阅读说明：本技术 鲁棒的量子逻辑门 (Robust quantum logic gate ) 是由 蒋良 史蒂文·M·格文 布里安·莱斯特 伊冯娜·高 罗伯特·J·舍尔科普夫三世 于 2019-01-05 设计创作，主要内容包括：提供并描述了用于实现鲁棒的量子逻辑门的技术。在一些方面,在包括第一腔和第二腔的多个腔之间的量子逻辑门通过以下操作实现：使用色散耦合至第一腔和第二腔两者的耦合transmon在第一腔与第二腔之间执行第一分束器操作；以及在第二腔与色散耦合至第二腔但并未色散耦合至第一腔的辅助transmon之间执行受控相移操作。(Techniques for implementing robust quantum logic gates are provided and described. In some aspects, a quantum logic gate between a plurality of cavities including a first cavity and a second cavity is implemented by: performing a first beam splitter operation between the first and second cavities using a coupling transmon that is dispersion coupled to both the first and second cavities; and performing a controlled phase shift operation between the second cavity and an auxiliary transmon dispersion coupled to the second cavity but not dispersion coupled to the first cavity.)

1. A method for implementing a quantum logic gate between a plurality of cavities including a first cavity and a second cavity, the method comprising:

performing a first Beam Splitter (BS) operation between the first cavity and the second cavity using a coupling transmon that is dispersion coupled to both the first cavity and the second cavity; and

performing a Controlled Phase Shift (CPS) operation between the second cavity and an auxiliary transmon dispersion coupled to the second cavity but not dispersion coupled to the first cavity.

2. The method of claim 1, further comprising performing a second BS operation between the first cavity and the second cavity using the coupling transmon, wherein the first BS operation is performed before the CPS operation and the second BS operation is performed after the CPS operation.

3. The method of claim 2 or any other preceding claim, wherein the quantum logic gate comprises a controlled SWAP (c-SWAP) operation.

4. The method of claim 3 or any other preceding claim, wherein the quantum logic gate comprises an exponential SWAP (e-SWAP) gate, and the method comprises:

performing a first c-SWAP operation between the first chamber and the second chamber;

performing a second c-SWAP operation between the first chamber and the second chamber; and

performing a rotation operation on the auxiliary transmon between the first c-SWAP operation and the second c-SWAP operation.

5. The method as recited in claim 2 or any other preceding claim, wherein the quantum logic gate comprises an exponential SWAP (e-SWAP) gate, the CPS operation is a first CPS operation, and the method comprises:

performing a second CPS operation between the second cavity and the auxiliary transmon after the first CPS operation and before the second BS operation; and

a rotation operation is performed on the auxiliary transmon between the first CPS operation and the second CPS operation.

6. The method of claim 5 or any other preceding claim, further comprising:

performing a first Hadamard operation before the first CPS operation and after the first BS operation; and

performing a second Hadamard operation after the second CPS operation and before the second BS operation.

7. The method of claim 6 or any other preceding claim, further comprising measuring the auxiliary transmon.

8. The method of claim 7 or any other preceding claim, further comprising determining a dephasing error that occurs based on a result of measuring the auxiliary transmon.

9. The method of claim 7 or any other preceding claim, wherein:

using at least four energy levels of the auxiliary transmon;

a first energy level of the auxiliary transmon is associated with a first decay error type;

a second energy level of the auxiliary transmon is associated with a second decay error type; and is

The method further comprises correcting decay errors based on measuring results of the auxiliary transmon.

10. The method as recited in claim 1 or any other preceding claim, where at least one local quantum logic gate is performed on the auxiliary transmon before and/or after the CPS operation.

11. The method of claim 10 or any other preceding claim, wherein:

the at least one local quantum logic gate comprises a first hadamard operation performed after the BS operation and before the CPS operation; and is

The method also includes measuring a state of the auxiliary transmon to compare a first quantum state of the first cavity to a second quantum state of the second cavity.

12. The method of claim 2 or any other preceding claim, wherein:

the plurality of lumens further comprising a third lumen and a fourth lumen;

the coupling transmon is a first coupling transmon;

the auxiliary transmon dispersion couples to the third cavity but not to the fourth cavity;

the CPS operation is a first CPS operation; and is

The method further comprises the following steps:

performing a third BS operation between the third cavity and the fourth cavity using a second coupling transmon that is dispersion coupled to both the third cavity and the fourth cavity; and

a second CPS operation is performed between the third chamber and the auxiliary transmon.

13. The method of claim 12 or any other preceding claim, wherein the method further comprises:

performing a fourth BS operation between the third cavity and the fourth cavity using the second coupling transmon, wherein the fourth BS operation is performed before the second CPS operation and the second BS operation is performed after the second CPS operation;

performing a third CPS operation between the third cavity and the auxiliary transmon after the second CPS operation and before the fourth BS operation;

performing a rotation operation on the auxiliary transmon between the second CPS operation and the third CPS operation; and

after the third CPS operation and before the second BS operation, a fourth CPS operation is performed between the second cavity and the auxiliary transmon.

14. The method of claim 13 or any other preceding claim, wherein the method further comprises:

performing a first Hadamard operation before the first CPS operation and after the first BS operation and the third BS operation; and

performing a second Hadamard operation after the fourth CPS operation and before the second BS operation and the fourth BS operation.

15. A multi-cavity quantum information system comprising:

a first cavity configured to support microwave radiation;

a second cavity configured to support microwave radiation;

a first transmon dispersion coupled to the first cavity;

a second transmon dispersion coupled to the second cavity;

a third transmon dispersion coupled to both the first and second cavities; and

a microwave source configured to:

applying microwave radiation to the third transmon to effect beam splitter operation and transmon rotation operation between the first and second cavities, and

applying microwave radiation to the first and second transmon to effect controlled phase shift operation on the first and second cavities, respectively.

16. A multi-cavity quantum information system according to claim 15 wherein the microwave source is further configured to apply microwave radiation to the first and second transmon to effect arbitrary rotational operation on the first and second transmon, respectively.

17. A multi-cavity quantum information system according to claim 16 or any other preceding claim wherein the microwave source is further configured to apply microwave radiation to the first, second and third transmon to achieve a set of common quantum gates between the first and second cavities.

18. A multi-cavity quantum information system according to claim 15 or any other preceding claim wherein the microwave source is further configured to apply microwave radiation to the first, second and third transmon to achieve a controlled SWAP gate between the first and second cavities.

19. A multi-cavity quantum information system according to claim 15 or any other preceding claim wherein the microwave source is further configured to apply microwave radiation to the first, second and third transmon to achieve an exponential SWAP gate between the first and second cavities.

20. A multi-cavity quantum information system according to claim 15 or any other preceding claim wherein the microwave source is further configured to:

applying microwave radiation to the first and/or second transmon to measure the first and/or second transmon; and is

Applying microwave radiation to the first, second, and/or third transmon to achieve a quantum gate based on a result of measuring the first and/or second transmon.

Background

Quantum information processing uses quantum mechanical phenomena such as energy quantization, superposition, and entanglement to encode and process information in a manner that is not utilized by conventional information processing. For example, it is known that certain computational problems can be solved more efficiently using quantum computation rather than conventional classical computation. However, to be a viable computing option, quantum computing requires the ability to precisely control a large number of qubits (called "qubits") and the interactions between these qubits. In particular, the qubit should: have long coherence times, can be individually manipulated, can interact with one or more other qubits to implement multi-qubit gates, can be efficiently initialized and measured, and can be extended to a large number of qubits.

Qubits can be formed from any physical quantum mechanical system having at least two orthogonal states. Two states of the system used to encode information are referred to as "computational bases". For example, photon polarization, electron spin, and nuclear spin are two-stage systems that can encode information, and thus can be used as qubits 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 multiple-qubit gates.

Different types of superconducting qubits using Josephson junctions (Josephson junctions) have been proposed, including: "phase qubits", where the calculation base is the quantized energy state of the Cooper pair (Cooper pair) in a josephson junction; "flux qubits" (flux qubits), where the computational basis is the direction of circulating current flow in the superconducting loop; and "charge qubits" where the computational base is the presence or absence of a cooper pair on the superconducting island. Superconducting qubits are an advantageous choice of qubits because the coupling between two qubits is strong, making the dual qubit gate relatively easy to implement, and are scalable because they are mesoscopic components that can be formed using conventional electronic circuit techniques. In addition, superconducting qubits exhibit strong nonlinearity associated with the josephson effect and excellent quantum coherence. All superconducting qubit designs use at least one josephson junction as the nonlinear non-dissipative element.

Disclosure of Invention

According to some aspects, there is provided a method for implementing a quantum logic gate between a plurality of cavities, the plurality of cavities including a first cavity and a second cavity, the method comprising: performing a first Beam Splitter (BS) operation between the first cavity and the second cavity using a coupling transmon that is dispersion coupled to both the first cavity and the second cavity; and performing a Controlled Phase Shift (CPS) operation between the second cavity and an auxiliary (ancilla) transmon, the auxiliary transmon being dispersion coupled to the second cavity but not to the first cavity.

According to some embodiments, the method further comprises performing a second BS operation between the first cavity and the second cavity using a coupling transmon, wherein the first BS operation is performed before the CPS operation and the second BS operation is performed after the CPS operation.

According to some embodiments, the quantum logic gate includes a controlled SWAP (c-SWAP) operation.

According to some embodiments, the quantum logic gate comprises an exponential SWAP (e-SWAP) gate, and the method comprises: performing a first c-SWAP operation between the first chamber and the second chamber; performing a second c-SWAP operation between the first chamber and the second chamber; and performing a rotation operation on the auxiliary transform between the first c-SWAP operation and the second c-SWAP operation.

According to some embodiments, the quantum logic gate comprises an exponential SWAP (e-SWAP) gate, the CPS operation is a first CPS operation, and the method comprises: performing a second CPS operation between the second cavity and the auxiliary transmon after the first CPS operation and before the second BS operation; and performing a rotation operation on the auxiliary transmon between the first CPS operation and the second CPS operation.

According to some embodiments, the method further comprises: performing a first hadamard operation before the first CPS operation and after the first BS operation; and performing a second hadamard operation after the second CPS operation and before the second BS operation.

According to some embodiments, the method further comprises measuring the secondary transmon.

According to some embodiments, the method further comprises determining a dephasing error, which occurs based on the results of measuring the auxiliary transmon.

According to some embodiments, at least four energy levels of the auxiliary transmon are used; the first energy level of the auxiliary transmon is associated with a first decay error type; the second energy level of the auxiliary transmon is associated with a second decay error type; and the method further comprises correcting decay errors based on the results of measuring the secondary transmon.

According to some embodiments, at least one local quantum logic gate is executed on the auxiliary transmon before and/or after CPS operation.

According to some embodiments, the at least one local quantum logic gate comprises a first hadamard operation performed after the BS operation and before the CPS operation; and the method further includes measuring a state of the auxiliary transmon to compare the first quantum state of the first cavity with the second quantum state of the second cavity.

According to some embodiments, the plurality of lumens further comprises a third lumen and a fourth lumen; the coupling transmon is a first coupling transmon; the secondary transmon is dispersion coupled to the third cavity but not dispersion coupled to the fourth cavity; the CPS operation is a first CPS operation; and the method further comprises: performing a third BS operation between the third cavity and the fourth cavity using a second coupling transmon dispersion-coupled to both the third cavity and the fourth cavity; and performing a second CPS operation between the third chamber and the auxiliary transmon.

According to some embodiments, the method further comprises: performing a fourth BS operation between the third cavity and the fourth cavity using a second coupling transmon, wherein the fourth BS operation is performed before the second CPS operation and the second BS operation is performed after the second CPS operation; performing a third CPS operation between the third cavity and the auxiliary transmon after the second CPS operation and before the fourth BS operation; performing a rotation operation on the auxiliary transmon between the second CPS operation and the third CPS operation; and performing a fourth CPS operation between the second cavity and the auxiliary transmon after the third CPS operation and before the second BS operation.

According to some embodiments, the method further comprises: performing a first hadamard operation before the first CPS operation and after the first BS operation and the third BS operation; and performing a second hadamard operation after the fourth CPS operation and before the second BS operation and the fourth BS operation.

According to some aspects, there is provided a multichamber quantum information system comprising: a first cavity configured to support microwave radiation; a second cavity configured to support microwave radiation; a first transmon dispersion coupled to the first cavity; a second transmon dispersion coupled to the second cavity; a third transmon dispersion coupled to both the first and second cavities; and a microwave source configured to: microwave radiation is applied to the third transmon to effect beam splitter operation and to effect transmon rotation operation between the first and second cavities, and to the first and second transmon to effect controlled phase shift operation on the first and second cavities, respectively.

According to some embodiments, the microwave source is further configured to apply microwave radiation application to the first and second transmon to effect arbitrary rotation operations on the first and second transmon, respectively.

According to some embodiments, the microwave source is further configured to apply microwave radiation to the first, second, and third transmon to implement a set of common quantum gates between the first and second cavities.

According to some embodiments, the microwave source is further configured to apply microwave radiation to the first, second, and third transmon to implement a controlled SWAP gate between the first and second cavities.

According to some embodiments, the microwave source is further configured to apply microwave radiation to the first, second, and third transmon to achieve an exponential SWAP gate between the first and second cavities.

According to some embodiments, the microwave source is further configured to: applying microwave radiation to the first and/or second transmon to measure the first and/or second transmon; and based on the results of measuring the first and/or second transmon, applying microwave radiation to the first, second, and/or third transmon to implement the quantum gate.

Drawings

Various aspects and embodiments of the disclosed technology will be described with reference to the following drawings. It should be understood that the drawings are not necessarily drawn to scale.

FIG. 1A is a schematic diagram of a multi-cavity quantum information system, according to some embodiments;

FIG. 1B is a schematic diagram of an example cavity and transmon of the multi-cavity quantum information system of FIG. 1A, according to some embodiments;

FIG. 2A illustrates a controlled SWAP (c-SWAP) gate according to some embodiments;

FIG. 2B illustrates a method of implementing the c-SWAP gate of FIG. 2A, according to some embodiments;

FIG. 2C shows a schematic diagram of a linear optics based C-SWAP gate, according to some embodiments;

FIG. 3A illustrates an exponential SWAP (e-SWAP) gate between two cavities, according to some embodiments;

FIG. 3B illustrates a method for implementing the e-SWAP gate of FIG. 3A, according to some embodiments;

FIG. 3C illustrates a method for implementing the e-SWAP gate of FIG. 3A, according to some embodiments;

FIG. 4A illustrates an exponential SWAP (e-SWAP) gate between four cavities, according to some embodiments;

FIG. 4B illustrates a method for implementing the e-SWAP gate of FIG. 4A, according to some embodiments;

FIG. 4C illustrates a method for implementing the e-SWAP gate of FIG. 4A, according to some embodiments;

FIG. 5A illustrates a four-level transmon energy structure, according to some embodiments; and

FIG. 5B illustrates a six-stage transmon energy structure, according to some embodiments; and

fig. 6 illustrates a method of performing a quantum state comparison according to some embodiments.

Detailed Description

In conventional approaches to quantum information processing using superconducting circuits, information is stored in josephson junction-based qubits (e.g., transmon) that are coupled together via exchange of microwave photons in a bus resonator or via direct capacitive coupling. The inventors have recognized and appreciated that different and complementary architectures in which quantum information is stored and manipulated in the microwave photonic state of a high Q resonator and with transmon acting as an assist to provide universal quantum control of photonic qubits have significant advantages over conventional approaches. First, the coherence time of the microwave cavity is longer than that of the superconducting qubit. For example, in aluminum, a three-dimensional superconducting microwave cavity can achieve a very high figure of merit approaching 109, and still be high in niobium, and the cavity coherence time of the aluminum cavity exceeds that of transmon by two orders of magnitude. Second, the number of states into which information can be encoded is larger in the microwave cavity than in superconducting qubits. For example, the higher excitation levels of the cavity modes extend the available Hilbert space (Hilbert space) such that single cavity modes can be used to encode informative qubits in a manner compatible with quantum error correction to address various defects including cavity dephasing (cavity phasing), excitation loss, or thermal heating (thermal phasing).

The inventors have recognized and appreciated that these advantages of using a microwave cavity may be used in a unique approach to quantum information processing that stores quantum information as quantum states of the microwave cavity while using transmon as an aid to assist in quantum information processing and quantum error correction. This "photonic qubit" approach is in contrast to conventional approaches, in which quantum information is stored in transmon, and the transmon is coupled using a microwave photonic mode as the quantum bus. The resulting superconducting cavity-transmon system implements deterministic coupling gates between cavities assisted by a transmon controller, universal quantum control for cavity modes, quantum error correction to reach equilibrium points, and joint parity measurements for both cavities.

The inventors have further recognized and appreciated that superconducting cavity-transmon systems may encounter limited fidelity of quantum operation due to transmon decoherence during cavity-transmon coupling. The inventors have recognized and appreciated that, in addition to experimentally improving the coherence properties of transmon, the fidelity of quantum operation may be improved by implementing robust methods for suppressing or even actively correcting errors due to de-coherence of other josephson nonlinear devices and transmon.

The inventors have further recognized and appreciated that a generalized set of quantum logic gates is "encoding independent" (encodingagnostic), meaning that the set of quantum logic gates remains generalized regardless of how quantum information is selected to be logically encoded in quantum states of a cavity (e.g., cat state encoding, coherent state encoding, Fock state encoding, etc.). Furthermore, the hardware used to implement the generic set of quantum logic gates is always unchanged, regardless of the encoding scheme selected.

According to some embodiments, one or more josephson nonlinear devices, such as transmon, are used to control and/or couple together multiple microwave cavities. In some embodiments, a coupling transmon is used to perform operations between the two chambers. Examples of such operations include a Beam Splitter (BS) operation coupling a quantum state of a first cavity to a quantum state of a second cavity and a Beam Splitter (BS) operation coupling a quantum state of the second cavity to a quantum state of the first cavity. These operations are referred to as beam splitter operations because they function the same as beam splitters in Linear Optical Quantum Computing (LOQC) schemes and achieve the same unitary operation as beam splitters do.

In some embodiments, Controlled Phase Shift (CPS) operation between a single cavity and an auxiliary transmon is achieved using the auxiliary transmon coupled to the single cavity. In some embodiments, the auxiliary transmon may also be controlled to effect rotation of the quantum state of the auxiliary transmon itself.

The inventors have recognized and appreciated that BS operation, CPS operation, and assisted rotation of transmon are sufficient to enable general gate based quantum computation. In some embodiments, the above-described operations are used to implement a c-SWAP gate and/or an e-SWAP gate. The e-SWAP gate couples the cavity modes while preserving the boson coding space. In this way, the entanglement operation between the bose sub-modes can be achieved independently of the logical encoding used. Some embodiments utilize a robust e-SWAP gate design that can predict defects due to transmon decoherence. Some embodiments use c-SWAP and e-SWAP gates for one or more applications, such as quantum routers, quantum Random Access Memories (RAMs), quantum principal component analysis, and gate-based general quantum computing.

Note that while transmon superconducting qubits are described herein as components of various embodiments, other types of superconducting devices may also be used in some embodiments.

Fig. 1 is a schematic diagram of a multi-cavity quantum information system 100 according to some embodiments. The multi-cavity quantum information system 100 includes a first cavity 101, a second cavity 102, a coupling transmon103, a first auxiliary transmon104, a second auxiliary transmon 105, and a microwave source 106. Although only two chambers and two auxiliary transmon are shown in the example system 100 of fig. 1, other embodiments may include additional chambers and transmon.

The first cavity 101 and the second cavity 102 may be any type of cavity that supports quantum states of microwave radiation. For example, in some embodiments, the first cavity 101 and the second cavity 102 may be transmission line resonators or three-dimensional cavities formed from superconducting materials such as aluminum.

The coupling transmon103 may be a transmon dispersion coupled to both the first cavity 101 and the second cavity 102. Coupling transmon103 provides interaction between the quantum states of the two cavities, allowing interaction such as BS interaction to be performed between the first cavity 101 and the second cavity 102.

The first auxiliary transmon104 and the second auxiliary transmon 105 are dispersion-coupled to the first cavity 101 and the second cavity 102, respectively. Each auxiliary transmon is coupled to a single cavity of system 100 and not to any other cavity for storing quantum information. The auxiliary transmon104 and 105 can be controlled to achieve rotation of the quantum states of the transmon104 and 105 themselves. Additionally, auxiliary transmon104 and 105 can be controlled to interact with associated cavities 101 and 102, respectively, to achieve controlled interactions, such as CPS interactions.

The microwave source 106 may be coupled to the first cavity 101, the second cavity 102, the first auxiliary transmon104, the second auxiliary transmon 105, and the coupling transmon 103. The coupling between microwave source 106 and the other components provides a way for the microwave source to apply microwave radiation to each component of multi-cavity quantum information system 100. In some embodiments, the microwave source 106 may be capacitively coupled to each component.

Fig. 1B is a schematic diagram of a particular example system 110 (microwave source 106 not shown for simplicity) that may function as the multi-cavity quantum information system 100 of fig. 1A, according to some embodiments. The system 110 includes a first three-dimensional (3D) cavity 111, a second 3D cavity 121, a coupling device 131, a first auxiliary device 141, and a second auxiliary device 151.

The first and second 3D cavities 111 and 112 serve as 3D versions of λ/4 transmission line resonators between the center stub 114 and the outer wall 115 and between the center stub124 and the outer wall 125, respectively. For example, the diameter of the center stubs 114 and 124 may be 3.2mm, and the diameter of the outer walls 115 and 125 may be 9.5 mm. Note, however, that embodiments are not limited to any particular size. The resonant frequency of each of the cavities 101 and 102 may be determined by the height of the center stub 114 and 124 within its respective cavity. For example, center stub 114 may have a height of 4.8mm, and second center stub124 may have a height of 5.6 mm. The first 3D cavity 111 supports microwave radiation 112 at a first frequency and the second 3D cavity 121 supports microwave radiation 122 at a second frequency different from the first frequency. In some embodiments, the first and second cavities 111 and 121 include ports 113 and 123, respectively, through which microwave radiation from the microwave source 106 may be applied. Applying microwave radiation to the cavity may, for example, effect a displacement operation of the quantum state of the cavity.

The coupling device 131 comprises a coupling transmon 134, the coupling transmon 134 providing a non-linear interaction between the first cavity 111 and the second cavity 121. the transmon 134 is coupled to a first antenna 135 that is at least partially inserted into the first cavity 111 and a second antenna 136 that is at least partially inserted into the second cavity 121 such that at least a portion of each antenna protrudes into its respective cavity. The first antenna 135 and the second antenna 136 may be, for example, circular pads providing capacitive coupling to the first cavity 111 and the second cavity 121, respectively.

Coupling device 131 also includes a resonator 138, resonator 138 providing the ability to sense the state of transmon 134. The third antenna 137 couples the resonator 138 to the resonator 138. In some embodiments, the resonator 138 is a quasi-planar resonator having a lower Q value than the first cavity 111 or the second cavity 121. In some embodiments, the transmon 134 and resonator 134 are fabricated on a single sapphire substrate. A readout pulse of microwave radiation may be received by the pump port 132 and the resulting microwave signal may be received from the readout port 133.

Transmo of coupling device 131The non-linearity of n 134 enables four-wave mixing, which is used to perform frequency-translating bilinear coupling between the first cavity 111 and the second cavity 121. By using a frequency matching condition omega₁-ω₂＝ω_p2-ω_p1Via a pump port 132 to pump transmon 134 to control four-wave mixing, where ω is₁Is the resonant frequency, ω, of the first cavity 111₂Is the resonant frequency, ω, of the second cavity 121_p1Is the frequency of the first pump associated with mode c, and ω_p2Is the frequency of the second pump associated with mode d. This coupling enables efficient time-dependent BS interaction between the cavity modes. As is known from conventional optics, the unitary evolution of a beam splitter is described by a unitary operator:

wherein

And

for θ ═ π/2, the beam splitter unitary performs a SWAP operation that SWAPs states between the two cavity modes associated with the annihilation operators a and b, respectively. For θ ═ π/4 and θ ═ π/4, the unitary operator corresponds to 50/50 beam splitters. Unlike ordinary optics, the microwave cavity intrinsic modes have different frequencies and do not couple to each other without nonlinearity. However, josephson nonlinearity and an additional parametric pump may be used to compensate for the cavity frequency difference so that modes with different frequencies can be coherently coupled. For example, based on four-mode coupling from Josephson nonlinearity(wherein h.c. denotes the Hermitian conjugation of the first term, and g₄Is a four-mode couplingConstant), modulo c and modulo d can be pumped so that the amplitude can be varied<c>＝A_c(τ) and<d>＝A_dthe classical coherent states of (τ) are approximated to estimate modes c and d, which yields the effective splitter coupling Hamiltonian in equation (2), where g (τ) g₄A_c ^*(τ)A_d(τ). Note that A can be controlled by_c(τ) and A_dThe amplitude and phase of (τ) tune g (τ). In this way, some embodiments can easily turn on/off the splitter coupling at extremely high on/off ratios. This is a distinct advantage over "always-on" capacitive or other fixed couplings between qubits. In addition, by pumping modes c and d such that their sum of frequencies equals the sum of cavity mode frequencies, this can be achievedA form of bilinear coupling. With these two operations, any linear symplectic transformation between the two chambers can be performed.

In some embodiments, the above-described methods of implementing a unitary beam splitter operator with josephson nonlinearity can be made robust to defects in josephson devices. For example, if the intrinsic nonlinearity of the device is weak but the parametric pumping is strong, then the amplitude a of the coherent state relative to the large can be determined by thermal noise in modes c and d_cAnd A_dThe fact that is small suppresses the influence of the noise. Operation in this regime may also increase the linearity of the beam splitter so that SWAP operations can be performed for quantum states containing a wide range of photon numbers.

The beam splitter unitary transformation is an element that is very useful for quantum information processing. For example, while the evolution of gaussian input states (e.g., coherent states, squeezed states) as acted upon by a beam splitter unitary can be effectively modeled with a classical computer, the evolution of non-gaussian input states (e.g., Fock states) can also result in non-trivial output states. For example, the complexity of bose sampling shows a nontrivial statistical property of the output state that is difficult to model with a classical computer. Furthermore, a beam splitter unit incorporating both a single photon source and a photon detector can enable universal linear photon computation (LOQC), although major challenges include the probabilistic nature of entanglement gates and the extremely feared resource overhead.

In some embodiments, not limited to linear light quantum computation, additional nonlinear elements may be used to facilitate quantum information processing. For example, a physical platform using superconducting circuits with microwave photons not only provides the functionality of a single photon source and photon detector, but also includes at least one highly controllable transmon that can be used as a two or more stage assist. In some implementations, the quantum information processing operation is performed using quantum operations that combine a beam splitter unitary and a cavity-transmon dispersion coupling gate. In some embodiments, the dispersion-coupled gate is still a linear optical transformation controlled (e.g., based on and/or conditioned) by a transmon (or other) assisted quantum state. This combination of capabilities for linear optics and gate-based quantum computing is powerful and allows the gating operation to be performed on qubits that are logically encoded into photonic states independently of the particular logical encoding. Thus, in some embodiments, the logical encoding of information may be changed while using the same hardware with the same operations.

The first auxiliary device 141 is similar to the coupling device 131, but is only coupled to the first cavity 111 instead of two cavities. The first auxiliary device comprises a pumping port 142 for driving the transmon144 with read-out pulses and pumping of microwave radiation and a read-out port 143 for receiving a read-out microwave signal from the transmon 144. the transmon144 is coupled to the first cavity 111 via a first antenna pad 145 that protrudes at least partially into the first cavity 111. A second antenna pad 146 couples the transmon144 to a quasi-planar resonator 147.

The second auxiliary device 151 is similar to the first auxiliary device 141, but is coupled only to the second cavity 121 and not to the first cavity 111. The second auxiliary device comprises a pumping port 152 for driving the transmon154 with read-out pulses and pumping of microwave radiation and a read-out port 153 for receiving a read-out microwave signal from the transmon 154. the transmon154 is coupled to the first cavity 111 via a first antenna pad 155 that protrudes at least partially into the first cavity 111. A second antenna pad 156 couples the transmon144 to the quasi-planar resonator 147.

The first and second auxiliary devices 141 and 152 may be used to implement a CPS operation, which is represented as:

whereinIs a numerical operator of the boson mode of the particular cavity coupled to transmon. In some embodiments, the phase shift is π, and the operation is due to photon-number parityThus enabling controlled parity operations. In some embodiments, for a duration t ═ pi/χ and a coupling strength χ, the CPS gate may be obtained by time evolution at the hamiltonian of the dispersive coupling between the auxiliary transmon and the respective cavity, expressed as:

an example set of parameters for implementing the quantum information system 110 is as follows: the first cavity 111 may have a kerr/2 pi-2 KHz and ω₁5.45GHz,/2 pi; the second chamber 121 may have lerr/2 pi-3 KHz and ω₂A 6.55 GHz/2 pi coupling device 131 may have a frequency of α/2 pi 74MHz, w/2 pi 5.95GHz, χ_c10.26MHz and χ for/2 π_c2The first auxiliary transmon 141 may have α/2 pi 180MHz, ω/2 pi 4.5GHz, χ/2 pi 0.76MHz, and the second auxiliary transmon 151 may have α/2 pi 180MHz, ω/2 pi 5.0GHz, χ/2 pi 1.26 MHz.

In some embodiments, the c-SWAP gate may be implemented using a combination of BS and CPS operations. The c-SWAP gate operates in one of the auxiliary transmon and two cavity modes, swapping the state of the two cavities based on the state of the auxiliary transmon. Thus, the unitary c-SWAP operator can be written as:

wherein, | g>And | e>Representing the ground state and the first excited state of the auxiliary transmon, respectively. Because (c-SWAP)²I, so c-SWAP is reversible. The c-SWAP gate is sometimes referred to as a freidkin gate, which is common to classical calculations. For quantum computation, c-SWAP and single-quantum bit rotation form a set of quantum gates that enable general purpose computation.

Fig. 2A shows a quantum circuit diagram 200 of a c-SWAP gate. Line 201, line 202 and line 203 represent the first chamber 111, the second chamber 121 and the auxiliary transmon 131. In the quantum circuit diagram, operations are performed on various components according to time from left to right. The symbol of the c-SWAP gate includes an "X" at each cavity line 201 and 202 and a point on the auxiliary transmon line 203.

FIG. 2B illustrates an example quantum circuit diagram 210 for implementing a c-SWAP gate using BS operation and CPS operation. First, a first BS operation 211 is performed between the first cavity 201 and the second cavity 202, where θ ═ pi/4. As described above, the BS operation may be performed using the coupling transmon 134. After the first BS operation 211, a CPS operation 213 is performed between the auxiliary transmon 203 and the second cavity 202. Finally, a second BS operation 215 is performed between the first cavity 201 and the second cavity 202, where θ ═ pi/4.

As shown in fig. 2C, the C-SWAP gate can be represented by a conventional linear optical diagram, in which there is a first optical mode 231, a second optical mode 232, and a transmon 232. The figure is a mach-zehnder interferometer 230 comprised of a first beam splitter 234, a second beam splitter 235, and a controlled phase shifter 236 that achieves a 0 or pi phase shift on one arm controlled by the state of transmon. For the transmon state | g >, there is a 0 phase shift, and thus the two bosch sub-modes do not swap. For the transmon state | e >, there is a pi phase shift (for each and every excitation coupled to transmon) that causes a complete exchange (SWAP) between the two optical modes 231 and 232. Thus, in some embodiments, c-SWAP can be used as a special quantum control router that uses quantum states (e.g., the states of an auxiliary transmon) to control the path of quantum signals (carried by the optical mode).

In some embodiments, a combination of c-SWAP gates and auxiliary transmon rotations may be used to perform e-SWAP operations. The e-SWAP operates on both chambers and is represented by a unitary operator:

U_eSWAP(θ)＝exp[iθSWAP]＝cosθ+isinθSWAP (7)

for θ ═ π/2, the e-SWAP gate is equivalent to a SWAP gate, in which case the resulting global phase shift (i) is not observable. For theta to pi/4,is a coherent combination of an identity operator and a SWAP operator, and is sometimes denoted asSingle qubit rotation sum on qubitsThe operators form a set of generic quantum gates.

There are similarities and differences between the e-SWAP operator and the beam splitter unitary. For a waver subspace with zero and one total excitation, the two operators are equivalent (e.g., U)_BS(θ)|0_a，1_b>＝cosθ|0_a，1_b>+isinθ|1_a，0_b>＝U_eSWAP(θ)|0_a，1_b>). For subspaces with more than a total excitation, however, the two operators behave differently (e.g.,this is different from U_eSWAP(θ)|1_a，1_b>＝|1_a，1_b>)。

One feature of the e-SWAP operator is that it preserves the logical subspace for single-mode boson encoding containing an arbitrary number of bosons. Logic states for arbitrary single mode boson codinge-SWAP operationThe code space for any boson code and any parameter θ is preserved. This important property enables quantum information processing to be performed with different choices of boson encoding using the same hardware. This powerful feature provides great flexibility to the hardware and allows experiments to be performed with different encodings for quantum error correction implemented via "software updates" on fixed hardware.

In some embodiments, the e-SWAP operator between two cavity modes may be implemented using a two-stage assist transmon. For example, fig. 3A is a quantum circuit diagram 300 between a first cavity 301, a second cavity 302, and an auxiliary transmon 303. The illustrated method for implementing e-SWAP operation is as follows: (1) auxiliary transmon303 is initialized to quantum state | +>＝1/√2(|g>+|e>) (ii) a (2) Performing a first c-SWAP operation 311 between the first and second chambers 301 and 302, which is controlled based on the state of the auxiliary transition 303; (3) with auxiliary transmon303 about the X-axisRotating by 312 degrees 2 theta; and (4) performing a second c-SWAP operation 313 between the first and second chambers 301 and 302 that is controlled based on the state of the auxiliary transition 303. After the above method, auxiliary transmon is restored to initial state | +>And decoupled from the two cavity modes; at the same time, both cavity modes undergo e-SWAP operation, U_eSWAP(theta). If auxiliary transmon303 is measured 314, assuming no error, the result is initial state | +>。

FIG. 3B shows a quantum circuit diagram 320 for implementing the e-SWAP operation of FIG. 3A using BS operation and CPS operation. In this example, the c-SWAP gate is simply replaced with the c-SWAP method shown in FIG. 2B. Accordingly, a method of implementing an e-SWAP gate comprises: (1) auxiliary transmon303 is initialized to quantum state | +>＝1/√2(|g>+|e>) (ii) a (2) In the first chamber 301 anda first BS operation 321 of θ ═ pi/4 is performed between the second chambers 302; (3) a first CPS operation 322 is performed between the second chamber 302 and the auxiliary transition 303; (4) a second BS operation 323 of θ ═ pi/4 is performed between the first cavity 301 and the second cavity 302; (5) about the X-axis for auxiliary transmon303Rotating by 324 degrees 2 theta; (6) a third BS operation 325 of θ ═ pi/4 is performed between the first cavity 301 and the second cavity 302; (7) a second CPS operation 326 is performed between the second chamber 302 and the auxiliary transition 303; (8) a fourth BS operation 327 of θ ═ pi/4 is performed between the first cavity 301 and the second cavity 302. As shown in FIG. 3A, if auxiliary transmon303 is measured 328, assuming no error, the result is initial state | +>。

In some embodiments, a simplified and more robust implementation of the e-SWAP operator may be obtained by decomposing the c-SWAP operator into a beam splitter operator and CPS gates in a manner different from simply substituting the quantum circuit diagram 210 of FIG. 2B into the quantum circuit diagram 300 of FIG. 3A. The simplification of the quantum circuit diagram 320 comes from the following recognition: two of the beam splitter operations (e.g., the second beam splitter operation 323 and the third beam splitter operation 325) are redundant in that they cancel each other out and can therefore be eliminated. Additionally, transmon may be initialized to the ground state and additional hadamard gates may be added to act on transmon just before the first CPS gate 322 and after the second CPS gate 326 so that transmon is held in the ground state during the beam splitter unitary operation.

With the two changes to the quantum circuit described above, a more robust quantum circuit diagram 340 for e-SWAP operation is achieved, as shown in fig. 3C. The method for realizing the e-SWAP gate comprises the following steps: (1) initialization of the auxiliary transmon303 to the Quantum State | g>(ii) a (2) A first BS operation 341 of θ ═ pi/4 is performed between the first cavity 301 and the second cavity 302; (3) performing a first hadamard operation 342 on the secondary transmon 303; (4) a first CPS operation 343 is performed between the second chamber 302 and the auxiliary transition 303; (5) performing auxiliary transmon303 about the X-axisRotate 344 degrees 2 θ; (6) a second CPS operation 345 is performed between the second chamber 302 and the auxiliary transition 303; (7) performing a second hadamard operation 346 on the auxiliary transmon 303; (8) a second BS operation 347 of θ ═ pi/4 is performed between the first cavity 301 and the second cavity 302. If the auxiliary transmon303 is measured 348, assuming no error, the result is the initial state | g >.

The simplified quantum circuit diagram 340 includes the above-described variations because the beam splitter operation may be relatively slow compared to other operations. Thus, the quantum circuit in fig. 3C has both a shorter overall duration (which reduces the risk of errors occurring in the cavity state) and a shorter duration of the period in which the auxiliary transmon is in the excited state (which reduces the risk of errors occurring on the auxiliary). In the quantum circuit designs of fig. 3A and 3B, transmon is never in the ground state, making it susceptible to decoherence throughout operation, especially during relatively slow beam splitter unitary operation. In contrast, the quantum circuit diagram 340 of FIG. 3C enables transmon to be implemented at relatively fast Hadamard gates, CPS gates, and X_θThe ground state is maintained outside the gate period. Thus, in some embodiments, the quantum circuit design effectively mitigates defects due to transmon decoherence, thereby enabling the error of the entire quantum gate to be reduced from O [ gamma ]_t(t_BS+t_CPS+t_Trans)]Reduced to O [ gamma ]_t(t_CPS+t_Trans)]Wherein γ is_tIs the transmon decoherence rate, t_BS、t_CPSAnd t_TransRespectively associated with a splitter gate, a CPS gate and a transmon turnstile (e.g. X)_θAnd H) an associated time. In some embodiments, t is_BS(～10μs)＞＞t_CPS(～1μs)＞＞t_Trans(. about.10 ns) thus making it advantageous to eliminate t_BSDuring which transmon breaks the weak point of coherence.

In some embodiments, the e-SWAP operator may be extended to operate on more than two chambers. For example, FIG. 4A shows the use of four lumens through the U_eSWAP ^{(a，b；a’，b’)}(θ)＝exp[iθSWAP_a，bSWAP_a’，b’]A quantum circuit diagram 400 of e-SWAP is performed. TheThe four-chamber e-SWAP method 400 includes, after initializing the assist: proceed the following operation (1) initialize auxiliary transmon403 to quantum state | +>＝1/√2(|g>+|e>) (ii) a (2) Performing a first c-SWAP operation 411 between the first chamber 401 and the second chamber 402, controlled by the state of the auxiliary transition 403; (3) a second c-SWAP operation 412 between the third chamber 404 and the fourth chamber 404, controlled by the state of the auxiliary transmon403, is performed; (4) performing auxiliary transmon403 about the X-axisRotating 415 degrees 2 theta; (5) a third c-SWAP operation 413 between the third chamber 404 and the fourth chamber 404, controlled by the state of the auxiliary transmon 403; (6) a fourth c-SWAP operation 414 between the first chamber 401 and the second chamber 402, controlled by the state of the auxiliary transmon403, is performed. If auxiliary transmon403 is measured 415, assuming no error, the result is initial state | +>。

Similar to the process of e-SWAP between two modes in fig. 3B-3C, the e-SWAP operation for four modes can be decomposed into splitter operation and CPS gates (fig. 4B) and converted into a more robust quantum circuit (fig. 4C). Thus, a method of implementing the e-SWAP gate illustrated by quantum circuit diagram 420 in fig. 4B includes: (1) auxiliary transmon303 is initialized to quantum state | +>＝1/√2(|g>+|e>) (ii) a (2) A first BS operation 421 of θ ═ pi/4 is performed between the first chamber 401 and the second chamber 402; (3) a second BS operation of θ ═ pi/4 is performed between the third cavity 404 and the fourth cavity 405; (4) a first CPS operation 423 is performed between the second chamber 402 and the auxiliary switch 403; (5) performing a second CPS operation 423 between the third chamber 404 and the auxiliary transmon 403; (6) performing a third BS operation 425 of θ ═ pi/4 between the first cavity 401 and the second cavity 402; (7) a fourth BS operation 426 of θ ═ pi/4 is performed between the third cavity 404 and the fourth cavity 405; (8) performing auxiliary transmon403 about the X-axisRotate 427 by angle 2 θ; (9) a fifth BS operation 428 of θ ═ pi/4 is performed between the first cavity 401 and the second cavity 402; (10) performing theta pi between the third cavity 404 and the fourth cavity 405Sixth BS operation 429 of/4; (11) a third CPS operation 430 is performed between the third chamber 404 and the transmon 403; (12) a fourth CPS operation 431 is performed between second chamber 402 and transmon 403; (13) a seventh BS operation 432 of θ ═ pi/4 is performed between the first cavity 401 and the second cavity 402; (7) an eighth BS operation 433 of θ ═ pi/4 is performed between the third cavity 404 and the fourth cavity 405. If the auxiliary transmon403 is measured 415, assuming no error, the result is the initial state | + >.

It should be noted that not all operations need be performed in the order shown. For example, the first BS operation 421 and the second BS operation 422 are shown to be performed simultaneously. In some embodiments, either BS operation may be performed before the other BS operation. The same is true for the third and fourth BS operations 425 and 426, the fifth and sixth BS operations 428 and 429, and the seventh and eighth BS operations 432 and 433. Also, the first CPS operation 423 is shown as occurring before the second CPS operation 424. In some implementations, the second CPS operation 424 may be performed before the first CPS operation 423. Similarly, the fourth CPS operation 431 may be performed before the third CPS operation 430.

After simplifying the quantum circuit diagram of fig. 3C and making the method more robust to errors, the quantum circuit diagram 450 of fig. 4C includes: (1) initializing auxiliary transmon303 to quantum state | g >; (2) a first BS operation 451 of θ ═ pi/4 is performed between the first cavity 401 and the second cavity 402; (3) a second BS operation 452 of θ ═ pi/4 is performed between the third cavity 404 and the fourth cavity 405; (4) performing a first hadamard operation 453 on the auxiliary transmon 403; (5) a first CPS operation 454 is performed between the second chamber 402 and the auxiliary transition 403; (6) a second CPS operation 455 is performed between the third cavity 404 and the auxiliary transmon 403; (7) performing auxiliary transmon403 about the X-axisRotating by 456 degrees 2 theta; (8) performing a third CPS operation 457 between the third cavity 404 and the auxiliary transmon 403; (9) a fourth CPS operation 458 is performed between the second chamber 402 and the auxiliary transition 403; (10) performing a second hadamard operation 459 on the auxiliary transmon 403; (11) performing θ between the first cavity 401 and the second cavity 402-pi/4 third BS operation 460; (12) a fourth BS operation 461 of θ ═ pi/4 is performed between the third cavity 404 and the fourth cavity 405. If the auxiliary transmon403 is measured 415, assuming no error, the result is the initial state | g>。

It should be noted that not all operations need be performed in the order shown. For example, the first BS operation 451 and the second BS operation 452 are shown to be performed simultaneously. In some embodiments, either BS operation may be performed before the other BS operation. The same is true for the third BS operation 460 and the fourth BS operation 461. Also, the first CPS operation 454 is shown to occur before the second CPS operation 455. In some implementations, the second CPS operation 455 may be performed before the first CPS operation 454. Similarly, fourth CPS operation 458 may be performed before third CPS operation 457.

Turning now to the quantum circuit diagram 340 of fig. 3C, the existence of dephasing and decay errors is discussed. Due to t_CPS＞＞t_TransTherefore, focus on the transmon error during two CPS gates and ignore X_θThe error in the period.

In some embodiments, the quantum circuit diagram 340 for e-SWAP operation shown in fig. 3C may be pre-illustrated to reduce phase-dephasingA transmon dephasing error occurs. The transmon dephasing error in the g-e subspace can be characterized by the following quantum channels:

whereinFor transmon dephasing error probability during each CPS gate, and σ Z ═ g><g|-|e><e | for | g>State and | e>The pi relative phase jumps between states. Transmon de-phasing during either of two CPS gates results in a transmon | e measured at measurement 348>State of which is in conjunction with | g>Are orthogonal and can be detected without ambiguityAnd (4) measuring. Thus, in some embodiments, any first order (i.e., single occurrence) transmon dephasing error during detection of the CPS gate is detected, but as described above, it is not possible to determine which of the two errors occurred, and therefore, the error is only predicted and cannot be corrected.

In some embodiments, in addition to the ground state | g > and the first excited state | e >, an additional energy level to assist transmon may be used to improve the robustness of the e-SWAP operation. For example, the first order error associated with spontaneous decay of the transmon state can be both predicted and corrected. In some embodiments, the cavity-transmon coupling hamilton is changed to "error transparent" such that the transmon decayed leading order (leading order) error is swapped with the error transparent hamilton for all logic states of the cavity. The transmon decay that occurs during evolution can then be identified as the transmon decay that occurs at the end of evolution, which can significantly simplify error analysis.

In some embodiments, four transmon energy levels (| g >, | r >, | r '>, | e >) are used, where | g > - | e > subspace is used to encode the information qubit and | r > - | r' >, the | subspace is used to detect the error. Fig. 5A shows a diagram 500 of the energy level of transmon. State | e >501 represents some higher excitation level (e.g., higher than the first excitation state) that does not decay directly to state | g >502, but rather decays as a lead decay error to state | r > 503. By introducing additional energy levels, the transmon decay error from | g > - | e > subspace can be characterized by the following quantum channels

Wherein the content of the first and second substances,the subspace spanned is used to encode the qubits, and | e>、|r>And | r'>The spanned subspace has a uniform dispersive coupling strength with the cavity mode.

In some implementations, a quantum circuit similar to that shown in fig. 3C is used, but the unitary operation is generalized to a four-level transmon. In some embodiments, the dispersion coupling between the cavity and the transmon is designed

Controllable reactionary to cavity modes is controlled so that states | e >, | r >, and | r' > have the same dispersion shift χ, so that the CPS gate is

In some implementations, the unitary associated with the generalized hadamard gate becomes a unitary

Thus, Hadamard rotation is performed within the g-e subspace and generally works on the r-r' subspace.

In some embodiments, the transmon rotation becomes

It rotates within the g-e subspace and swaps | r > and | r' >.

In some embodiments, CPS operation becomes

By extending the gates above for four levels of transmon, one order transmon dephasing errors can be detected and corrected.

Transmon decay during the first CPS operation 343 causes a transition state | r'>While the cavity mode continues to evolve without error under CPS gates, since the decay does not change the dispersion coupling. T during the second CPS operation 345The ransmon decay causes a response to the transmon state r>While the cavity mode evolves to the desired U_eSWAP(theta). Since the transmon decay error (r 'associated with the transmon decay during the first and second CPS gates, respectively) can be unambiguously distinguished'>Sum of states | r>States), a first order transmon decay error during the CPS gate may be actively corrected in some embodiments.

In some embodiments, by folding the r-r' subspace to a single state | r>And reduces the operations within the r-r' fold to the pair state | r>By trivial operation of (e.g. X)_θ|r>＝|r>) Three transmon levels (| g) may be used>，|r>，|e>). In some embodiments, such a detection scheme may also detect transmon thermal heating (e.g., | g)>→|r>). Thus, the three-stage transmon assist mode is sufficient to enable detection of a first-order transmon dephasing/decay/heating error during the CPS gate.

In some embodiments, both transit decay and heating errors are corrected by deploying additional transit stages. For example, the dispersion coupling in equation (10) and six transmon stages 511 to 516(| g >, | e >, | r >, | r '>, | s >, | s' >) with decay transitions shown in the energy level diagram 510 of fig. 5B may be used. In some embodiments, qubits of information are encoded in the g-e subspace. Based on the measurement of the transmon state that causes | r' > or | r, > the transmon decay error during the first CPS gate or the second CPS gate may be corrected. Based on the measurement of the transacton state for | s' > or | s >, a transacton heating error during the first CPS gate or the second CPS gate may be corrected.

Some embodiments use the above-described techniques in various applications. Such as quantum routers, quantum RAMs, quantum state comparisons, quantum principal component analysis, or general quantum computation.

In a modular architecture for information processing, a router has an indispensable role in connecting different modules and different components within the modules. There are many different types of routers used for classical and quantum information processing. As listed in table 1, routers may be classified based on the classical/quantum (C/Q) properties of the input and control signals. Different types of routers and example related applications. Classical-classical routers are only classical switches. The classical-quantum router sends quantum information to the classical address. The quantum-quantum router sends quantum information to the quantum overlay of the address. The quantum-classical router sends classical information to the sub-address. However, since the information is classical, the quantum state collapse causes only one address to receive a classical signal. This occurs in quantum measurements, where a semi-classical "meter" is entangled with a quantum state, and the quantum state "collapses" when the meter is "read".

TABLE 1

In some embodiments, the Q-Q router is implemented using c-SWAP operation by: (1) storing the quantum input and control signals in the cavity mode and the transmon mode, respectively, (2) applying c-SWAP operation on the cavity mode conditioned on the transmon mode, and (3) retrieving the quantum signals from the cavity mode and the transmon mode.

In some embodiments, a quantum Random Access Memory (RAM) may be implemented using c-SWAP operations. Quantum RAM can perform memory accesses using coherent quantum superposition of addresses. To build an efficient quantum RAM, a Q-Q router with three levels (transmon) of memory (labeled "0", "L", and "R") and three cavity modes (labeled "input", "left", and "right") is used. The (transmon) memory is initialized to the "0" state and all three chambers are initialized to the vacuum | vac > state. The goal of a Q-Q router is to process or route a quantum signal, which may be a superposition of three possible states, vacuum | vac >, left | L >, or right | R >.

In some embodiments, the Q-Q router has five different functional settings: (1) idle, (2) store control signals, (3) forward route signals, (4) reverse route signals, and (5) retrieve control signals. Note that steps (2) and (3) can be used to open up a path towards the target memory, while steps (4) and (5) can be used to do a reverse unitary to decouple the quantum RAM.

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

(1) when the Q-Q router is idle, no application operation is required.

(2) To store the control signal, the Q-Q router stores the incoming control signal in the input cavity and then transmits the control signal from the input cavity to the transmon mode.

(3) To forward route the signal, the Q-Q router temporarily stores the signal in the input cavity. Then, the Q-Q router (via c-SWAP operation) directs the quantum signal to the "input" cavity, the "left" cavity, or the "right" cavity, respectively, conditioned on the (transmon) memory being in states "0", "L", or "R". Finally, the Q-Q router releases the signals in the "left" and "right" cavities to allow it to move on to the next stage.

(4) To route the signals in reverse, the Q-Q router performs the reverse of (3). For example, a Q-Q router stores incoming signals to a "left" cavity and a "right" cavity. Then, conditioned on the (transmon) memory state "0", "L", or "R", the Q-Q router directs the quantum signal in the "input" cavity, the "left" cavity, or the "right" cavity (via c-SWAP operation) to the "input" cavity, respectively. Finally, the Q-Q router retrieves the signal from the input cavity.

(5) To receive the control signal, the Q-Q router performs the reverse of (2). For example, a Q-Q router transmits control signals from (transmon) memory to an input cavity, and then retrieves signals from the input cavity.

In some embodiments, the overlap of two quantum states may be estimated using a c-SWAP operation. For example, if transmon is initialized to | g>+|e>States and quantum states of the two subsystems (cavities) are represented by a density matrix ρ^AAnd ρ^BIf so, the initial input state is:

after the c-SWAP operation, the state changes to:

wherein the reduced density matrix takes the form

Wherein the characteristics of use

Thus, in some embodiments, by measuring ρ_transmonPhase coherence (e.g., Ramseyfinger (Ramseyfinger) contrast) of (A), the overlapping O (ρ) can be extracted^A，ρ^B)≡Trρ^Aρ^BIt is always real.

In some embodiments, transmon is measured projectively on the basis of | g > + | e >. If transmon is measured as | g > + | e >, then both subsystems are projected into a symmetric subspace (i.e., the eigensubspace for the SWAP operator's eigenvalue + 1),

II therein₊Is a projection operator. If transmon is measured as | g>+|e>Then the two subsystems are projected into an anti-symmetric subspace (i.e., the eigensubspace for eigenvalues-1 of the SWAP operator),

wherein the projection operator pi-1-pi₊. Subsequent measurements should give the same result, which is consistent with the quantum non-destructive nature of these measurements.

In some embodiments, for purposes of quantum state comparison,the more robust quantum circuit 600 as shown in fig. 6 acting on the first 601, second 602 and transmon 603 comprises the following modifications: (1) removing the second beam splitter that is not necessary for quantum state comparison purposes (see fig. 2B); (2) initialization of transmon to | g>And rotated to | g just before the CPS door>+|e>To avoid transmon decoherence during operation of the first splitter. The simplified circuit 600 minimizes the use of splitter operations and decoherence of the transmon probe and includes: (1) a first beam splitter operation 605 between the first cavity 601 and the second cavity 602; (2) a first hadamard operation 606 on transmon 603; (3) CPS operation 607 between the second chamber 602 and transmon 603; (4) a second hadamard operation 608 on transmon 603; and (5) measurement 609 of transmon 603 state. Note that this simplified circuit may also be interpreted as an astronomical measurement of the second cavity mode after the beam splitter operation. Pi for symmetric subspace based on projection operator₊Can easily be justified, the symmetric subspace pi₊From symmetrical states having non-negative integers n, mSpan (similarly, antisymmetric subspace Π_-From an anti-symmetric stateCross over). Recall that, after 50/50 beam splitters,creating operators corresponding to the first cavity mode and the second cavity mode, respectively. Thus, in the second cavity mode the symmetric state always has 2n even excitations, and the anti-symmetric state always has 2n +1 odd excitations.

For p^AAnd ρ^BTo obtain a binary result of 0 or 1 associated with a transmon measurement, N- ∈ is required^-2For rho^AAnd ρ^BTo reliably estimate the overlap O (p) with accuracy ∈^A，ρ^B)。

In some implementations, quantum principal component analysis (qPCA) can be performed using e-SWAP operations, which can be used to implement machine learning from large datasets. More specifically, qPCA may reveal the largest eigenvalues of the ensemble of similarly prepared density matrices. The key idea is to use the following characteristics of e-SWAP

It effectively models hamiltonian evolution, where hamiltonian is the hermitian density matrix H ═ ρ, for small durations Δ t. Using n identical copies of the density matrix ρ, e can be constructed^-iρnΔtσe^iρnΔt. In some embodiments, together with a quantum phase estimation algorithm, the phase associated with the largest few eigenvalues of the density matrix ρ may be efficiently estimated. An e-SWAP gate according to some embodiments herein will implement a physical implementation of the key steps of qPCA.

In some embodiments, e-SWAP gates are used to implement generic quantum computation with boson sub-system, which can be encoded with any single-mode boson (denoted asAndand (4) compatibility. For example, if four such boson modes are used to encode a logical qubit, a four-track logical qubit basis is

With sub-index marking patterns. Due to the fact thatAre the eigenstates of the SWAP operator with eigenvalues + -1, respectively, so that a four-rail logic Z rotation is an e-SWAP operation

Z_Q(θ)＝exp[-iθSWAP_1，2]＝exp[iθSWAP_3，4](23)

In addition, there is SWAP_1，3SWAP_2，4|0_Q>＝|1_Q>＝X_Q|0_Q>This yields that the four-track logical X rotation is a four-modulus e-SWAP operation

X_Q(θ)＝exp[-iθSWAP_1，3SWAP_2，4](24)

Finally, a controlled Z-gate between two encoded four-track logic qubits is another four-mode e-SWAP operation

CZ(θ)＝exp[-iθSWAP_1，2SWAP_1’，2’](25)

With sub-indices 1 'and 2' labeling the first two boson modes from another four-orbital logical qubit. Given any logic Z and X rotations and controlled Z gates, it is sufficient to implement any general quantum computation. Since the gate does not depend on orthogonal basisAndso the scheme can be used for any single mode boson coding, including cat codes, binomial codes, GKP codes, etc. Thus, the robust design described above for the e-SWAP implementation (see fig. 4C) is sufficient to implement the key components of a general quantum computation.

Having thus described several aspects and embodiments of the technology set forth in this disclosure, it is to be appreciated various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the functions 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 embodiments described herein. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, embodiments of the invention may be practiced otherwise than as specifically described. 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 scope of the present disclosure.

Further, as described, some aspects may be implemented as one or more methods. The actions performed as part of the method may be ordered in any suitable way. Thus, 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.

As defined and used herein, all definitions 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 in this specification and the claims should be understood to mean "at least one" unless expressly indicated to the contrary.

The phrase "and/or" as used in the specification and claims should be understood to mean "either or both" of the elements so combined, i.e., elements that are present in combination in some cases and are present in isolation in other cases. Multiple elements listed with "and/or" should be interpreted in the same manner, i.e., "one or more" of the elements so combined. In addition to the elements specifically identified by the "and/or" clause, other elements may optionally be present, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, when used in conjunction with open language such as "including," references to "a and/or B" may refer in one embodiment to only a (optionally including elements other than B); and in another embodiment may refer to B only (optionally including elements other than a); may refer to both a and B (optionally including other elements) in yet another embodiment; and the like.

As used herein in the specification and claims, the phrase "at least one," when referring 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 in the list of elements, and not excluding any combinations of elements in the list of elements. The definition also allows that elements other than those specifically identified in the list of elements to which the phrase "at least one" refers may optionally be present, 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") may refer, in one embodiment, to at least one a (optionally including more than one a) without B (and optionally including elements other than B); in another embodiment, may refer to at least one B (optionally including more than one B) without a (and optionally including elements other than a); in yet another embodiment, may refer to at least one a (optionally including more than one a) and at least one B (optionally including more than one B) (and optionally including other elements); and the like.

Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of "including," "comprising," or "having," "containing," "involving," and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

In the claims, as well as in the specification above, all transitional phrases such as "comprising," "including," "carrying," "having," "containing," "involving," "holding," "consisting of," and the like are to be understood to be open-ended, i.e., to mean including but not limited to. The transition phrases "consisting of" and "consisting essentially of" alone shall be closed or semi-closed transition phrases, respectively.