Patent Description:
A qubit may be formed from any physical quantum mechanical system with at least two orthogonal states. The two states of the system used to encode information are referred to as the "computational basis. " For example, photon polarization, electron spin, and nuclear spin are two-level systems that may encode information and may therefore be used as a qubit for quantum information processing. Different physical implementations of qubits have different advantages and disadvantages. For example, photon polarization benefits from long coherence times and simple single qubit manipulation, but suffers from the inability to create simple multi-qubit gates.

Different types of superconducting qubits using Josephson junctions have been proposed, including "phase qubits," where the computational basis is the quantized energy states of Cooper pairs in a Josephson Junction; "flux qubits," where the computational basis is the direction of circulating current flow in a superconducting loop; and "charge qubits," where the computational basis is the presence or absence of a Cooper pair on a superconducting island. Superconducting qubits are an advantageous choice of qubit because the coupling between two qubits is strong making two-qubit gates relatively simple to implement, and superconducting qubits are scalable because they are mesoscopic components that may be formed using conventional electronic circuitry techniques. Additionally, superconducting qubits exhibit excellent quantum coherence and a strong non-linearity associated with the Josephson effect. All superconducting qubit designs use at least one Josephson junction as a non-linear non-dissipative element. Article entitled "<NPL>) disclose examples of quantum error-correcting solutions.

According to a first aspect a method is provided as defined by the independent claim <NUM>. According to a second aspect a multi-cavity quantum information system is provided as defined by the independent claim <NUM>. The dependent claims specify preferred embodiments.

Various aspects and embodiments of the disclosed technology will be described with reference to the following figures.

In conventional approaches to quantum information processing with superconducting circuits, the information is stored in Josephson-junction based qubits (e.g., transmons) coupled together via exchange of microwave photons in bus resonators or via direct capacitive coupling. The inventors have recognized and appreciated that a different and complementary architecture in which the quantum information is stored and manipulated in microwave photon states of high-Q resonators with transmons acting as ancillae to give universal quantum control of the photonic qubits has significant advantages over the conventional approaches. First, coherence times of microwave cavities are longer than coherence times of superconducting qubits. For example, three-dimensional superconducting microwave cavities can achieve extremely high quality factors approaching <NUM> in aluminum, and still higher in niobium, with cavity coherence times for aluminum cavities exceeding that of the transmon by two orders of magnitude. Second, the number of states in which information can be encoded is larger in a microwave cavity than in a superconducting qubit. For example, the higher excitation levels of the cavity mode expand the available Hilbert space so that a single cavity mode can be used to encode quantum bits of information in a manner compatible with quantum error correction against various imperfections, including cavity dephasing, excitation loss or thermal heating.

The inventors recognized and appreciated that these advantages of using microwave cavities can be utilized in a unique approach to quantum information processing where quantum information is stored in the quantum state of a microwave cavity while using the transmon as an ancilla to assist quantum information processing and quantum error correction. This 'photonic qubit' approach is the reverse of the conventional approach in which quantum information is stored in the transmons and microwave photon modes are used as quantum busses to couple the transmons. The resulting superconducting cavity-transmon system allows universal quantum control over the cavity mode, quantum error correction reaching the break-even point, joint parity measurement over two cavities, and deterministic coupling gates between the cavities assisted by a transmon controller.

The inventors have further recognized and appreciated that the superconducting cavity-transmon system may encounter limited fidelity of quantum operations, due to the transmon decoherence during the cavity-transmon coupling. The inventors have recognized and appreciated that, besides experimentally improving the coherence properties of the transmon, the fidelity of quantum operations can be improved by implementing robust approaches to suppress or even actively correct the errors due to the decoherence from transmon and other Josephson nonlinear devices.

The inventors have further recognized and appreciated that the universal set of quantum logic gates are "encoding agnostic," meaning no matter how the quantum information is chosen to be logically encoded in the quantum state of the cavity (e.g., cat state encoding, coherent state encoding, Fock state encoding, etc.), the set of quantum logic gates remains universal. Moreover, the hardware used to implement the universal set of quantum logic gates remains the same, independent of the chosen encoding scheme.

According to some embodiments, multiple microwave cavities are controlled and/or coupled together using one or more Josephson nonlinear devices such as a transmons. In some embodiments, a coupling transmon is used to perform operations between two cavities. Examples of such operations include beam splitter (BS) operations that couple the quantum state of a first cavity to the quantum state of a second cavity and vice versa. These operations are referred to as beam splitter operations because they play the same role and implement the same unitary operation as beam splitters do in linear optics quantum computation (LOQC) schemes.

In some embodiments, an ancilla transmon that is coupled to a single cavity is used to implement controlled phase shift (CPS) operations between the single cavity and the transmon. In some embodiments, the ancilla transmon may also be controlled to implement rotations on the quantum state of the ancilla transmon itself.

The inventors have recognized and appreciated that the BS operations, CPS operations, and rotations of the ancilla transmon are sufficient to implement universal gate based quantum computation. In some embodiments, the above operations are used to implement c-SWAP and/or e-SWAP gates. The e-SWAP gate couples cavity modes while preserving the bosonic code space. In this way, entangling operations between bosonic modes can be achieved regardless of the logical encoding used. Some embodiments utilize a robust design of the e-SWAP gate that can herald imperfections due to transmon decoherence. Some embodiments utilized the c-SWAP and e-SWAP gates for one or more applications, such as quantum routers, quantum random access memory (RAM), quantum principle component analysis, and gate-based universal quantum computation.

It is noted that, while a transmon superconducting qubit is described herein as a component of multiple embodiments, other types of superconducting devices may be used in some embodiments.

<FIG> is a schematic diagram of a multi-cavity quantum information system <NUM>, according to some embodiments. The multi-cavity quantum information system <NUM> includes a first cavity <NUM>, a second cavity <NUM>, a coupling transmon <NUM>, a first ancilla transmon <NUM>, a second ancilla transmon <NUM> and a microwave source <NUM>. While only two cavities and two ancilla transmon are shown in the example system <NUM> of <FIG>, other embodiment may include additional cavities and transmons.

The first cavity <NUM> and the second cavity <NUM> may be any type of cavity that supports quantum states of microwave radiation. For example, in some embodiments, the first cavity <NUM> and the second cavity <NUM> may be a transmission line resonator or a three-dimensional cavity formed from a superconducting material, such as aluminum.

The coupling transmon <NUM> may be a transmon that is dispersively coupled to both the first cavity <NUM> and the second cavity <NUM>. The coupling transmon <NUM> provides an interaction between the quantum states of the two cavities, allowing for interactions such as BS interactions to be performed between the first cavity <NUM> and the second cavity <NUM>.

The first ancilla transmon <NUM> and the second ancilla transmon <NUM> are dispersively coupled to the first cavity <NUM> and the second cavity <NUM>, respectively. Each ancilla transmon is coupled to a single cavity of the system <NUM> and not to any other cavity used to store quantum information. The ancilla transmons <NUM> and <NUM> can be controlled to implement rotations of the quantum state of the transmons <NUM> and <NUM> themselves. Additionally, the ancilla transmons <NUM> and <NUM> can be controlled to interact with the associated cavities <NUM> and <NUM>, respectively, to implement controlled interactions such as a CPS interaction.

The microwave source <NUM> may be coupled to the first cavity <NUM>, the second cavity <NUM>, the first ancilla transmon <NUM>, the second ancilla transmon <NUM>, and the coupling transmon <NUM>. The coupling between the microwave source <NUM> and the other components provides a way for the microwave source to apply microwave radiation to each of the components of the multi-cavity quantum information system <NUM>. In some embodiments, the microwave source <NUM> may be capacitively coupled to each of the components.

<FIG> is a schematic diagram of a particular example system <NUM> that may play the role of the multi-cavity quantum information system <NUM> of <FIG> (for simplicity the microwave source <NUM> is not shown), according to some embodiments. The system <NUM> includes a first three-dimensional (3D) cavity <NUM>, a second 3D cavity <NUM>, a coupling device <NUM>, a first ancilla device <NUM>, and a second ancilla device <NUM>.

The first and second 3D cavities <NUM> and <NUM> acts as a 3D version of a λ/<NUM> transmission line resonator between a central stubs <NUM> and <NUM>, respectively, and outer walls <NUM> and <NUM>, respectively. For example, the diameter of central stubs <NUM> and <NUM> may be <NUM> and the diameter of the outer walls <NUM> and <NUM> may be <NUM>. It is noted, however, that embodiments are not limited to any particular dimensions. The resonant frequency of each of the cavities <NUM> and <NUM> may be determined by the height of the central stub <NUM> and <NUM> within their respective cavity. For example the central stub <NUM> may have a height of <NUM> and the second central stub <NUM> may have a height of <NUM>. The first 3D cavity <NUM> supports microwave radiation <NUM> of a first frequency and the second 3D cavity <NUM> supports microwave radiation <NUM> of a second frequency that is different from the first frequency. In some embodiments, the first cavity <NUM> and the second cavity <NUM> include ports <NUM> and <NUM>, respectively, through which microwave radiation from the microwave source <NUM> may be applied. Applying microwave radiation to a cavity may, for example, implement a displacement operation on the quantum state of the cavity.

The coupling device <NUM> includes a coupling transmon <NUM> that provides a nonlinear interaction between the first cavity <NUM> and the second cavity <NUM>. The transmon <NUM> is coupled to a first antenna <NUM> that is inserted at least partially into the first cavity <NUM> and a second antenna <NUM> that is inserted at least partially into the second cavity <NUM> such that at least a portion of each antenna protrudes into its respective cavity. The first and second antennas <NUM>/<NUM> may be, for example, circular pads that provide capacitive coupling to the first and second cavities <NUM>/<NUM>, respectively.

The coupling device <NUM> also includes a resonator <NUM> that provides the ability to readout the state of the transmon <NUM>. A third antenna <NUM> couples the resonator <NUM> to the resonator <NUM>. In some embodiments, the resonator <NUM> is a quasi-planar resonator with a lower Q value than either the first cavity <NUM> or the second cavity <NUM>. In some embodiments, the transmon <NUM> and the resonator <NUM> are fabricated on a single sapphire substrate. A readout pulse of microwave radiation may be received by a pump port <NUM> and a resulting microwave signal may be received from readout port <NUM>.

The nonlinearity of the transmon <NUM> of the coupling device <NUM> enables four wave mixing, which is used to perform a frequency-converting bilinear coupling between the first cavity <NUM> an the second cavity <NUM>. The four-wave mixing is controlled by pumping the transmon <NUM> via a pump port <NUM> with microwave radiation that satisfies the frequency matching condition ω<NUM> - ω<NUM> = ωp<NUM> - ωp<NUM>, where ω<NUM> is the resonant frequency of the first cavity <NUM>, ω<NUM> is the resonant frequency of the second cavity <NUM>, ωp<NUM> is the frequency of the first pump associated with a mode c, and ωp<NUM> is the frequency of the second pump associated with a mode d. This coupling implements an effective time-dependent BS interaction between the cavity modes. As is known from conventional optics, the unitary evolution of the beam splitter is described by the unitary operator: <MAT> where <MAT> and <MAT>.

For θ = π/<NUM>, the beam splitter unitary operator implements the SWAP operation that exchanges the states between the two cavity modes associated with the annihilation operators a and b, respectively. For θ = π/<NUM> and θ = -π/<NUM> the unitary operator corresponds to a <NUM>/<NUM> beam splitter. Different from ordinary optics, the microwave cavity eigenmodes have different frequencies and do not couple to each other without a nonlinearity. However, the Josephson nonlinearity and additional parametric pumps can be used to compensate for the cavity frequency difference, so that modes with different frequencies can be coherently coupled. For example, based on the four-mode coupling g<NUM>a†bc†d + h. represents the Hermitian conjugate of the first term and g<NUM> is the four mode coupling constant) from the Josephson non-linearity, the modes c and d may be pumped so that they can be approximated by classical coherent states with amplitudes 〈c〉 = Ac(τ) and 〈d〉 = Ad(τ), which leads to an effective beam-splitter coupling Hamiltonian in Eqn. (<NUM>) with g(τ) = g<NUM>Ac*(τ)Ad(τ). Note that g(τ) may be tuned by controlling the amplitudes and phases of Ac(τ) and Ad(τ). In this way, some embodiments can easily switch on/off the beam-splitter coupling with extremely high on/off ratio. This is a distinct advantage over 'always-on' capacitive or other fixed couplings between qubits. In addition, by pumping modes c and d so that the sum of their frequencies is equal to the sum of the cavity mode frequencies, one can realize a bi-linear coupling of the form HS=f(τ)a†b† +f*(τ)ab. With these two operations one can perform any linear symplectic transformation between the two cavities.

In some embodiments, the above approach of implementing the unitary beam splitter operator using the Josephson non-linearity can be made robust against imperfections in the Josephson device. For example, if the intrinsic non-linearity of the device is weak but the parametric pumping is strong, the effect of thermal noise in modes c and d may be suppressed by the fact that this noise is small relative to the large coherent state amplitudes Ac and Ad. Operation in this regime may also increases the linearity of the beam splitter so that SWAP operations can be carried out for quantum states containing a wide range of photon numbers.

The beam splitter unitary transformation is a very useful element for quantum information processing. For example, while evolution of a Gaussian input state (e.g., coherent states, squeezed states) acted upon by the beam splitter unitary operator can be efficiently simulated with a classical computer, evolution of a non-Gaussian input state (e.g., Fock states) may lead to non-trivial output states. For example, the complexity of boson sampling illustrates the non-trivial statistical properties of the output state which are hard to simulate with classical computers. Moreover, the beam splitter unitary combined with both single-photon sources and photon detectors can achieve universal linear optical quantum computation (LOQC), albeit with major challenges that include the probabilistic nature of entangling gates and extremely daunting resource overhead.

In some embodiments, rather than being restricted to linear optical quantum computing, additional nonlinear elements may be used to facilitate quantum information processing. For example, using the physical platform of superconducting circuits with microwave photons not only provides the capabilities of single-photon sources and photon detectors, but also includes at least one highly controllable transmon that can be used as two-level or multi-level ancillae. In some embodiments, quantum operations that combine the beam splitter unitary operator and cavity-transmon dispersive coupling gates are used to perform quantum information processing operations. In some embodiments, the dispersive coupling gates are still linear optics transformations that are controlled by (e.g., based upon and/or conditioned on) the quantum state of a transmon (or other) ancilla. This merging of the capabilities of linear optics and gate-based quantum computation is powerful and allows one to carry out gate operations on qubits logically encoded in photon states in a manner that is independent of the particular logical encoding. Thus, in some embodiments, the logical encoding of the information can be changed while using the same hardware with the same operations.

First ancilla device <NUM> is similar to the coupling device <NUM>, but only couples to a the first cavity <NUM>, not both cavities. The first ancilla device includes a pump port <NUM> for driving a transmon <NUM> with pump and readout pulses of microwave radiation and a readout port <NUM> for receiving readout microwave signals from the transmon <NUM>. The transmon <NUM> is coupled to the first cavity <NUM> via a first antenna pad <NUM> that at least partially protrudes into the first cavity <NUM>. A second antenna pad <NUM> couples the transmon <NUM> to a quasi-planar resonator <NUM>.

The second ancilla device <NUM> is similar to the first ancilla device <NUM>, but is coupled to only the second cavity <NUM>, not the first cavity <NUM>. The second ancilla device includes a pump port <NUM> for driving a transmon <NUM> with pump and readout pulses of microwave radiation and a readout port <NUM> for receiving readout microwave signals from the transmon <NUM>. The transmon <NUM> is coupled to the first cavity <NUM> via a first antenna pad <NUM> that at least partially protrudes into the first cavity <NUM>. A second antenna pad <NUM> couples the transmon <NUM> to a quasi-planar resonator <NUM>.

The first and second ancilla devices <NUM> and <NUM> may be used to implement a CPS operation, which is represented as: <MAT> where n̂ = a†a is the number operator of the bosonic mode of the particular cavity coupled with the transmon. In some embodiments, the phase shift is π and resulting in the implementation of a controlled-Parity operation since the photon number parity operation is P̂ = (—<NUM>)a†a. In some embodiments, the CPS gate can be obtained from the time evolution under the Hamiltonian with dispersive coupling between the ancilla transmon and the respective cavity <MAT> for a time duration t = π/χ and coupling strength χ.

An example set of parameters for implementing the quantum information system <NUM> is as follows: the first cavity <NUM> may have a kerr/2π = <NUM> and ω<NUM>/<NUM>π = <NUM>; the second cavity <NUM> may have a kerr/<NUM>π = <NUM> and ω<NUM>/<NUM>π = <NUM>; the coupling device <NUM> may have α/2π = <NUM>, ω/<NUM>π = <NUM>, χc<NUM>/<NUM>π = <NUM>, and χc<NUM>/<NUM>π = <NUM>; the first ancilla transmon <NUM> may have α/<NUM>π = <NUM>, ω/<NUM>π = <NUM>, χ/<NUM>π = <NUM>; and the second ancilla transmon <NUM> may have α/<NUM>π = <NUM>, ω/<NUM>π = <NUM>, χ/<NUM>π = <NUM>.

A c-SWAP gate is implemented using a combination of BS operations and CPS operations. A c-SWAP gate operates on two cavity modes and one of the ancilla transmons, swapping the states of the two cavities based on the state of the ancilla transmon. The unitary c-SWAP operator can therefore be written as: <MAT>
where |g〉 and |e〉 represent the ground state and the first excited state of the ancilla transmon, respectively. Because (c-SWAP)<NUM> = I, c-SWAP is reversible. The c-SWAP gate is sometimes called the Fredkin gate, which is universal for classical computation. For quantum computation, c-SWAP and single-qubit rotations form a set of quantum gates capable of universal computation.

<FIG> illustrates a quantum circuit diagram <NUM> for a c-SWAP gate. The lines <NUM>, <NUM>, and <NUM> represent the first cavity <NUM>, the second cavity <NUM> and an ancilla transmon <NUM>. In quantum circuit diagrams, operations are performed on the various components as a function of time, from left to right. The symbol for the c-SWAP gate includes an "X" at each of the cavity lines <NUM> and <NUM> and a dot on the ancilla transmon line <NUM>.

<FIG> illustrates an example quantum circuit diagram <NUM> for implementing a c-SWAP gate using BS operations and CPS operations. First, a first BS operation <NUM> with θ = π/<NUM> is performed between the first cavity <NUM> and the second cavity <NUM>. As described above, the BS operation may be performed using the coupling transmon <NUM>. After the first BS operation <NUM>, a CPS operation <NUM> between the ancilla transmon <NUM> and the second cavity <NUM> is performed. Finally, a second BS operation <NUM> with θ = -π/<NUM> is performed between the first cavity <NUM> and the second cavity <NUM>.

As shown in <FIG>, the c-SWAP gate can be represented in terms of conventional linear optical diagram with a first optical mode <NUM> a second optical mode <NUM> and a transmon <NUM>. The diagram is a Mach-Zehnder interferometer <NUM> built from a first beam splitter <NUM>, a second beam splitter <NUM>, and a controlled phase shifter <NUM> that implements a <NUM> or a π phase shift on one arm controlled by the state of the transmon. For transmon state |g〉, there is a <NUM> phase shift and thus no exchange of the two bosonic modes. For transmon state |e〉, there is a π phase shift (for each and every excitation coupled with the transmon) leading to full exchange (SWAP) between the two optical modes <NUM> and <NUM>. Hence, in some embodiments, the c-SWAP can be used as a special quantum-controlled router, which uses a quantum state (e.g., the state of an ancilla transmon) to control the pathway of quantum signals (carried by the optical modes).

An e-SWAP operation is performed using a combination of c-SWAP gates and ancilla transmon rotations. The e-SWAP operates on two cavities and is represented by the unitary operator: <MAT>.

For θ = π/<NUM>, an e-SWAP gate is equivalent to a SWAP gate, where the resulting global phase shift (i) is non-observable. For θ = π/<NUM>, <MAT> is a coherent combination of the identity operator and the SWAP operator, and is sometimes denoted as <MAT>. Single-qubit rotations and the <MAT> operator operating on qubits form a set of universal quantum gates.

There are similarities and differences between the e-SWAP operator and the beam-splitter unitary operator. For the bosonic subspace with zero and one total excitations, the two are equivalent (e.g., UBS(θ)|<NUM>a, <NUM>b〉 = cosθ|<NUM>a, <NUM>b〉 + isinθ|<NUM>a, <NUM>b〉 = UeSWAP(θ)|<NUM>a, <NUM>b〉). However, for the subspace with more than one total excitations, the two operators behave differently (e.g., <MAT>, which is distinct from UeSWAP(θ)|<NUM>a, <NUM>b〉 = |<NUM>a, <NUM>b〉).

One feature of e-SWAP operator is that it preserves the logical subspace with respect to single-mode bosonic encodings that contain arbitrary numbers of bosons. For logical states of arbitrary single-mode bosonic encoding |ϕ<NUM>〉,
<MAT>
the e-SWAP operation UeSWAP(θ)|ϕ<NUM>〉a|ϕ<NUM>〉b = cosθ |ϕ<NUM>〉a|ϕ<NUM>〉b + isinθ |ϕ<NUM>〉a|ϕ<NUM>〉b preserves the code space for any bosonic codes and for any parameter θ. This important property enables one to carry out quantum information processing with different choices of bosonic encoding using the same hardware. This powerful feature gives great flexibility to the hardware and allows experimentation with different encodings for quantum error correction via 'software updates' on fixed hardware.

In some embodiments, the e-SWAP operator between two cavity modes can be implemented using a two-level ancilla transmon. For example, <FIG> is a quantum circuit diagram <NUM> between a first cavity <NUM>, a second cavity <NUM> and an ancilla transmon <NUM>. The illustrated method for implementing the e-SWAP operation is as follows: (<NUM>) initialize the ancilla transmon <NUM> to the quantum state |+〉 = <NUM>/√<NUM>(|g〉 +|e〉); (<NUM>) perform a first c-SWAP operation <NUM> between the first cavity <NUM> and the second cavity <NUM> controlled based on the state of the ancilla transmon <NUM>; (<NUM>) rotate <NUM> the ancilla transmon <NUM> by angle <NUM>θ around the X axis Xθ = eiθσx; and (<NUM>) perform a second c-SWAP operation <NUM> between the first cavity <NUM> and the second cavity <NUM> controlled based on the state of the ancilla transmon <NUM>. After the foregoing method, the ancilla transmon is restored to the initial state | + 〉 and decoupled from the two cavity modes; meanwhile, the two cavity modes undergo the e-SWAP operation, UeSWAP(θ). If the ancilla transmon <NUM> is measured <NUM>, the result, assuming no errors, is the initial state | + 〉.

<FIG> illustrates a quantum circuit diagram <NUM> for implementing the e-SWAP operation of <FIG> using BS operations and CPS operations. In this example, the c-SWAP gates are simply replaced with the c-SWAP method shown in <FIG>. Thus, the method of implementing the e-SWAP gate includes: (<NUM>) initializing the ancilla transmon <NUM> to the quantum state | + 〉 = <NUM>/√<NUM>(|g〉 +|e〉); (<NUM>) performing a first BS operation <NUM> with θ = π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a first CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a second BS operation <NUM> with θ = -π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a rotation <NUM> on the ancilla transmon <NUM> by angle <NUM>θ around the X axis Xθ = eiθσx; (<NUM>) performing a third BS operation <NUM> with θ = π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a second CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a fourth BS operation <NUM> with θ = -π/<NUM> between the first cavity <NUM> and the second cavity <NUM>. As in <FIG>, if the ancilla transmon <NUM> is measured <NUM>, the result, assuming no errors, is the initial state | + 〉.

In some embodiments, a simplified and more robust implementation of the e-SWAP operator can be obtained by decomposing the c-SWAP operators into beam splitter operators and CPS gates in a way that is different from simply substituting the quantum circuit diagram <NUM> of <FIG> into the quantum circuit diagram <NUM> of <FIG>. The simplification of the quantum circuit diagram <NUM> comes from the realization the two of the beam splitter operations are redundant (e.g., the second and third beam splitter operations <NUM> and <NUM>) as they cancel each other and can therefore be removed. In addition, the transmon can be initialized in the ground state and additional Hadamard gates can be added to act on the transmon just before the first CPS gate <NUM> and after the second CPS gate <NUM>, so that the transmon is kept in the ground state during the beam splitter unitary operations.

With the aforementioned two changes to the quantum circuit, a more robust quantum circuit diagram <NUM> for the e-SWAP operation is achieved, as illustrated in <FIG>. The method of implementing the e-SWAP gate includes: (<NUM>) initializing the ancilla transmon <NUM> to the quantum state | g 〉; (<NUM>) performing a first BS operation <NUM> with θ = π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a first Hadamard operation <NUM> on the ancilla transmon <NUM>; (<NUM>) performing a first CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a rotation <NUM> on the ancilla transmon <NUM> by angle <NUM>θ around the X axis Xθ = eiθσx; (<NUM>) performing a second CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a second Hadamard operation <NUM> on the ancilla transmon <NUM>; (<NUM>) performing a second BS operation <NUM> with θ = -π/<NUM> between the first cavity <NUM> and the second cavity <NUM>. If the ancilla transmon <NUM> is measured <NUM>, the result, assuming no errors, is the initial state | g 〉.

The simplified quantum circuit diagram <NUM> include the aforementioned changes because the beam splitter operation may be relatively slow compared to the other operations. Thus, the quantum circuit in <FIG> has both a shorter total time duration (which reduces the risk of an error occurring in the cavity states) and a shorter duration for the period in which the ancilla transmon is in the excited state (which reduces the risk of an error occurring on the ancilla). In the quantum circuit designs of <FIG>, the transmon is never in the ground state, making the transmon vulnerable to decoherence throughout the entire operation, especially during the relatively slow beam splitter unitary operation. In contrast, the quantum circuit diagram <NUM> of <FIG> keeps the transmon in the ground state, except during the relatively fast Hadamard, CPS and Xθ gates. Therefore, in some embodiments, the quantum circuit design efficiently mitigates imperfections due to transmon decoherence, reducing the error of the overall quantum gate from O[γt(tBS+tCPS+tTrans)] to O[γt(tCPS + tTrans)], where γt is the transmon decoherence rate, tBS, tCPS, and tTrans are times associated with the beam splitter, CPS, and transmon rotation gates (e.g., Xθ and H), respectively. In some embodiments, tBS( ~ <NUM>µs)»tCPS( ~ <NUM>µs)»tTrans( ~ <NUM>ns), and making it advantageous to eliminate the vulnerability to transmon decoherence during tBS.

In some embodiments, the e-SWAP operator can be extended to operate over more than two cavities. For example, <FIG> illustrates a quantum circuit diagram <NUM> where an e-SWAP is performed using four cavities with UeSWAP(a,b;a',b')(θ) = exp[iθSWAPa,b SWAPa',b']. The four-cavity e-SWAP method <NUM> includes, after initilizing the ancilla in the : (<NUM>) initializing the ancilla transmon <NUM> to the quantum state | + 〉 = <NUM>/√<NUM>(|g〉 + |e〉); (<NUM>) performing a first c-SWAP operation <NUM> between the first cavity <NUM> and the second cavity <NUM> controlled by the state of the ancilla transmon <NUM>; (<NUM>) performing a second c-SWAP operation <NUM> between the third cavity <NUM> and the fourth cavity <NUM> controlled by the state of the ancilla transmon <NUM>; (<NUM>) performing a rotation <NUM> on the ancilla transmon <NUM> by angle <NUM>θ around the X axis Xθ= eiθσx; (<NUM>) performing a third c-SWAP operation <NUM> between the third cavity <NUM> and the fourth cavity <NUM> controlled by the state of the ancilla transmon <NUM>; and (<NUM>) performing a fourth c-SWAP operation <NUM> between the first cavity <NUM> and the second cavity <NUM> controlled by the state of the ancilla transmon <NUM>. If the ancilla transmon <NUM> is measured <NUM>, the result, assuming no errors, is the initial state | +.

Similar to the procedure of <FIG> for the e-SWAP between two modes, the e-SWAP operation for four modes can be decomposed into beam splitter operations and CPS gates (<FIG>) and converted to a more robust quantum circuit (<FIG>). Thus, the method of implementing the e-SWAP gate illustrated by the quantum circuit diagram <NUM> in <FIG> includes: (<NUM>) initializing the ancilla transmon <NUM> to the quantum state | + 〉 = <NUM>/√<NUM>(|g〉 + |e〉); (<NUM>) performing a first BS operation <NUM> with θ = π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a second BS operation <NUM> with θ = π/<NUM> between the third cavity <NUM> and the fourth cavity <NUM>; (<NUM>) performing a first CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a second CPS operation <NUM> between the third cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a third BS operation <NUM> with θ = -π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a fourth BS operation <NUM> with θ = -π/<NUM> between the third cavity <NUM> and the fourth cavity <NUM>; (<NUM>) performing a rotation <NUM> on the ancilla transmon <NUM> by angle <NUM>θ around the X axis Xθ= eiθσx; (<NUM>) performing a fifth BS operation <NUM> with θ = π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a sixth BS operation <NUM> with θ = π/<NUM> between the third cavity <NUM> and the fourth cavity <NUM>; (<NUM>) performing a third CPS operation <NUM> between the third cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a fourth CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a seventh BS operation <NUM> with θ = -π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a eighth BS operation <NUM> with θ = -π/<NUM> between the third cavity <NUM> and the fourth cavity <NUM>. If the ancilla transmon <NUM> is measured <NUM>, the result, assuming no errors, is the initial state | + 〉.

It should be noted that not all operations have to be performed in the order shown. For example, the first BS operation <NUM> and the second BS operation <NUM> are illustrated to be performed at the same time. In some embodiments, either BS operation may be performed before the other. The same is true for the third BS operation <NUM> and the fourth BS operation <NUM>; the fifth BS operation <NUM> and the sixth BS operation <NUM>; and the seventh BS operation <NUM> and the eighth BS operation <NUM>. Also, the first CPS operation <NUM> is illustrated as occurring before the second CPS operation <NUM>. In some embodiments, the second CPS operation <NUM> may be performed before the first CPS operation <NUM>. Similarly, the fourth CPS operation <NUM> may be performed before the third CPS operation <NUM>.

Following the example of simplifying the quantum circuit diagram of <FIG> and making the method more robust to errors, the quantum circuit diagram <NUM> of <FIG> includes: (<NUM>) initializing the ancilla transmon <NUM> to the quantum state | g 〉; (<NUM>) performing a first BS operation <NUM> with θ = π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a second BS operation <NUM> with θ = π/<NUM> between the third cavity <NUM> and the fourth cavity <NUM>; (<NUM>) performing a first Hadamard operation <NUM> on the ancilla transmon <NUM>; (<NUM>) performing a first CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a second CPS operation <NUM> between the third cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a rotation <NUM> on the ancilla transmon <NUM> by angle <NUM>θ around the X axis Xθ = eiθσx; (<NUM>) performing a third CPS operation <NUM> between the third cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a fourth CPS operation <NUM> between the second cavity <NUM> and the ancilla transmon <NUM>; (<NUM>) performing a second Hadamard operation <NUM> on the ancilla transmon <NUM>; (<NUM>) performing a third BS operation <NUM> with θ = -π/<NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) performing a fourth BS operation <NUM> with θ = -π/<NUM> between the third cavity <NUM> and the fourth cavity <NUM>. If the ancilla transmon <NUM> is measured <NUM>, the result, assuming no errors, is the initial state | g 〉.

It should be noted that not all operations have to be performed in the order shown. For example, the first BS operation <NUM> and the second BS operation <NUM> are illustrated to be performed at the same time. In some embodiments, either BS operation may be performed before the other. The same is true for the third BS operation <NUM> and the fourth BS operation <NUM>. Also, the first CPS operation <NUM> is illustrated as occurring before the second CPS operation <NUM>. In some embodiments, the second CPS operation <NUM> may be performed before the first CPS operation <NUM>. Similarly, the fourth CPS operation <NUM> may be performed before the third CPS operation <NUM>.

Returning now to the quantum circuit diagram <NUM> of <FIG>, the presence of dephasing and decay errors are discussed. Since tCPS»tTrans, the focus is on transmon errors during the two CPS gates, while neglecting the errors during Xθ.

In some embodiments, the quantum circuit diagram <NUM> for the e-SWAP operation illustrated in <FIG> can herald transmon dephasing errors, which occur with a dephasing rate γϕ. The transmon dephasing error in the g-e subspace can be characterized by the quantum channel
<MAT>
where <MAT> for transmon dephasing error probability during each CPS gate and σZ = |g〉〈g| - |e〉〈e| for π relative phase jump between the |g) and |e〉 states. Transmon dephasing during either of the two CPS gates results in a measured |e〉 state for the transmon at the measurement <NUM>, which is orthogonal to |g) and can be detected without ambiguity. Hence, in some embodiments, any first-order (i.e. single-occurrence) transmon dephasing errors during the CPS gates are detected, though as noted above, which of the two errors occurred cannot be determined and, therefore, the errors are only heralded and cannot be corrected.

In some embodiments, additional levels of the ancilla transmon, beyond the ground state |g〉 and the first excited state |e〉, may be used to improve the robustness of the e-SWAP operation. For example, first-order errors associated with spontaneous decay of the transmon state may be both heralded and corrected. In some embodiments, the cavity-transmon coupling Hamiltonian is altered to be "error transparent", so that the leading order error of transmon decay commutes with the error transparent Hamiltonian for all logical states of the cavities. The transmon decay that occurred during the evolution can then be identified as the transmon decay that occurred at the end of the evolution, which can significantly simplify the error analysis.

In some embodiments, four transmon levels (|g〉, |r〉, |r'〉, |e〉) are used, where the |g〉-|e〉 subspace is used to encode a qubit of information, while the |r〉-|r'〉, |subspace is used to detect errors. <FIG> illustrates an energy level diagram <NUM> for the transmon. The state |e〉 <NUM> represents some higher excited level (e.g., higher than the first excited state), that does not directly decay to state |g〉 <NUM>, but decays to state |r〉 <NUM> as a leading decay error. By introducing additional levels, the transmon decay error from the |g〉-|e〉 subspace can be characterized by the quantum channel <MAT> where <MAT> and K<NUM> = e-γ<NUM>tCPS/<NUM> |r〉〈e|. The subspace spanned by |g〉-|e〉 is used to encode the qubit, whereas the subspace spanned by |e〉, |r〉, and |r'〉 has uniform strength of dispersive coupling with the cavity mode.

In some embodiments, a quantum circuit similar to the circuit shown in <FIG> is used, but the unitary operations are generalized to the four-level transmon. In some embodiments, controllable back-action to the cavity modes is controlled by engineering the dispersive coupling between the transmon and cavity <MAT> to have identical dispersive shift χ for states |e〉, |r〉, and |r'〉, so that the CPS gate is <MAT>.

In some embodiments, the unitary operator associated with a generalized Hadamard gate becomes <MAT>.

Thus, the Hadamard rotation is performed within the g-e subspace and acts trivially over the r-r' subspace.

In some embodiments, the transmon rotation becomes <MAT> which rotates within the g-e subspace and swaps states |r〉 and |r'〉.

In some embodiments, the CPS operation becomes <MAT>.

With the above extended gates over the four-level transmon, it is possible to detect first-order transmon dephasing errors and correct first-order transmon decay errors.

Transmon decay during the first CPS operation <NUM> results in measurement <NUM> of |r'〉 for the transmon state, while the cavity modes continue to evolve under the CPS gate without error because the decay does not change the dispersive coupling. Transmon decay during the second CPS operation <NUM> results in measurement of |r〉 for the transmon state, while the cavity modes evolve as the desired UeSWAP(θ). Because the transmon decay errors (|r'〉 and |r〉 states associated with transmon decay during the first and second CPS gates, respectively) can be unambiguously distinguished, the first-order transmon decay errors during the CPS gates can be actively corrected in some embodiments.

In some embodiments, three transmon levels (|g〉, |r〉, |e〉) may be used by collapsing the r-r' subspace to a single state |r〉 and reducing the operations within the r-r' manifold to trivial operation on state |r〉 (e.g., Xθ |r〉 = |r〉). In some embodiments, such a detection scheme can also detect transmon thermal heating (e.g., |g〉 → |r〉). A three-level transmon ancilla mode is therefore sufficient to achieve detection of first-order transmon dephasing/decay/heating errors during the CPS gates.

In some embodiments, both decay and heating errors of the transmon are corrected by deploying additional transmon levels. For example, six transmon levels <NUM>-<NUM> (|g〉, |e〉, |r〉, |r'〉, |s〉, |s'〉) with decay transitions shown in the energy level diagram <NUM> of <FIG> and dispersive coupling in Eqn. (<NUM>) can be used. In some embodiments, a qubit of information is encoded in the g-e subspace. Based on a measurement of the transmon state resulting in |r'〉 or |r, 〉 a transmon decay error during the first or second CPS gates can be corrected. Based on a measurement of the transmon state of |s'〉 or |s〉, a transmon heating error during the first or second CPS gates can be corrected.

Some embodiments use the above-discussed techniques in a variety of applications. Such as a quantum router, quantum RAM, quantum state comparison, quantum principal component analysis, or universal quantum computing.

In a modular architecture for information processing, routers play an indispensable role in connecting different modules and different components within a module. There are many different types of routers for classical and quantum information processing. As listed in Table <NUM>, routers may be classified based on the classical/quantum (C/Q) nature of the input signals and control signals. Different types of routers and example associated applications. A classical-classical router is simply a classical switch. A classical-quantum router sends quantum information to a classical address. A quantum-quantum router sends quantum information to a quantum superposition of addresses. The quantum-classical router sends classical information to a quantum address. However since the information is classical the quantum state collapses so that only one address receives the classical signal. This is what happens in a quantum measurement where a semi-classical 'meter' is entangled with a quantum state and the state 'collapses' when the meter is 'read'.

In some embodiments, a Q-Q router is implemented using the c-SWAP operation by: (<NUM>) storing quantum input and control signals in the cavity and transmon modes, respectively, (<NUM>) applying the c-SWAP operation over the cavity modes conditioned on the transmon modes, and (<NUM>) retrieving the quantum signals from the cavity and transmon modes.

In some embodiments, the c-SWAP operation can be used to implement a quantum Random Access Memory (RAM). The quantum RAM can perform memory accesses using a coherent quantum superposition of addresses. To build an efficient quantum RAM, a Q-Q router with a three-level (transmon) memory (labeled "<NUM>", "L", and "R") and three cavity modes (labeled "input", "left", and "right") is used. The (transmon) memory is initialized in the "<NUM>" state, and all three cavities are initialized in the vacuum |vac〉 state. The goal of the Q-Q router is to process or route the quantum signal(s), which can be a superposition of three possible states: vacuum |vac〉, left |L〉, or right |R〉.

In some embodiments, a Q-Q router has five different function settings: (<NUM>) idle, (<NUM>) store control signal, (<NUM>) route signal forward, (<NUM>) route signal backward, and (<NUM>) retrieve control signal. Note that steps (<NUM>) & (<NUM>) can be used to carve out the pathway towards the target memory, while steps (<NUM>) and (<NUM>) can be used for the inverse unitary to decouple the quantum RAM.

In some embodiments, the operation of each of these five function settings is as follows:.

In some embodiments, the c-SWAP operation can be used to estimate the overlap of two quantum states. For example, if the transmon is initialized in the state |g〉 + |e〉 and the quantum state for two subsystems (cavities) are represented by the density matrices ρA and ρB, the initial input state is: <MAT>.

After the c-SWAP operation, the state becomes: <MAT> with reduced density matrix taking the form <MAT> where we have used the property <MAT>.

Thus, in some embodiments, by measuring the phase coherence (e.g., Ramsey fringe contrast) of ρtransmon, the overlap O(ρA, ρB) ≡ TrρAρB, which is always a real number, can be extracted.

In some embodiments, the transmon is projectively measured in the |g〉±|e〉 basis. If the transmon is measured as |g) + |e〉, the two subsystems are projected to the symmetric subspace (i.e., the eigen-subspace with eigenvalue + <NUM> for the SWAP operator), <MAT> where Π+is a projection operator. If the transmon is measured in |g〉 - |e〉, the two subsystems are projected to the anti-symmetric subspace (i.e., the eigen-subspace with eigenvalue - <NUM> for the SWAP operator), <MAT> where projection operator Π- = <NUM> - Π+. A subsequent measurement should give the same outcome, which is consist with the quantum non-demolition nature of these measurements.

In some embodiments, a more robust quantum circuit <NUM> acting on a first cavity <NUM>, a second cavity <NUM>, and a transmon <NUM>, as illustrated in <FIG>, for the purpose of quantum state comparison includes the following modifications: (<NUM>) remove the second beam-splitter (See <FIG>) that is dispensable for the purpose of quantum state comparison; (<NUM>) initialize the transmon in |g〉 and rotate to |g〉 + |e〉 right before the CPS gate to avoid the transmon decoherence during the first beam-splitter operation. The simplified circuit <NUM> minimizes the use of the beam splitter operation and decoherence of the transmon probe and includes: (<NUM>) a first beam splitter operation <NUM> between the first cavity <NUM> and the second cavity <NUM>; (<NUM>) a first Hadamard operation <NUM> on the transmon <NUM>; (<NUM>) a CPS operation <NUM> between the second cavity <NUM> and the transmon <NUM>; (<NUM>) a second Hadamard operation <NUM> on the transmon <NUM>; and (<NUM>) a measurement <NUM> of the transmon <NUM> state. Note that the simplified circuit can also be interpreted as the parity measurement of the second cavity mode after the beam-splitter operation. This interpretation can be easily justified based on the property of the projection operator to the symmetric subspace Π+, which is spanned by the symmetric states (a† - b†)<NUM>n(a† + b†)m|vac〉 with non-negative integers n, m (similarly, the anti-symmetric subspace Π- is spanned by the anti-symmetric states (a† - b†)<NUM>n+<NUM>(a† + b†)m|vac〉. Recall that after the <NUM>/<NUM> beam splitter, a†±b† correspond to the creation operators of the first and second cavity modes, respectively. Therefore, the symmetric states always have 2n even excitations in the second cavity mode, while the antisymmetric states always have <NUM>n + <NUM> odd excitations.

For each of ρA and ρB, a binary outcome <NUM> or <NUM> associated with the transmon measurement is obtained. N ~ ε-<NUM> pairs of ρA and ρB are needed to reliably estimate the overlap O(ρA, ρB) with precision ε.

In some embodiments, an e-SWAP operation is used to perform quantum principal component analysis (qPCA), which may be used to perform machine learning from large data sets. More specifically, qPCA can reveal the largest eigenvalues of an ensemble of identically prepared density matrices. The key idea is to use the following property of e-SWAP <MAT> which effectively simulates the Hamiltonian evolution with Hamiltonian being the Hermitian density matrix H = ρ for small duration Δt. With n identical copies of the density matrix ρ, one can construct e-iρnΔtσeiρnΔt. In some embodiments, together with a quantum phase estimation algorithm, the phase associated with the largest few eigenvalues of the density matrix ρ can be efficiently estimated. The e-SWAP gate according to some embodiments herein will enable the physical implementation for the key step of the qPCA.

In some embodiments, the e-SWAP gate is used to achieve universal quantum computing with bosonic systems, which can be compatible with arbitrary single-mode bosonic encoding (denoted as |<NUM>̃〉 and |<NUM>̃〉). For example, if one encodes one logical qubit using four such bosonic modes, with quad-rail logical qubit basis as <MAT> where the sub-indices label the modes. Since |<NUM>̃<NUM>̃〉±|<NUM>̃<NUM>̃〉 are respectively eigenstates of the SWAP operator with eigenvalues ±<NUM>, the quad-rail logical Z-rotation is the e-SWAP operation <MAT>.

In addition, we have SWAP<NUM>, <NUM> SWAP<NUM>, <NUM>|<NUM>Q〉 = |<NUM>Q〉 = XQ|<NUM>Q〉, which implies that the quad-rail logical X-rotation is a four-mode e-SWAP operation <MAT>.

Finally, the controlled-Z gate between the two encoding quad-rail logical qubits is another four-mode e-SWAP operation <MAT> where the sub-indices <NUM>' and <NUM>' label the first two bosonic modes from the other quad-rail logical qubit. Given an arbitrary logical Z- and X-rotations and controlled-Z gate, it is sufficient to achieve arbitrary universal quantum computation. Because the above gates do not depend on the details of the choice of orthogonal basis |<NUM>̃〉 and |<NUM>̃〉, this scheme can work for any single-mode bosonic encoding, including a cat code, a binomial code, a GKP code, etc. Therefore, the aforementioned robust design of e-SWAP implementations (see <FIG>) is sufficient to implement the key ingredients for universal quantum computation.

Claim 1:
A method for implementing a set of universal quantum gates, wherein the set of universal quantum gates comprises a controlled-SWAP (c-SWAP) gate and/or an exponential-SWAP (e-SWAP) gate between a plurality of cavities comprising a first cavity (<NUM>) and a second cavity (<NUM>), a first qubit being encoded in the first cavity (<NUM>) and a second qubit being encoded in the second cavity (<NUM>), the method comprising:
performing, by applying microwave radiation to a coupling transmon (<NUM>) that is dispersively coupled to both the first cavity (<NUM>) and the second cavity (<NUM>), a first and a second beam splitter (BS) operation between the first cavity (<NUM>) and the second cavity (<NUM>);
performing, by applying microwave radiation to an ancilla transmon (<NUM>) that is dispersively coupled to the second cavity (<NUM>) but not coupled to the first cavity (<NUM>), a first and/or a second controlled phase shift (CPS) operation between the second cavity (<NUM>) and the ancilla transmon (<NUM>) ; and
performing, by applying microwave radiation to the ancilla transmon (<NUM>), a rotation operation on the ancilla transmon (<NUM>).