Systems and methods for simulating quantum circuits using decoupled Hamiltonians

Abstract

The present application provides a method and system for simulating quantum circuits. A system may perform operations including generating a transformed Hamiltonian corresponding to a quantum circuit. The transformed Hamiltonian may include transformed local and coupled Hamiltonians. The generation of the transformed Hamiltonian may include obtaining a charge coupling matrix and a flux coupling matrix corresponding to the original Hamiltonian of the quantum circuit, and at least partially diagonalizing the charge coupling matrix and the flux coupling matrix. These operations may also include determining a finite eigenbasis including a plurality of eigenvectors of the transformed local Hamiltonian, projecting the transformed coupled Hamiltonian and the transformed local Hamiltonian onto a finite eigenbasis, and generating an at least partially decoupled Hamiltonian by combining the transformed coupled Hamiltonian and the projection of the local Hamiltonian. These operations may also include using the at least partially decoupled Hamiltonian to simulate the behavior of the quantum circuit.

Description

Technical Field

[0001] The present disclosure relates to quantum computing, and more particularly, to simulating quantum circuits by classical computers using at least partially decoupled Hamiltonians. Background Art

[0002] The design and verification of quantum computers must be performed using classical computers, which poses a problem because quantum computers are constructed to perform certain tasks that classical computers cannot. For example, as quantum computer designs become more complex and involve larger quantum circuits, simple simulation techniques become computationally infeasible. For example, a qubit design like the 0-π qubit involves three degrees of freedom, or modes, so simulating multiple such qubits would involve six or more modes. In addition, components of a quantum computer design beyond qubits may also need to be simulated. As a result, this simple simulation approach to quantum circuits leads to high-dimensional Hamiltonians that are difficult to diagonalize or computationally exponentially expensive to simulate the behavior of a quantum computer design. Summary of the invention

[0003] The disclosed systems and methods involve simulating a quantum circuit using an at least partially decoupled Hamiltonian. The at least partially decoupled Hamiltonian can be generated using a linear transformation of an original Hamiltonian associated with the quantum circuit.

[0004] The disclosed embodiments include a method for simulating a quantum circuit using a computer that processes bits. The method may include: obtaining a representation of the quantum circuit. The method may also include generating a transformed Hamiltonian corresponding to the quantum circuit. The transformed Hamiltonian may include a transformed local Hamiltonian and a transformed coupled Hamiltonian. The method may also include determining a finite eigenbasis including a plurality of eigenvectors of the transformed local Hamiltonian. The method may also include projecting the transformed coupled Hamiltonian onto a finite eigenbasis, the transformed coupled Hamiltonian being represented in a pattern of the transformed local Hamiltonian. The method may also include projecting the transformed local Hamiltonian onto a finite eigenbasis. The method may also include generating an at least partially decoupled Hamiltonian by combining a projection of the transformed coupled Hamiltonian and a projection of the transformed local Hamiltonian. The method may also include simulating the behavior of the quantum circuit using the at least partially decoupled Hamiltonian by the computer.

[0005] The disclosed embodiments include a system for simulating a quantum circuit using a computer that processes bits. The system may include at least one processor and at least one computer-readable medium. The computer-readable medium may contain instructions that, when executed by at least one processor, cause the system to perform operations. The operations may include generating a transformed Hamiltonian corresponding to the quantum circuit. The transformed Hamiltonian may include a transformed local Hamiltonian and a transformed coupled Hamiltonian. The generation of the transformed Hamiltonian may include obtaining a charge coupling matrix and a flux coupling matrix corresponding to the original Hamiltonian of the quantum circuit, and at least partially diagonalizing the charge coupling matrix and the flux coupling matrix. The operations may also include determining a finite eigenbasis including a plurality of eigenvectors of the transformed local Hamiltonian. The operations may also include projecting the transformed coupled Hamiltonian onto a finite eigenbasis, the transformed coupled Hamiltonian being represented as a pattern of the transformed local Hamiltonian. The operations may also include projecting the transformed local Hamiltonian onto a finite eigenbasis. The operations may also include generating an at least partially decoupled Hamiltonian by combining a projection of the transformed coupled Hamiltonian and a projection of the transformed local Hamiltonian. These operations may also include simulating the behavior of quantum circuits using at least partially decoupled Hamiltonians.

[0006] The disclosed embodiments include a non-transitory computer-readable medium containing instructions that are executable by at least one processor of a system to cause the system to perform operations. The operations may include generating a transformed Hamiltonian corresponding to a quantum circuit. The transformed Hamiltonian includes a transformed local Hamiltonian and a transformed coupled Hamiltonian. The generation of the transformed Hamiltonian may include obtaining a charge coupling matrix and a flux coupling matrix corresponding to the original Hamiltonian of the quantum circuit, and at least partially diagonalizing the charge coupling matrix and the flux coupling matrix. The operations may also include determining a finite eigenbasis including a plurality of eigenvectors of the transformed local Hamiltonian. The operations may also include projecting the transformed coupled Hamiltonian onto a finite eigenbasis, the transformed coupled Hamiltonian being represented as a pattern of the transformed local Hamiltonian. The operations may also include projecting the transformed local Hamiltonian onto a finite eigenbasis. The operations may also include generating an at least partially decoupled Hamiltonian by combining a projection of the transformed coupled Hamiltonian and a projection of the transformed local Hamiltonian. These operations may also include simulating the behavior of quantum circuits using at least partially decoupled Hamiltonians by computers processing bits.

[0007] It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosed embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] The accompanying drawings, which constitute a part of this specification, illustrate several embodiments and, together with the description, are used to explain the principles and features of the disclosed embodiments. In the drawings:

[0009] Figure 1A Depicted are example diagrams representing quantum circuits in accordance with disclosed embodiments.

[0010] Figure 1B and 1C Depicted is a schematic diagram of a system according to the disclosed embodiment. Figure 1A The values ​​of the components of the quantum circuit.

[0011] Figure 1D and 1E Depicted is a method for Figure 1A Exemplary flux coupling matrices and charge coupling matrices of a quantum circuit.

[0012] Figure 2A and 2B Depicted are graphs of state and transition energies as a function of per-mode energy levels for a quantum circuit of 1A in accordance with disclosed embodiments.

[0013] Figure 3A Depicted is a flow chart of an exemplary process for simulating a quantum circuit using a classical computer in accordance with the disclosed embodiments.

[0014] Figures 3B to 3J Depicted are exemplary equations involved in the construction of an at least partially decoupled Hamiltonian for simulating a quantum circuit in accordance with the disclosed embodiments.

[0015] Figure 4 Depicted is a flow chart of an exemplary process for selecting a transformed Hamiltonian for simulating a quantum circuit in accordance with the disclosed embodiments.

[0016] Figure 5A Depicted is a flow chart of an exemplary simultaneous approximate diagonalization process for generating a transformed Hamiltonian in accordance with disclosed embodiments.

[0017] Figures 5B to 5D Depicted is a method according to the disclosed embodiment of Figure 5A The process is applied to Figure 1A The transformation matrix and the transformed charge and flux coupling matrix generated by the quantum circuit.

[0018] Figure 5E and 5F Depicted is a method for using a Figures 5B to 5D A transformed Hamiltonian generates a plot of state and transition energies as a function of per-mode energy levels for an at least partially decoupled Hamiltonian.

[0019] Fig. 6A Depicted is a flow chart of an exemplary inductor-only symplectic diagonalization process for generating a transformed Hamiltonian in accordance with disclosed embodiments.

[0020] Figures 6B to 6J Depicted are exemplary equations involved in inductor-only symplectic diagonalization in accordance with disclosed embodiments.

[0021] Figures 6M to 6O Depicted is a method according to the disclosed embodiment of Fig. 6A The process is applied to Figure 1A The transformation matrix and the transformed charge and flux coupling matrix generated by the quantum circuit.

[0022] Figure 6P and 6Q Depicted is a method for using a Figures 6M to 6O A transformed Hamiltonian generates a plot of state and transition energies as a function of per-mode energy levels for an at least partially decoupled Hamiltonian.

[0023] Fig. 7A Depicted is a flow chart of an exemplary fully symplectic diagonalization process for generating a transformed Hamiltonian in accordance with the disclosed embodiments.

[0024] Figures 7B to 7H Depicted are exemplary equations involved in inductor-only symplectic diagonalization in accordance with disclosed embodiments.

[0025] Figures 7I to 7K Depicted is a method according to the disclosed embodiment of Fig. 7A The process is applied to Figure 1A The transformation matrix and the transformed charge and flux coupling matrix generated by the quantum circuit.

[0026] Figure 7L and 7M Depicted is a method for using a Figures 7I to 7K A transformed Hamiltonian generates a plot of state and transition energies as a function of per-mode energy levels for an at least partially decoupled Hamiltonian.

[0027] Figure 8 A classical computer suitable for simulating quantum circuits using an at least partially decoupled Hamiltonian in accordance with the disclosed embodiments is depicted.

[0028] Fig. 9 An exemplary quantum circuit simulator according to some embodiments of the present disclosure is shown. DETAILED DESCRIPTION

[0029] Reference will now be made in detail to the exemplary embodiments discussed with reference to the accompanying drawings. In some cases, the same reference numerals will be used throughout all drawings and the following description to refer to the same or similar parts. Unless otherwise defined, technical or scientific terms have the meanings commonly understood by those of ordinary skill in the art. The disclosed embodiments are described in sufficient detail to enable those skilled in the art to practice the disclosed embodiments. It should be understood that other embodiments may be utilized and may be changed without departing from the scope of the disclosed embodiments. Therefore, materials, methods, and examples are illustrative only and are not meant to be necessarily restrictive.

[0030] Quantum computers offer the ability to perform certain tasks (equivalently, solve certain problems) that are thought to be intractable for classical computers, including any possible future classical computers. To understand the advantages of quantum computers, it is useful to understand how quantum computers contrast with classical computers. Classical computers operate according to digital logic. Digital logic refers to a logic system that operates on units of information called bits. A bit can have two values, usually represented as 0 and 1, and is the smallest unit of information in digital logic. Operations are performed on bits using logic gates, which take one or more bits as input and give one or more bits as output. Typically, a logic gate usually has only one bit as output (although this single bit can be sent as input to multiple other logic gates), and the value of this bit usually depends on the value of at least some of the input bits. In modern computers, logic gates are usually made up of transistors, and bits are usually represented by the voltage levels of wires connected to the transistors. A simple example of a logic gate is an AND gate, which (in its simplest form) takes two bits as input and gives one bit as output. The output of the AND gate is 1 if both inputs have a value of 1, otherwise it is zero. By wiring together the inputs and outputs of various logic gates in specific ways, classical computers can implement arbitrarily complex algorithms to accomplish a wide variety of tasks.

[0031] On the surface, quantum computers operate in a similar way to classical computers. Quantum computers operate according to a logic system that operates on units of information called qubits (a combination of "quantum" and "bit"). A qubit is the smallest unit of information in a quantum computer, and a qubit can have any linear combination of two values, which are usually represented as |0> and |1>. In other words, for any combination of α and β, the value of the qubit is represented as |ψ>, which can be equal to α|0>+β|1>, where α and β are complex numbers and |α| 2 +|β| 2= 1. Operations are performed on qubits using quantum logic gates, which take one or more qubits as input and give one or more qubits as output. Given the low-energy nature of most current quantum systems, quantum algorithms are often expressed in terms of their underlying quantum circuits. Quantum circuits, in turn, consist of quantum gates, which are the basic components that directly manipulate qubits.

[0032] Superconducting quantum circuits are one of the leading platforms for realizing quantum computing. While there are many existing designs, designing new types of circuits to store and process quantum information remains an active area of ​​research. An important step in the design process is to be able to simulate the dynamics of the quantum circuit on a classical computer. However, the simulation of large quantum circuits is quite challenging. In fact, this is why quantum computers can perform computational tasks that are difficult for classical computers. Despite this obstacle, there is still a need to simulate small-scale quantum circuits in order to understand how the qubits behave and interact.

[0033] Currently, for many existing designs, circuits of one or a few qubits are simple enough to be implemented with simple numerical simulation techniques. In particular, the Hamiltonian of the circuit is easily diagonalized. However, as qubit designs become more complex and involve larger circuits, these simple simulation techniques are no longer applicable. For example, more complex qubit designs (e.g., 0-π qubits) involve three degrees of freedom or modes, so simulating multiple such qubits will involve six or more modes. In addition, resonant cavities often need to be included in the simulation as well. Approaches that simply discretize the Hilbert space of the circuit lead to very high-dimensional Hamiltonians that are difficult to diagonalize or computationally exponentially expensive to time evolve. The problem becomes how to analyze quantum circuits in a computationally efficient way.

[0034] Perturbation theory provides a computationally efficient way to analyze weakly coupled quantum circuits. Figure 3B As shown, the Hamiltonian H can be expressed as H 局部 (consisting of all local terms) and H 耦合 (composed of all coupled terms). When the magnitude of the coupled terms is smaller than the local terms (e.g., H 局部 The norm is much larger than H 耦合 norm), then H 耦合 can be considered as a disturbance.

[0035] According to this method, H 局部 can be diagonalized. In some cases, each local Hamiltonian can be diagonalized using standard numerical methods for single-degree-of-freedom subsystems. The low-energy eigenstates of H then approximately involve only H 局部 The low energy eigenstate of H to H 局部The projection of the low-energy eigenspace of approximately preserves the low-energy spectrum and eigenstates of H. 耦合 can be expressed as a sum of tensor products of local operators, which in turn can be expressed as an eigenbasis of the corresponding local Hamiltonian. The H involving high-energy eigenstates can be discarded. 耦合 Finally, to H 局部 The projections of the first few eigenstates of can be added to the truncated H 耦合 The resulting Hamiltonian can approximate H in the low-energy eigenspace.

[0036] How closely the Hamiltonian approximates H can depend on the coupling between the modes of the Hamiltonian and the number of eigenstates included in the eigenbasis. In general, a greater degree of coupling requires the inclusion of more eigenstates to achieve the desired accuracy. However, the more eigenstates included in the approximating Hamiltonian, the more computational resources are required to simulate the quantum circuit.

[0037] The disclosed embodiments relate to methods for linearly transforming circuit modes to reduce inter-mode coupling. Perturbation theory is applied to the Hamiltonian of the transformed mode, and the transformed Hamiltonian can be expressed as a transformed local Hamiltonian and a transformed coupled Hamiltonian. As described above, the transformed coupled Hamiltonian can be projected onto the eigenbasis of the transformed local Hamiltonian. Then, numerical methods (e.g., the Lanczos algorithm or another suitable method) can be used to find or calculate low-energy states, thereby allowing the low-energy characteristics of the quantum circuit to be well approximated.

[0038] Linear transformations of the disclosed quantum circuit modes can be computed from the Hamiltonian of the quantum circuit (which can be obtained by standard circuit quantization techniques). The Hamiltonian can include contributions from capacitive, inductive, and (Josephson) junction terms. The inter-mode coupling terms are all capacitive and inductive terms, which can be summarized by a coupling matrix that describes the charge and flux coupling between modes, including self-coupling (e.g., local terms). The algorithm computes different linear transformations of the circuit modes, effectively transforming the coupling matrix to reduce its off-diagonal terms, i.e., the coefficients of the coupling terms.

[0039] The disclosed linear transformations affect the flux and charge operators of the modes. The transformed operators still obey canonical commutation relations. One disclosed transformation performs the same orthogonal transformation on the flux and charge operators of the linear inductance modes using an optimization loop to reduce coupling. Another disclosed transformation diagonalizes a submatrix of the coupling matrix corresponding to the linear inductance modes. This is possible through Williamson's theorem. A third disclosed transformation fully diagonalizes both coupling matrices at the expense of introducing flux coupling between the transformed modes via the knot term.

[0040] Figure 1AAn example diagram showing a quantum circuit suitable for simulation according to the disclosed embodiments is depicted. The quantum circuit includes two strongly inductively coupled qubits. For clarity, the capacitances on the edges [1,3], [1,4], [2,3], [2,4], [1,5], [4,6] are not included. Figure 1B and Figure 1C The capacitance and inductance values ​​of the components of this quantum circuit are given in . With these parameters, and after removing the free modes, as disclosed in the following paragraphs, it is possible to obtain Figure 3C The Hamiltonian given in .

[0041] As will be appreciated by those skilled in the art, the original Hamiltonian of a quantum circuit may be derived in a variety of ways. As a non-limiting example, the Hamiltonian of a general superconducting quantum circuit may be derived using the method disclosed in "Circuit Theory for Decoherence in Superconducting Charge Qubits" by G. Burkard, Physical Review B, April 2005, the entire contents of which are incorporated herein by reference and will be referred to as Burkard in this disclosure. In Burkard, the derived dissipative Hamiltonian of a general superconducting quantum circuit (e.g., quantum circuit 201) takes the following form:

[0042]

[0043] Here, and is the vector of flux operator and charge operator of the circuit mode of quantum circuit 201 . represents the externally applied magnetic flux, φ0 is the magnetic flux quantum, φ i is the flux variable. is the voltage bias vector in quantum circuit 201, C -1 , M0, N and C V They are charge coupling matrix, flux coupling matrix, external flux coupling matrix and voltage coupling matrix. J is the number of Josephson junctions, E J，i is the characteristic energy of each Josephson junction.

[0044] In some embodiments, the number F of free modes of quantum circuit 201 may be given as:

[0045] F≡dim(ker(M0)∩ker(N T )∩V L ) (Equation 2)

[0046] Here, V Lis the subspace of inductor flux generation. As shown in Equation 2, the number of free modes F can be defined as the dimension of the subspace, which is the kernel of the flux coupling matrix M0, the transposed outer flux coupling matrix N T The kernel and the subspace V generated by the inductor flux of the quantum circuit 201 L In some embodiments, a mode in quantum circuit 201 may have a very small potential energy term. Although in this case, no mode is free, a mode that meets the threshold criteria may be considered a free mode. For example, a mode in Hamiltonian with a potential energy value less than a threshold value may be considered a free mode, although the potential energy value of the mode may not be zero.

[0047] According to some embodiments of the present disclosure, the flux operator and charge operator Some form may be taken so that the free modes are explicit in the derived Hamiltonian (e.g., expressed as Equation 1). In some embodiments (e.g., via an appropriate transformation that diagonalizes the intersection of the subspaces in Equation 2), the flux operator It can be expressed as Among them, Φ1,···,Φ F is the flux operator of the free mode, Φ F+1 ,···,Φ n is the flux operator for the non-free modes, and n is the total number of modes in the Hamiltonian. Similarly, the charge operator can be expressed as Among them, Q1, ···, Q F is the charge operator of the free mode, Q F+1 ,···,Q n is the charge operator for non-free modes. and charge operator Consistent with this representation, the elements of the first F rows of the external flux coupling matrix N and the first F rows and first F columns of the flux coupling matrix M0 may all be zero.

[0048] In some embodiments, a transformed Hamiltonian that decouples the free mode from the non-free mode can be obtained, for example, by linearly transforming the circuit mode of the quantum circuit 201 to effectively transform the charge coupling matrix C -1 Gaussian elimination is performed on the inverse matrix of (e.g., C). The same Gaussian elimination can then be performed on the flux coupling matrix M0 according to the canonical transformation requirements. As a result, the transformed Hamiltonian will have the same number of free modes as the original Hamiltonian. The extracted Hamiltonian can then be obtained by removing the free modes from the transformed Hamiltonian.

[0049] According to the disclosed embodiments, the transformation matrix W may be defined such that the free mode components can be transformed from the charge coupling matrix C-1 The non-free mode components in are decoupled. The charge coupling matrix C -1 W may be the inverse matrix of the effective capacitance matrix C of the quantum circuit 201, which may be positive definite. For f∈{1, 2, ..., F}, the matrix W f and C f Can be defined iteratively. The matrix W f can be defined as an n×n identity matrix, excluding f columns, with the following entries:

[0050]

[0051] Among them, the matrix C f Defined as C f ≡W f C f-1 W f T The matrix C0 can be defined as the effective capacitance matrix C of the quantum circuit 201. Since the matrix C f-1 is positive definite, so we can prove by induction that the matrix W f is well defined (so the element (C f-1 ) ff is not zero). First, the matrix C0≡C can be positive definite according to Burkard's requirement. Second, suppose the matrix C f-1 is positive definite, the matrix W f is well defined because the element (W f ) ff =-1. Therefore, the matrix W f The fth column of the matrix W f The other columns of f The other columns of form the identity matrix by definition). Therefore, W f has full rank, which means that the matrix C f ≡W f C f-1 W f T It is also positive.

[0052] Final matrix:

[0053] C′≡WCW T (Equation 3)

[0054] in, The off-diagonal elements of the first F rows and first F columns of vanish, which can be verified as follows. f The off-diagonal elements of the fth column of can be calculated as:

[0055]

[0056] Since the matrix Cf For the symmetry of the f-th row matrix C f the non-diagonal elements in also disappear, i.e., they are zero. It can also be seen that for matrix C f the non-diagonal terms in the first to f - 1 rows and the first to f - 1 columns are also disappearing, which can be determined by induction. First, matrix C1 is like this. Second, assume that for matrix C f it is also the case, which means the element (C f ) if+1 = 0 (for i < f + 1), which means for matrix W f+1 it is also the case, i.e., (W f+1 ) if+1 = 0 (for i < f + 1). According to symmetry, the same applies to the f + 1-th row of matrix C f and also to matrix W f+1 . Therefore, matrices C f and W f+1 and matrix W f+1 T are all block diagonal matrices with block dimensions 1, ···, 1, n - f, which means matrix C f+1 is also the case.

[0057] As described above, each matrix W f is full rank and thus invertible, which means the transformation matrix W is invertible. Therefore, the transformed charge-coupled matrix

[0058] C′ -1 = (W T ) -1 C -1 W -1 (Equation 4)

[0059] is well-defined. Since the transformed charge matrix C’ is a block diagonal matrix with block dimensions 1, ···, 1, n - F, the transformed charge-coupled matrix C’ -1 is also a block diagonal matrix with block dimensions 1, ···, 1, n - F.

[0060] In addition, the submatrices of the charge-coupled matrix C -1 corresponding to the exponents greater than F (the number of free modes) and the transformed charge-coupled matrix C’ -1 are the same. This can be proven as follows. W f -1 = W f holds because when i = j:

[0061]

[0062] When i ≠ j:

[0063]

[0064] Among them, δ jf ((W f ) if -(W f ) if ) Then, because i≠j and i≠f. Here, when j=f, δ if =1, and when j≠f, δ jf =0.

[0065] For i, j>F, the following relationship holds:

[0066]

[0067] Therefore, the charge coupling matrix C corresponding to an exponent greater than F -1 And the transformed charge coupling matrix C' -1 The submatrices of are identical. Therefore, the linear transformation does not affect the charge coupling of the non-free modes of the original Hamiltonian.

[0068] Charge Operator The linear transformation of can be defined as:

[0069]

[0070] In order to maintain the following canonical commutation relation between canonical conjugate quantities in the Hamiltonian,

[0071] [Φ i , Φ i ]=0

[0072] [Q i , Q i ]=0

[0073] [Φ i , Q i ]=ihδ ij

[0074] Flux operator It can also be transformed into:

[0075]

[0076] This preserves the canonical commutation relation. For i>F, Φ i =∑ j W ji Φ j ′=Φ i ',in, is the transformed flux operator. Thus, according to the disclosed embodiments, the transformed flux includes the flux of the original non-free mode. Thus, by removing the free modes in the Hamiltonian according to the transformed modes, the Hamiltonian on the original non-free modes can be obtained explicitly. Furthermore, any Josephson knot terms in the Hamiltonian are preserved and maintained as local terms (e.g., terms involving a single mode can be expressed as a sum of tensor products of local operators, compared to general terms involving multiple modes).

[0077] According to the disclosed embodiments, a linear transformation of the flux pattern implies a corresponding transformation of the flux coupling matrix:

[0078] M0→WM0W T (Equation 7)

[0079] However, this transformation does not affect the element values ​​of the flux coupling matrix, which is shown below:

[0080]

[0081] Because the fth row and fth column of M0 are both zero. Therefore:

[0082] M0=WM0W T

[0083] Therefore, the transformed flux coupling matrix is ​​the same as the original flux coupling matrix.

[0084] Similar results apply to the external flux coupling matrix N. A linear transformation of the flux pattern implies a corresponding linear transformation of the external flux coupling matrix:

[0085] N→WN (Equation 8)

[0086] However,

[0087]

[0088] Therefore, N=WN, and the external flux coupling matrix is ​​not affected by the linear transformation of the flux mode. The first F modes in the transformed external flux coupling matrix N are free modes, and the flux and external flux coupling of the remaining modes (ie, non-free modes) remain unchanged.

[0089] The linear transformation means that the voltage coupling matrix C V The corresponding linear transformation of :

[0090] C V →WC V (Equation 9)

[0091] According to some embodiments of the present disclosure, based on equations 3 to 9, the Hamiltonian of equation 1 can be expressed in a transformation mode as follows:

[0092]

[0093] Here, the transformed Hamiltonian represented as Equation 10 describes a system of n modes with F free modes independent of all other modes. Therefore, according to some embodiments of the present disclosure, the original non-free mode Hamiltonian can be extracted by eliminating the term corresponding to the free mode charge from the Hamiltonian of Equation 10.

[0094] refer to Fig. 9 , the Hamiltonian extraction unit 222 can be configured to generate an extracted Hamiltonian of the quantum circuit 201, for example, by removing the component of the transformed Hamiltonian corresponding to the free mode. The extracted Hamiltonian can be expressed as follows:

[0095]

[0096] In Equation 11, the subscript \F indicates that the component corresponding to the free mode has been removed from the corresponding operator or matrix. According to some embodiments of the present disclosure, for the external flux coupling matrix N and the transformed voltage coupling matrix C V ', the symbol \F may mean removing the rows corresponding to the free modes of the transformed Hamiltonian.

[0097] According to the disclosed embodiment, the extracted Hamiltonian H of Equation 11 \F Can not include V 2 The proportional identity term (where V is the vector of voltage biases in the circuit). In some embodiments, this term may only have a negligible effect on the Hamiltonian.

[0098] According to the disclosed embodiment, the extracted Hamiltonian H \F The driving term proportional to V can be equal to As shown below, the transformed charge coupling matrix C' -1 The block diagonal property of the charge coupling matrix C -1 The submatrix of and the transformed charge coupling matrix C' corresponding to the non-free mode of the original Hamiltonian -1 As support for this relationship, consider the following driver terms including the free mode:

[0099]

[0100] Removing the free mode from this drive term is equivalent to removing the transformed charge coupling matrix C' -1 The first F columns and the transformed charge operator The first F entries of . Because the transformed charge coupling matrix C' -1is a block diagonal matrix, so once the transformed charge coupling matrix C' is removed -1 The first F columns and the first F rows of are all zero. Therefore, the transformed charge coupling matrix C' can also be removed -1 The first F rows and the transformed voltage coupling matrix C' V As mentioned above, after removing the free modes, the transformed charge coupling matrix C' -1 The remaining submatrix and the charge coupling matrix C -1 The submatrices of are the same, so the extracted Hamiltonian H \F The driving term of can be expressed in Equation 11.

[0101] According to the disclosed embodiments, and according to the derivation of the extracted Hamiltonian provided herein, the voltage coupling matrix C can be obtained by removing the free mode terms in the original Hamiltonian and transforming the voltage coupling matrix C V to obtain the extracted Hamiltonian, as shown in Equation 9. In some embodiments, transformation of the voltage coupling matrix can ensure that when a voltage source is used, analysis using the extracted Hamiltonian instead of the original Hamiltonian will provide correct results.

[0102] Figure 1D and Figure 1E The coupling matrix is ​​given by and M0. The coupling terms in the Hamiltonian correspond to the off-diagonal terms in these matrices. The sum of the squares of the off-diagonal terms is 9.61e10. In this non-limiting example, the constant E JA and E JB The definition is as follows: JA =3.0GHZ·h and E JB = 3.2GHZ·h. The Hamiltonian can be directly diagonalized, and the energy of the low-energy state (in Figure 2A The difference between the medium and low energy states (in Figure 2B (center) is plotted as a function of the local Hamiltonian low-energy eigenspace onto which the total Hamiltonian is projected. The local dimension d is for each individual mode, so the total dimension is d 5 (Because there are 5 modes in this system).

[0103] If the coupling is small, the energy converges quickly as the local dimensionality increases. In this non-limiting example, no decoupling techniques were applied, and the state energies did not converge quickly (e.g., the energy continued to decrease as the number of eigenvectors in the eigenbasis increased beyond seventeen). Because the state energies did not converge, the differences between low energy states were unreliable. The systems and methods disclosed herein provide improved approaches based on this basic result, allowing at least partially decoupled Hamiltonians to have state energies and energy transforms that converge faster as the number of eigenvectors in the eigenbasis increases.

[0104] Figure 3A Depicted is a process 300 for simulating the behavior of a quantum circuit using an at least partially decoupled Hamiltonian according to the disclosed embodiments. Process 300 can enable a classical computer to simulate the behavior of a quantum circuit, even when the quantum circuit is configured to perform a calculation that a classical computer cannot actually perform. Thus, process 300 provides a technical improvement in the design or verification of quantum circuits. In some embodiments, in addition to simulating the behavior of a quantum circuit, a computer can be configured to perform at least some steps of process 300. The computer can be configured to perform these steps automatically or at least in part through interaction with a user.

[0105] In step 301 of process 300, a representation of a quantum circuit may be obtained. In some embodiments, the representation may be a design of a quantum circuit (e.g., a circuit diagram, etc.). The representation may include data or instructions that specify the components of the quantum circuit and how these components are interconnected. In some embodiments, the representation may be obtained by a computer configured to perform at least some steps of process 300. The representation may be obtained as an input to a program for performing process 300. This input may come in a variety of forms, how the input is represented (e.g., the data structures involved) and what the input represents (e.g., what quantum circuit is used to represent the input). In other embodiments, the quantum circuit may be created directly in a program for performing process 300. In some embodiments, the quantum circuit may be received from another system or retrieved from a computer-accessible memory. The circuit may include one or more qubits.

[0106] In step 303 of process 300, a transformed Hamiltonian corresponding to the quantum circuit may be generated. The transformed Hamiltonian may be generated from the original Hamiltonian. In some embodiments, the pattern of the transformed Hamiltonian may be a linear combination of the patterns of the original Hamiltonian. As described herein, the transformed Hamiltonian may include at least a partially diagonal charge and flux coupling matrix. The transformed Hamiltonian may include a transformed local Hamiltonian and a transformed coupling Hamiltonian. In some embodiments, a single transformed Hamiltonian may be generated. In various embodiments, multiple transformed Hamiltonians may be generated, and one of the transformed Hamiltonians may be selected (e.g., as described with respect to Figure 4 described in ). Figure 5A , 6A and 7A to generate the transformed Hamiltonian.

[0107] It should be understood that the original Hamiltonian of a quantum circuit can be derived in a variety of ways. As a non-limiting example, the Hamiltonian of a general superconducting quantum circuit can be derived using the method disclosed in "Circuit Theory for Decoherence in Superconducting Charge Qubits" by G. Burkard, Physical Review B, April 2005, the entire contents of which are incorporated herein by reference. In this article, the derived dissipative Hamiltonian adopts Figure 3C The form shown, where are the vectors of flux and charge operators for different modes of the circuit, is the vector of voltage bias in the circuit, M0, N, C V are the matrices of charge, flux, external flux and voltage coupling, n J is the number of Josephson junctions, E J，i is the characteristic energy of each knot, φ x is the external magnetic flux, and φ0 is the magnetic flux quantum. In general, quantum information is encoded in non-driven circuits, so when When is set to 0, it corresponds to the target Hamiltonian. For ease of explanation, this notation will be used to describe the disclosed embodiments. However, they are not limited to a particular representation of the Hamiltonian of a quantum circuit.

[0108] like Figure 3B As shown, the original Hamiltonian of the quantum circuit can be divided into local terms and coupling terms. Here, H 局部 You can Figure 3D As shown, where n L H can be the number of inductors included in the spanning tree. 耦合 You can Figure 3E As shown, where n≡n J +n L is the total number of modes. H describes the coupling term Quantum many-body systems with weak coupling when the local terms are small.

[0109] Assume H 耦合 is purely quadratic, and linear transformations on the modes can be used to reduce and the off-diagonal elements in M0. More specifically, this can be achieved Figure 3F The transformation shown in , where This linear transformation can maintain Figure 3G The canonical commutation relation shown in . It is easy to check that the first two types are preserved. For the third type, equivalence can be shown as Figure 3H Shown to prove.

[0110] This transformation can be used Fig. 3IThe secondary coupling matrix is ​​transformed by the mapping shown in Figure 3J Note that even if the undriven Hamiltonian is considered, it may still be necessary to V A transformation is performed to obtain the drive Hamiltonian according to the transformed mode. As discussed in the application, in some embodiments, the resultant terms will also be transformed and may affect the determination of the at least partially decoupled Hamiltonian.

[0111] In step 305, a finite eigenbase may be determined for the transformed local Hamiltonian. Multiple low-energy or important eigenvectors of the transformed local Hamiltonian may be selected as a finite eigenbase. For example, the number may be between 2 and 20, or higher. As a non-limiting example, 5, 10, or 20 lowest energy eigenvectors of the transformed local Hamiltonian may be selected as a finite eigenbase. In some embodiments, the number may be predetermined. In other embodiments, the number may depend on a convergence criterion (e.g., a convergence rate of energy levels of the local transformed Hamiltonian, etc.).

[0112] In step 307, the transformed coupled Hamiltonian can be projected onto a finite eigenbasis. The transformed coupled Hamiltonian can be represented as a sum of tensor products of local operators. The local operators can then be represented in the finite eigenbasis (e.g., terms of expressions containing eigenstates that are not included in the finite eigenbasis can be truncated).

[0113] In step 309, an at least partially decoupled Hamiltonian may be generated. The at least partially decoupled Hamiltonian may combine the projection of the transformed coupled Hamiltonian and the projection of the transformed local Hamiltonian. In some cases, the at least partially decoupled Hamiltonian may be the sum of these projections.

[0114] In step 311, a classical computer can simulate the behavior of a quantum circuit using the at least partially decoupled Hamiltonian. In some cases, the classical computer can simulate the evolution of a state of the quantum circuit over time (e.g., the time evolution of a state of a mode of the quantum circuit). In various cases, the classical computer can simulate the response of the quantum circuit to an input or other perturbation. The disclosed embodiments are not limited to any particular simulation performed using the at least partially decoupled Hamiltonian.

[0115] Figure 4 A flow chart of an exemplary process 400 for selecting a transformed Hamiltonian for simulating a quantum circuit according to the disclosed embodiments is depicted. In some embodiments, process 400 may be performed as part of step 303 of process 300, as described above with respect to Figure 3AAs described. Process 400 can generate multiple transformed Hamiltonians. One of these transformed Hamiltonians can be selected based on a selection criterion. The selection criterion can involve the degree of coupling between modes of the transformed Hamiltonian. The selected Hamiltonian can be used to generate an at least partially decoupled Hamiltonian for simulating quantum circuits. As described above, Hamiltonians with reduced inter-mode coupling are more suitable for approximation using perturbation theory. Therefore, process 400 can support the generation of an improved at least partially decoupled Hamiltonian that supports more accurate simulation of quantum circuits. In some embodiments, a computing device (e.g., such as Figure 8 4 (shown) may be configured to perform at least some steps of process 400. The computer may perform these steps automatically or at least in part through interaction with a user.

[0116] In step 401 of process 400, a spanning tree may be selected for a quantum circuit. A spanning tree may be a subset of the elements in the quantum circuit. In some embodiments, a spanning tree may include all junctions, all voltage sources, and at least some inductors in the quantum circuit. The observability of these components may completely determine the state of the entire quantum circuit. It should be understood that a quantum circuit may include multiple spanning trees, each associated with a different Hamiltonian. Using different spanning trees may result in different sizes of coupling terms in the transformed Hamiltonian.

[0117] In some embodiments, a set of spanning trees may be determined for a quantum circuit. A spanning tree that was not previously selected may then be selected from this set. The set may include all possible spanning trees for a quantum circuit or a subset of possible spanning trees for a quantum circuit. In some embodiments, the selection may be performed automatically by a computing device. In various embodiments, the selection may be performed manually by a user (e.g., through interaction between a computing device and a user). The disclosed embodiments are not limited to any particular method of selecting a spanning tree. Process 400 may be repeated until all spanning trees in the set have been selected.

[0118] In step 403 of process 400, based on the selected spanning tree, a raw Hamiltonian of quantum circuit 403 may be determined. The raw Hamiltonian may be determined according to the method described above with respect to step 303 of process 300. The raw Hamiltonian may include a charge coupling matrix and a flux coupling matrix.

[0119] In step 405 of process 400, one or more linear transformations of the pattern of the original Hamiltonian may be determined. Each of the one or more linear transformations may be a linear transformation described herein. In some cases, such linear transformations may achieve simultaneous approximation of diagonalization (e.g., with respect to Figure 5AIn some embodiments, the linear transformation may depend on a block diagonal symplectic matrix. The block diagonal symplectic matrix may include a first submatrix and a second submatrix, the second submatrix being a function of the first submatrix. For example, in various cases, the linear transformation may implement a symplectic diagonalization of only the inductor (e.g., as described with respect to Fig. 6A As described above). In this case, the block diagonal symplectic matrix can be configured as a submatrix of the diagonalized flux coupling matrix, which corresponds to the linear inductance mode of the Hamiltonian. As another example, in some cases, the linear transformation can achieve full symplectic diagonalization (e.g., as described with respect to Fig. 7A In this case, the block diagonal symplectic matrix can be configured to diagonalize the charge coupling matrix and the flux coupling matrix.

[0120] In step 407 of process 400, one or more linear transformations of the pattern of the original Hamiltonian may be used to generate an at least partially decoupled Hamiltonian. In some embodiments, each possible linear transformation may be performed and the resulting transformed Hamiltonians compared. In various embodiments, a subset of the possible linear transformations may be performed.

[0121] In step 409 of process 400, a coupling value may be determined for the at least partially decoupled Hamiltonian. The coupling value may indicate the degree of coupling between the modes of the at least partially decoupled Hamiltonian. When the linear transformation achieves simultaneous approximate diagonalization or symplectic diagonalization of only the inductor, the coupling value may be a function of the non-diagonal elements of the transformed charge coupling matrix and the transformed flux coupling matrix (e.g., the sum of the squares of the non-diagonal elements of the transformed charge coupling matrix and the transformed flux coupling matrix, etc.). When the linear transformation achieves complete symplectic diagonalization, the coupling value may be a function of certain rows of the first submatrix of the block diagonal symplectic matrix (e.g., those rows corresponding to the coupling modes of the original Hamiltonian). Such a function may include the sum of the squares of the elements of these rows, etc.

[0122] In step 411 of process 400, the at least partially decoupled Hamiltonian generated in step 407 can be selected as the transformed Hamiltonian for use in process 300. The selection can be based on the coupling value associated with the at least partially decoupled Hamiltonian. In some embodiments, before selecting the transformed Hamiltonian, one or more at least partially decoupled Hamiltonians can be generated for all spanning trees in the set. In such embodiments, the selection can depend on the coupling values ​​associated with all of these at least partially decoupled Hamiltonians. For example, the at least partially decoupled Hamiltonian with the smallest magnitude coupling value can be selected as the transformed Hamiltonian.

[0123] In some embodiments, the at least partially decoupled Hamiltonian can be selected based on a coupling value associated with an at least partially decoupled Hamiltonian. For example, the selection criteria can involve a threshold coupling value (e.g., a predetermined value, etc.) or a threshold reduction in the coupling value (e.g., compared to the same coupling value calculated for the original Hamiltonian). Continuing with this example, such a threshold reduction can be two or more orders of magnitude (e.g., the coupling value of the at least partially decoupled Hamiltonian is two or more orders of magnitude less than the coupling value of the original Hamiltonian). When the at least partially decoupled Hamiltonian is generated to meet the criteria (e.g., the coupling value is below the threshold, the coupling value reduction is greater than the coupling value reduction threshold, etc.), process 400 can terminate and the at least partially decoupled Hamiltonian can be selected as the transformed Hamiltonian.

[0124] Figure 5A A flow chart of an exemplary simultaneous approximate diagonalization process 500 for generating a transformed Hamiltonian according to the disclosed embodiments is depicted. The process 500 generates a transformed Hamiltonian using an orthogonal transformation. For such a transformation, the transformation matrix W = (W T ) -1 Transforming the original Hamiltonian using such an orthogonal transformation is equivalent to simultaneously diagonalizing and M0. However, in general, these two matrices do not commute, so exact diagonalization is not possible. An alternative approach is to define the optimization task: find an orthogonal matrix W such that the sum of the squares of the off-diagonal entries of the transformation matrix is ​​minimized. However, orthogonal transformations are limited to n L The cosine-Josephson knot terms in the Hamiltonian do not contain linear combinations of flux operators, which would make them coupled terms. This technique is referred to in this application as simultaneous approximate diagonalization. Process 500 can perform simultaneous approximate diagonalization based on the method outlined in "Jacobian Angles for ," by J. F. Cardoso and A. Souloumiac, SIAM Journal on Matrix Analysis and Applications, January 1996, which is incorporated herein by reference in its entirety.

[0125] In step 501, process 500 may begin. Process 500 may begin as part of process 400 or 300. For example, process 500 may be used to determine a transformation in step 405 of process 400. Process 400 may begin with a rotation matrix having the same dimensions as the flux coupling matrix and the charge coupling matrix of the original Hamiltonian. Process 400 may include continuously updating the rotation matrix to generate a transformation matrix for transforming the original Hamiltonian. The rotation matrix may be updated by iteratively determining a rotation about a selected axis of the rotation matrix. In some embodiments, a computing device (e.g., such as Figure 8) may be configured to perform at least some steps of process 500. The computer may perform these steps automatically or at least in part through interaction with a user.

[0126] In step 503 of process 500, an axis may be selected. The axis may be the next axis in a list or ordering of axes, or may be selected from a set of axes (e.g., randomly or deterministically). The selected axis may correspond to a linear inductance mode in the original Hamiltonian.

[0127] In step 505 of process 500, a rotation value may be determined. The rotation value may be an angle of rotation about a selected axis. The disclosed embodiments are not limited to any particular method of determining the rotation angle. In various embodiments, the rotation angle may be obtained using a closed form equation disclosed in "Jacobian Angles for Simultaneous Diagonalization" or another suitable method.

[0128] The rotation matrix can be updated to reflect the rotation of the rotation angle around the selected rotation axis. The transformed charge coupling matrix and the transformed flux coupling matrix can be determined using the updated rotation matrix. The termination value can be determined based on the off-diagonal elements of the transformed charge coupling matrix and the transformed flux coupling matrix. For example, the termination value can be the sum of the squares of the off-diagonal elements of these matrices. In some embodiments, although only the rotation axis corresponding to the linear inductance mode is iterated, the termination value can be calculated on all modes, including the Josephson junction mode.

[0129] In step 507 of process 500, it may be determined whether a stop condition is satisfied. In some embodiments, the stop condition may depend on a termination value associated with the rotation value (e.g., the termination value is less than an absolute or relative threshold) or a trend of the termination value associated with the determined rotation value (e.g., a difference or derivative of a sequence of termination values ​​that satisfies a convergence criterion, e.g., less than a convergence threshold, etc.). In various embodiments, the stop condition may depend on elapsed time, number of iterations, computational usage, etc. If the stop condition is satisfied, process 500 may proceed to step 509. If the stop condition is not satisfied, process 500 may proceed to step 503, and another axis may be selected.

[0130] In step 509 of process 500, process 500 may stop. In some embodiments, a transformed Hamiltonian is available (e.g., the transformed Hamiltonian may have been used to generate the final termination value). In such embodiments, process 500 may include steps 405 and 407 of process 400. In some embodiments, a rotation matrix may be provided for generating the transformed Hamiltonian.

[0131] Figure 5B Describes the orthogonal transformation matrix W, by Figure 1A The quantum circuit performs Figure 5A This matrix is ​​obtained by the simultaneous diagonalization technique described in . Note that the first two modes are Josephson junction modes and therefore do not transform. Figure 5C and 5D Depicts the use of transformation matrices W and Figure 1D and 1E The transformed charge and flux coupling matrices generated from the original charge and flux coupling matrices depicted in Fig. 3I With this transformation, the sum of the squares of the off-diagonal terms is reduced to 3.15e3, a reduction of seven orders of magnitude.

[0132] Figure 5E Plots depicting the state energies of multiple energy levels as a function of the local basis dimension. Figure 5E A graph depicting the transition energy of different transitions as a function of the local basis dimension is shown. Figures 5B to 5D The transformed Hamiltonian generates an at least partially decoupled Hamiltonian to estimate the state energy and the curve diagram. Figure 2A and 5E , the state energy converges faster than if the decoupling technique was not applied. In addition, the energy itself is lower, which demonstrates the technical improvement provided by process 500.

[0133] Fig. 6A 1 depicts a flow chart of an exemplary inductor-only symplectic diagonalization process 600 for generating a transformed Hamiltonian in accordance with the disclosed embodiments. Process 600 uses a more general linear transformation than process 500. In some embodiments, the linear transformation applied to the pattern may be a symplectic transformation (e.g., given Figure 6B The matrix S shown, where 0 n is an n×n dimensional zero matrix and can be Fig.6D The block matrix Ω shown in Figure 2 is T ΩS = Ω). Symplectic transformations can be performed on the charge and flux vectors of the Hamiltonian (e.g., Figure 6C In some embodiments, a computing device (e.g., Figure 8 ) may be configured to perform at least some steps of process 600. The computer may perform these steps automatically or at least in part through interaction with a user.

[0134] In step 601 of process 600, a block diagonal symplectic matrix may be determined. The block diagonal symplectic matrix diagonalizes the coupling submatrix of the linear inductance mode corresponding to the original Hamiltonian of the quantum circuit. For example, in some embodiments, the flux coupling matrix M0 may be positive definite. As a non-limiting example, when each Josephson junction in the quantum circuit is shunted by an inductor, the flux coupling matrix M0 is positive definite. Then, Fig. 6EThe block matrix Q depicted in (which encodes the quadratic part of the Hamiltonian) is positive definite. Since Q is positive definite, Q L It is also positive. Fig. 6F The block matrix depicted in includes only the sub-matrices corresponding to the L linear inductance modes in the Hamiltonian. Therefore, Williamson's theorem can be applied: given any positive definite matrix There exists a symplectic matrix Make S T MS=diag(Λ,Λ), where Λ is a diagonal matrix with positive diagonal elements.

[0135] In the proof of Williamson's theorem, the matrix i can be M -1 / 2 Ω -1 / 2 Construct a normalized basis of eigenvectors in, v1 is iM -1 / 2 ΩM -1 / 2 The eigenvector of with corresponding eigenvalue λ, and It is v1 and iM -1 / 2 ΩM -1 / 2 The complex conjugate of the eigenvector of with the corresponding eigenvalue λ. The orthogonal matrix B can be constructed using its eigenvectors. in, Then, in, and D = diag(λ1, ..., λ n ) is iM -1 / 2 ΩM -1 / 2 The diagonal of the positive eigenvalues ​​of is in the order corresponding to the order of the eigenvectors in B. Given this definition of S, it can be shown that S T ΩS=Ω, so S is symplectic.

[0136] Extending this proof, we can show a stronger statement: given Figure 6G Any block diagonal positive definite matrix M shown, where M1, M2∈) m×m , exists as Figure 6H The symplectic matrix S' shown, such that S' T MS′=diag(Λ,Λ), where Λ is a diagonal matrix with positive diagonal elements. By applying Williamson's theorem, the matrix Q L It can be represented by the block diagonal symplectic matrix S L Diagonalization, such as Fig.6I As shown, Matrix S L It can be a first sub-matrix (L×L matrix) and the second submatrix is a block diagonal matrix.

[0137] As the determination matrix S L The first step is to construct the basis B of eigenvectors given M. It is possible to determine the real positive definite matrix with eigenvalues ​​greater than zero A set of n eigenvectors {w i ,…,w n}. The corresponding set of n vectors {w′1,…,w′ n},in, It can be proved that the vector It's iM -1 / 2 ΩM -1 / 2 Therefore, as mentioned above, the matrix iM -1 / 2 ΩM -1 / 2 Constructing feature vectors The normalized basis of You can construct a matrix Among them, D=diag(λ1,...,λ n ) is iM -1 / 2 ΩM -1 / 2 The diagonal of the positive eigenvalues ​​of corresponds to the order of the eigenvectors in B. The eigenvectors of B can then be used to construct an orthogonal matrix in as well as However, given n vectors {w′1,…,w′ n} construction, as well as Therefore, the matrix O has the form And the matrix S = The form is Then, the matrix in as well as

[0138] In step 603 of process 600, a block diagonal symplectic matrix S L To generate the transformation matrix. In some embodiments, the linear transformation can be Figure 6J The matrix W is depicted in FIG. As shown in the figure, the matrix is ​​a block diagonal matrix, including the first submatrix S nL , block identity matrix (n J ×n J Matrix, where n J is the number of Josephson junction modes), the zero matrix and

[0139] In step 605 of process 600, the flux and charge coupling matrices of the original Hamiltonian may be transformed using a transformation matrix W. Applying this transformation may at least partially diagonalize the quadratic coupling matrix: the flux coupling matrix M0 may be as follows Figure 6K The charge coupling matrix is ​​transformed as shown in You can Figure 6L . In these figures, the subscripts J and JL correspond to the submatrices of the Josephson junction modes and the coupling coefficients between the Josephson junction modes and the linear inductance modes, respectively. Recall that Λ is an L×L diagonal matrix, which effectively completely decouples the linear inductance modes from each other. However, there may still be coupling between the linear inductance modes and the Josephson junction modes. The coupling between the Josephson junction modes can remain intact.

[0140] As a non-limiting example, Fig. 6A Only the inductor symmetric diagonal decoupling can be applied to Figure 1A The original Hamiltonian of the quantum circuit. The matrix Q L A submatrix M0(3:5, 3:5) may also be included, the submatrix including the third to fifth rows and columns of M0. L You can also include sub-matrices This sub-matrix includes The third to fifth rows and columns of . It can be found that the block diagonal symplectic matrix S L Q L Diagonalization. S L The first submatrix S nL Can be used to generate Figure 6M As shown in the figure, W is a block diagonal, including sub-matrices I2 (e.g., 2×2 identity matrices) in rows and columns corresponding to the Josephson junction patterns. Figure 6N and 6O As shown, using W to transform M0 and This results in a diagonalization of the modes corresponding to the inductive terms of the original Hamiltonian.

[0141] In this non-limiting example, the sum of the squares of the off-diagonal terms is 1.35e3. By comparison, the simultaneous approximate diagonalization method generates the transformed M0 and matrix The sum of squares of the off-diagonal terms is three times larger (1.35e3 vs. 3.15e3) and the largest element in the off-diagonal terms is twice as large (17.1 vs. 39.30). Thus, this technique further reduces the magnitude of the off-diagonal terms. Figure 6P As shown in , the reduction in the magnitude of the off-diagonal terms matches the improvement in convergence to the low energy states, and as Figure 6Q As shown, it matches its difference. Figure 5E and 5F The simultaneous approximate diagonalization technique shown and Figure 2Aand 2B The state and transition energies converge faster than the case without the decoupling technique shown.

[0142] Fig. 7A A flow chart of an exemplary fully symplectic diagonalization process 700 for generating a transformed Hamiltonian according to the disclosed embodiments is depicted. The process 700 can be configured to fully diagonalize the quadratic part of the original Hamiltonian of the quantum circuit. As previously described, when the flux coupling matrix M0 is positive definite, according to the Williamson theorem for block diagonal matrices, there exists Figure 7B The symplectic matrix shown includes the first diagonal submatrix (S n ) and the second diagonal submatrix Make S T QS=diag(Λ,Λ). In some embodiments, a computing device (e.g., Figure 8 ) may be configured to perform at least some steps of process 700. The computer may perform these steps automatically or at least in part through interaction with a user.

[0143] In step 701 of process 700, according to the process described above with respect to step 601 of process 600, it can be determined as follows: Figure 7B The symplectic matrix shown in FIG. However, unlike process 600, the matrix M in process 700 may include the entire flux coupling and charge coupling matrices. Therefore, the linear transformation W of the mode of the original Hamiltonian may be as follows: Figure 7C Defined as shown.

[0144] In step 703 of process 700, a linear transformation W may be used to fully diagonalize the quadratic portion of the Hamiltonian (e.g., according to Fig. 3I and 3J ), thereby removing all quadratic coupling terms. However, the transformed modes can still couple via the transformed junction terms. Fig.7D As shown, the transformation flux coupling matrix Φ→S n Φ′ causes the junction terms to depend on the linear inductance mode. In addition, there may be additional local terms in the power series expansion of the cosine.

[0145] In step 705 of process 700, these local terms can be separated from the coupled terms, such as Fig. 7E As shown, the transformed coupled Hamiltonian can be expressed as Figure 7F As described in the embodiment of the present invention, the transformed Hamiltonian can be represented by the transformed local Hamiltonian and the transformed coupled Hamiltonian, as shown in Figure 7G As shown in the figure, the transformed local Hamiltonian includes the knot term, such as Figure 7H shown.

[0146] As a non-limiting example, Fig. 7A The fully symplectic diagonalization technique can be applied to Figure 1A The original Hamiltonian of the quantum circuit. First, we can generate Figure 7B The symplectic matrix shown includes the first diagonal submatrix (S n ). The first submatrix can be used to generate a linear transformation W, such as Fig.7I As shown. Fig. 6A In contrast to the partial diagonalization method of , the Josephson junction patterns are also transformed to fully diagonalize the coupling matrices. Linear transformation matrices can be used to diagonalize the charge and flux coupling matrices. In such an embodiment, Figure 7J As shown, the diagonalized charge and flux coupling matrices can be the same.

[0147] However, all coupled terms may now be included in the resultant terms. Figure 7F The equation described in Figure 7K S described in n =W -1 The first n J = 2 rows can be used to determine these coupled terms.

[0148] In this non-limiting example, Figure 7L and 7M As shown, the state and transition energies are Figure 2A and 2B The original Hamiltonian shown here is either Figure 5A and 6A The method shown generates an at least partially decoupled Hamiltonian that converges faster. As shown in these figures, the state and transition energies converge at 5 to 7 local basis vectors.

[0149] For the fully symplectic diagonalization technique, we can take it one step further by first performing a Taylor expansion on the cosine Josephson knot terms and adding the quadratic terms to M0. That is, the mapping

[0150]

[0151] in, Yes J ×n J Diagonal matrix, the junction energy is on the diagonal. If M0 is positive definite, then due to E J，i > 0, the mapping will remain positive definite, and we can then perform a fully symplectic diagonalization using the new coupling matrix. This will effectively eliminate Figure 7F The quadratic coupling terms in . However, the flux of the local system may not be localized near zero, so we may not be able to perturbatively treat the low-order terms in the cosine expansion. Therefore, eliminating the quadratic terms may not be what we want. This can be corrected by expanding the cosines around where the flux of the local system is localized.

[0152] Figure 8 is a description of an example system 801 suitable for performing the disclosed method according to the disclosed embodiments. Figure 8 Although depicted as a server in FIG. 8 , system 800 may include any computer, such as a desktop computer, a laptop computer, a tablet computer, etc., which is configured to use the above Figure 5A , 6A 7A, generating an at least partially decoupled Hamiltonian for a quantum circuit, and simulating the quantum circuit using the at least partially decoupled Hamiltonian. Figure 8 As shown, system 801 may have a processor 802. Processor 802 may include a single processor or multiple processors. For example, processor 802 may include a CPU, a GPU, a reconfigurable array (e.g., an FPGA or other ASIC), etc. Processor 802 may communicate with memory 803, an input / output module 807, and a network interface controller (NIC) 809.

[0153] The memory 803 may include a single memory or a plurality of memories. In addition, the memory 803 may include a volatile memory, a non-volatile memory, or a combination thereof. Figure 8 As shown, the memory 803 can store one or more operating systems 804 and optimizers 805. For example, the optimizer 805 can include instructions for optimizing quantum circuits (e.g., as described above). Therefore, the optimizer 805 can simulate and optimize one or more quantum circuits according to any of the methods described above. The input / output module (I / O) 807 can store and retrieve data from one or more databases 808. For example, according to the disclosed embodiments, the database 808 can include a data structure describing a quantum circuit, and a decoupled Hamiltonian can be generated for the data structure. The NIC 809 can connect the system 801 to one or more computer networks. As shown in FIG. Figure 8 As shown, NIC 809 can connect system 801 to network 810. The network can be or include a wide area network (e.g., the Internet), a local area network, etc. The network can be implemented using wired, wireless, cellular or other communication technologies. The disclosed embodiments are not limited to any particular type of network or network implementation. System 801 can use NIC 809 to receive data and instructions over the network, and can use NIC 809 to transmit data and instructions over the network.

[0154] The disclosed embodiments are not limited to implementations using a single computing device. For example, a system including multiple computing devices (eg, a cluster or cloud computing platform) similar to system 801 can be configured to interoperate to perform the disclosed methods.

[0155] In some embodiments, a non-transitory computer-readable storage medium including instructions is also provided, and the instructions can be executed by a device (e.g., the disclosed encoder and decoder) for performing the above method. Common forms of non-transitory media include, for example, floppy disks, disks, hard disks, solid-state drives, tapes or any other magnetic data storage media, CD-ROMs, any other optical data storage media, any physical media with hole patterns, RAM, PROM and EPROM, FLASH-EPROM or any other flash memory, NVRAM, cache, registers, any other storage chip or cassette memory and network versions thereof. The device may include one or more processors (CPU), input / output interfaces, network interfaces and / or memories.

[0156] The foregoing description is provided for the purpose of illustration. These descriptions are not exhaustive and are not limited to the precise forms or embodiments disclosed. In view of the detailed description and practice of the disclosed embodiments, modifications and adjustments of the embodiments will be apparent. For example, the described implementations include hardware, but the systems and methods consistent with the present disclosure can be implemented with hardware and software. In addition, although some components have been described as being coupled to each other, these components can be integrated with each other or distributed in any suitable manner.

[0157] In addition, although illustrative embodiments have been described herein, the scope includes any and all embodiments based on the present disclosure having equivalent elements, modifications, omissions, combinations, adjustments, or variations (e.g., across aspects of the various embodiments). The elements in the claims are to be interpreted broadly based on the language used in the claims, and are not limited to the examples described in this specification or during the prosecution, which are to be interpreted as non-exclusive. In addition, the steps of the disclosed methods may be modified in any manner, including reordering steps or inserting or deleting steps.

[0158] It should be noted that the relational terms (e.g., "first" and "second") in this document are used only to distinguish one entity or operation from another entity or operation, and do not require or imply any actual relationship or order between these entities or operations. In addition, the words "include," "have," "contain," and "comprises" and other similar forms are intended to be equivalent in meaning and open-ended, in that one or more items following any of these words are not meant to be an exhaustive list of the one or more items, nor are they meant to be limited to the one or more items listed.

[0159] According to the detailed description, the features and advantages of the present disclosure are obvious, and therefore, the appended claims are intended to cover all systems and methods that fall within the true spirit and scope of the present disclosure. As used herein, the indefinite articles "a" and "an" mean "one or more". Similarly, the use of plural terms does not necessarily mean a plurality unless it is clear in a given context. In addition, since many modifications and changes will easily occur through the study of the present disclosure, it is not desired to limit the present disclosure to the exact construction and operation shown and described, and therefore, all suitable modifications and equivalents may be considered to fall within the scope of the present disclosure.

[0160] As used herein, unless expressly stated otherwise, the term "or" includes all possible combinations unless not feasible. For example, if it is specified that a database may include A or B, then unless otherwise specifically stated or not feasible, the database may include A, B, or A and B. As a second example, if it is specified that a database may include A, B, or C, then unless otherwise specifically stated or not feasible, the database may include A, or B, or C, or A and B, or A and C, or B and C, or A, B, and C.

[0161] It should be understood that the above embodiments can be implemented by hardware, software (program code), or a combination of hardware and software. If implemented by software, it can be stored in the above-mentioned computer-readable medium. When executed by a processor, the software can execute the disclosed method. The computing unit and other functional units described in this disclosure can be implemented by hardware, software, or a combination of hardware and software. It will also be understood by those of ordinary skill in the art that a plurality of the above-mentioned modules / units can be combined into one module / unit, and each of the above-mentioned modules / units can be further divided into a plurality of submodules / subunits.

[0162] In the foregoing description, embodiments have been described with reference to many specific details, which may vary depending on the implementation. Certain adjustments and modifications may be made to the described embodiments. In view of the detailed description and practice of the invention disclosed herein, other embodiments will be apparent to those skilled in the art. The description and embodiments are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the appended claims. The order of steps shown in the figures is also intended to be used for illustrative purposes only and is not intended to be limited to any particular order of steps. Therefore, it will be appreciated by those skilled in the art that these steps may be performed in different orders when implementing the same method.

[0163] The embodiments may be further described using the following terms:

[0164] 1. A method for simulating a quantum circuit using a computer that processes bits, the method comprising: obtaining a representation of the quantum circuit; generating a transformed Hamiltonian corresponding to the quantum circuit, the transformed Hamiltonian comprising a transformed local Hamiltonian and a transformed coupled Hamiltonian; determining a finite eigenbasis comprising a plurality of eigenvectors of the transformed local Hamiltonian; projecting the transformed coupled Hamiltonian onto a finite eigenbasis, the transformed coupled Hamiltonian being represented as a pattern of the transformed local Hamiltonian; projecting the transformed local Hamiltonian onto the finite eigenbasis; generating an at least partially decoupled Hamiltonian by combining a projection of the transformed coupled Hamiltonian and a projection of the transformed local Hamiltonian; and simulating the behavior of the quantum circuit using the at least partially decoupled Hamiltonian by the computer.

[0165] 2. A method according to clause 1, wherein generating the transformed Hamiltonian comprises: repeatedly generating an at least partially decoupled Hamiltonian and a corresponding coupling value, the repetition comprising: selecting a spanning tree for the quantum circuit; determining an original Hamiltonian of the quantum circuit using the spanning tree, the original Hamiltonian comprising a charge coupling matrix and a flux coupling matrix; determining a linear transformation of a mode of the original Hamiltonian; generating the at least partially decoupled Hamiltonian using the linear transformation; and determining a corresponding coupling value for the at least partially decoupled Hamiltonian; and selecting the at least partially decoupled Hamiltonian as the transformed Hamiltonian based on the corresponding coupling value.

[0166] 3. A method according to clause 2, wherein: the linear transformation depends on a block diagonal symplectic matrix, the block diagonal symplectic matrix includes a first submatrix and a second submatrix, and the second submatrix is ​​a function of the first submatrix.

[0167] 4. A method according to clause 3, wherein: generating an at least partially decoupled Hamiltonian using a linear transformation comprises diagonalizing a charge coupling matrix and a flux coupling matrix using a block diagonal symplectic matrix; and the corresponding coupling values ​​depend on the rows of the first submatrix corresponding to the Josephson junction pattern of the original Hamiltonian.

[0168] 5. A method according to clause 3, wherein: generating an at least partially decoupled Hamiltonian using a linear transformation comprises: generating a first transformation matrix using a block diagonal symplectic matrix; transforming the charge coupling matrix by diagonalizing a submatrix of the charge coupling matrix using the first transformation matrix, the submatrix of the charge coupling matrix corresponding to the linear inductance mode of the original Hamiltonian; generating a second transformation matrix using a block diagonal symplectic matrix; and transforming the flux coupling matrix by diagonalizing a submatrix of the flux coupling matrix, the submatrix of the flux coupling matrix corresponding to the linear inductance mode; and the corresponding coupling values ​​depend on the non-diagonal elements of the transformed charge coupling matrix and the transformed flux coupling matrix.

[0169] 6. A method according to clause 2, wherein: determining the linear transformation includes: generating a rotation matrix by iteratively determining rotations around an axis of the rotation matrix, the axis corresponding to the linear inductance mode in the original Hamiltonian.

[0170] 7. A method according to any one of clauses 3 to 5, wherein: the method also includes generating a block diagonal symplectic matrix, the generation comprising: determining an initial block diagonal matrix comprising a flux coupling matrix and a charge coupling matrix; determining a Hermitian matrix based on the initial block diagonal matrix; determining an eigenbasis of the Hermitian matrix and an eigenvalue matrix corresponding to the eigenbasis; and determining the block diagonal symplectic matrix using the initial block diagonal matrix, the eigenbasis of the Hermitian matrix and a matrix of corresponding eigenvalues.

[0171] 8. The method of clause 1, wherein generating the transformed Hamiltonian comprises at least partially decoupling an original Hamiltonian corresponding to the quantum circuit.

[0172] 9. The method of clause 8, wherein at least partially decoupling the original Hamiltonian comprises diagonalizing at least one linear inductive mode of a quadratic portion of the original Hamiltonian.

[0173] 10. A system for simulating a quantum circuit using a computer that processes bits, comprising: at least one processor; and at least one computer-readable medium containing instructions, which, when executed by the at least one processor, cause the system to perform operations, the operations comprising: generating a transformed Hamiltonian corresponding to the quantum circuit, the transformed Hamiltonian comprising a transformed local Hamiltonian and a transformed coupled Hamiltonian, the generation comprising: obtaining a charge coupling matrix and a flux coupling matrix corresponding to the original Hamiltonian of the quantum circuit; at least partially diagonalizing the charge coupling matrix and the flux coupling matrix; determining a finite eigenbasis comprising a plurality of eigenvectors of the transformed local Hamiltonian; projecting the transformed coupled Hamiltonian onto a finite eigenbasis, the transformed coupled Hamiltonian being represented as a pattern of the transformed local Hamiltonian; projecting the transformed local Hamiltonian onto a finite eigenbasis; generating an at least partially decoupled Hamiltonian by combining the projection of the transformed coupled Hamiltonian and the projection of the transformed local Hamiltonian; and simulating the behavior of the quantum circuit using the at least partially decoupled Hamiltonian.

[0174] 11. A system according to claim 10, wherein: at least partially diagonalizing the charge coupling matrix and the flux coupling matrix includes: generating a rotation matrix by iterating the axis of the rotation matrix, the axis corresponding to the linear inductance mode of the original Hamiltonian, and the iteration around an axis includes: updating the rotation matrix to achieve a rotation around an axis.

[0175] 12. A system according to clause 11, wherein: the axes of the rotation matrix are iterated until the function values ​​of the non-diagonal terms of the charge coupling matrix and the flux coupling matrix meet the termination condition.

[0176] 13. A system according to clause 10, wherein: at least partially diagonalizing the charge coupling matrix and the flux coupling matrix includes: generating a block diagonal matrix using the charge coupling matrix and the flux coupling matrix; generating a block diagonal symplectic matrix of the diagonalized block diagonal matrix; generating a transformation matrix using the block diagonal symplectic matrix; and transforming the charge coupling matrix and the flux coupling matrix using the transformation matrix.

[0177] 14. A system according to clause 13, wherein: generating a block diagonal symplectic matrix includes: determining a Hermitian matrix based on the block diagonal matrix; determining an eigenbasis of the Hermitian matrix and a matrix of eigenvalues ​​corresponding to the eigenbasis; and determining the block diagonal symplectic matrix using the block diagonal matrix, the eigenbasis of the Hermitian matrix and a matrix of corresponding eigenvalues.

[0178] 15. A system according to any of clauses 10 to 13, wherein each Josephson junction in the quantum circuit is shunted by an inductor.

[0179] 16. The system of clause 13, wherein: in response to determining that the flux coupling matrix is ​​positive definite, generating a transformation matrix.

[0180] 17. A system according to any of clauses 13 or 16, wherein: the transformation matrix comprises a block diagonal matrix comprising two sub-matrices: an identity sub-matrix; and an inverse of a sub-matrix of the block diagonal symplectic matrix.

[0181] 18. A system according to any of clauses 13 to 17, wherein the block diagonal matrix comprises only sub-matrices of the charge coupling matrix and the flux coupling matrix corresponding to the linear inductance modes of the original Hamiltonian.

[0182] 19. A system according to any one of clauses 13 to 16, wherein: the transformed local Hamiltonian includes transformed Josephson knot terms; or the flux coupling matrix and the charge coupling matrix of the transformed Hamiltonian are the same.

[0183] 20. A non-transitory computer-readable medium containing instructions, the instructions being executable by at least one processor of a system to cause the system to perform operations, the operations comprising: generating a transformed Hamiltonian corresponding to a quantum circuit, the transformed Hamiltonian comprising a transformed local Hamiltonian and a transformed coupled Hamiltonian, the generation comprising: obtaining a charge coupling matrix and a flux coupling matrix corresponding to the original Hamiltonian of the quantum circuit; at least partially diagonalizing the charge coupling matrix and the flux coupling matrix; determining a finite eigenbasis comprising a plurality of eigenvectors of the transformed local Hamiltonian; projecting the transformed coupled Hamiltonian onto a finite eigenbasis, the transformed coupled Hamiltonian being represented as a pattern of the transformed local Hamiltonian; projecting the transformed local Hamiltonian onto a finite eigenbasis; generating an at least partially decoupled Hamiltonian by combining a projection of the transformed coupled Hamiltonian and a projection of the transformed local Hamiltonian; and simulating the behavior of the quantum circuit using the at least partially decoupled Hamiltonian by a computer processing bits.

[0184] In the drawings and the specification, exemplary embodiments have been disclosed. However, many variations and modifications may be made to these embodiments. Therefore, although specific terms are employed, these terms are used only in a general and descriptive sense and are not intended to limit or restrict the scope of the embodiments, which is defined by the appended claims.

Claims

1. A method of simulating a quantum circuit using a computer that processes bits, the method comprising: Obtain a representation of the quantum circuit; generating a transformed Hamiltonian corresponding to the quantum circuit, the transformed Hamiltonian comprising a transformed local Hamiltonian and a transformed coupled Hamiltonian; determining a finite eigenbasis comprising a plurality of eigenvectors of the transformed local Hamiltonian; Projecting the transformed coupled Hamiltonian onto the finite characteristic basis, the transformed coupled Hamiltonian being represented in the mode of the transformed local Hamiltonian; Projecting the transformed local Hamiltonian onto the finite characteristic basis; generating an at least partially decoupled Hamiltonian by combining a projection of the transformed coupled Hamiltonian and a projection of the transformed local Hamiltonian; as well as The at least partially decoupled Hamiltonian is used by a computer to simulate the behavior of the quantum circuit.

2. The method according to claim 1, wherein: The generating the transformed Hamiltonian comprises: Repeatedly generating an at least partially decoupled Hamiltonian and corresponding coupling values, the repeating comprising: selecting a spanning tree for the quantum circuit; Determine the original Hamiltonian of the quantum circuit using the spanning tree, wherein the original Hamiltonian includes a charge coupling matrix and a flux coupling matrix; determining a linear transformation of the mode of the original Hamiltonian; generating an at least partially decoupled Hamiltonian using the linear transformation; and determining a corresponding coupling value for the at least partially decoupled Hamiltonian; and An at least partially decoupled Hamiltonian is selected based on the corresponding coupling value as the transformed Hamiltonian.

3. The method according to claim 2, wherein: The linear transformation depends on a block diagonal symplectic matrix, which includes a first submatrix and a second submatrix, and the second submatrix is ​​a function of the first submatrix.

4. The method according to claim 3, wherein: Generating the at least partially decoupled Hamiltonian using the linear transformation includes diagonalizing the charge coupling matrix and the flux coupling matrix using the block diagonal symplectic matrix; and The corresponding coupling values ​​depend on the rows of the first submatrix corresponding to the Josephson junction patterns of the original Hamiltonian.

5. The method according to claim 3, wherein: Generating an at least partially decoupled Hamiltonian using the linear transformation comprises: generating a first transformation matrix using the block diagonal symplectic matrix; transforming the charge coupling matrix by diagonalizing a sub-matrix of the charge coupling matrix using the first transformation matrix, the sub-matrix of the charge coupling matrix corresponding to a linear inductance mode of the original Hamiltonian; generating a second transformation matrix using the block diagonal symplectic matrix; and transforming the flux coupling matrix by diagonalizing a sub-matrix of the flux coupling matrix using the second transformation matrix, the sub-matrix of the flux coupling matrix corresponding to the linear inductance mode; and The corresponding coupling values ​​depend on the off-diagonal elements of the transformed charge coupling matrix and the transformed flux coupling matrix.

6. The method according to claim 2, wherein: Determining the linear transformation comprises: A new rotation matrix is ​​generated by iteratively determining rotations about axes of the rotation matrix corresponding to linear inductance modes in the original Hamiltonian.

7. The method according to claim 3, wherein: The method further includes generating the block diagonal symplectic matrix, the generating comprising: determining an initial block diagonal matrix including the flux coupling matrix and the charge coupling matrix; determining a Hermitian matrix based on the initial block diagonal matrix; determining an eigenvalue matrix corresponding to the eigenvalue matrix of the Hermitian matrix; and The block diagonal symplectic matrix is ​​determined using the initial block diagonal matrix, the eigenbasis of the Hermitian matrix, and the eigenvalue matrix of the corresponding eigenvalues.

8. The method according to claim 1, wherein: Generating the transformed Hamiltonian includes at least partially decoupling an original Hamiltonian corresponding to the quantum circuit.

9. The method according to claim 8, wherein: At least partially decoupling the original Hamiltonian includes diagonalizing at least one linear inductive mode of a quadratic portion of the original Hamiltonian.

10. The method according to claim 6, wherein: Iterations around one of the axes include: The rotation matrix is ​​updated to achieve the rotation about the one axis.

11. The method according to claim 10, wherein: The axis of the rotation matrix is ​​iterated until function values ​​of non-diagonal terms of the charge coupling matrix and the flux coupling matrix satisfy a termination condition.

12. The method of claim 1, wherein: Generating a transformed Hamiltonian corresponding to the quantum circuit comprises: Obtaining a charge coupling matrix and a flux coupling matrix corresponding to the original Hamiltonian of the quantum circuit; The charge coupling matrix and the flux coupling matrix are at least partially diagonalized.

13. The method according to claim 12, wherein: The at least partially diagonalizing the charge coupling matrix and the flux coupling matrix comprises: generating a block diagonal matrix using the charge coupling matrix and the flux coupling matrix; generating a block diagonal symplectic matrix that diagonalizes the block diagonal matrix; generating a transformation matrix using the block diagonal symplectic matrix; and The charge coupling matrix and the flux coupling matrix are transformed using the transformation matrix.

14. The method according to claim 13, wherein: Each Josephson junction in the quantum circuit is shunted by an inductor.

15. The method of claim 13, wherein: In case it is determined that the flux coupling matrix is ​​positive definite, the transformation matrix is ​​generated.

16. The method of claim 13, wherein: The transformation matrix comprises a block diagonal matrix comprising two sub-matrices: the identity submatrix; and The inverse of a submatrix of the block diagonal symplectic matrix.

17. The method of claim 13, wherein: The block diagonal matrix includes only sub-matrices of the charge coupling matrix and the flux coupling matrix corresponding to linear inductance modes of the original Hamiltonian.

18. The method of claim 13, wherein: The transformed local Hamiltonian includes the transformed Josephson knot term; or The flux coupling matrix and the charge coupling matrix of the transformed Hamiltonian are identical.

19. A system for simulating a quantum circuit using a computer that processes bits, comprising: at least one processor; as well as At least one computer-readable medium containing instructions, when the instructions are executed by the at least one processor, causes the system to perform the operations included in the method of any one of claims 1 to 18.

20. A non-transitory computer-readable medium containing instructions, wherein the instructions are executable by at least one processor of a system to cause the system to perform the operations included in the method of any one of claims 1 to 18.

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