阅读说明：本技术 广义量子通道 (sense quantum channel ) 是由 蒋良 沈超 诺耿朱 维克托·V·艾伯特 斯特凡·克拉斯塔诺夫 米歇尔·德沃尔特 罗伯特· 于 2017-11-10 设计创作，主要内容包括：根据一些方面,提供了一种量子信息系统,该量子信息系统包括：辅助量子比特；量子数位,其耦合到该辅助量子比特；检测器,其被配置成基于辅助量子比特的量子态来生成检测结果；以及驱动源,其耦合到量子数位和辅助量子比特,并且被配置成基于该检测结果将至少一个量子数位驱动信号施加至量子数位以及基于该检测结果将至少一个量子比特驱动信号施加至量子数位。(According to aspects, there is provided a quantum information system that includes an ancillary qubit, a qubit coupled to the ancillary qubit, a detector configured to generate detection results based on quantum states of the ancillary qubit, and a drive source coupled to the qubit and the ancillary qubit and configured to apply at least qubit drive signals to the qubit based on the detection results and at least qubit drive signals to the qubit based on the detection results.)

1, a quantum information system, comprising:

an auxiliary qubit;

a qubit coupled to the ancillary qubit;

a detector configured to generate a detection result based on a quantum state of the ancillary qubit; and

a drive source coupled to the qubit and the auxiliary qubit and configured to apply at least qubit drive signals to the qubit based on the detection results and at least qubit drive signals to the qubit based on the detection results.

2. The quantum information system of claim 1, further comprising a controller coupled to the drive source and the detector, wherein the controller is configured to:

receiving a detection signal indicative of the detection result from the detector;

controlling the drive source to drive the qubit using the at least qubit drive signals, and

controlling the drive source to drive the qubit using the at least qubit drive signals.

3. The quantum information system of claim 2, wherein the controller is further configured to:

obtaining an indication of a desired quantum channel;

determining the at least qubit drive signals based on the indication of the desired quantum channel, and

determining the at least qubit drive signals based on the indication of the desired quantum channel.

4. The quantum information system of claim 3, wherein the indication of the desired quantum channel is received from a user.

5. The quantum information system of claim 3, wherein the indication of the desired quantum channel comprises a plurality of Kraus operators.

6. The quantum information system of claim 5, wherein the plurality of Kraus operators constitutes a minimum Kraus representation of the desired quantum channel.

7. The quantum information system of claim 6, wherein obtaining the indication of the desired quantum channel comprises determining the plurality of Kraus operators that constitute the smallest Kraus representation of the desired channel from or more other operators.

8. The quantum information system of claim 7, wherein the controller is further configured to determine a plurality of joint unitary operators, each of the plurality of joint unitary operators configured to operate on a joint system comprising the ancillary qubits and the qubits.

9. The quantum information system of claim 8, wherein the controller is further configured to create a binary tree structure associated with the plurality of unitary operators.

10. The quantum information system of claim 9, wherein the controller is further configured to determine each of the plurality of joint unitary operators based on diagonalization of a sum of a subset of the plurality of Kraus operators rooted at an associated node of the binary tree structure.

11. The quantum information system of claim 10, wherein the controller is further configured to determine a qubit unitary operator, a qubit unitary operator, and a second qubit unitary operator associated with each of the plurality of joint unitary operators and based on each of the plurality of joint unitary operators.

12. The quantum information system of claim 11, wherein the controller is further configured to:

determining the at least qubit drive signals based on the qubit unitary operator and the second qubit unitary operator, and

determining the at least qubit drive signals based on the qubit unitary operator.

13. The quantum information system of claim 3, wherein the desired quantum channel comprises an initialization channel for the qubit.

14. The quantum information system of claim 3, wherein the desired quantum channel comprises a stabilization channel for the quantum digit.

15. The quantum information system of claim 3, wherein the desired quantum channel comprises a quantum error correction channel for the quantum digital bit.

16. A quantum information system according to claim 3, wherein the desired quantum channel comprises a positive operator value measurement of the qubit.

17. The quantum information system of claim 3, wherein the desired quantum channel comprises a quantum instrument channel.

18. The quantum information system of claim 1, wherein the ancillary qubit is a superconducting qubit.

19. The quantum information system of claim 18, wherein the auxiliary qubit is included at least on a josephson junction.

20. The quantum information system of claim 19, wherein the ancillary qubit comprises a transmon qubit.

21. The quantum information system of claim 1, wherein the quantum digital bit comprises a quantum oscillator.

22. The quantum information system of claim 21, wherein the quantum oscillator comprises electromagnetic radiation in a cavity.

23. The quantum information system of claim 22, wherein the cavity is a stripline cavity.

24. The quantum information system of claim 22, wherein the cavity comprises a three-dimensional metal cavity.

25. A quantum information system according to claim 1, wherein the drive source comprises a source of electromagnetic radiation.

26. A quantum information system according to claim 25, wherein said electromagnetic radiation source comprises a microwave pulse generator.

27. The quantum information system of claim 1, wherein the detector comprises a readout cavity.

28, a method of operating a quantum information system, the quantum information system including a qubit coupled to an ancillary qubit, thereby forming a qubit-qubit system, the method comprising:

applying th unitary operation to the qubit system;

generating a detection result based on the quantum state of the ancillary qubit; and

applying a second unitary operation to the qubit system based on the detection result.

29. The method of claim 28 further comprising initializing said ancillary qubits prior to applying said unitary operation.

30. The method of claim 28, further comprising:

obtaining an indication of a desired quantum channel;

determining at least qubit drive signals based on the indication of the desired quantum channel, and

determining at least qubit drive signals based on the indication of the desired quantum channel.

31. The method of claim 30, wherein the indication of the desired quantum channel is received from a user.

32. The method of claim 30, wherein the indication of the desired quantum channel comprises a plurality of Kraus operators.

33. The method of claim 32, wherein the plurality of Kraus operators constitutes a minimum Kraus representation of the desired quantum channel.

34. The method of claim 33, wherein obtaining the indication of the desired quantum channel comprises determining the plurality of Kraus operators that constitute the minimum Kraus representation of the desired channel from or more other operators.

35. The method of claim 34 further comprising determining a plurality of joint unitary operators, each of the plurality of joint unitary operators configured to operate on a joint system comprising the ancillary qubits and the qubits.

36. The method of claim 35, further comprising: a binary tree structure associated with the plurality of unitary operators is determined.

37. The method of claim 36, further comprising determining each of the plurality of joint unitary operators based on diagonalization of a sum of a subset of the plurality of Kraus operators rooted at an associated node of the binary tree structure.

38. The method of claim 37 further comprising determining a qubit unitary operator, a qubit unitary operator, and a second qubit unitary operator associated with each of the plurality of joint unitary operators and based on each of the plurality of joint unitary operators.

39. The method of claim 38, further comprising:

determining the at least qubit drive signals based on the qubit unitary operator and the second qubit unitary operator, and

determining the at least qubit drive signals based on the qubit unitary operator.

40. The method of claim 30, wherein the desired quantum channel comprises an initialization channel for the qubit.

41. The method of claim 30, wherein the desired quantum channel comprises a stabilization channel for the quantum digit.

42. The method of claim 3, wherein the desired quantum channel comprises a quantum error correction channel for the quantum digital bit.

43. The method of claim 30, wherein the desired quantum channel comprises a positive operator value measurement of the qubit.

44. The method of claim 30, wherein the desired quantum channel comprises a quantum instrument channel.

45. The method of claim 28, wherein the ancillary qubit is a superconducting qubit.

46. The method of claim 45, wherein the auxiliary qubit is included at least on a Josephson junction.

47. The method of claim 46, wherein the ancillary qubit comprises a transmon qubit.

48. The method of claim 28, wherein the quantum digital bit comprises a quantum oscillator.

49. The method of claim 48, wherein the quantum oscillator comprises electromagnetic radiation in a cavity.

50. The method of claim 49, wherein the cavity is a stripline cavity.

51. The method of claim 49, wherein the cavity comprises a three-dimensional metal cavity.

52. The method of claim 28, wherein the drive signal comprises an electromagnetic radiation signal.

53. The method of claim 52, wherein the electromagnetic radiation signal comprises a microwave radiation signal.

54. The method of claim 28, wherein the detector comprises a readout cavity.

55. At least non-transitory storage media encoded with executable instructions that, when executed by at least processors, cause the at least processors to perform a method of creating a -sense quantum channel, wherein the method comprises:

obtaining a plurality of Kraus operators associated with a desired quantum channel;

generating a plurality of unitary joint operations associated with a joint qubit system based on the Kraus operator;

determining th unitary joint operation of a plurality of unitary joint operations to be performed on the joint qubit system, and

determining two unitary qubit-only operations and unitary qubit-only operations based on the th unitary joint operation.

56. The at least non-transitory storage media of claim 55, wherein obtaining the plurality of Kraus operators comprises:

receiving an indication of a desired quantum channel; and

determining a minimum Kraus representation of the quantum channel based on the indication.

57. The at least non-transitory storage media of claim 56, wherein the indication of the desired channel includes a plurality of non-minimum Kraus operators.

58. The at least non-transitory storage media of claim 56, wherein the indication of the desired channel comprises a super operator matrix, a Choi matrix, or a Jamiolkowski matrix.

59. The at least non-transitory storage media of claim 55, wherein the method further comprises generating a binary tree structure associated with the plurality of unitary joint operations.

60. The at least non-transitory storage media of claim 59, wherein each node of the binary tree structure is associated with a respective unitary joint operation of the plurality of unitary joint operations.

61. The at least non-transitory storage media of claim 60, wherein each leaf of the binary tree structure is associated with a Kraus operator of the desired quantum channel.

62. The at least non-transitory storage media of claim 55, wherein the unitary qubit-only operation is a selective number-dependent arbitrary phase operation.

Technical Field

The technology described herein relates generally to quantum information systems. In particular, the present application relates to systems and methods for controlling quantum mechanical systems.

Background

As in in conventional information processing where information is stored in or more bits, quantum information may be stored in or more qubits called "qubits". The qubits may be physically implemented in any two-state Josephson mechanical system such as photon polarization, electron spin, nuclear spin, or various characteristics of superconducting Severson junctions (Josephson junctions), such as charge, energy, or current direction.

In addition, "quantum digital" (qudit), a quantum system having a number "d" of discrete quantum states, may be used to store and process quantum information. A qubit is a specific example of a qubit with d-2. A quantum digital bit may be implemented using a physical quantum system having multiple states with multiple energy levels, such as a quantum oscillator.

Disclosure of Invention

The present application relates generally to systems and methods for controlling quantum mechanical systems.

According to aspects, there is provided a qubit information system including an ancillary qubit, a qubit coupled to the ancillary qubit, a detector configured to generate detection results based on quantum states of the ancillary qubit, and a drive source coupled to the qubit and the ancillary qubit and configured to apply at least qubit drive signals to the qubit based on the detection results and at least qubit drive signals to the qubit based on the detection results.

According to embodiments, the quantum information system further includes a controller coupled to the drive source and the detector, wherein the controller is configured to receive a detection signal from the detector indicative of a detection result, control the drive source to drive the qubit using at least qubit drive signals, and control the drive source to drive the qubit using at least qubit drive signals.

According to embodiments, the controller is further configured to obtain an indication of a desired quantum channel, determine at least qubit drive signals based on the indication of the desired quantum channel, and determine at least qubit drive signals based on the indication of the desired quantum channel.

According to , an indication of a desired quantum channel is received from a user.

According to embodiments, the indication of the desired quantum channel includes a plurality of Kraus operators.

According to embodiments, the multiple Kraus operators constitute a minimum Kraus representation of the desired quantum channel.

According to embodiments, obtaining the indication of the desired quantum channel includes determining, from or more other operators, a plurality of Kraus operators that constitute a minimum Kraus representation of the desired channel.

According to embodiments, the controller is further configured to determine a plurality of joint unitary operators, each of the plurality of joint unitary operators configured to operate on a joint system comprising an ancillary qubit and a qubit.

According to embodiments, the controller is further configured to create a binary tree structure associated with the plurality of unitary operators.

According to embodiments, the controller is further configured to determine each of the plurality of joint unitary operators based on diagonalization of a sum of a subset of the plurality of Kraus operators rooted at an associated node of the binary tree structure.

According to embodiments, the controller is further configured to determine a qubit unitary operator, a qubit unitary operator, and a second qubit unitary operator associated with each of the plurality of joint unitary operators and based on each of the plurality of joint unitary operators.

According to embodiments, the controller is further configured to determine at least qubit drive signals based on a qubit unitary operator and a second qubit unitary operator and to determine at least qubit drive signals based on the qubit unitary operator.

According to embodiments, it is desirable that the quantum channel include an initialization channel for the qubit.

According to embodiments, it is desirable that the quantum channel include a stabilization channel for the quantum digit.

According to embodiments, it is desirable that the quantum channel include a quantum error correction channel for the quantum digit.

According to , wherein the desired quantum channel comprises a positive operator value measurement of a quantum digit.

According to embodiments, it is desirable that the quantum channel comprise a quantum instrument channel.

According to , the ancillary qubits are superconducting qubits.

According to , the auxiliary qubits are included at least on a josephson junction.

According to , the ancillary qubits include a transmon qubit.

According to , the quantum digital bit includes a quantum oscillator.

According to embodiments, a quantum oscillator includes electromagnetic radiation in a cavity.

According to the cavity is a stripline cavity.

According to embodiments, the cavity comprises a three-dimensional metal cavity.

According to , the drive source includes a source of electromagnetic radiation.

According to , the electromagnetic radiation source includes a microwave pulse generator.

According to embodiments, the detector includes a readout cavity.

aspects relate to methods of operating a quantum information system that includes a qubit coupled to a qubit to form a qubit-qubit system that includes applying a th unitary operation to the qubit-qubit system, generating a detection result based on a quantum state of the qubit, and applying a second unitary operation to the qubit-qubit system based on the detection result.

According to embodiments, the method further includes initializing the ancillary qubits prior to applying the unitary operation.

According to embodiments, the method further includes obtaining an indication of a desired quantum channel, determining at least qubit drive signals based on the indication of the desired quantum channel, and determining at least qubit drive signals based on the indication of the desired quantum channel.

According to , an indication of a desired quantum channel is received from a user.

According to embodiments, the indication of the desired quantum channel includes a plurality of Kraus operators.

According to embodiments, the multiple Kraus operators constitute a minimum Kraus representation of the desired quantum channel.

According to embodiments, obtaining the indication of the desired quantum channel includes determining, from or more other operators, a plurality of Kraus operators that constitute a minimum Kraus representation of the desired channel.

According to embodiments, the method further includes determining a plurality of joint unitary operators, each of the plurality of joint unitary operators configured to operate on a joint system including an ancillary qubit and a qubit.

According to embodiments, the method further includes determining a binary tree structure associated with the plurality of unitary operators.

According to embodiments, the method further includes determining each of the plurality of joint unitary operators based on diagonalization of a sum of a subset of the plurality of Kraus operators rooted at an associated node of the binary tree structure.

According to embodiments, the method further includes determining a qubit unitary operator associated with each of the plurality of unitary joint operators and based on each of the plurality of unitary joint operators, a qubit unitary operator, and a second qubit unitary operator according to embodiments, the method further includes determining at least qubit drive signals based on the qubit unitary operator and the second qubit unitary operator, and determining at least qubit drive signals based on the qubit unitary operator.

According to embodiments, it is desirable that the quantum channel include an initialization channel for the qubit.

According to embodiments, it is desirable that the quantum channel include a stabilization channel for the quantum digit.

According to embodiments, it is desirable that the quantum channel include a quantum error correction channel for the quantum digit.

According to , the quantum channel is expected to include a positive operator value measurement of a quantum digit.

According to embodiments, it is desirable that the quantum channel comprise a quantum instrument channel.

According to , the ancillary qubits are superconducting qubits.

According to , the auxiliary qubits are included at least on a josephson junction.

According to , the ancillary qubits include a transmon qubit.

According to , the quantum digital bit includes a quantum oscillator.

According to embodiments, a quantum oscillator includes electromagnetic radiation in a cavity.

According to the cavity is a stripline cavity.

According to embodiments, the cavity comprises a three-dimensional metal cavity.

According to , the drive signal includes an electromagnetic radiation signal.

According to , the electromagnetic radiation signal includes a microwave radiation signal.

According to embodiments, the detector includes a readout cavity.

aspects relate to at least non-transitory storage media encoded with executable instructions that, when executed by at least processors, cause the at least processors to perform a method of creating a sense quantum channel, wherein the method includes obtaining a plurality of Kraus operators associated with a desired quantum channel, generating a plurality of unitary joint operations associated with a joint qubit-qubit system based on the Kraus operators, determining a th unitary joint operation of the plurality of unitary joint operations to be performed on the joint qubit-qubit system, and determining two unitary qubit-only operations and unitary qubit-only operations based on a th unitary joint operation.

According to embodiments, obtaining a plurality of Kraus operators includes receiving an indication of a desired quantum channel and determining a minimum Kraus representation of the quantum channel based on the indication.

According to embodiments, the indication of the desired channel includes a plurality of non-minimum Kraus operators.

According to embodiments, the indication of the desired channel includes a super operator matrix, a Choi matrix, or a Jamiolkowski matrix.

According to embodiments, the method further includes generating a binary tree structure associated with the plurality of unitary joint operations.

According to embodiments, each node of the binary tree structure is associated with a respective unitary joint operation of a plurality of unitary joint operations.

According to embodiments, each leaf of the binary tree structure is associated with a Kraus operator of the desired quantum channel.

According to embodiments, the unitary qubit-only operation is a selective number-dependent arbitrary phase operation.

The foregoing is a non-limiting summary of the invention, which is defined by the appended claims.

Drawings

Various aspects and embodiments are described with reference to the following drawings, which are not to scale.

Fig. 1 is a block diagram of a quantum information system according to embodiments.

Fig. 2 depicts a quantum circuit for constructing an arbitrary quantum channel according to embodiments.

Fig. 3 depicts a binary tree structure for constructing arbitrary quantum channels according to embodiments.

Fig. 4 is a flow diagram of a method of operating a quantum information system according to embodiments.

Fig. 5 is a block diagram of a quantum information system based on cavity quantum electrodynamics, according to embodiments.

Fig. 6 depicts an exemplary spectrum of a transmon qubit of a quantum oscillator coupled to a storage cavity in accordance with embodiments .

Fig. 7 depicts an energy level diagram of a transmon qubit of a quantum oscillator coupled to a storage cavity in accordance with embodiments.

FIG. 8 is a block diagram of a computer system according to embodiments.

Fig. 9 is a flow diagram of a method of operating a quantum information system according to embodiments.

Fig. 10 is a block diagram of different types of quantum channels classified according to their outputs.

Detailed Description

Conventional quantum information processing schemes encode information in or more two-level quantum systems (i.e., "qubits"), the state of a single qubit may be represented by a quantum state | ψ >, which may be an arbitrary superposition of two quantum states |0> and |1>, e.g., | ψ > - α |0> + β |1>, where α and β are complex numbers representing the probability magnitudes of the logical qubits in states |0> and |1>, respectively.

To perform useful quantum information processing, conventional quantum information systems initialize sets of qubits to specific quantum states, implement sets of quanta 0 on the qubits, and measure the final quantum state of the qubits after implementing the quanta 2. 1 type of conventional quanta 3 are single-quantum bits 4, which transition the quantum state of the single-quantum bit from the 7 quantum state to a second quantum state.an example of a single-quantum-bit quantum 5 includes a rotating set of qubits on a Bloch sphere.a second type of conventional quanta 6 is a double-quantum bit , which transitions the quantum state of the quantum bit based on the quantum state of the second qubit.an example of a double-quantum bit includes a Controlled Not (CNOT) and a controlled phase . conventional single-quantum-bit and double-quantum-bit evolve the quantum state of the qubits from the quantum state to the second unitary quantum state.

Conventional quantum information systems typically perform detection of qubits by measuring which quantum state of sets of possible quantum states each qubit is in.this type of measurement is referred to as a projection measurement (sometimes referred to as a projection). an example of a projection measurement includes measuring the quantum state of a qubit on a particular basis to determine a detection result of |0> or |1 >.

The inventors have recognized and appreciated that unique and powerful quantum information processing may be achieved using a more general type of quantum operation known as sense quantum channel and also known as fully positive apodization (CPTP) mapping, which includes not only the unitary quantum and projection measurements described above, but also non-unitary quantum state evolution and sense quantum measurements known as positive operator valued measures (pomms)>Conversion from density matrix

Mixed states of representation, wherein i marks each pure quantum state forming the mixed state, and the coefficient p_iNon-negative and sum to 1.

However, the inventors have recognized and appreciated that any arbitrary CPTP mapping may be implemented for a qubit and an additional single ancillary qubit only, as well as circuit depths that vary logarithmically with the dimension of the qubit.

Referring to fig. 1, a quantum information system 101 for constructing a quantum channel according to embodiments includes a qubit 110, a single ancillary qubit 120, a qubit state detector 130, a controller 140, and a drive source 150. the ancillary qubit 120 is coupled to the qubit 110 such that the state of the qubit 110 can affect the state of the ancillary qubit 120, and vice versa for example, the ancillary qubit 120 and the qubit 110 can be dispersedly coupled — meaning that the detuning between the ancillary qubit 120 and the qubit 110 (e.g., a quantum oscillator in a cavity) is much greater (e.g., orders) than the coupling strength between the ancillary qubit 120 and the qubit 110, the detuning being the frequency difference between the transition frequency of the ancillary qubit 120 and the or more supported modes of the cavity.

Quantum digit 110 may be any suitable d-dimensional quantum system having d quantum states. For example, qubit 110 may include a quantum oscillator (e.g., harmonic or non-harmonic). For example, a quantum digital bit may be physically implemented using a cavity that supports electromagnetic radiation, such as a stripline cavity or a three-dimensional conductive cavity (e.g., made of a metal such as aluminum). The subset of photon number states of the oscillator may be d quantum states of a quantum number bit.

Examples of superconducting qubits include superconducting charge qubits where the two quantum states relate to the charge of the superconductor, superconducting flux qubits where the two quantum states are the direction of current flow, and superconducting phase qubits where the two quantum states are two energy eigenstates.

A drive source 150 is coupled to qubit 110 and to the ancillary qubits 120 to enable the drive source 150 to control the quantum states of the qubits 110 and the ancillary qubits 120. For example, the driving source 150 may implement a unitary operation on the qubit 110 and the auxiliary qubit 120. For example, in an embodiment where qubit 110 is a quantum oscillator and the auxiliary qubit 120 is a transmon qubit, the drive source 150 may generate an electromagnetic signal for driving the qubit 110 and the auxiliary qubit 120.

Qubit state detector 130 measures the state of the auxiliary qubit 120 in embodiments, the measurement of the auxiliary qubit 120 does not interfere with the state of the qubit 110. qubit state detector 130 communicates the detection results to controller 140.

The controller 140 is configured to control the drive signal generated by the drive source 150 in embodiments, the controller 140 may determine the drive signal based on the detection results received from the qubit state detector 130.

As will be described in more detail below, the qubits 110 and the ancillary qubits 120 will operate through multiple unitary operations, controlled by the controller 140 and implemented by the drive source 150. after each unitary operation, the qubit state detector 130 performs a measurement of the ancillary qubits 120. the controller uses the detection results from the qubit state detector 130 to determine a subsequent unitary operation to be performed on the joint qubit-qubit system. in embodiments, the quantum states of the ancillary qubits 120 are reset to an initial state (e.g., the ground state) after each measurement and before performing the unitary operation on the qubits 110 and the ancillary qubits 120.

The above process is discussed in theoretical detail below, followed by an example implementation based on a particular implementation using a cQED device.

Obtaining a minimum Kraus representation of a quantum channel

Quantum channels (i.e., CPTP) can be represented using the Kraus representation:

in the formula 1, the first and second groups,

representing a CPTP mapping acting on a density matrix ρ representing the manipulated quantum states. Using N different Kraus operators K_iTo represent CPTP mappings, where the dagger operator represents the Hermitian conjugate of the Kraus operator is not intended to be a unitary, Hermitian or square matrix

The Kraus representation is not -only, since unitary matrices can be used to form any different set of Kraus operators, and the resulting Kraus operator represents the same CPTP mapping as the original Kraus operator.

To efficiently construct CPTP mappings according to embodiments, it is convenient to perform the Kraus representation with the minimum number of Kraus operators, referred to as the Kraus rank of the CPTP mapping²A linear independent operator, so that the Kraus rank is not greater than d². Efficient computational techniques known in the art can be used to convert the non-minimal representation of the CPTP map to a minimal Kraus representation. For example, the Kraus representation may be converted to a Choi matrix (d)²×d²Hermitian matrix) and derived therefrom the minimum Kraus representation as described in m.d. choi, Linear Algebra appl, 10,285(1975), the entire content of which and at least for its discussion of techniques for determining the minimum Kraus representation is incorporated herein by reference. (in the event that any term used herein conflicts with the use of that term in Choi, the term should be given the most consistent meaning to those of ordinary skill in how to understand its use herein.) the second approach is to compute an overlap matrix

And then to make it diagonalized,

if the original representation is redundant, the new Kraus accounts forIs the most economical representation, where of them are zero matrices.

In some implementations , the quantum channel may be provided in a representation other than the Kraus representation (e.g., a super operator matrix representation, a Jamiolkowski/Choi matrix representation). these alternative representations may also be converted to a minimum Kraus representation.

Since the CPTP mapping is linear in the density matrix ρ, ρ can be considered as a vector and the super-operator acting on the quantum states represented by the density matrix

The matrix form of (a) can be written as:

or

Wherein the content of the first and second substances,

is the state of the quantum system after application of the quantum channel.

The matrix form of the quantum channel also allows the quantum channel to be characterized using the determinant of the matrix.

Obtaining a super operator matrix representation of a quantum channel from a Kraus representation is relatively simple compared to obtaining the Kraus representation from a super operator matrix representation. Given the channel in Kraus, the super operator matrix T can be obtained as follows:

wherein, K_iIs the Kraus operator (where there are N different Kraus operators), and K_i ^*Is the Kraus operator K_iComplex conjugation of (a). However, obtaining the Kraus representation from the super operator matrix T uses channel state duality (i.e., Jamiolkowski-Choi isomorphism), known from it for having a d-dimensional Hilbert space

Each channel of the system of

As follows and with respect to having a hilbert space

The states (density matrix) () of the two subsystems correspond to:

wherein the content of the first and second substances,

is the maximum entangled state of the two subsystems and τ is the Jamiolkowski matrix representation of the quantum channel. ChoThe i-matrix M is simply the Jamiolkowski matrix τ multiplied by the dimension of the hilbert space, i.e., a constant multiple of the constant d. The relationship between Choi matrix M and super operator matrix T is as follows:

T_ij，mn＝M_im，jnballoon catheter (formula 5)

As a density matrix, τ is Hermite. Furthermore, if and only if

Fully positive, τ is semi-positive; if it is not

Is apodized, then we rank τ the Choi matrix M can be converted to a Kraus representation using the following fact if M is diagonalized:

wherein vi is d of τ²A dimensional feature vector. Then, by mixing

The d × d matrix is rearranged to obtain the Kraus operator. Non-zero eigenvalue λ_i, numerical calculations are performed, wherein the data may be obtained by, for example, having a value of less than 10^-10All eigenvalues of the value of (c) are set to the value 0 to truncate the eigenvalues.

General structural theory of quantum channels

Having described the above techniques for obtaining a minimum Kraus representation of a particular quantum channel, in accordance with embodiments, techniques for physically constructing a desired quantum channel are described embodiments, any arbitrary CPTP mapping is constructed using a binary tree scheme

Are associated. For sheetsA plurality of auxiliary qubits to be applied to the qubit,is the lowest, which is the meaning herein when the structure of the equivalent sub-channel is referred to as "active".

Referring to FIG. 2, sense quantum channel is shown

Quantum circuit 200 of construction(s) of (a) includes series of unitary operations 251 to 253 on qubit 110 and auxiliary qubit 120 over time quantum circuit 200 also includes measurements 231 to 233 after every of unitary operations 251 to 253 each of measurements 231 to 233 may be taken at different times, for example, by a single qubit state detector 130 although fig. 2 shows a specific example of a binary tree depth of L ═ 3, it should be understood that the technique may be extended to any binary tree depth that is a function of the Kraus rank of the desired quantum channel.

The quantum state of qubit 110 may begin in any arbitrary state represented by the density matrix ρ in embodiments, the ancillary qubit 120 is prepared in a predetermined initial state the initial state of the ancillary qubit 120 may be represented by state |0, for example>The illustrated ground state, an th unitary operation 251 to the joint qubit system is performed after initialization of the auxiliary qubits 120, after completion of the th unitary operation 251, a th measurement 231 of the auxiliary qubits 120 is performed resulting in a detection result 241. the th detection result 241 may be output to the controller 140 (not shown in fig. 2) to determine a second unitary operation 252 to be performed on the joint qubit system, after performing the second unitary operation 252, a second measurement 232 of the auxiliary qubits 120 is performed resulting in a second detection result 242. the second detection result 243 may be output to the controller 140. then, a th detection result 241 (which may be stored by the controller) and the second detection result 242 are used to determine a third unitary operation 253 to be performed on the joint qubit systemThe row assists in a third measurement 233 of the qubit 120 resulting in a third detection result 243. The output state of qubit 120 after all operations described is

In the application of quantum channels

In , the classical information contained in the detection results 241 through 243 is output and can be used for additional processing.

Referring to fig. 3, a binary tree representation 300 is used to create the quantum channel described by the quantum circuit 200 of fig. 2. Is shown as

Of the Kraus operators 351 to 358 and the binary tree 300 denoted b^(L)∈{0，1}^LAre associated with different leaves. For example, the Kraus operators 351 to 352, K₀₀₀And K₀₀₁Is identical to the th unitary operation 251 (which is the same for each Kraus operator and is denoted as) A second unitary operation 320 (denoted as U)₀) And is represented as U₀₀Is associated with a third unitary operation 330; kraus operators 353 to 354, namely K₀₁₀And K₀₁₁With th unitary operation 251, 320 th unitary operation and 331 th unitary operation (denoted as U₀₁) Associating; kraus operators 355 to 356, i.e. K₁₀₀And K₁₀₁Corresponding to th unitary operation 251, 321 th unitary operation (denoted as U)₁) And a sixth unitary operation 332 (denoted as U)₁₀) Associating; and Kraus operators 357 to 358, i.e., K₁₁₀And K₁₁₁Corresponding to th unitary operation 251, 321 th unitary operation (denoted as U)₁) And a seventh unitary operation 333 (denoted as U)₁₁) And (4) associating. Thus, for a binary tree with a depth of L ═ 3, there are seven different unitary operations determined to achieve the desired quantum channel.

Applying series of joint qubit-qubit unitary operations, as shown in FIG. 2. which unitary operation is applied in the first round (by

Representing) based on the most recent detection result of the secondary qubit 120 binary tree 300 is used to identify the unitary operation used at each stage based on the detection result of the secondary qubit 120, which is always of two results ("0" or "1". for example, referring to fig. 2 through 3 simultaneously, the unitary operation 251 is not dependent on any measurement result.the second unitary operation 252 is dependent on the detection result 241. if the detection result 252 is "0", the controller 140 uses the second unitary operation 320 of fig. 3 as the second unitary operation 252 of fig. 2. in another aspect , if the detection result 252 is "1", the controller 140 uses the fifth unitary operation 321 of fig. 3 as the second unitary operation 252 of fig. 2. this same technique can be applied to subsequent unitary operations of fig. 2, such as the third unitary operation 252. at each leaf of the tree 300, the branch selected by the controller 140 is based on the most recent detection result "0", the upper branch is selected, if the lower branch is selected, the lower branch is selected.

Any arbitrary quantum channel can be constructed and effectively implemented using quantum circuit 200 and binary tree 300.

Before describing the details of how to generate sense quantum channels according to embodiments, a simplified example of a simple case with a tree depth of L ═ 1 is described, which corresponds to a quantum channel with a Kraus rank less than 2₀And K₁. In this case, the quantum circuit of fig. 2 is simplified to the following steps: (1) initializing auxiliary qubits 120 to state |0>The auxiliary qubits 120 need not be measured, since there are only rounds of operation, and thus no adaptive control or feedbackIn an embodiment, in addition to creating an sense quantum channel of rank 2, the ancillary qubit 120 may be measured for other reasons.

The joint unitary operation U can be represented by a 2d × 2d matrix as follows:

wherein the content of the first and second substances,<0|U|0>＝K₀and<1|U|0>＝K₁are all d × d sub-matrices and the asterisks (×) indicate other sub-matrices that are not relevant in case U is unitary. Thus, the left column of matrix U in equation 7 is a 2d × d matrix, which is isomorphic (isomorphism), meaning that the following condition is satisfied:

the isomorphic condition of equation 7 is guaranteed by the apodization property of the CPTP mapping. When the ancillary qubits 120 are discarded (traced), a quantum channel is realized

Thus, any quantum channel of Kraus rank 2 that acts on qubit 110 can be formed by implementing the single joint unitary operation of equation 7 and a single ancillary qubit 120.

As described below, if the auxiliary qubit 120 is measured instead of being ignored, information of "which track" is obtained in the form of the detection result. This information may be used to determine additional operations to be performed when constructing quantum channels with a Kraus rank greater than 2.

Having thus described a simplified implementation of constructing an arbitrary Kraus rank 2 quantum channel, a similar but more complex technique can be used to construct an arbitrary complex quantum channel having any Kraus rank N in implementations using a circuit depth of

To implement a quantum channel of Kraus rank N, the electricityThe quantum circuit repeats multiple "rounds" of operations, each rounds including (1) initializing the ancillary qubits, (2) performing a unitary operation on the joint qubit-qubit system based on detection results from the first rounds (except rounds, where unitary is not based on measurement results), (3) detecting the ancillary qubits 120, and (4) storing classical detection result information for use in subsequent rounds ^L1 possible unitary operations (2 of a binary tree with a depth L)^L-1 node associated) and 2^LStrip possible trajectories (2 from binary tree)^LWith each leaf associated). For example, in fig. 3, the depth L ═ 3, resulting in seven unitary operations (251, 320, 321, 330, 331, 332, 333) and eight possible trajectories to the leaves 351 to 358, which leaves 351 to 358 are not represented by eight different Kraus operators of quantum channels of Kraus rank eight.

As discussed above in connection with fig. 2, the ith unitary operation consists of

Represents and is compared with node b of a binary tree^(l)＝(b₁b₂…b_L)∈{0，1}^l) And (c) correlating, wherein L is 0. For L1, for

There are only unitary operations, which are

As determined by equation 7 above, unitary is determined in a manner similar to equation 7

In the auxiliary state always in the ground state |0>In the first embodiment, a d × d sub-matrix is specified

It suffices, wherein | b_l+1>Is directed to b_l+1The projected measurement state of the auxiliary qubit of 0, 1. Each leaf b of the binary tree^(L)∈{0，1}^LWith Kraus operator marked in binary representation

Wherein i ═ (b)₁b₂…b_L) +1 and K _i＞N0, where N is the Kraus rank of quantum channels, as shown by leaves 351-358 of fig. 3. Each Kraus operator has an associated singular value decomposition Kraus operator

In embodiments, the dxd sub-matrixThe following can be constructed from the known Kraus operator represented by the minimum Kraus. For each node b with L-1^(l)The non-negative Hermite matrix is determined and diagonalized as follows:

wherein the content of the first and second substances,

is a unitary matrix of the matrix,

is a diagonal matrix with non-negative elements, an

Is to satisfy

The hermitian matrix of. For ease of labelling, the matrix is arranged

Substituted and defined as:

wherein sgn (0) is defined as zero, so thatAnd

matrix array

Is defined as an orthogonal projectionAnd associated projection

The correlated projection

Is defined as

Further, the inverse matrix is defined as:

in addition, the matrix

Moore-Penrose pseudo-inverse ofIs defined as:

finally, for l ═ 0, the following values are fixed:

and

based on the above definitions and relationships, the explicit expression of the correlation sub-matrix of the unitary matrix is for L0.

Wherein, b^(l+1)＝(b^(l)，b_(l+1)) And for L ═ L-1 is

Therefore, the unitary matrix can be completely determined based on equations 15 and 16 using the various definitions described above

And isIs to ensure that the equidistant condition is satisfied

Is used to generate the unitary matrix. Since each of equations 15 and 16 can be determined from the Kraus operator represented by the minimum Kraus, each unitary operation required to construct quantum circuits and binary trees similar to the examples shown in fig. 2 and 3 can be determined from the Kraus operator represented by the minimum Kraus. Note that for L ═ 1, the above equation reduces to:

which is the result of for the quantum channel with Kraus rank 2 as discussed above in connection with equation 7.

FIG. 4 depicts a method 400 of operating a quantum information system including a qubit coupled to an auxiliary qubit to form a qubit-qubit system according to embodiments the method 400 may be applied to, for example, the system 100 shown in FIG. 1 and discussed above, but may also be applied to any suitable quantum system in which the auxiliary qubit 120 is coupled to the qubit 110. in embodiments, the quantum circuit 200 shown in FIG. 2 is implemented using the method 400.

In embodiments, the ancillary qubit 120 is initialized to a ground state of the ancillary qubit 120 the initialization of the quantum state of the ancillary qubit 120 may be performed by driving a state of the ancillary qubit 120 with a drive signal from the drive source 150 and/or performing a measurement of the ancillary qubit 120 using a qubit state detector to project the ancillary qubit 120 to a particular state.

At act 404, if available, the drive source applies a unitary operation to the qubit bit system based on previous detection results, for round , there is no previous detection result, and thus the unitary operation is independent of the measurement results, for all other rounds of operation, or more previous measurement results may be used to determine the unitary operation applied to the joint qubit bit-qubit system, for all other rounds of operation, a binary tree structure such as binary tree structure 300 shown in FIG. 3 may be used to determine the selection of the unitary operator, in embodiments, or more drive signals acting on the qubits 110 and the auxiliary qubits 120 at different times are used to implement the unitary operation such that the unitary operation is decomposed into a plurality of simpler operations.

In some embodiments, the qubit state detector 130 may measure whether the ancillary qubit 120 is in a ground state or an excited state, and in other embodiments, the qubit state detector 130 may measure the ancillary qubit 120 on a basis including a superposition of the ground state and the excited state.

If so, the method 400 returns to act 402. if not, the method 400 ends.

Circuit QED-based implementation

The top section describes how series of unitary operations and measurements can be used to create arbitrary quantum channels (i.e., CPTP maps), where the unitary operations used are adaptively controlled based on the detection results from the measurements.

Referring to FIG. 5, an exemplary cQED-based quantum information system 500 includes a storage cavity 510 and a transmon qubit 520 dispersively coupled from . the storage cavity 510 may be a stripline cavity or a three-dimensional cavity. the storage cavity 510 supports electromagnetic radiation, such as microwave radiation, to produce a quantum oscillator. a predetermined number d of photon number states of the quantum oscillator stored within the storage cavity 510 is used to implement the qubit 110 of FIG. 1. the transmon qubit 520 is used as the auxiliary qubit 120.

The operation between the storage cavity 510 and the transmon qubit 520 may be used to perform entanglement operations between the two quantum systems. These operations may be implemented using drive signals generated by an electromagnetic pulse generator 550 controlled by the controller 540.

The quantum information system includes a readout cavity 532 also coupled to a transmon qubit 520. operations between readout cavity 532 and transmon qubit 510 may map the quantum states of the transmon qubits to the states of a quantum oscillator within readout cavity 532. these operations may be controlled by controller 540 controlling drive signals that control operations performed on the readout cavity and transmon qubit 520. in operation, readout cavity 532 may operate as a fast "readout" oscillator while storage cavity 510 may operate as a "storage" oscillator. in embodiments, readout cavity 532 may have a shorter decay time (and lower quality factor) than storage cavity 510. when using cavity state detector 534 to detect the state of the readout oscillator, the state of storage cavity 510 remains unmeasured.a measurement is passed from transmon qubit 520 to readout cavity 532, then using the quantum state detector of readout cavity, which may be used to detect the state of storage cavity 510. by passing quantum state information from transmon qubit 520 to readout cavity 532, storage cavity 510 may be used to determine the state of storage cavity 520 in a manner known as a measurement of storage cavity 520. this state detector may be used to determine the state of storage cavity 520.

In embodiments, the electromagnetic drive pulses generated by the electromagnetic pulse generator 550 are used to implement unitary operation on the quantum states of the quantum oscillator and the transmon qubit 520 stored in the storage cavity 510.

According to , the joint qubit-qubit system of quantum information system 500 (e.g., the joint system of a quantum oscillator with transmon qubit 520) may be described using the following Hamiltonian (Hamiltonian):

wherein higher is omittedThe term of the order. In formula 18,. omega._qIs the ground state | g of the transmon qubit 520>(sometimes referred to as | 0)>) And excited state | e>(sometimes referred to as | 1)>) Qubit transition frequency in between; omega_cIs the resonant frequency of the cavity; χ is the dispersion coupling constant between the transmon qubit 520 and the oscillator;

and

respectively, a creation operator and an annihilation operator for photons within the storage chamber 510. As a result of the dispersive coupling, the qubit transition frequency changes χ when a photon is added to the cavity. Thus, the drive signal may be passed at a frequency ω_q+ n χ drives the transmon qubit 520 (i.e., by applying an electromagnetic pulse to the transmon qubit 520) to modify a particular Fock state | n of the oscillator>According to embodiments, such a drive signal can modify the Fock state | n by changing the phase of the state>。

By way of illustrative but non-limiting example, the transmon qubit 520 may have a transition frequency ω between 5GHz and 10GHz (such as between 7GHz and 8GHz, or about 7.6GHz)_q(ii) a The quantum mechanical oscillator may have a transition frequency ω between 6GHz and 11GHz, such as between 8GHz and 9GHz or about 8.2GHz_cThe dispersion shift χ may be between 1MHz and 10MHz, such as between 4MHz and 9MHz, or such as about 8.2MHz in embodiments, the dispersion shift χ may be three orders of magnitude greater than the dissipation of the transmon qubit 520 and the storage cavity 510, which allows for greater unitary control of the combined system.

FIG. 5 depicts an illustrative frequency spectrum 500 of a transmon qubit coupled to a quantum oscillator according to embodiments As described above, dispersive coupling between a physical qubit and a quantum mechanical oscillator causes the number state | n > of the oscillator to be decomposed into different frequencies of the transmon qubit.

FIG. 6 is a plan view ofExample of a qubit spectrum 600 for qubits dispersively coupled to a resonant cavity having an average photon number

The horizontal axis of the graph represents the shift in qubit transition frequency for exciting different Fock states of the coupled resonator. In other words, the figure illustrates that the transition frequency of a transmon qubit depends on the number of photons within the cavity.

In the example spectrum 600 of FIG. 5, the different Fock states of the oscillator |0>, |1>, |2, |3>, |4> and |5> are each associated with a different transition frequency of the transmon qubit for example, the transition frequency of a qubit for which no photon is present in the cavity is defined as a detuning of 0MHz (and equal to the surface qubit transition frequency, which as described above can be between 5GHz and 10GHz in embodiments.) when the cavity includes a single photon, the transition frequency of a qubit is detuned by about 10MHz, when the cavity includes two photons, the transition frequency of a qubit is detuned by about 17MHz, when the cavity includes three photons, the transition frequency of a qubit is detuned by about 26MHz, when the cavity includes four photons, the transition frequency of a qubit is detuned by about 34MHz, and when the cavity includes five photons, the transition frequency of a qubit is detuned by about 43 MHz.

FIG. 7 depicts an energy level diagram 700 for a combined system including a transmon qubit 520 dispersively coupled to a storage cavity 510 based on this amount-dependent detuning of the transition frequency of the transmon qubit 520, the qubit can be selectively addressed using drive pulses that are narrow in spectral width and whose center frequency is tuned to match the transition frequency that is detuned for a particular number of excitations.

In some embodiments, the transmon qubit 520 may be driven independently of the storage cavity 510, causing a rotation of the quantum states of the transmon qubit 520. the amount of rotation of the quantum states may depend on the quantum states of the storage cavity 510 (e.g., the rotation may be photon-number dependent). such a rotation induces a photon-number dependent belie phase to the quantum states of the transmon qubit 520 while leaving the transmon states unmodified. different phases θ are shown qualitatively in FIG. 6_iThis type OF operation is referred to as Selective Number-dependent Arbitrary Phase (SNAP) operation and is described in detail in U.S. patent application No. 15/552,998 entitled "TECHNIQUES OF OSCILLATOR CONTROL FOR QUANTUM INFORMATION PROCES AND RELATED SYSTEMS AND METHOD" and filed on 2017, 8/23, the entire contents OF which are incorporated herein by reference FOR at least the purpose OF enabling the discussion OF SNAP (in the event that any term used herein conflicts with the use OF that term in U.S. patent application No. 15/552,998, that term should be given a meaning that most consistent with the use herein as understood by a skilled artisan.)

In embodiments, SNAP can be used to implement the following entangled unitary operations:

wherein, Y_n≡-i|g，n><e, n | + H.c. is the Pauli Y operator for the two-dimensional subspace associated with n photons in the storage cavity 510, H.c. represents the Hermite conjugate, and d is the use of quantumThe d-level of the oscillator is used to physically implement the dimensionality of the quantum digital bits. Entanglement operation U_entAnd the quantum channel { S ] described by the Kraus operator₀，S₁Are associated. Related entanglement operation U'_entCan be operated by first utilizing unitary

Acting solely on the qubit (e.g. storage chamber 510) and executing U_entFollowed by separately performing an adaptive unitary operation W on the storage chamber 510₀Or W₁Wherein a unitary operation W₀Or W₁Depending on the detection results from previous measurements of the transmon qubit 520. Thus:

the decomposition of equation 20 is called "cosine-sine decomposition" and matches the two sub-matrices of the correlation of the desired unitary operation:

wherein the content of the first and second substances,

and is

Based on this, a quantum circuit similar to that of fig. 2 for the sense quantum channel in cQED can be identified for different rounds of unitary operation

Matrix W of₀、S₀、W₁、S₁And V. In this manner, a potentially complex joint unitary operation (e.g., of fig. 2)) Become three simplerSingle unitary operation-two unitary operations acting only on the oscillator and unitary operations acting only on the transmon qubit 520.

In embodiments, the entangled unitary operation U 'is determined in the following manner'_ent. First, singular value decomposition is determined

And

where the W matrix and the S matrix are set to their desired values based on equation 20. Then, V is ensured₀＝V₁To decompose exclusively at , embodiments may require singular values of S₀In descending order such that (S)₀)_j，j≥(S₀)_j+1，J+1And S is₁Are arranged in ascending order such that (S)₁)_j，j≤(S₁)_j+1，j+1. Condition of equal distance

Ensure that

Due to the fact that

Andboth are diagonals of the element in ascending order, and so

Must be the same, which means V₀＝V₁V. Thus, obtain and satisfy

Andu's'_entAll of the components of (a).

In the case of quantum circuits, the techniques described herein for cQED systems reduce the complex 2 d-dimensional unitary operation to two unitary operations, i.e., an entanglement operation and a measurement, that act separately on the quantum digits (e.g., quantum oscillators), where the unitary operation used may be based on the detection results from the measurement.

Computer and software aspects

In embodiments, at least non-transitory storage media are encoded with executable instructions that when executed by at least processors cause at least processors to perform a method of creating 0-sense quantum channels in embodiments controller 140 and/or controller 540 may comprise a computer system that performs such a method, referring to FIG. 8, example computer system 800 may comprise a processor 810, a memory 820, a storage device 830, and an input/output device 840. System bus 801 couples various components of computer system 800 to allow information exchange between the components. in embodiments, at least non-transitory storage media encoded with executable instructions that when executed by at least processors cause at least processors to perform a method of creating -sense quantum channels.

Computer system 800 may include a variety of non-transitory computer-readable media including memory 820 and storage 830. Computer readable media can be any available media that includes both volatile and nonvolatile media, removable and non-removable media. Examples of computer readable media include storage media such as RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, Digital Versatile Disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can accessed by computer system 800.

The memory 820 may include computer storage media in the form of volatile and/or nonvolatile memory such as Read Only Memory (ROM) and Random Access Memory (RAM). A basic input/output system BIOS, containing the basic routines that help to transfer information between elements within computer system 800, such as during start-up, is typically stored in ROM. RAM typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processor 810. By way of example, computer system 800 includes software 822 stored in memory 820 that is executable by processor 810.

The computer 800 may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only, FIG. 8 illustrates storage 830. Storage 830 may be a hard disk drive that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive that reads from or writes to a removable, nonvolatile magnetic disk, and an optical disk drive that reads from or writes to a removable, nonvolatile optical disk such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like.

For example, computer system 800 may include a plurality of input/output devices 840. for ease of operation in a networked environment, computer system 800 may use a network interface to store information in a network storage drive or to receive information from an external computer system connected to a network.Another example of input/output device 840 includes a user interface that allows a user of computer system 800 to input information (such as an indication of a desired quantum channel) and receive feedback.A further example of input/output device 840 may include a keyboard, a touch screen interface, a mouse, a microphone, a speaker, and/or a display is a communication interface that allows the computer system to send data to and receive data from other devices such as qubit state detector 130 and/or drive source 150. for example, computer system 800 may receive detection results from qubit state detector 130 via input/output device 840, store the detection results in memory 820 and/or storage device 830, process data using processor 810 with the detection results, and then transmit information via input/output device 840 to control drive signals implemented by drive source 150.

FIG. 9 depicts an example method 900 for creating a sense quantum channel, by way of example, the acts of method 900 may be performed by a processor 810 of computer system 800, processor 810 being part of controller 140.

The indication may be input using a user interface, for example, the user may input the indication via a user interface, alternatively, the indication may be received from a different computer system via a network interface, in embodiments the indication may be sets of Kraus operators associated with the desired quantum channel, in another embodiment the indication may be a super operator matrix, a Choi matrix, or a Jamiolkowski matrix.

At act 904, the processor 810 determines a minimum Kraus representation of the desired quantum channel. The exact procedure for achieving this action depends on the form of indication of the desired quantum channel. For example, if the indication of the desired quantum channel obtained in act 902 includes a Kraus operator of the minimum Kraus representation of the desired quantum channel, act 902 may simply include verifying that the obtained Kraus operator is indeed the minimum Kraus representation. Alternatively, if the indication of the desired quantum channel is a super operator matrix, a Choi matrix, or a Jamiolkowski matrix, the above-described technique is used to convert the indication to a minimum Kraus representation.

As described above, in embodiments, each node of the binary tree structure is associated with a respective unitary joint operation of a plurality of unitary joint operations, and each leaf of the binary tree structure is associated with a Kraus operator represented by a minimum Kraus of a desired quantum channel.

At act 908, processor 908 converts each unitary joint operation into two unitary oscillator-only operations (unitary-only operations) and unitary qubit-only operations (unitary qubit-only operations). in embodiments, the unitary qubit-only operations are SNAP operations in embodiments, equation 20 is used to determine the unitary oscillator-only operations and the unitary qubit-only operations.

At act 910, the processor 810 determines drive signals associated with two oscillator-only operations and unitary qubit-only operations in embodiments the drive signals for the unitary qubit-only operations have spectral characteristics based on photon-number-dependent transition frequencies described in fig. 7 in embodiments an indication of the determined drive signals is provided to the drive source 150, where the drive signals are generated and directed to the qubit number 110 and the auxiliary qubit 120.

Example applications

The -sense quantum channels (i.e., CPTP maps) described in this application may include a variety of physical operations including cooling, quantum , measurement and dissipation dynamics (discrete dynamics). the ability to construct arbitrary CPTP maps provides a method of system for many aspects of quantum technology.

The application of to construct sense quantum channels is the initialization and/or stabilization of the quantum states of the quantum digits many quantum information processing tasks require working with well-defined (usually pure) initial states common methods are to cool the system to the ground state quiescently by coupling to a cold bath (cold bath) or optically pumping to a specific dark state and then performing a unitary operation to cool the system to the ground state to bring it to the desired initial state.

Quantum tubeRoad

The quantum state of the quantum digit is stabilized at the target state sigma. If the target state has a diagonal representation sigma ═ sigma_μλ_μ|ψ_μ><ψ_μL, where λ_μIs more than or equal to 0 and sigma_μλ_μ1, forms of the Kraus operator representing a stabilized quantum channel are

Wherein, | i>In contrast to the conventional approach discussed in the previous paragraph , the dissipation map bundles the cooling and state preparation steps and pumps any state to the target state σ.

A second application of the -sense quantum channel construction techniques described herein is in Quantum Error Correction (QEC). in this application, multiple stable quantum states or even stable subspace may be stabilized.A plurality of quantum states may be used to encode useful classical or quantum information.in embodiments, a stable subspace using QECs may include a recovery map that implements QECs.

For having N_sA qubit based stabilizer code for a stabilizer generator, the recovery being of Kraus rank

, all N may be measured sequentially using the ancillary qubits_sA stabilizer generator extracts the syndrome and finally performs a correcting unitary operation conditioned on the syndrome pattern. Since the stabilizer generators are swapped with each other, their ordering does not change the syndrome. Furthermore, the stabilizer measurement need not be conditioned on previous measurements, since the unitary operation of the l-th round is only:

wherein the content of the first and second substances,

for the first stabilizer, run in the month

Independent of previous measurement results b^(l-1). Finally, with the syndrome

Conditional on performing corrective unitary operations

In embodiments, QEC codes that satisfy quantum error correction conditions associated with sets of error operations can be used for which Kraus representations of QEC recovery maps can be obtained and efficiently implemented using the construction of -sense quantum channels described herein.

For a small loss probability γ per shot, the coding scheme can correct up to O (γ)²) Which includes the following four correlation processes: identifying evolution

Discard excitationsDiscarding two excitations

And reaction induced dephasingBased on Kraus representation of QEC recovery (Kraus rank 4), the following sets of unitary operations were obtained

For constructing a QEC recovery channel with white-adapted quantum circuits:

wherein the projection is defined as

And

and for sigma ═ ↓, unitary operator

(wherein,

) Will error the state

Transition back to | W_σ>. In other words:

wherein, U^⊥, the first two rounds of projection measurements are performed to extract the error syndrome, in the last rounds, a corrective unitary operation is applied to restore the logical state⁽²⁾If (0, 0), there is no error and the operation is identified

It is sufficient. If b is⁽²⁾With (0, 1), there is a counter-effect induced dephasing error that changes the coefficients of the Fock state, so we need to use

This is corrected. If b is⁽²⁾If (1, 1), then there is a single excitation loss, which can be used

To make the correction. If b is⁽²⁾If (1, 0), then there are two excitation losses, which can be used

To make a full correction. Repeatedly applying the QEC to restore the channel can be performed at the point of time of the channel from | W_↑>And | W_↓>The system is stabilized in the spanned code space. Note that for more complex QEC codes (e.g., GKP code [ GKP _ PRA _2001 ]]) And QEC.

For an approximated QEC code, analytically obtaining an optimal QEC recovery map is challenging, but a semi-deterministic plan may be used to numerically optimize the entanglement fidelity and obtain an optimal QEC recovery map.

In another application of the techniques described herein, if the intermediate measurements are part of of the output, along with the state of the quantum system, the construction of the quantum channel can be further expanded , which results in an interesting class of quantum channels called Quantum Instruments (QI).

Wherein, | mu><μ | is the orthogonal projection of the measuring device with M classical results, ε_μIs a full positive non-incremental mapping, andthe traces are preserved. Note that_μ(ρ) gives the measured state associated with the result μ.

In embodiments, QI is implemented as follows (1) for J1, 2_μTo find epsilon_μ(ρ) with Kraus operator K_μ，jMinimum Kraus representation of ( each with rank J)_μ). (2) Introducing these Kraus operators

Wherein the length of the binary label is L ═ L₁+L₂Wherein, the front

To encode mu, and the remaining previous ones

Bit to encode j (padded with zero operator so that there is always 2 in total)^LOne Kraus operator). (3) Quantum circuits with L-round adaptive evolution and auxiliary measurements are used. (4) Final state of output quantity subsystem and encoding mu associated with M possible classical resultsIn this way, any QI. described in configuration 25 was constructed in embodiments, QI was used to achieve complex condition evolution of the system in embodiments QI was used for quantum information processing tasks that require measurement and adaptive control.

In such embodiments, the constructed quantum channel is actually a Positive Operator Value Measure (POVM), also known as sense quantum measurement.

It is characterized by groups of hermitian semideterminants

Which adds to the identifier in embodiments, positive half pi_μIs decomposed into sets of Kraus operators

Is/are as followsThus, in embodiments, the quantum circuit of the quantum instrument also implements POVM if the qubit state is removed from the QI output, in embodiments, this reduces the binary tree construction scheme of POVM.

In some embodiments, POVM is used for quantum state discrimination any detector may not perfectly discriminate sets of non-orthogonal quantum states however, an optimal detector may achieve a so-called Hellstrom boundary by properly designing the POVM to optimize discrimination between the non-orthogonal states.

Summarizing the above application, there are three different classifications of CPTP mappings based on the output of the mappings, shown in FIG. 10 (a) standard quantum channel 1010 with quantum system (e.g., qubit 1011) as output and discarding all detection results from measurements of the ancillary qubits, (b) POVM 1020 with classical measurement results 1021 through 1023 as output and discarding quantum system, and (c) QI with both quantum system 1031 and at least the portion of classical measurement results 1032 through 1033 being used for output₁Bit-encoded result μ. Discarding the remainder of the measurement recordResidual L₂A bit. In QI 1030 of FIG. 10, L ₁2 and L₂However, in some embodiments, it is more resource efficient to use a classical memory for the classical measurement, so the quantum system does not expand unnecessarily and become overly complex.

Other considerations

Having thus described several aspects of at least embodiments of the invention, it is to be understood that various alterations, modifications, and improvements will readily occur to those skilled in the art, which alterations, modifications, and improvements are intended to be part of this disclosure , and are intended to be within the spirit and scope of the invention.

Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements not specifically discussed in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings, for example, aspects described in the embodiments may be combined in any manner with aspects described in other embodiments.

The use of ordinal terms such as "," "second," "third," etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of claim elements over another claim element or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish claim elements having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

All definitions, as defined and used herein, should be understood to control dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.

The indefinite articles " (" a "and" an ") as used herein in this specification and the claims are to be understood as meaning" at least "unless expressly indicated to the contrary.

As used herein in the specification and claims, the phrase "at least ," when referring to a list of or more elements, is understood to mean at least elements selected from any or more elements in the list of elements, without specifying at least of every elements specifically listed in the list of elements and without excluding any combination of elements in the list of elements.

As used herein in the specification and claims, the phrase "equal" or "the same" when referring to two values (e.g., distance, width, etc.) means that the two values are the same within manufacturing tolerances. Thus, two values being equal or identical may mean that the two values differ from each other by ± 5%.

As used herein in the specification and claims, the phrase "and/or" should be understood to mean "any or two" of the elements so joined (i.e., " or more" of the elements so joined.) the use of "and/or" listing of a plurality of elements should be interpreted in the same manner (i.e., " or more" of the elements so joined).

As used herein in the specification and claims, "or" should be understood to have the same meaning as "and/or" as defined above, "or" and/or "should be interpreted as inclusive, i.e., containing more than elements of at least , but also including more than elements of the multiple or lists, and optionally other unlisted items, when separating items in a list, terms such as" only "or" exactly "that are only explicitly indicated to the contrary, or" consisting of "when used in a claim, such as" consisting of "would mean elements of the multiple or lists , whereas terms used herein" or "in the immediate field such as" , "," " only" or "539" should be interpreted as indicating only exclusive when used in the general sense of "," but not be interpreted as having the meaning of "," or "as used in the general term" .

Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of "including" and "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.