Patent Publication Number: US-2022215281-A1

Title: Optically Heralded Entanglement of Superconducting Systems in Quantum Networks

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
     This application claims the priority benefit, under 35 U.S.C. § 119(e), of U.S. Application No. 63/122,741, which was filed on Dec. 8, 2020, and is incorporated herein by references in its entirety for all purposes. 
    
    
     BACKGROUND 
     A quantum computer uses the superposition and entanglement of quantum states to perform computations, including factoring large integers. The computations are typically represented as a set of quantum logic gates, such as the controlled NOT (CNOT) gate, that operate on quantum bits (qubits). Just like the bit is the basic unit of information in a classical computer, the qubit is the basic unit of information in a quantum information. Like a classical bit, a qubit is represented by a physical system (e.g., an electron or a photon) and has two states (e.g., representing logical 1 or 0). Unlike a classical bit, however, a qubit can be in a superposition of both states simultaneously and can be entangled without other qubits. 
     In a superconducting quantum computer, qubits and quantum logic gates are implemented as superconducting electronic circuits. Examples of superconducting qubits include the charge qubit, which is a superconducting island between a capacitance and a Josephson junction; the radio-frequency (RF) superconducting quantum interference device (SQUID) qubit, which is a superconducting loop interrupted by a Josephson junction; the phase qubit, which is a Josephson junction biased by a current; and the transmission line shunted plasma oscillation (transmon) qubit, which is a Cooper-pair box where the two superconductors are also capacitively shunted to reduce their sensitivity to charge noise, while maintaining a sufficient anharmonicity for selective qubit control. In operation, superconducting qubits are typically contained in refrigerators or cryogenic coolers that cool them to cryogenic temperatures (temperatures of about 0 K to about 120 K, e.g., about 1 K). 
     SUMMARY 
     A central challenge in quantum information science is to transfer quantum states between superconducting systems, potentially over long distances. This is especially challenging for transferring quantum states between superconducting qubits cooled to cryogenic temperatures in separate refrigerators, possibly separated by kilometers or more. Superconducting electrical connections between separate refrigerators would have to be cooled to cryogenic temperatures, which is impractical over distances larger than a few meters. 
     One way to circumvent the need for superconducting connections between widely separated superconducting qubits is to transduce the quantum information encoded in a superconducting qubit into photons, which can be transmitted at room temperature over long distances with low loss. This type of transduction is called microwave-optical (M-O) quantum state transduction and has been investigated widely. However, despite concerted effort and tremendous progress in direct M-O transduction, it remains extremely challenging to achieve a high transduction efficiency without adding significant noise. Moreover, the efficiency and noise challenges are compounded because a full state transfer between two quantum devices involves sequential M-O and O-M transduction steps. 
     Here, we replace these M-O-M transduction steps with one round of optically heralded microwave-microwave (M-M) entanglement (involving the detection of a single photon), followed by state teleportation between the quantum devices. Heralding indicates that the quantum devices have been entangled successfully. In contrast to direct M-O transduction, this photon-heralded entanglement scheme exploits low M-O coupling efficiency to eliminate added noise, while assuring on-demand state teleportation by heralding and distilling M-M Bell pairs faster than their decoherence rates. Specifically, on present-day hardware, the entanglement rates could exceed 100 kHz per channel, the entanglement fidelity could exceed 0.99, and the entanglement purification (reduction or removal of effects of decoherence) could reach a fidelity of 0.999. Leveraging standard telecom equipment, dense wavelength division multiplexing can boost entanglement rates by an order of magnitude. Our approach also allows efficient and high-fidelity heralded state transfer to other physical modalities including trapped ions, cold atoms, solid-state spin systems, or another traveling photon (corresponding to heralded M-O quantum state transduction). Our approach unifies and simplifies entanglement generation between superconducting devices and other physical modalities in quantum networks. 
     Our approach can be implemented as a method of entangling a first superconducting qubit in a first node and a second superconducting qubit in a second node. The first and second nodes can be cooled to 120 K in first and second refrigerators, respectively, and can be coupled by an optical fiber or other optical link that is at a temperature greater than 120 K (e.g., at about 300 K). This method comprises generating a microwave photon and an optical photon at the first node, then coupling the microwave photon to the first superconducting qubit. The optical photon is coupled to a beam splitter that is in optical communication with the first and second nodes. The beam splitter erases information about which node generated the optical photon. A photodetector coupled to an output of the beam splitter senses the optical photon to herald entanglement of the first and second superconducting qubits. 
     Generating the microwave photon and the optical photon can include coupling pulses from a pump laser to a transducer in the first node and to a transducer in the second node. The transducer in the first node converts a pump pulse to the microwave photon and the optical photon. The transducer generates the microwave photon at a resonance frequency of a microwave resonator coupled to the first qubit and to the transducer. It generates the optical photon at a resonance frequency of an optical resonator coupled to the transducer. The transducer can generate the microwave and optical photons via an interaction between a nonlinear medium and a pump photon at a frequency equal to a sum of a frequency of the microwave photon and a frequency of the optical photon. 
     Coupling the optical photon to the beam splitter can comprise guiding the optical photon through an optical fiber with a length of at least 1 km. A reconfigurable add-drop multiplexer (ROADM) can couple the optical photon from the first node to the beam splitter. In some cases, the first node generates another microwave photon and another optical photon, and the ROADM couples the other optical photon to the beam splitter. The photodetector coupled to the output of the beam splitter detects this other optical photon to herald entanglement of the first superconducting qubit with a qubit in the third node. The third qubit can be a third superconducting qubit, a color center qubit, or a trapped ion qubit. 
     This method can be carried out in a quantum optical network that includes a pump laser, first node, second node, beam splitter, first photodetector, and second photodetector. The first node is in optical communication with the pump laser and comprises a first superconducting qubit and a first transducer in electromagnetic communication with each other. Similarly, the second node is in optical communication with the pump laser and comprises a second superconducting qubit and a second transducer in electromagnetic communication with each other. In operation, the pump laser generates at least a first pump photon and a second pump photon. The first and second transducers generate a first microwave photon and a first optical photon from the first pump photon and a second microwave photon and a second optical photon from the second pump photon, respectively. The beam splitter is in optical communication with the first and second nodes. The beam splitter is configured to receive the first optical photon from the first node, to receive the second optical photon from the second node, and to erase information about which of the first node and the second node generated the optical photon. The first and second photodetectors are in optical communication with the first and second output ports, respectively, of the beam splitter and are configured to detect the first or second optical photon. Detection of the first and/or second optical photons heralds entanglement of the first and second superconducting qubits. 
     The first node can comprise a first optical resonator coupled to the transducer and the pump laser is blue-detuned from resonance with the first optical resonator. The first node may also include a first microwave resonator coupled to the first transducer. The second node can include a second optical resonator having a resonance frequency equal to a resonance frequency of the first optical resonator. 
     The quantum optical network can include a ROADM, in optical communication with the first node, the second node, and the beam splitter, to couple the first optical photon from the first node to the beam splitter and/or to couple the second optical photon from the second node to the beam splitter. A third node, in optical communication with the ROADM, can generate a third optical photon for heralding entanglement of a qubit in the third node with one of the first superconducting qubit or the second superconducting qubit. 
     Another implementation of a quantum optical network may include a pump laser to generate pump pulses, a ROADM, a plurality of nodes, and a plurality of path-erasure photodetectors. The nodes are in optical communication with the pump laser via the ROADM. At least one node includes superconducting qubits configured to generate pairs of spectrally multiplexed optical photons and microwave photons from the pump pulses. The path-erasure photodetectors are in optical communication with the plurality of nodes via the ROADM. The path-erasure photodetectors detect the optical photons while erasing information about paths taken by the optical photons. Detection of the optical photons herald entanglement of the superconducting qubits with qubits in other nodes in the plurality of nodes. 
     All combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the inventive subject matter disclosed herein. Terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein. 
    
    
     
       BRIEF DESCRIPTIONS OF THE DRAWINGS 
       The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar elements). 
         FIG. 1A  shows a superconducting quantum node with transduction hardware, here utilizing a χ (2)  process whereby a classical pump enables a beam-splitter Hamiltonian between an optical mode supported by an optical resonator and a microwave mode supported by a microwave resonator coupled to the optical resonator by a transducer comprising χ (2)  nonlinear material. 
         FIG. 1B  illustrates superconducting quantum nodes coupled to each other and to a pump that is red-detuned from the optical mode. This enables a beam splitter interaction that can transduce a microwave state into an optical state, which is then transmitted over a fiber to a distant node where the same process is used to transduce it back. This is a deterministic, low-fidelity operation. 
         FIG. 1C  illustrates superconducting quantum nodes that are coupled to pump that is blue- or red-detuned from the optical mode of the nodes&#39; optical resonators and to a beam splitter for optically heralded entanglement. A blue-detuned pump creates pairs of microwave/optical photons. By detecting the optical photons after erasing the path information we can herald entanglement between the microwave oscillators. This is a high-fidelity, low-efficiency probabilistic operation. 
         FIG. 2A  shows the infidelity of a microwave Bell pair obtained using the nodes of  FIG. 1C  in different regimes. The blue-detuned “squeezing” implementation has higher infidelity due to populating higher-than-one-photon states (upper solid line). The red-detuned “beam splitter” implementation involves preparing the microwave resonator in the 1) state but does not suffer from the aforementioned excitations. The other lines represent the red-detuned approach followed by purification and/or storage for on-demand use. Besides the ˜g/γ infidelity in the solid traces, storage at low generation-rates causes an infidelity floor, due to finite microwave lifetime. Similarly, purification can be detrimental at low rates due to having to wait for a second pair to be generated. 
         FIG. 2B  shows the rates of entanglement generation (upper solid and dashed lines) and equivalent ebit generation, i.e., the hashing yield (lower solid and dashed lines) for the red-tuned regimes of  FIG. 2B . Purification causes a drop in the entanglement rate and that the ebit rate suffers at high infidelities. At high entanglement rates, the curves flatten out due to the finite time for resetting the microwave resonators in the nodes of  FIG. 1C , e.g., by projective measurement of algorithmic cooling. 
         FIG. 2C  gives the in-fridge heating due to the intrinsic loss of the optical resonators in the nodes of  FIG. 1C . Evaluated at coupling g 0 =1 kH, optical extrinsic and intrinsic loss γ e =γ i =100 MHz, microwave loss γ MW =1 kHz, pump wavelength λ p =1500 nm, resonator material LiNbO 3  of n=2.3, microwave gate fidelity 0.999, and microwave resonator reset time of 1 μs, which are state of the art values. Mismatches in the coupling rates of the two nodes and detector dark count rates are assumed negligible. Errors in the 1) microwave state preparation before heralding is also not included. 
         FIG. 3  illustrates connectivity among quantum nodes of different types and in different locations, including in different refrigerators. A reconfigurable optical add-drop multiplexer (ROADM) allows routes a frequency comb from a pump laser to qubits in the different quantum nodes. By connecting the ROADM to a bank of path-erasure detectors for each frequency channel, this design enables multiplexed heralded entanglement generation between multiple fridges of different quantum modalities. (Electro-optic modulators and attenuators are omitted for clarity.) 
     
    
    
     DETAILED DESCRIPTION 
     Our technology shares entanglement between distant quantum processing units (QPUs) or other entities by heralding entanglement generation between remote microwave photons. Our scheme uses spontaneous parametric down-conversion (SPDC) with electro-optic transducers tuned to the low-coupling regime, such that low-efficiency generation of a microwave- and optical-photon pair is the primary transduction process. By heralding the optical photons from two QPUs, we can entangle the accompanying microwave photons remaining in the microwave cavities in the QPUs. Hence, our scheme allows for high-fidelity entanglement in the same hardware used today for low-fidelity microwave-to-optical transduction (and vice versa). In other words, our technology replaces the high-probability, low-fidelity microwave-to-optical transduction process with a low-probability, high-fidelity heralding process. Our architecture can connect multiple QPUs across several quantum modalities to provide on-demand, multiplexed entanglement for a quantum network. 
     Optically Heralded Entanglement 
       FIGS. 1A-1C  illustrate nodes with transduction hardware for low-efficiency, high-fidelity optically heralded entanglement.  FIG. 1A  shows a node  100  that includes a QPU  110  with several qubits  112  coupled to a microwave resonator  120 , which can have a resonant frequency in the 1-10 GHz range and a cavity lifetime of tens of microseconds or more. In this case, the qubits  112  are superconducting qubits  112  that are inside a cryostat or refrigerator  102  that cools the qubits  112  to cryogenic temperatures (e.g., about 1 K). The microwave resonator  120  is also inside the refrigerator  102 , as are a transducer  130  and an optical resonator  150 , which can have a resonant frequency in the visible or infrared (e.g., in the telecom band) and a quality factor of about 10 4  to 10 5 . 
     The transducer  130  couples the microwave resonator  120  to the optical resonator  150  and can take any of a variety of forms; for instance, it could include a piezoelectric material or a χ (2)  nonlinear material. The transducer  130  supports an optical mode and a microwave mode in a material that enables interaction of the optical mode with the microwave mode. Typically, this interaction is made possible through three-wave mixing, where the third wave is strong classical laser light (here, light  141  from a pump laser  140 , described below). Modulating the strength of the classical light permits modulating the strength of the transduction coupling. 
     The transducer  130  and optical resonator  150  are also optically (e.g., evanescently) coupled to an optical waveguide  142  that runs through the refrigerator  102 . The optical waveguide  142  connects the transducer  130  and optical resonator  150  to an optical pump laser  140  outside the refrigerator  102  (the optical pump laser  140  could also be inside the refrigerator  102 , though this could increase the heat load on the refrigerator  102 ). The other end of the optical waveguide  142  splits into two, with one path terminating at an optical beam dump  144  and the other end providing an optical port  146 . 
     In operation, the optical pump laser  140  emits a pulsed pump beam  141 . The pump beam  141  can have a peak pulse power from 10-100 mW and a repetition rate of 100 kHz for superb entanglement quality (0.999 fidelity). Orders of magnitude improvements can reasonably be expected as the pump laser technology matures. The pulse durations should be high to provide a high duty cycle, with a limit imposed by a microwave reset operation performed between each pulse. This microwave reset can take many microseconds, so in typical operation the pulse duration can be microseconds or higher. 
     The waveguide  142  guides this pump beam  141  to the transducer  130 , which transduces the pump photons into microwave photons and optical photons  151 , also called heralding photons, as explained in greater detail below. The microwave photons and optical photons  151  may circulate in the microwave resonator  120  and optical resonator  150 , respectively. The coupling between the optical waveguide  142  and optical resonator  150  is passive. The optical resonator  150  and the input/output waveguide  142  are structurally connected, which enables crosstalk between the two. The classical pump light  141  enters the optical resonator  150  because there is a lot of it in the waveguide  142 . And the single heralding photon  151  simply leaks out of the optical resonator  150  into the waveguide  142  due to the finite lifetime of the optical resonator  150 . The optical waveguide  142  guides the optical photons  151  from the optical resonator  150  and any excess pump photons  141 ′ out of the refrigerator  102 , where a dichroic beam splitter or other filter directs the excess pump photons  141 ′ to the beam dump  144  and the optical photons  151  to the optical port  146 . 
       FIG. 1B  shows how two nodes  100   a  and  100   b  can be connected by an optical fiber  148  or other optical link to form at least part of a quantum network. The nodes  100   a  and  100   b  are in separate refrigerators  102   a  and  102   b  that may be in different parts of the same room, different parts of the same building, or even different countries or continents. The optical fiber  148  connects the nodes&#39; optical ports  146   a  and  146   b . The optical fiber  148  can be at temperatures above cryogenic temperatures (temperatures above 120 K, e.g., about 300 K) instead of being cooled to cryogenic temperature and can have a length of meters to kilometers. 
     Each node  100   a ,  100   b  includes its own QPU  110   a ,  110   b , optical qubits  112   a ,  112   b , microwave resonator  120   a ,  120   b , transducer  130   a ,  130   b , and optical resonator  150   a ,  150   b . (The beam dumps are omitted for clarity.) The nodes  100   a  and  100   b  are coupled to the same optical pump laser  140 ′, e.g., by optical fibers. The optical pump laser  141 ′ is red-detuned with respect to the optical cavities  150   a  and  150   b  (i.e., the optical pump laser  141 ′ emits light at a carrier frequency that is lower than the resonance frequencies of the optical carriers  150   a  and  150   b ) and can be collocated with either node  100   a  or  100   b  or located in separate location, possibly meters to kilometers away from the nodes  100   a  and  100   b.    
     The optical nodes  100   a  and  100   b  in  FIG. 1B  are configured to exchange quantum information using microwave-to-optical and optical-to-microwave transduction. In  FIG. 1B , the transducer  130   a  in node  100   a  converts a microwave photon  121   a  from qubits  112   a  into an optical photon  151 ′ via a χ (2)  nonlinear interaction (sum-frequency generation) between the microwave photon  121   a  and the red-detuned pump beam  141 ′. The resulting optical photon  151 ′ propagates through the waveguide  142   a  to the optical fiber  148 , which guides it to the optical port  146   b  of the other node  100   b . Waveguide  142   b  guides the optical photon  151 ′ from the optical port  146   b  to the transducer  130   b , which converts the optical photon  151 ′ to another microwave photon  121   b  via a difference-frequency interaction with red-detuned pump photons  141 ′. This microwave photon  121   b  is encoded with the quantum information from the qubits  112   a  in the first node  100   a  and couples to the qubits  112   b  in the second node  100   b  via the microwave resonator  120   b.    
     As explained above, the microwave-to-optical and optical-to-microwave transduction processes carried out by the transducers  130   a  and  130   b  in  FIG. 1B  tend to be efficient but noisy. Worse, increasing the efficiency of the transduction processes tends to increase the noise. The noise degrades the fidelity of the quantum information transfer. 
       FIG. 1C  illustrates the nodes  100   a  and  100   b  configured for optically heralded entanglement instead of quantum information transfer by M-O-M transduction. Again, the nodes  100   a  and  100   b  are coupled to the same optical pump laser  140 ″, which in this case can be blue- or red-detuned from (i.e., emits light at a higher or lower carrier frequency than) the resonance frequencies of the optical resonators  150   a  and  150   b  in the nodes  100   a  and  100   b , respectively. The optical resonators  150   a  and  150   b  are tuned to the same (optical) resonance frequency, and the microwave resonators  120   a  and  120   b  are tuned to the same (microwave) resonance frequency. If desired, the microwave resonators  120   a  and  120   b  and the optical resonators  150   a  and  150   b  can be locked to the pump beam  141 , which serves as a global frequency reference. The nodes&#39; optical ports  146   a  and  146   b  are coupled to different input ports of a 2×2 beam splitter  190  via respective optical fibers  148   a  and  148   b . The beam splitter&#39;s output ports are coupled to respective photodetectors  192   a  and  192   b . The beam splitter  190  and photodetectors  192   a  and  192   b  can be at a temperature greater than 120 K (e.g., around 300 K). 
     The nodes  100   a  and  100   b  exchange quantum information using a modified version of the DLCZ protocol, which is named after Duan, Lukin, Cirac and Zoller, the authors who proposed it. The original DLCZ protocol exploits the simultaneous spontaneous Raman emission of photons from and creation of spin excitations in a pair of atomic ensembles to create entanglement between the pair of atomic ensembles. Either or both of the ensembles emits a photon, which is guided to a beam splitter whose outputs are coupled to respective detectors as in  FIG. 1C . The beam splitter couples the photon to one of the detectors, which senses the photon, heralding excitation of one atomic ensemble. Coupling the photon to the detectors via the beam splitter erases information about which path the photon traveled to the beam splitter and hence which atomic ensemble emitted the photon and is excited. Simultaneous detection of a photon at each beam splitter output heralds excitation and entanglement of both atomic ensembles. For more on the original DLCZ protocol, please see L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413 (2001), which is incorporated herein by reference in its entirety. 
     In the modified version of the DLCZ protocol executed by the nodes  100   a  and  100   b , the transducers  130   a  and  130   b  convert each blue-detuned pump photon  141 ″ into a microwave photon  121   a ′,  121   b ′ and a correlated optical photon  151   a ,  151   b  via a nonlinear interaction as described below. Each optical photon  151   a ,  151   b  exits its optical resonator  150   a ,  150   b  and propagates to its corresponding input port on the beam splitter  190  via the corresponding waveguide  142   a ,  142   b ; optical port  146   a ,  146   b ; and optical fiber  148   a ,  148   b . The optical photon  151   a ,  151   b  exits one of the beam splitter&#39;s output ports and is detected by the corresponding photodetector  192   a ,  192   b . However, the mapping between the beam splitter&#39;s input and output ports is not deterministic; rather, this mapping varies in an unknown fashion over time, making it impossible to determine which input port the optical photon  151   a ,  151   b  arrived at from the output port at which the optical photon  151   a ,  151   b  is detected. Put differently, the beam splitter  190  erases information about which path the optical photon  151   a ,  151   b  took to the beam splitter  190  and hence which node  100   a ,  100   b  generated the optical photon  151   a ,  151   b.    
     Detection of a single optical photon  151   a ,  151   b  by one of the photodetectors  192   a ,  192   b  heralds creation of a microwave photon  121   a ′,  121   b ′ at one of the nodes  100   a ,  100   b . Detection of an optical photon  151   a ,  151   b  without the beam splitter  190  simply heralds the creation of a microwave photon at the corresponding node  100   a ′,  100   b ′. Detection of an optical photon  151   a ,  151   b  after the beam splitter  190  heralds the creation of a microwave photon  121   a ′,  121   b ′, but without knowledge about the location of the microwave photon  121   a ′,  121   b ′. The microwave photon  121   a ′,  121   b ′ is in a superposition of being at the left node  100   a ′ or at the right node  100   b ′. The probability of two microwave photons  121   a ′ and  121   b ′ being created at the same time is very low and is a source of infidelity. 
     The operation of the nodes  100   a  and  100   b  shown in  FIG. 1C  can be described more formally as follows. A typical microwave-optical quantum transducer (e.g., transducer  130   a  or  130   b ) can use a χ (2)  nonlinear interaction between a classical optical pump mode  3  (e.g., pump beam  141 ″), an optical mode â (e.g., optical photon  151   a  or  151   b ), and a microwave mode b (e.g., microwave photon  121 ). The classical mode is red-detuned with respect to â leading to the Hamiltonian 
         Ĥ=ĥg   0   {circumflex over (p)}   †   â   †   {circumflex over (b)}+H.c ., 
     where g 0  is the single-photon nonlinear interaction rate and H.c. is the Hermitian conjugate. The pump mode {circumflex over (p)} is in a coherent state with an amplitude high enough that it can be replaced by a classical field with the same amplitude {circumflex over (p)}→√{square root over ( n p   )} where  n p    is the average number of photons in the pump mode. This leads to a beam splitter-type Hamiltonian â † {circumflex over (b)}+H.c. that can be used for transduction. This transduction is deterministic, but its fidelity is low in practice due to the relatively low value of g 0  (e.g., about 1 kHz) compared to the various loss rates in the system. 
     If the pump mode {circumflex over (p)} is instead blue-detuned with respect to the optical mode â the result is two-mode squeezing. This interaction generates pairs of optical and microwave photons via spontaneous parametric down-conversion (SPDC): 
         Ĥ=ĥgâ{circumflex over (b)}+H.c ., 
         g=g   0 √{square root over (   n   p   )}.
 
     In either case the pump power, P, is related to the number of photons in the pump mode as 
     
       
         
           
             
               
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     where γ e  is the extrinsic loss rate of the optical mode and γ=γ e +γ i  is the sum of the extrinsic and intrinsic loss rates. Here ω denotes the frequency of the pump and ω MW  denotes the frequency of the microwave resonator. 
     If the optical mode is coupled to a waveguide ending with a photodetector, this system can herald the production of a single microwave photon by detecting a single optical photon. First, we disclose the performance of this procedure, and then we disclose how it enables the heralding of entanglement between two remote microwave systems (e.g., nodes  100   a  and  100   b  in  FIG. 1C ). 
     The collapse operator that describes the detection of the optical photon is ĉ=√{square root over (γ e )}â, which leads to a stochastic master equation with non-Hermitian effective Hamiltonian given by 
     
       
         
           
             
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     To start, consider the intrinsic loss rate γ i  to be negligible and investigate the waveguide coupling rate γ e  (non-zero γ i  reduces the detection efficiency but not the fidelity). The loss rate γ is typically much larger than g, which simplifies the dynamics. This hardware constraint leads to poor fidelity in even state-of-the-art transduction devices. On the other hand, in our heralding protocol, g&lt;&lt;γ is used for high-fidelity operation, as it ensures the SPDC process is not polluted by higher-number excitations. However, it also limits the rate of photon generation. The lifetime of the microwave oscillator is orders of magnitude longer than the characteristic times of the dynamics studied here and can be considered infinite in the initial analysis. Once we have the desired heralded state in the microwave mode, we swap it out into one of the qubits of the superconducting QPU (e.g., qubits  112   a  or  112   b  in QPU  110   a  or  110   b )—a nonlinear operation at which transmon-based devices are very capable. 
     We denote a Fock state with n a  photons in optical mode â and n b  in microwave mode {circumflex over (b)} as n a n b   . To obtain a  11 ) pair on which we can herald the single microwave photon we simply pump the system and let the Hamiltonian evolve, while waiting for a click at the detector (i.e., the detection of a photon  151   a  or  151   b  by either photodetector  192   a  or  192   b  in  FIG. 1C ). A click heralds the creation of single photon (e.g., microwave photon  121   a ′ or  121   b ′) in the microwave mode (a correct approximation as long as g&lt;&lt;γ e ). Solving the dynamics gives a rate of photon generation under a continuous pump of 
     
       
         
           
             
               
                 
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                         e 
                       
                       + 
                       
                         γ 
                         i 
                       
                     
                     ) 
                   
                   2 
                 
               
             
             , 
           
         
       
     
     due to the γ e /(γ e +γ i ) drop in efficiency. 
     One way to derive this form for r 0  is to restrict oneself to the basis of {∨00),∨11)}. For a state ψ =c 0 ∨00)+c 1 ∨11) we get the following ordinary differential equation (ODE): 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               c 
                               . 
                             
                             0 
                           
                         
                       
                       
                         
                           
                             
                               c 
                               . 
                             
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             
                               - 
                               ig 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 ig 
                                 * 
                               
                             
                           
                           
                             
                               - 
                               
                                 
                                   γ 
                                   e 
                                 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               c 
                               0 
                             
                           
                         
                         
                           
                             
                               c 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     which leads to 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       
                         c 
                         0 
                       
                     
                   
                   
                     
                       
                         c 
                         1 
                       
                     
                   
                 
                 ) 
               
               = 
               
                 
                   e 
                   
                     
                       - 
                       
                         
                           γ 
                           e 
                         
                         4 
                       
                     
                     ⁢ 
                     t 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               γ 
                               
                                 4 
                                 ⁢ 
                                 
                                   g 
                                   ′ 
                                 
                               
                             
                             ⁢ 
                             
                               cosh 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     g 
                                     ′ 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             sinh 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   g 
                                   ′ 
                                 
                                 ⁢ 
                                 t 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             i 
                           
                           ⁢ 
                           
                             g 
                             
                               g 
                               ′ 
                             
                           
                           ⁢ 
                           
                             sinh 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   g 
                                   ′ 
                                 
                                 ⁢ 
                                 t 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     The stochastic master equation formalism gives the probability density of photon detections as 
     
       
         
           
             
               
                 〈 
                 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   | 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
                 〉 
               
               ∼ 
               
                 e 
                 
                   
                     ( 
                     
                       
                         2 
                         ⁢ 
                         g 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ′ 
                       
                       - 
                       
                         
                           γ 
                           e 
                         
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   t 
                 
               
               ∼ 
               
                 e 
                 
                   
                     - 
                     
                       
                         4 
                         ⁢ 
                         
                           
                              
                             g 
                              
                           
                           2 
                         
                       
                       
                         γ 
                         e 
                       
                     
                   
                   ⁢ 
                   t 
                 
               
             
             , 
           
         
       
     
     which corresponds to a Poissonian process of rate r 0 . 
     If we use the same coherent pump (e.g., pump beam  141 ″ from pump laser  140 ″) to drive two separate copies of this system (e.g., nodes  100   a  and  100   b ), we can herald the generation of a single microwave photon in either one of the systems. Erasing the which-path information carried by the heralding optical photon yields a superposition of the heralded microwave photon being in either node. Thus, we obtain the distributed microwave Bell pair |01 ±|10 . The path-erasure is performed by the beam splitter  190  in  FIG. 1C  and the sign of the Bell pair is determined by which detector  192   a  or  192   b  clicks (detects an optical photon). 
     The rate at which these Bell pairs are generated is 
     
       
         
           
             
               
                 r 
                 e 
               
               = 
               
                 2 
                 ⁢ 
                 
                   r 
                   0 
                 
                 ⁢ 
                 
                   e 
                   
                     
                       - 
                       
                         r 
                         0 
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     t 
                   
                 
                 ⁢ 
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     t 
                   
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                     + 
                     
                       t 
                       r 
                     
                   
                 
               
             
             , 
           
         
       
     
     where Δt is the duration of each pump pulse and t r  is the time to reset of the microwave cavity after each attempt (typically on the order of 1 μs). 
     Optical Heralding Performance 
       FIGS. 2A-2C  illustrate simulations of the performance of this heralding technique.  FIG. 2A  is a plot of the entanglement infidelity versus the pump power/number of pump mode photons for the nodes in  FIG. 1C . The upper solid line in  FIG. 2A  represents the fidelity of the generated entanglement (extremely high by any standards). The lower solid, dashed, dotted, and dashed-dotted lines represent the generated entanglement with additional purification steps and/or considering memory infidelities for providing entanglement on-demand to independent consumer hardware. The dotted curve includes both purification and infidelity effects. 
       FIG. 2B  is a plot of the entanglement rate r e  after optimizing with respect to the pump pulse duration Δt versus the pump power/number of pump mode photons for the nodes in  FIG. 1C . This is a figure of merit that represents at the same time the fidelity of each generated microwave/optical photon pair and the rate at which such microwave/optical photon pairs are generated. The “raw pairs” curves present the rate at which entangled pairs are generated, and the “ebit” curves present a theoretical measure of equivalent rate of generation of perfect entangled pairs. Many low-quality microwave/optical photon pairs are in a way equivalent to one high-quality microwave/optical photon pair (this statement is not rigorous but it provides correct intuition). 
       FIG. 2C  shows the in-fridge heating caused by absorption of the optical power that leaks out of the optical resonator as a function of the pump power/number of pump mode photons for the nodes in  FIG. 1C . It should be balanced against the entanglement curves in  FIG. 2A . 
     As discussed above, the rate of entanglement generation drops if the intrinsic optical loss is non-negligible, by a factor of γ e /(γ e +γ i ), without degrading the fidelity of the obtained Bell pair. The fidelity may be less than unity outside of the g&lt;&lt;γ approximation for two reasons: (1) the SPDC process can excite higher-than-one-photon states (i.e., there is a small chance of generating two microwave photons and two optical photons from the same node); and (2) the microwave resonator that was not heralded upon may have a small amplitude c 1 /√{square root over (|c 0 | 2 +|c 1 | 2 )} of being excited too. Both of these infidelities scale as g/γ. 
     While the excitation of the non-heralded cavity may be unavoidable, the higher-photon-number error can be side-stepped using the following method: instead of a blue-detuned pump, use a red-detuned pump as used in transduction; however, reset the microwave cavity to the |1  state. In other words, use the pump to transform a microwave photon into an optical photon using the red-detuned pump and the microwave resonator initialized in the |1  state. This leads to the same ODE as seen in Eq. (1); however, the basis for the evolving state is |ψ =c 0 |01 +c 1 |10 . Given that this Hamiltonian preserves the total photon number, it should be impossible to excite states outside of the {|01 , |10 } subspace. The same protection can also be implemented by keeping the blue-detuned version of the Hamiltonian and using a strongly anharmonic microwave resonator, such that two-photon excitation is suppressed. As seen in  FIG. 2A , this leads to a notable drop in infidelity; however, the issue of partially exciting the non-heralded resonator persists leading to a residual infidelity scaling as g/γ. 
     With typical hardware parameters seen in today&#39;s state-of-the-art devices, we can obtain pair generation rates of 100 kHz at fidelities of 0.99, while suffering 0.1 mW of in-fridge heating due to leakage from the pump. This estimate accounts for the finite lifetimes of the optical and microwave cavities. Through simple single-stage purification performed on the microwave superconducting quantum computer, the infidelity can be lowered by an order of magnitude with a rate decrease of slightly more than 2×. The fidelity after purification may be chiefly limited by the gate-fidelity of the superconducting quantum computer performing the purification. At gate fidelities of 99% the usability of purification is rather limited as the raw entanglement already reaches that level; however,  FIG. 2A  shows that with gate fidelities of 99.9% the purification provides for drastically higher entanglement fidelity while lowering entanglement rate by only a factor of approximately two. Lastly, at low entanglement rates the entanglement fidelity reaches a floor and starts to worsen because at such low rates the entangled microwave state should be stored for a time comparable to the lifetime of the microwave cavity. 
     The entanglement rate scales as 
     
       
         
           
             
               
                 γ 
                 e 
                 2 
               
               
                 
                   
                     ( 
                     
                       
                         γ 
                         e 
                       
                       + 
                       
                         γ 
                         i 
                       
                     
                     ) 
                   
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       ω 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         W 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             γ 
                             e 
                           
                           + 
                           
                             γ 
                             i 
                           
                         
                         ) 
                       
                       2 
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     which is maximal at γ e ≈γ i . However, this also leads to only γ e /(γ e +γ i )=50% of generated photons reaching the photodetector. Due to missing half of the heralding events, the microwave cavity should be reset after each attempt, resulting in a roughly 1 μs delay that limits the maximal rate as seen in the rate plot in  FIG. 2B . If we use a two-cavity resonator, thus having a dedicated resonator for the pump mode, we can have γ e =γ i  for the classical pump mode, while having γ e &gt;&gt;γ i  for the heralding mode. In other words, in addition to physically distinct optical and microwave resonators, a node could include an extra optical resonator, dedicated just to the pump beam (which can be strong enough to couple into the system even if it is not perfectly resonant), to permit more freedom in choosing the coupling parameters. This would let us forgo resetting the microwave resonator in the blue-detuned version of our protocol, but it could also significantly lower the entanglement rate. 
     In summary, at high entanglement rates (high pump powers) the fidelity of entanglement falls due to undesired excitations in the non-heralded cavity, while at low rates the fidelity falls because the entangled pair is stored for a long time in the microwave resonator. If we want higher rates of entanglement generation, we can counteract the drop in fidelity by performing purification. However, pushing for rates higher than 100 kHz could cause heating due to leakage from the pump higher than 0.1 mW. 
     To improve the resilience against this heating we can operate the microwave-optical transducer at a temperature of 1 K, while radiatively coupling the microwave cavity mode to a lower-temperature 10 mK thermal bath. If the microwave mode is over-coupled to the cold bath, then the thermal mode occupancy may be dominated by the 10 mK occupancy rather than the 1 K occupancy. This allows us to utilize the greater cooling power of a 1 K refrigerator (e.g., 360 mW of cooling power at 1 K, compared to typical cooling power of 12 μW at 20 mK) while the noise is dominated by that of the colder stage. 
     This analysis neglects dark count detector errors because they are orders of magnitude lower than other error floors, with dark count rates much lower than 1 kHz. Mismatches in the γ e  coupling rates between the two nodes could lead to a coherent error in which the heralded Bell pair is not a perfectly equal superposition of 01  and 10 , so a high-fidelity hardware implementation may benefit from in situ calibration of the coupling rates. 
     Heralding Entanglement Among Diverse Quantum Devices 
     Our approach extends to heralding entanglement between a diverse set of quantum devices. To do this, we can multiplex the entanglement heralding over multiple frequency channels, thus improving the entanglement generation rate. We can also use our techniques to entangle different quantum modalities, for example, entangling a superconducting device (where the pump, under a χ (2)  interaction, creates a state |00 +ε|11 ) and a trapped ion device (where a conditional reflection of an attenuated pump from a |+  ion state creates a state |0+ +ε|1− ) in which after path erasure we obtain the entangled microwave-ion pair |0+ ±|1− . It can also herald the entanglement of the microwave cavity with a flying photon: one of the nodes is of the architecture shown in  FIGS. 1A-1C , while the other node uses the pump in a SPDC photon pair generation experiment calibrated to have the same generation rate. In other words, one of the nodes can employ a microwave-optical χ (2)  process while the other node employs a purely optical χ (2)  process, leading to heralding the entanglement of a microwave qubit and a flying optical qubit (in the single-rail basis). 
       FIG. 3  shows a quantum optical network that can implement the DLCZ protocol in many different modalities, with multiplexing over time, space (optical fiber), pump wavelength, and/or heralding photon wavelength, provided that the pump frequency equals the sum of the frequencies of the generated optical/microwave photon pair. The quantum optical network includes nodes  300   a - 300   d  of different types connected via optical fibers  348  and a reconfigurable optical add-drop multiplexer (ROADM)  380 , which switches photons of different color among the different optical fibers  348 . 
     Each node  300   a - 300   d  includes its own set of qubits, e.g., qubits  312   a - 1  through  312   a - 4  in node  300   a . The nodes  300   a - 300   d  can be separated from each other by kilometers or more any may have qubits implemented using different technologies. In  FIG. 3 , for example, nodes  300   a  and  300   b  include superconducting qubits  312   a  and  312   b  with microwave resonators, transducers, and optical resonators in separate refrigerators like the nodes  100   a  and  100   b  in  FIG. 1C . Node  300   c  includes a color-center computer with qubits  312   c  that are represented by the spin states of color centers or other spin defect centers in a solid-state host (e.g., nitrogen vacancies in diamond). And node  300   d  includes an ion-trap computer with qubits  312   d  represented as spin states of ions trapped in an optical lattice. Other potential qubit implementations include optically trapped Rydberg atoms. In this example, superconducting qubits  312   a - 1  and  312   b - 1  have optical resonators tuned to the same resonance frequency (or more precisely, blue- or red-detuned by the same amount from that resonance frequency), with color center qubit  312   c - 1  and trapped ion qubit  312   d - 1  also resonant at that resonance frequency. Likewise, qubits  312   a - 2  through  312   d - 2  are resonant at the same frequency, and so on. 
     The ROADM  380  also connects to a mode-locked laser  340  or bank of wavelength-division-multiplexed lasers and a bank of path-erasure detectors  390 - 1  through  390 - 4 . The mode-locked laser  340  and/or path-erasure detector bank  390  can be located together or separately, with or separate from any of the nodes  300   a - 300   d . (Electro-optic modulators, filters, attenuators, and/or other components are omitted for clarity. These components may shape optical photon wave packets from the different nodes to ensure that the optical photons are indistinguishable when detected for higher fidelity.) Each path-erasure detector  390 - 1  through  390 - 4  includes a beam splitter with a pair of input ports coupled to different outputs from the ROADM  380  and a pair of output ports coupled to respective photodetectors. In this example, there is one path-erasure detector per heralding photon frequency, which can be as close as a fraction of a gigahertz apart, hence we can have tens or even hundreds of them on a single telecom band. 
     In operation, the mode-locked laser  340  generates frequency comb with components or comb lines at frequencies of ω 1 , ω 2 , . . . ω N . There can be one comb line per path-erasure detector  390  for spectrally multiplexing by the frequencies of the heralding photons or one comb line per pair of qubits for spectrally multiplexing by the frequencies of the pump pulses. If the pump pulses are at the same frequency and the heralding photons are at different frequencies, then qubits in the different nodes are also spectrally multiplexed. 
     The ROADM  380  can be configured to route pump pulses to two of any four qubits (e.g., qubits  312   a - 1  and  312   c - 1 ) at the same spectral channel for generating a heralding photon from one of the connected qubits. The ROADM  380  routes the heralding photon to path-erasure detector (e.g., path-erasure detector  390 - 1 ) for detecting entanglement between the connected pair of qubits. The beam splitters and photodetectors detect the photons without sensing the photons&#39; origins (only that there are two possible source nodes  300   a - 300   d  for each photon). Each detection heralds entanglement of the pair of qubits coupled via the ROADM  380 . By connecting the ROADM  380  to the path-erasure detector  390  for the corresponding frequency channel, this design enables multiplexed heralded entanglement generation between multiple nodes  300   a - 300   d  of different quantum modalities. 
     CONCLUSION 
     While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize or be able to ascertain, using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. 
     The foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure. 
     Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. 
     All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms. 
     The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” 
     The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc. 
     As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e., “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law. 
     As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc. 
     In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.