down-conversion of the first and second optical excitations, which generates respective microwave modes and optical modes containing correlated and entangled microwave-optical photon pairs. The microwave modes and optical modes are generated at a certain quantum entanglement generation rate. In some instances, the transducer devices can be implemented as the transducer devices  522 A,  522 B in  FIG. 5  or in another manner. 
     At  606 , the generated optical modes and microwave modes are transmitted. In some implementations, the generated microwave modes are transmitted within the respective quantum processing units for performing quantum operation or other types of quantum information processing; and the optical modes are transmitted out of the respective quantum processing units for identifying the successful transmission of quantum entanglement between the two qubit devices. Operation  606  includes two sub-operations  606 A,  606 B. 
     At  606 A, the microwave modes are transmitted to the respective qubit devices in the two quantum processing units. In some implementations, the microwave modes containing microwave photons are transmitted in respective superconducting quantum circuits of the respective quantum processing units. Each of the respective superconducting quantum circuits includes microwave transmission lines, filters, resonators, qubit devices, coupler devices, readout resonator devise, control signal lines (e.g., qubit drive control lines and flux-bias control lines), and other types of quantum circuit elements. In some instances, the superconducting quantum circuits for transmitting the microwave modes can be implemented as the superconducting quantum circuits  320 ,  526  in  FIGS. 3 and 5 , or in another manner. In some implementations, the quantum entanglement in a form of an entangled microwave state can be transmitted to qubits defined by the respective qubit devices in the respective quantum processing units for further processing (e.g., quantum entanglement distillation). 
     At  606 B, the optical modes are transmitted out of the quantum processing units. In some implementations, optical output signals from the respective transducer devices include unconverted first and second optical excitations and the generated optical modes. In some implementations, prior to transmitting the optical modes out of the quantum processing units, the optical output signals are filtered by operation of filter pumps to remove the unconverted first and second optical excitations from the generated optical modes. In some instances, the optical filters may be implemented as the optical filters  524 A/ 524 B,  712  in  FIGS. 5 and 7B , or in another manner. In some instances, the optical modes can be transmitted on optical fibers back to the global controller system, where the optical modes may be processed and detected. 
     At  608 , output signals on output channels of an interferometer device are detected. In some implementations, the optical modes are transmitted from the respective quantum processing units to the interferometer device. One of the optical modes in the output signal from one of the quantum processing units (e.g., the first output optical signal  536 A from the first quantum processing unit  502 A) can be modulated by a phase shifter (e.g., the phase shifter  510  in  FIG. 5 ). The two optical modes are interfered and combined by operation of another beam splitter device of the interferometer device to erase “which-path” information. In some instances, the interferometer device can be implemented as the interferometer device  540  of the global controller system  508  in  FIG. 5 , or in another manner. In some implementations, the beam splitter device of the interferometer device includes two output channels (e.g., the output channels  542 A,  542 B of the second beam splitter device  506 B in  FIG. 5 ). The output signals on the two output channels of the interferometer device can be received and detected by two photodetector devices (e.g., the photodetector devices  512 A,  512 B in  FIG. 5 , or in another manner). 
     At  610 , based on the output signals, the quantum entanglement transferred to the two qubit devices in the two quantum processing units is identified. In some instances, optical photons in at least one of the optical modes generated by the transducer devices of the respective quantum processing units can be detected by the two photodetector devices. In response to a successful detection of the optical photons in the optical modes by the photodetector devices, a creation of the entangled microwave-optical photon pairs and a successful transmission of the quantum entanglement to the respective qubit devices in the respective quantum processing units by the microwave photons in the microwave modes is identified. In some implementations, transmission of the quantum entanglement to the respective qubit devices includes an application of entangled microwave states to qubits defined by the respective qubit devices. 
       FIG. 7A  is a table  700  showing quantum entanglement generation rates obtained by operation of two example types of transducer devices. Each type of the electro-optic transducer devices is used in a quantum computing network to create a single link of remote entanglement, between two qubit devices and two transducer devices. For example, each type of the electro-optic transducer devices can be implemented in the example quantum computing network  500  in  FIG. 5  or in another manner. The quantum entanglement generation rates are estimated according to properties and geometrical designs of lithium niobate-based electro-optic transducer devices reported in literatures. Examples of transducer devices of Generations I and II are described in the publication entitled “Cavity Electro-optics in Thin-film Lithium Niobate for Efficient Microwave-to-optical Transduction” by J. Holzgrafe, et al. (arXiv: 2005.00939V2 [quantu-ph] May 2, 2020). 
     In Generation I (Gen I), an optical quality factor (Q i ) of the transducer devices is 10 6 . An electro-optic interaction rate (the single photon coupling rate) (g 0 ) of the transducer devices is 0.5 kHz. An attempt rate (R) of the transducer devices is 1 MHz. A total estimated detection efficiency (η opt ) of the transducer devices is −28 dB, which includes a fiber-chip coupling loss of −10 dB, an insertion loss of −15 dB, which is from commercial fabry-perot filters, and an additional loss of −3 dB, including detector efficiency, fiber loss, and another loss mechanism. An average power of the incident pump laser beam is 2.5 μW. Based on these parameters, the quantum entanglement generation rate (R ent ) based on transducer devices of Gen I is 10 −5  kHz. 
     In Generation II (Gen II), an optical quality factor (Q i ) of the transducer devices is 10 7 . An electro-optic interaction rate (g 0 ) of the transducer devices is 1.5 kHz. In some instances, the optical quality factor (Q i ) can be improved by using wider waveguides in the transducer devices with weaker sidewall scattering. In certain instances, the electro-optic interaction rate (g 0 ) can be improved by optimizing electrode geometries of the transducer devices. An attempt rate (R) of the transducer devices is 1 MHz. A total estimated detection efficiency (η opt ) of the transducer devices is −6 dB, which includes a reduced fiber-chip coupling loss of −3 dB, a negligible insertion loss, and an additional loss of −3 dB. An average power of the incident pump laser beam is 2.5 μW. Based on these parameters, the quantum entanglement rate (R ent ) based on the transducer devices of Generation II can be improved to 7.5 kHz. 
       FIG. 7B  is a plot  710  of simulated transmission in decibels (dB) as a function of detuning in Giga Hertz (GHz) of an example pump filter  712 . A schematic diagram of the example pump filter  712  is shown as an inset in  FIG. 7B . In some implementations, the example pump filter  712  is an on-chip pump filter which can be fabricated on the same substrate as a transducer device in a quantum processing unit, e.g., the transducer devices  304 ,  522 A,  522 B of the example quantum processing units  300 ,  502 A,  502 B in  FIGS. 3 and 5 . In some implementations, the pump filter  712  can be operated at a cryogenic temperature with qubit devices and other quantum circuit elements in a quantum processing unit, for example on the lowest-temperature thermal stage in a fridge. 
     In some implementations, the pump filter  712  includes a sequence of optical ring resonators coupled to a common optical waveguide  716 . As shown in  FIG. 7B , the example pump filter  712  includes five parallel-coupled optical ring resonators  714  (e.g., N filter =5) In certain instances, the pump filter  712  may include a different number of optical ring resonators and the optical ring resonators  714  can be configured relative to the common optical waveguide  716  in a different manner. As further shown in  FIG. 7B , each of the five optical ring resonators  714  are associated with a tuning capacitor  718 . In some instances, a tuning capacitor  718  includes superconducting materials and is configured to tune the resonant frequency of the associated optical ring resonator  714 . The independent electro-optic tunability of the individual optical ring resonators  714  enables that small fabrication-induced variations in the resonant frequencies of the optical ring resonators  714  can be reduced without significant additional heat load to the cryostat. 
     As shown in  FIG. 7B , each of the five optical ring resonators  714  is tunable (e.g., has a tunable resonant frequency) and has a low loss rate. In some instances, the five optical ring resonators  714  in the pump filter  712  have the same geometrical design. In some instances, the low loss rates of optical ring resonators in the lithium niobate platform allow high-extinction low-loss filtering using the pump filter  712 . For example, the optical ring resonators  714  in the pump filters  712  may also include lithium niobate material. For example, the pump filter  712  may be off-chip filters or another type of optical filter. 
     In order to obtain the simulated transmission of the pump filter  712 , parameters of each of the five optical ring resonators  714  are assigned. For example, an intrinsic loss rate is 70 Hz, e.g., κ i /2π=70 Hz; an extrinsic coupling rate between waveguides in a single optical resonator  714  is 1 GHz, e.g., κ e /2π=1 GHz with a standard deviation of Δκ e /2π=0.1 GHz, caused by typical variations during fabrication; a standard deviation in a resonance frequency is 50 MHz, e.g., Δf 0 =50 MHz, caused by variations in tuning voltages applied on the tuning capacitors  718 . The estimated −95 dB extinction can be achieved over a stop bandwidth of SBW=200 MHz, with an insertion loss of −0.5 dB at a signal frequency of 7 GHz, e.g., the frequency of the incident pump laser signal. 
       FIG. 8  is a table  800  showing a raw link fidelity of a generated quantum entanglement using the example quantum computing network  500  shown in  FIG. 5 . In some implementations, a raw link fidelity can be determined by several mechanisms of independent errors or losses across the quantum computing network. For example, the raw link fidelity can be expressed as: 
         F= 1− p− 2 P   m −1.5 P   loss   −P   false   −P   phase ,  (3)
 
     where F represents the raw link fidelity of a generated quantum entanglement; p represents a probability of multi-pair generation; P m  represents the average thermal microwave occupancy; P loss  represents the microwave loss between a transducer device and a qubit device (e.g., between the transducer device  522 A/ 522 B and the qubit device in the superconducting quantum circuit  526 A/ 526 B in  FIG. 5 ); P false  represents a probability of false heralding due to dark counts in the photodetector devices (e.g., the photodetector devices  512 A/ 512 B in  FIG. 5 ); and P phase  is a probability of phase errors due to drift in the interferometer device (e.g., the interferometer device  540  in  FIG. 5 ). In some implementations, the microwave loss P loss  represents the total loss in the superconducting quantum circuit  526 A/ 526 B in  FIG. 5  between the transducer device and the qubit device of the quantum processing unit  502 A/ 502 B, including microwave transmission lines, filters, resonator devices, and other superconducting circuit devices. 
     In some instances, the independent errors described in Equation (3) may be reduced and the raw link fidelity (F) can be improved. A higher raw link fidelity of generated entanglement reduces overhead in quantum computing resources (e.g., number of qubit devices in quantum processing units) required for distillation. In some instances, multi-photon detection schemes can further reduce the sensitivity to some types of noise, at the cost of reduced quantum entanglement generation rate (R ent ) that scale as R ent ∝(η opt ·p) 2 , where η opt  represents the total estimated detection efficiency, and p represents the probability of multi-pair generation. 
     In some instances, the average power of the incident pump laser signal can be reduced so as to reduce the probability of multi-pair generation (p) to a negligible level (e.g., p&lt;&lt;1). In certain instances, techniques can be used to reduce the thermal microwave occupancy P m  at a given average power of the incident pump laser signal. For example, the transducer devices can be shielded from stray pump light; and alternative superconducting materials with weak optical absorption and short quasiparticle lifetime can be used. For another example, the transducer devices can be immersed in liquid helium or cooled by another cryogenic cooling technique (e.g., radiative cooling). 
     In some implementations, separately packaged transducer devices and superconducting quantum processor modules are connected by low-loss superconducting transmission lines which enable communication of quantum information. In some instances, techniques for light-tight shielding of superconducting qubit devices can be applied enabling operation of the superconducting quantum processor modules without added noise from the transducer devices (e.g., from stray optical light). In some instances, the techniques for light-tight shielding bring the transducer devices and other optical circuit elements closer to the superconducting quantum processor modules to reduce the microwave loss (P loss ). Examples of light-tight shielding are described in the publication entitled “Optimization of Infrared and Magnetic Shielding of Superconducting TiN and Al Coplanar Microwave Resonators” by J. M. Kreikebaum et al. (arXiv:1608.06273v1 [physics.ins-det], Aug. 22, 2016). 
     In some instances, the over-coupling of the transducer device (e.g., κ m,e &gt;&gt;κ m,i , where κ m,i  and κ m,e  represent the intrinsic and extrinsic coupling rates, respectively; and m refers to the microwave mode. And using low-loss superconducting interconnects can reduce the microwave loss P loss , between the transducer devices and the qubit devices. In some instances, the large loss rates κ m ˜10 MHz, of the transducer devices can be tuned to optimize the overall efficiency and to facilitate the over-coupling, where κ m  is the summation of the intrinsic and extrinsic loss rates κ m,i  and κ m,e . In certain examples, the microwave resonator of the transducer device can be actively controlled to shape the microwave photons in the microwave modes for optimal absorption, thereby reducing P loss . For example, one or more nonlinear superconducting elements can be integrated with the transducer device to achieve tunability of the microwave resonator. In some cases, the photodetector devices include superconducting nanowire single photon detectors with low dark count rates (e.g., &lt;10 Hz), which can be used to reduce the false heralding rate (P faulse ). In some instances, active stabilization of the optical path length of the interferometer device (e.g., the interferometer device  540  in  FIG. 5 ) or persistent calibration can be used to reduce the probability of phase errors (P phase )  FIG. 9A  is a schematic diagram showing an example quantum logic circuit  900  for quantum entanglement distillation. The example quantum logic circuit  900  shown in  FIG. 9A  represents a portion of a single-selection quantum entanglement distillation protocol that is used to improve fidelity of entangled states. The quantum entanglement distillation protocol is mapped onto logical qubits. In some implementations, quantum entanglement distillation is a probabilistic process which purifies a raw Bell pair of entangled states using multiple raw Bell pairs into one single Bell pair with an increased fidelity at a success probability of p d . Quantum entanglement distillation is used in order to achieve a desired fidelity. In some instances, the quantum entanglement distillation protocol shown in  FIG. 9A  can be used to boost a fidelity of entangled states from an initial value of 0.82 to 0.96 with a success probability p d  of greater than 65% for each link. 
     As shown in  FIG. 9A , each of the two raw Bell pairs  904 A,  904 B includes a pair of entangled microwave states, which are generated with respect to the operations in the example process  600  in  FIG. 6  and using the example quantum computing network  500  in  FIG. 5  or in another manner. Particularly, a first raw Bell pair  904 A is applied on two logical qubits  902 - 1 ,  902 - 2 ; and a second raw Bell pair  904 B is applied on two logical qubits  902 - 3 ,  902 - 4 . In some implementations, the logical qubits  902 - 1  and  902 - 3  are in one quantum processor module; and the logical qubits  902 - 2  and  902 - 4  are in a distinct quantum processor module. In some instances, the logical qubits  902 - 1 ,  902 - 2 ,  902 - 3 ,  902 - 4  may be defined on different quantum processor modules or in another manner. 
     The example quantum logic circuit  900  for single-selection quantum entanglement distillation shows one round  910  of quantum entanglement distillation. As shown in  FIG. 9A , a round  910  of quantum entanglement distillation includes two CNOT gates  906  and two Z measurements  908 . An example process for achieving quantum entanglement distillation using the quantum logic circuit shown in  FIG. 9A  is described in the publication entitled “Optimized Entanglement Purification”, by S. Krastanov, et al. (arXiv:1712.09762v3 Feb. 14, 2019). 
     In some implementations, the first raw Bell pair  904 A can be distilled using the second raw Bell pair  904 B to generate a distilled Bell pair  912  of entangled microwave states with an increased fidelity, with success heralded by the outcome of the measured qubits in the Z basis. In some implementations, this protocol can be run recursively to further boost fidelity of the distilled Bell pair towards a desired amount, where a previously distilled Bell pair from a preceding round can be further distilled by another raw Bell pair in a later round. For example, fidelity of the Bell pair  912  distilled from the first and second raw Bell pairs  904 A,  904 B can be further increased using a third raw Bell pair or another distilled Bell pair. 
       FIG. 9B  is a schematic diagram showing an example quantum logic circuit  920  for quantum entanglement distillation. The example quantum logic circuit  920  shown in  FIG. 9B  represents a portion of a single-selection quantum entanglement distillation protocol that is used to improve fidelity of entangled states. The quantum entanglement distillation protocol is mapped onto logical qubits. 
     As shown in  FIG. 9B , each of two raw Bell pairs  904 A,  904 B includes a pair of entangled microwave states, which is generated with respect to the operations in the example process  600  in  FIG. 6  and using the example quantum computing network  500  in  FIG. 5  or in another manner. A first raw Bell pair  930 A is generated and applied on two qubits defined by two ancilla qubit devices  924 A,  924 B. The qubits defined by the two ancilla qubit devices  924 A,  924 B are swapped with qubits defined by two respective data qubit devices of a patch of surface code in the same quantum processing units. In particular, the qubit of the ancilla qubit device  924 A is swapped with the qubit of a first data qubit device in the first quantum processing unit  922 A; and the qubit of the ancilla qubit device  924 B is swapped with the qubit of a first data qubit device in the second quantum processing unit  922 B. After the swapping operations, the first raw Bell pair  930 A is applied on the two data qubit devices of the respective quantum processing units. In some implementations, a GHZ state spanning the array distance (here taking initial distance d=3) is generated with each of the data qubit devices  926 A- 2  and  926 B- 1 . All local stabilizer measurement qubit devices (e.g., the local stabilizer measurement qubit devices  1014  in  FIG. 10A ) of a quantum processing unit are measured in a round of surface code to produce logical qubits  926 A- 2  and  926 B- 1 . In some implementations, the patch of surface code is expanded to a higher distance (e.g., d&gt;3), which is chosen to sustain memory for the time-duration required to execute a desired distillation strategy. Examples of a logical qubit encoding operation are described in the publication entitled “Surface Code Quantum Computing by Lattice Surgery” by C. Horsman et al. (New Journal of Physics 14 (2012) 123011 (27pp), Dec. 7, 2012). In some implementations, after performing the swapping operations, the first raw Bell pair  930 A is applied on the two logical qubits  926 A- 2  and  926 B- 1  of the respective quantum processing units  922 A,  922 B. 
     In some implementations, the logical qubit encoding operation is repeated with a separate patch of data qubit devices (e.g., a separate quantum processor module in a modular quantum processing unit, or a different subset of qubit devices in a quantum processing unit) to produce multiple pairs of remotely entangled logical qubits. As shown in  FIG. 9B , a second raw Bell pair  930 B is generated and further applied on the two qubits defined by the two ancilla qubit devices  924 A,  924 B. The qubits defined by the two ancilla qubit devices  924 A,  924 B are swapped with qubits defined by two different data qubit devices in the same quantum processing units  922 . In particular, the qubit of the ancilla qubit device  924 A is swapped with the qubit of a second data qubit device in the first quantum processing unit  922 A; and the qubit of the ancilla qubit device  924 B is swapped with the qubit of a second data qubit device in the second quantum processing unit  922 B. After performing the third and fourth logical qubit encoding operations  929 A,  929 B, the second raw Bell pair  930 B is applied on the two logical qubits  926 A- 1  and  926 B- 2 . 
     In some implementations, quantum operations that are implemented in a fault-tolerant manner with the surface code can be performed to distill the raw Bell pairs. As shown in  FIG. 9B , logical CNOT gates  932  and logical qubit measurements  934  within the quantum processing units  922 A,  922 B are performed. In some instances, other quantum logic operations including H gates, X basis measurement can be performed to obtain a distilled Bell pair. Examples of surface codes are described in the publication entitled “Surface codes: Towards practical large-scale quantum computations” by A. Fowler et al. (arXiv:1208.0928v2 [quant-ph], Oct. 27, 2012). 
     In some implementations, a distilled Bell state can be used to orchestrate distributed computation. For example, distilled Bell states with increased fidelity can be consumed to perform multi-qubit parity measurements. Examples of multi-qubit parity measurements are described in the publication entitled “Surface codes quantum communication” by A. Fowler et al. (arXiv:0910.4074v3 [quant-ph], Feb. 5, 2010). In some implementations, a logical qubit state can be teleported from one quantum processing unit in a quantum computing system to another in a remote, distinct quantum computing system. 
     To determine a suitable configuration of superconducting qubit devices within a quantum processor module for the purposes of deploying the above protocol, we consider the estimated parameters of a single transducer device as outlined above as well as multiplexing factors assumed to provide fully parallel attempts at remote entanglement. Taking into account the simulated power dissipation of 2 μW per transducer device and reasonable cooling powers in a dilution system at suitable temperatures for the functioning optical mechanism (e.g., 1 mW at 100 mK); upwards of M=500 individual elements are anticipated to be available within a single quantum processing unit or a single processor module of a quantum processing unit for networking purposes, where M is a multiplexing factor, resulting in a total quantum entanglement generation rate of 2 MHz (e.g., R=M×r=2 MHz) for a quantum entanglement generation rate of an individual element of 4 kHz (e.g., r=4 kHz). In some implementations, each individual element include a transducer device and an associated qubit device. 
     In some implementations, to determine the size of individual logical qubits (e.g., code distance d) required within the quantum entanglement distillation protocol as described in  FIG. 9B , a number of raw Bell pairs required, initial fidelity of each of the raw Bell pairs, and a final fidelity can be estimated. For a target remote operation of logical qubit teleportation, a number of raw Bell pairs that scales as D 2  for D the distance of the main processor surface code patches, since there are 2D parity measurements that are repeated D times to correct for measurement errors to achieve the logical distance of the main processor. Assuming a physical error rate of p rate =10 −3  (corresponding to the error associated with gates between the processor qubits) and considering standard fault-tolerant applications, such as Shor&#39;s algorithm, a distance of the main processor surface code patches of D=27 is expected to suffice for acceptable execution success criteria. Therefore, assuming the raw Bell pairs have high fidelity &gt;92-96%, the time to produce the raw Bell pairs without distillation is approximately one millisecond (1 ms). Thus, coherence times exceeding 10 ms would be necessary to maintain these entangled microwave states with high fidelity, which is orders of magnitude beyond the reach of superconducting qubit devices. 
     In some instances, a smaller intermediate logical qubit with a code distance d can be used for storing each state in a raw Bell pair. Assuming a physical error rate of p rate =10 −3  and using a threshold error for planar surface code of p th =0.57%, storing a state for 10 ms can be achieved with a logical suppression P L ≈0.03(p gate /p th ) de , where the effective error suppression de=└(d+1)/2┘. Examples of surface codes are described in the publication entitled “Surface codes: Towards practical large-scale quantum computations” by A. Fowler et al. (arXiv:1208.0928v2 [quant-ph], Oct. 27, 2012). In some instances, a code distance of d=7, which corresponds to approximately 200 physical qubit devices per stored state, can reduce memory errors by a significant factor. In this case, in order to store all the required raw Bell pairs for fault-tolerant lattice surgery at a distance D can require a cache memory of approximately C=3×10 5  physical qubit devices and ancilla qubit devices for data movement. The size of the cache memory is equivalent to about 10 logical qubits on a quantum processing unit with D=27. In some implementations, the size of the cache memory of C=3×10 5  represents an upper bound, since these entangled states are consumed in a round-by-round fashion and not consumed instantaneously. In some implementations, when entangled states are consumed in a round-by-round fashion, the raw Bell pairs that are used for one round of stabilizer measurements are pre-fetched, which results in a reduced cache memory size. For example, at a code distance of d=7, the cache memory size is C=2D×(2d) 2 =10 4 . For another example, in order to tolerate some memory degradation, at a code distance of d=5, the cache memory size is C=5×10 3 . In some implementations, adding quantum entanglement distillation into these physical requirements scales as the width of the distillation quantum logic circuit (e.g., as shown in  FIGS. 9A-9B ). For a networking error threshold anticipated at 10% due to the error syndrome decoding graphs requiring a topology having dimension being only 2 (e.g., errors occur only on the one physical on the boundary, tracked over one time dimension), two rounds of single-selection quantum entanglement distillation operation can be used for an improved or optimized setup. 
       FIG. 10A  is a block diagram showing aspects of an example quantum computing network  1000 . The example quantum computing network  1000  includes multiple quantum computing systems and each of the quantum computing systems includes a quantum processing unit. In some instances, the quantum processing unit can be a modular quantum processing unit including multiple quantum processor modules. As shown in  FIG. 10A , two of the multiple quantum processor modules are shown, e.g., a first quantum processor module  1002 A and a second quantum processor module  1002 B. The first and second quantum processor module  1002 A,  1002 B are from two distinct modular quantum processing units of two distinct, remote quantum computing systems. Quantum entanglements between two respective qubit devices in the two respective quantum processor modules  1002 A,  1002 B can be created according to operations with respect to the example process  600  described in  FIG. 6  using the example quantum computing network  500  in  FIG. 5  or in another manner. Performing quantum entanglement distillation over many qubit devices, instead of logical qubits is another way to purify raw Bell pairs of entangled qubits. In some implementations, a lattice surgery can be performed via remote quantum entanglement using multiple qubit devices. 
     As shown in  FIG. 10A , each of the first and second quantum processor modules  1002 A,  1002 B includes a superconducting quantum circuit with multiple qubit devices. In some instances, each of the first and second quantum processor modules  1002 A,  1002 B includes other quantum circuit devices (e.g., coupler devices, connections, and control signal lines). Square tiles in  FIG. 10A  represent different patterns of parity checks, for instance ZZZZ (tiles in light gray) or XXXX (tiles in dark gray) parity checks in the case of the standard surface code. In particular, the first quantum processor module  1002 A includes ZZZZ parity checks  1008 A- 1  and XXXX parity checks  1008 A- 2 ; and the second quantum processor module  1002 B includes ZZZZ parity checks  1008 B- 1  and XXXX parity checks  1008 B- 2 . Each of the parity checks includes a parity check qubit device at the center of a tile connected to four data qubit devices at the corners of the same tile. The parity check qubit device is coupled to the data qubit devices via intra-chip connections. Couplings provided by intra-chip connections within a tile can be used to apply multi-qubit quantum logic gates or other types of operations to qubits within a quantum processor module. Quantum logic gates mediated by intra-chip connections can be used to create quantum entanglement between qubits defined by qubit devices within a quantum processor module. In some instances, the intra-chip connections may include circuit elements such as static capacitive coupling elements or tunable coupler devices, that support two-qubit/qudit quantum logic gates. The square tiles in the same color (dark or light gray) represent the prescribed pattern of quantum logic gates. 
     The first quantum processor module  1002 A further includes five boundary data qubit devices  1016 A- 1 ,  1016 A- 2 ,  1016 A- 3 ,  1016 A- 4 , and  1016 A- 5  (labeled as “unfilled” circles in  FIG. 10A ), and four local stabilizer measurement qubit devices  1014 A- 1 ,  1014 A- 2 ,  1015 A- 1 , and  1015 A- 2  (labeled as “filled” circles in  FIG. 10A ). The second quantum processor module  1002 B includes five boundary data qubit devices  1016 B- 1 ,  1016 B- 2 ,  1016 B- 3 ,  1016 B- 4 , and  1016 B- 5 , and four local stabilizer measurement qubit devices  1014 B- 1 ,  1014 B- 2 ,  1015 B- 1 , and  1015 B- 2 . The local stabilizer measurement qubit devices  1014 A- 1 ,  1014 A- 2  are X parity checks qubit device; and the local stabilizer measurement qubit devices  1015 A- 1 ,  1015 A- 2  are Z parity checks qubit devices. Similarly, the local stabilizer measurement qubit devices  1014 B- 1 ,  1014 B- 2  are X parity checks qubit device; and the local stabilizer measurement qubit devices  1015 B- 1 ,  1015 B- 2  are Z parity checks qubit devices. 
     As shown in  FIG. 10A , each of the local stabilizer measurement qubit devices  1014 A/ 1014 B,  1015 A/ 1015 B on a quantum processor module  1002 A/ 1002 B is communicably coupled to two neighboring boundary data qubit devices  1016 A/ 1016 B on the same quantum processor module  1002 A/ 1002 B via intra-chip connections  1010 . Each of the local stabilizer measurement qubit devices  1014 A,  1015 A on the first quantum processor module  1002 A is communicably coupled to respective local stabilizer measurement qubit devices  1014 B,  1015 B on the second quantum processor module  1002 B via respective links  1018 . Specifically, the X parity check qubit devices  1014 A- 1  of the first quantum processor module  1002 A is communicably coupled to the X parity check qubit devices  1014 B- 1  of the second quantum processor module  1002 B via a first link  1018 - 1 ; the Z parity check qubit devices  1015 A- 1  of the first quantum processor module  1002 A is communicably coupled to the Z parity check qubit devices  1015 B- 1  of the second quantum processor module  1002 B via a second link  1018 - 2 ; the X parity check qubit devices  1014 A- 2  of the first quantum processor module  1002 A is communicably coupled to the X parity check qubit devices  1014 B- 2  of the second quantum processor module  1002 B via a third link  1018 - 3 ; and the Z parity check qubit devices  1015 A- 2  of the first quantum processor module  1002 A is communicably coupled to the Z parity check qubit devices  1015 B- 2  of the second quantum processor module  1002 B via a fourth link  1018 - 4 . 
     As shown in  FIG. 10A , the links  1018 - 1 ,  1018 - 2 ,  1018 - 3 , and  1018 - 4  (labeled in dotted lines) represent quantum entanglement between qubit devices that reside on different quantum processor modules. In some instances, each link  1018  may include a global controller system  508 A,  508 B, two transducer devices  522 A,  522 B, pump filters  524 A,  524 B, and a portion of the superconducting quantum circuit  526 A,  526 B as shown in the example quantum computing network  500  in  FIG. 5 . Each pair of the parity check qubit devices that are coupled through a respective link  1018  are prepared in a raw Bell pair. In some implementations, a lattice merge operation is achieved by performing a ZZZZ parity measurement  1020  and a XXXX parity measurement  1040 , each of which can consume a remote Bell pair. 
       FIGS. 10B-10C  are schematic diagrams showing aspects of example quantum logic circuits  1120 ,  1140  for performing a ZZZZ parity measurement and a XXXX parity measurement with a remote Bell pair of entangled qubits. 
     As shown in  FIG. 10B , the remote Bell pair of entangled states  1022  are generated and transmitted to the local stabilizer measurement qubit device  1014 A- 1  of the first quantum processor module  1002 A and the local stabilizer measurement qubit device  1014 B- 1  of the second quantum processor module  1002 B. Two CNOT gates  1024  are applied between the local stabilizer measurement qubit device  1014 A- 1  and two respective data qubit devices  1016 A- 1 ,  1016 A- 2 . A Z-basis measurement  1026  is performed on the local stabilizer measurement qubit device  1014 A- 1 . Similarly, two CNOT gates  1024  are applied between the local stabilizer measurement qubit device  1014 B- 1  and two respective data qubit devices  1016 B- 1 ,  1016 B- 2 . A Z-basis measurement  1026  is performed on the local stabilizer measurement qubit device  1014 B- 1 . 
     As shown in  FIG. 10C , the remote Bell pair of entangled states  1042  are generated and transmitted to the local stabilizer measurement qubit device  1015 A- 2  of the first quantum processor module  1002 A and the local stabilizer measurement qubit device  1015 B- 2  of the second quantum processor module  1002 B. Two Hadamard gates  1044  are applied on the local stabilizer measurement qubit devices  1015 A- 2 ,  1015 B- 2 . Two CNOT gates  1046  are applied between the local stabilizer measurement qubit device  1015 A- 2  and two respective data qubit devices  1016 A- 4 ,  1016 A- 5 . A X-basis measurement  1048  is performed on the local stabilizer measurement qubit device  1015 A- 2 . Similarly, two CNOT gates  1046  are applied between the local stabilizer measurement qubit device  1015 B- 2  and two respective data qubit devices  1016 B- 4 ,  1016 B- 5 . A X-basis measurement  1048  is performed on the local stabilizer measurement qubit device  1015 B- 2 . In some implementation, the lattice merge operation described here allows for obtaining the requisite stabilizer measurement. In some instances, the lattice merge operation can be used to replace the CNOT gates  906  and Z measurements  908  as described in  FIG. 9A , for example. 
       FIG. 11  is a schematic diagram showing aspects of an example quantum logic circuit  1100  for quantum entanglement distillation. The example quantum logic circuit  1100  shown in  FIG. 11  represents a portion of a double-selection quantum entanglement distillation protocol that is used to improve fidelity of entangled qubits. The quantum entanglement distillation protocol shown in  FIG. 11  is mapped onto physical qubit devices. 
     In some implementations, a first Bell pair of entangled qubits  1102 - 1  transmitted to qubit devices  1104 - 1  and  1104 - 2  is distilled using four additional remotely prepared Bell pairs (e.g., a second Bell pair of entangled qubits  1102 - 2 , a third Bell pair of entangled qubits  1102 - 3 , a fourth Bell pair of entangled states  1102 - 4 , and a fifth Bell air of entangled states  1102 - 5 ). In some implementations, all five Bell pairs of entangled states are prepared with respect to the operations in the example process  600  using the quantum computing network  500  in  FIG. 5 . In some implementations, assuming each of the five Bell pairs has input raw fidelity of F=0.82, a finite probability of protocol success in achieving distillation p d =76%. 
     As shown in  FIG. 11 , the example quantum logic circuit  1100  for double-selection quantum entanglement distillation shows two rounds of double-selection quantum entanglement distillation operations  1110 - 1 ,  1110 - 2 . In some implementations, the quantum logic circuit representing a first round of double-selection quantum entanglement distillation operation  1110 - 1  includes four controlled-Z gates  1114 , and four X-basis measurements  1112 . Similarly, the quantum logic circuit representing a second round of double-selection quantum entanglement distillation  1110 - 2  includes four controlled-X gates  1116  and four X-basis measurements  1112 . In some instances, outcomes of the example quantum logic circuits  1100  can be sampled to determine the success probability for the input raw fidelity. 
     In the example presented here, the quantum logic circuit  1100  is expressed using a wire diagram as shown in  FIG. 11 . In certain instances, a quantum logic circuit may be described in another manner, for example, by a quantum instruction program, equivalently via tensor-network representation. When a quantum logic circuit is expressed using a quantum instruction program, the example quantum logic circuit  1100  can be expressed using the following Quil program adapted for Z-basis measurements:
         Def make_doubleesel_prog( ):   prog=Programs( )   ro=prog.declare(‘ro’, memory_type=‘BIT’, memory_size=10)   for k in range(3):   make wener(2*k, *k+1, prog. Raw_fid)   for k in range(4):   prog+=CNOT(k+2,k)   prog+=[MEASURE(k, ro[i]) for i in [2, 3, 4, 5]]   for k in [ 3 ,  4 ]:   make werner(2*k, 2*k+1, prog, raw_fid)   for k in [6, 8]:   prog+=H(k)   prog+=CNOT(0,k)   prog+=H(k)   for k in [7,9]:   prog+=H(k)   prog+=CNOT(0,k)   prog+=H(k)   prog+=[MEASURE(, ro(i)) for i in [6, 7, 8, 9]]   return prog       

       FIG. 12  is a log-log plot  1200  of a total number of links (e.g., a multiplexing factor m) as a function of raw entangling probability (p s ) of a single link at various numbers (k) of raw Bell pairs of entangled qubits. The total number of raw Bell pairs of entangled qubits used in a numerical study includes k=1, 10, 100, 1000, and 10000. The plot  1200  in  FIG. 12  shows a total number of links that are needed to allow a generation of k raw Bell pairs of entangled qubits with high reliability (p r =99%). In some implementations, the total numbers of raw Bell pairs of entangled qubits are generated with respect to the operations of the example process  600  in  FIG. 6  using the example quantum computing network  500  in  FIG. 5 . 
     As shown in  FIG. 12 , the dashed line  1202  represents a constant raw entangling probability (p s ) of 0.06 which can be reasonably obtained using a single link based on lithium niobate based electro-optic transducer devices in an example quantum computing network (e.g., the transducer devices  522 A,  522 B of the example quantum computing network  500  in  FIG. 5 ). As shown in  FIG. 12 , in order to achieve a total number of raw Bell pairs of k=1000 and each individual link has a constant raw entanglement probability of p s =0.06, a multiplexing factor of m=20000 (e.g., a total number of 20000 links) is required. 
     In some implementations, to perform one round of stabilizer measurements during lattice surgery, q distilled Bell pairs and at least 5q raw Bell pairs are required. Assuming the probability of reaching the 5q raw Bell pairs is p r , since each quantum entanglement generation (e.g., raw Bell pair generation) event is a Bernoulli trial, the probability of realizing k links from m trials is a binomial distribution (p k ) is: 
     
       
         
           
             
               
                 
                   
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     In some implementations, in order to obtain a sufficient number of raw Bell pairs for distillation, we need at least k=5q links, which means that 
     
       
         
           
             
               
                 
                   
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                   ) 
                 
               
             
           
         
       
     
     where P r  is the probability of reaching the 5q raw Bell pairs; m represents a total multiplexing factor, meaning that for n physical channels with c trials, m=nc. In some implementations, a link is a successful trial (e.g., a raw Bell pair) established across two physical channels, each of which includes a transducer device. This can be sustained with auxiliary qubit devices for distillation and re-use of the physical channels independent of prior success. The number of trials c that can be attempted in a single quantum entanglement distillation operation is limited by the finite coherence of the superconducting qubit devices. Assuming T2=350 μs, an attempt rate of R=1 MHz, about c=80 trials can be made before the incoherent memory error becomes comparable to the target distillation level. In some implementations, the logical clock rate requirement of 150 quantum bit per second (qb/s) for data movement, it is necessary to reduce the number of trials through physical multiplexing as much as possible. In some instances, the number of attempts can be limited by the cooling power at low temperatures. In some instances, most of a transducer device is thermalized at 100 mK; and a dilution refrigerator at this stage (e.g., 100 mK) can achieve a cooling power of 3 mW. Assuming an average dissipation of 2.5 μW per transducer device and the entire cooling power of a dilution refrigerator at this stage is dedicated to the active heat load of the transducer devices, a total number of n&lt;1200 physical channels is required. 
     In some instances, the number of physical channels is n=1200, and the number of trials c=17 may be used for minimizing latency. In certain instances, the number of physical channels n=250, and the number of trials c=80 may be used for minimizing power dissipation. In some implementations, frequency domain multiplexing is used to achieve additional efficiencies at a greater number of physical channels (n), and thus faster clock speeds are possible. In this case, multiple transducers of respective physical channels operate at respective distinct frequencies. These correspond to remote-mediated stabilizer rounds occurring at rates of R d =1 MHz per trial, which need to be repeated d times for fault-tolerance, that is R=d×R/c, where d is the code space, R is the quantum entanglement generation rate, and c is the number of trials. In some instances, a logical qubit teleportation clock rate of about 300 qb/s (c=17) can be multiplexing limited, and a logical qubit teleportation clock rate of about 62 qb/s (c=80) can be coherence-time limited. In certain instances, the parameters may have other values according to the types of applications. 
     In a general aspect, a photonic quantum networking for large superconducting qubit modules is disclosed. 
     In a first example, quantum entanglement between a first qubit device of a first quantum processing unit (QPU) and a second qubit device of a second QPU is generated. Microwave modes and optical modes are generated on the first and second QPUs by operation of a first transducer device of the first QPU and a second transducer device of the second QPU. The microwave modes are transmitted within the first and second QPUs from the first and second transducer devices to the respective first and second qubit devices. The optical modes are transmitted from the first and second QPUs to an interferometer device. By operation of the interferometer device, output signals are generated on respective output channels based on the optical modes from the first and second QPUs. The output signals are detected by operation of photodetector devices coupled to the respective output channels of the interferometer device. Based on the output signals, identifying the quantum entanglement transferred to the first and second qubit devices by the microwave modes. 
     Implementations of the first example may include one or more of the following features. When the microwave modes and optical modes are generated on the first and second QPUs, microwave modes and optical modes are generated by operation of the first transducer device in response to first optical excitations; and microwave modes and optical modes are generated by operation of the second transducer device in response to second optical excitations. The first and second optical excitations are generated by operation of a pump laser system and a beam splitter device; and the first and second optical excitations are transmitted from the beam splitter device to the respective first and second QPUs. The first QPU includes a first pump filter that filters at least a portion of the first optical excitations, and the second QPU includes a second pump filter that filters at least a portion of the second optical excitations. Each of the first and second pump filters includes two optical ring resonators coupled in series. Each of the first and second pump filters includes a sequence of optical ring resonators coupled in parallel. 
     Implementations of the first example may include one or more of the following features. Each of the first and second transducer devices is a direct electro-optic transducer device comprising a lithium niobate thin film. Each of the first and second transducer devices includes a first optical ring resonator, a second optical ring resonator, a superconducting microwave resonator configured to modulate a resonant frequency of the first optical ring, and a tuning capacitor configured to apply DC bias to the second optical ring resonator. The microwave modes include microwave photons. The superconducting microwave resonator is configured to generate the quantum entanglement. Each of the first and second QPUs includes a microwave transmission line capacitively coupled to the respective first and second transducer devices, a Purcell filter, and an intermediate resonator. When the microwave modes are transmitted within the first and second QPUs, the microwave photons are transmitted on the microwave transmission lines; the microwave photons are filtered by operation of the Purcell filters; and coherence of the microwave photons are maintained by operation of the intermediate resonator. 
     Implementations of the first example may include one or more of the following features. Each of the first and second qubit devices is a tunable transmon qubit device. When each of the first and second QPUs includes a plurality of qubit devices that collectively defines one or more logical qubits, the quantum entanglement is distilled by operation of the one or more logical qubits. When each of the first and second QPUs includes a plurality of qubit devices, the quantum entanglement is distilled by operation of the plurality of qubit devices. 
     In a second example, a quantum computing network includes a plurality of quantum processing units (QPUs) and a global controller system. Each of the plurality of QPUs includes a qubit device and a transducer device. The plurality of QPUs includes a first QPU and a second QPU. The first QPU includes a first qubit device and a first transducer device; and the second QPU includes a second qubit device and a second transducer device. Each of the plurality of QPUs is configured to perform operations including generating microwave modes and optical modes by operation of the transducer device; transmitting the microwave modes within the QPU from the transducer device to qubit device; and transmitting the optical modes from the QPU to an interferometer device. The global controller system includes the interferometer device and photodetector devices. The global controller is configured to perform operations including: by operation of the interferometer device, generating output signals on respective output channels based on the optical modes from the first and second QPUs; detecting the output signals by operation of the photodetector devices coupled to the respective output channels of the interferometer device; and based on the output signals, identifying quantum entanglement transferred to the first and second qubit devices by the microwave modes. 
     Implementations of the second example may include one or more of the following features. When the microwave modes and optical modes are generated on the first and second QPUs, microwave modes and optical modes are generated by operation of the first transducer device in response to first optical excitations; and microwave modes and optical modes are generated by operation of the second transducer device in response to second optical excitations. The global controller system further includes a pump laser system and a beam splitter device. The global controller system is configured to perform operations including: generating the first and second optical excitations by operation of the pump laser system and the beam splitter device; and transmitting the first and second optical excitations from the beam splitter device to the respective first and second QPUs. The first QPU includes a first pump filter configured to filter at least a portion of the first optical excitations; and the second QPU includes a second pump filter configured to filter at least a portion of the second optical excitations. Each of the first and second pump filters includes two optical ring resonators coupled in series. Each of the first and second pump filters includes a sequence of optical ring resonators coupled in parallel. 
     Implementations of the second example may include one or more of the following features. Each of the first and second transducer devices is a direct electro-optic transducer device comprising a lithium niobate thin film. Each of the first and second transducer devices includes a first optical ring resonator, a second optical ring resonator, a superconducting microwave resonator configured to modulate a resonant frequency of the first optical ring, and a tuning capacitor configured to apply DC bias to the second optical ring resonator. The microwave modes include microwave photons. The superconducting microwave resonator is configured to generate the quantum entanglement. Each of the first and second QPUs includes a microwave transmission line capacitively coupled to the respective first and second transducer devices, a Purcell filter, and an intermediate resonator. When the microwave modes are transmitted within the first and second QPUs, the microwave photons are transmitted on the microwave transmission lines; the microwave photons are filtered by operation of the Purcell filters; and coherence of the microwave photons is maintained by operation of the intermediate resonator. 
     Implementations of the second example may include one or more of the following features. Each of the first and second qubit devices is a tunable transmon qubit device. Each of the first and second QPUs includes a plurality of qubit devices that collectively defines one or more alogical qubits. Each of the first and second QPUs is configured to perform operations including: distilling, by operation of the one or more logical qubits, the quantum entanglement. Each of the first and second QPUs includes a plurality of qubit devices. Each of the first and second QPUs is configured to perform operations including: distilling, by operation of the plurality of qubit devices, the quantum entanglement. 
     Some of the operations described in this specification can be implemented as operations performed by a data processing apparatus on data stored on one or more computer-readable storage devices or received from other sources. 
     The term “data-processing apparatus” encompasses all kinds of apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, a system on a chip, or multiple ones, or combinations, of the foregoing. The apparatus can include special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). The apparatus can also include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, a cross-platform runtime environment, a virtual machine, or a combination of one or more of them. 
     A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, object, or other unit suitable for use in a computing environment. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. 
     Some of the processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform actions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). 
     While this specification contains many details, these should not be understood as limitations on the scope of what may be claimed, but rather as descriptions of features specific to particular examples. Certain features that are described in this specification or shown in the drawings in the context of separate implementations can also be combined. Conversely, various features that are described or shown in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. 
     Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single product or packaged into multiple products. 
     A number of embodiments have been described. Nevertheless, it will be understood that various modifications can be made. Accordingly, other embodiments are within the scope of the following claims.