Modeling exponential large classical physical systems using quantum computing

Abstract

Systems and methods for simulating a classical physical system are provided. In one example, a method may include initializing one or more qubits with an initial quantum state that encodes one or more physical characteristics of a classical physical system that includes a network of oscillators. An example method may include simulating, by one or more quantum computing devices, the classical physical system using the one or more qubits.

Description

Technical Field

[0001] The present disclosure generally relates to systems and methods for quantum computing.

[0002] CROSS-REFERENCE TO RELATED APPLICATIONS

[0003] This application is based upon and claims the benefit of priority from U.S. Provisional Patent Application No. 63 / 486,537, filed on February 23, 2023, the disclosure of which is hereby incorporated by reference in its entirety for all purposes. Background Art

[0004] Quantum computing is a computing method that uses quantum effects (such as basis state superposition and entanglement) to perform specific calculations more efficiently than classical digital computers. Compared to digital computers that store and manipulate information in the form of bits (e.g., "1" or "0"), quantum computing systems can use quantum bits ("qubits") to manipulate information. A qubit can refer to a quantum device that can superimpose multiple states (e.g., data in the "0" and "1" states), and / or refers to the superposition of data itself in multiple states. According to conventional terminology, the superposition of "0" and "1" states in a quantum system can be represented as, for example The "0" and "1" states of a digital computer are similar to the and Basis vector state. Summary of the Invention

[0005] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, or may be obvious from the description, or may be learned through practice of the embodiments.

[0006] Example aspects of the present disclosure provide an example method. In some implementations, the example method may include encoding one or more first properties of a classical physical system in the state of one or more quantum bits. In the example method, the classical physical system may include an oscillator network. The example method may include simulating the classical physical system using the one or more quantum bits by one or more quantum computing devices.

[0007] These and other features, aspects and advantages of various embodiments of the present disclosure will be better understood with reference to the following description and appended claims.The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain the relevant principles. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] With reference to the accompanying drawings, a detailed discussion of the embodiments for persons of ordinary skill in the art is set forth in this specification, in which:

[0009] Figure 1 Depicted is an example system of a generalized resonator according to example aspects of the present disclosure;

[0010] Figure 2 depicts an example generalized waveform according to example aspects of the present disclosure;

[0011] Figure 3 An example of a quantum computing system according to example aspects of the present disclosure is depicted;

[0012] Figure 4 depicts a flow chart of an example quantum computing method according to the present disclosure;

[0013] Figure 5 depicts a flow chart of an example quantum computing method according to example aspects of the present disclosure;

[0014] Figure 6 Depicted is a block diagram of an example computing system according to example aspects of the present disclosure. DETAILED DESCRIPTION

[0015] Overview

[0016] Example embodiments according to some aspects of the present disclosure relate to systems and methods for effectively simulating classical physical systems using quantum computing. More specifically, systems and methods according to examples of the present disclosure can simulate various classical physical systems (e.g., electromagnetic waves, sound waves, molecular vibrations, etc.) that can be modeled using harmonic approximation. For example, harmonic approximation can include approximating a classical physical system as a resonator system, which can be mathematically similar to a system of interconnected masses and spring oscillators. In some cases, the systems and methods of the present disclosure can calculate some properties of a classical physical system in a time that is logarithmic to the size of the classical physical system. For example, in this way, the methods of the present disclosure can model some classical physical systems that grow exponentially with respect to the complexity of the quantum computing used to model the system.

[0017] An example method may include, for example, initializing a plurality of qubits with an initial quantum state encoding a physical property of a classical physical system at a first time. The example method may include simulating a time evolution of a Hamiltonian to generate a second quantum state encoding a physical property of the classical physical system at a second time. The example method may include, for example, measuring an observable associated with the second quantum state, wherein the observable corresponds to a property of interest associated with the classical physical system at the second time.

[0018] For example, the initial quantum state can encode physical properties associated with one or more generalized momenta and generalized displacements (e.g., relative to a rest position) of a harmonic approximation of a classical physical system. The generalized properties of the harmonic approximation can be, for example, properties that are mathematically similar (e.g., mathematically identical) to corresponding properties of a spring and mass oscillator system corresponding to the harmonic approximation of the classical physical system. For example, a generalized oscillating mass of the spring and mass approximation can have a generalized mass, a generalized position, a generalized momentum, a generalized velocity, etc. The spring of the spring and mass approximation can have, for example, a generalized spring constant. As a non-limiting illustrative example, a circuit such as a series resistor-inductor-capacitor (RLC) circuit can generate an output waveform corresponding to a resonator, which can be mathematically similar to a spring and mass oscillator. In such a circuit, the charge can correspond to the generalized position of the oscillating mass; the current can correspond to the generalized velocity; the inductance can correspond to the generalized mass; and so on.

[0019] In some cases, the complexity of the method for encoding the initial quantum state may be logarithmic with respect to the size of the classical physical system being encoded. For example, in some cases, a classical physical system or harmonic approximation can be characterized by sparse connections between generalized oscillating masses. For example, each oscillating mass can be connected to only d other oscillating masses, where d can be a constant. Such a system can be called a "d-sparse" system. In such a case, a variety of efficient compilations can be achieved. Some example encoding implementations are further described below and in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0020] In some cases, the time evolution of the Hamiltonian can be based on a Hamiltonian configured to correspond to the time evolution of a classical physical system or a harmonic approximation thereof. In some cases, the Hamiltonian can be constructed in a time that is logarithmic with respect to the size of the classical physical system, and the evolution of the Hamiltonian can be simulated in a time that is logarithmic with respect to the size of the classical physical system. For example, in some cases where the classical physical system (or harmonic approximation) is d-sparse, a unitary operator can be provided that receives an index j indicating a particular oscillating mass and efficiently returns one or more of the following: the generalized mass of the oscillating mass; one or more of the d non-zero spring constants associated with the oscillating mass; and one or more indices k associated with the corresponding oscillating mass connected to the jth oscillating mass via a corresponding spring having a non-zero spring constant. In such a case, the Hamiltonian can be efficiently constructed based on the output of the provided unitary operator, and the evolution of the Hamiltonian can be efficiently simulated according to known methods. In some cases, the complexity of the simulation may also be sublinear with respect to d. Example implementation details are further described below and in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0021] In some cases, the property measured can be a global property of the entire classical physical system, and in some cases, the property can be measured in a time that is logarithmic with respect to the size of the classical physical system. For example, in some cases, the provided methods can efficiently estimate the generalized kinetic energy associated with the entire classical physical system. In some cases, physical properties of a subset of the classical physical system can be measured. For example, a subset of harmonically approximated generalized oscillating masses can be identified, and the generalized kinetic energy of the subset can be efficiently measured. Other example properties are also possible (e.g., potential energy, etc.).

[0022] In some cases, multiple simulations of a classical physical system can be performed to generate multiple measurements. In some cases, the target accuracy can be determined based on the Choose the number of times to simulate the classical physical system. For example, in some cases, you can get the target error probability δ and the target additive error In such cases, the generalized physical properties of interest can be estimated efficiently using quantum algorithms that use O( ) times to simulate and measure the quantum circuit of the classical physical system. In some cases, the number of times to simulate the classical physical system can be selected based on known statistical methods (e.g., classical statistical methods).

[0023] In some cases, the systems and methods according to examples of the present disclosure can be BQP-complete, meaning that any problem in the class of bounded-error quantum polynomial-time (BQP) problems can be mapped to harmonic approximations of classical systems of the present disclosure, and vice versa. In this way, for example, other BQP problems (e.g., other quantum algorithms) can be mapped to harmonic approximations of classical systems and efficiently solved according to the provided systems and methods. In addition, in some cases, the provided systems and methods can simulate BQP-class quantum algorithms using existing methods for simulating resonant oscillators. For example, BQP quantum problems can be mapped to quantum computations of the present disclosure; quantum computations of the present disclosure can be mapped to classical resonant oscillators; and resonant oscillators can be simulated according to classical methods (e.g., classical computing devices, etc.). Although in some cases the complexity of such classical simulations may be exponentially greater than the complexity of the corresponding quantum computations, such classical simulations can still be useful in some cases (e.g., for small to moderate problem sizes, etc.). For example, in some cases, classical computing systems may have technical advantages over corresponding quantum computing systems, such as reduced noise, lower computational cost (e.g., per bit or per qubit), etc. In such cases, classical simulations of BQP problems may be useful for some purposes (e.g., error rate benchmarking, etc.), even when classical simulations are associated with high computational complexity related to the problem size.

[0024] Example embodiments according to some aspects of the present disclosure can provide many technical effects and benefits, such as improvements to computing technology (e.g., quantum computing technology). For example, the systems and methods of the present disclosure can simulate classical physical systems more efficiently than alternative methods such as classical computing simulations. For example, in some cases, the complexity associated with the systems and methods of the present disclosure can be logarithmically related to the size of the classical physical system being simulated. For example, in this way, the classical physical system can grow exponentially relative to the complexity of the example quantum computing of the present disclosure. In contrast, alternative methods (e.g., classical computing methods) may in some cases require a complexity that is at least linearly related to the size of the classical physical system. Therefore, compared to the provided systems and methods, alternative methods may be associated with exponentially growing complexity.

[0025] In some cases, the exponential speedups associated with the systems and methods of the present disclosure can enable tasks that may be difficult and / or quite non-trivial to actually perform using a classical computing system. For example, a task with exponential complexity may become virtually impossible when the problem size becomes too large, even though the task was easy when the problem size was smaller. For example, a 256-bit RSA encryption can be decrypted ("cracked") in under a minute using brute force computing, but a problem only eight times larger (a 2048-bit RSA encryption) may take trillions or quadrillions of years to decrypt using today's classical computers. In contrast, a system or method with non-exponential complexity can, in some cases, scale to large problem sizes more efficiently. As an illustrative example, if a system or method has O(n) 2 ) and can perform a small calculation in one minute, the same system or method can perform an eight-times larger calculation in about 64 minutes instead of several quadrillions of years. Thus, the systems and methods of the present disclosure can, in some cases, enable classical calculations that would be difficult to perform without quantum methods.

[0026] Although the present disclosure describes some activities that can be performed in logarithmic time relative to classical physical systems, systems and methods with non-logarithmic efficiency can be used without exceeding the scope of the present disclosure. For example, in some cases, the quantum computations of the present disclosure can simulate classical physical systems with a size that is polynomial in the complexity of the quantum computation without exceeding the scope of the present disclosure.

[0027] Referring now to the accompanying drawings, example embodiments of the present disclosure will be discussed in further detail.

[0028] Example Harmonic Approximation

[0029] Figure 1An example harmonic approximation is depicted in which a classical physical system can be modeled as a network of oscillators including a plurality of generalized oscillating masses. The harmonic approximation can include, for example, mapping the classical physical system to a corresponding resonant subsystem, which can include a generalized oscillating mass 102, a generalized spring 104, and a generalized wall 106. The generalized oscillating masses 102A-H can be attached to each other and / or to one or more generalized walls 106 via one or more generalized springs 104A-N. Various classical physical systems (e.g., molecular vibrations, thermal expansion, various systems including waves, etc.) can be modeled or approximated according to the depicted harmonic approximation. Below is a description of Figure 2 Depicts a further example of harmonic approximation.

[0030] The generalized oscillating mass 102 may include, for example, an oscillating mass of a harmonic approximation, wherein the generalized oscillating mass 102 is configured to be mathematically similar to (e.g., identical to, approximately similar to, etc.) a component or property of a classical physical system. The generalized oscillating mass 102 may have, for example, a plurality of generalized properties, including, but not limited to, generalized position; a generalized mass property; and generalized momentum. For example, each generalized property may be configured to be mathematically similar to (e.g., identical to, approximately similar to, etc.) a corresponding physical property of a classical physical system, and mathematically similar to (e.g., identical to, etc.) a corresponding physical property of an oscillating mass in a mass and spring resonator system. For example, generalized velocity may be the rate of change of generalized position; generalized momentum may be the product of generalized mass and generalized velocity; and the generalized mass of the oscillating mass 102 may indicate the amount of generalized force required to accelerate or decelerate the generalized oscillating mass 102 at a particular rate. As a non-limiting illustrative example, a series resistor-inductor-capacitor (RLC) circuit may generate an output waveform corresponding to a resonator, where charge may correspond to the generalized position of the oscillating mass; current may correspond to the generalized velocity; inductance may correspond to the generalized mass; etc. As another example, a parallel RLC circuit may generate different output waveforms corresponding to different resonators, where flux linkage may correspond to the generalized position; voltage may correspond to the generalized velocity; capacitance may correspond to the generalized mass; and charge may correspond to the generalized momentum.

[0031] The generalized spring 104 can be, for example, a spring associated with a harmonic approximation, wherein the generalized spring 104 is configured to be mathematically similar to (e.g., identical to, approximate to, etc.) a component or property of a classical physical system. The generalized spring 104 can have, for example, multiple generalized properties, including, but not limited to, a generalized spring constant and a generalized displacement (e.g., a generalized distance compressed or stretched relative to a generalized rest position). For example, the generalized spring constant of the generalized spring 104 can correspond to the force exerted by the generalized spring 104 and the generalized displacement of the generalized spring 104 relative to a generalized rest position (e.g., a generalized distance compressed or stretched relative to the generalized rest position). As a non-limiting illustrative example, a series resistor-inductor-capacitor (RLC) circuit can generate an output waveform corresponding to a resonator, wherein the inverse capacitance can correspond to the generalized spring constant of the generalized spring 104 corresponding to the series RLC circuit.

[0032] The generalized wall 106 may correspond to a generalized immovable object (eg, having a fixed generalized position) associated with the harmonic approximation, for example, wherein the one or more generalized oscillating masses 102 may be attached to the generalized wall 106 via the one or more generalized springs 104 .

[0033] In general, generalized oscillating mass 102, generalized spring 104, and generalized wall 106 can possess any generalized physical properties (e.g., damping, driving force, etc.) corresponding to any physical properties that a corresponding classical oscillating system can possess. In some cases, the generalized physical properties of generalized objects 102, 104, 106 can be derived from other generalized physical properties of generalized objects 102, 104, 106 according to classical physical laws (e.g., Newton's laws). For example, the generalized kinetic energy of generalized oscillating mass 102 can correspond to , where m is the generalized mass value, and v is the generalized velocity of the generalized vibrating mass 102. Similarly, the generalized elastic potential energy of the generalized spring 104 may correspond to , where k may be the generalized spring constant and x may be the generalized displacement relative to the rest position of the generalized spring 104 .

[0034] Figure 2 An example harmonic approximation of waveform 208 is depicted, wherein a generalized displacement of waveform 208 with respect to time can be modeled as or mapped to a resonant oscillator system. Various classical physical waveforms can be harmonically approximated in a manner similar to (e.g., identical to) that depicted, including but not limited to acoustic waves, light waves, electromagnetic waves, etc.

[0035] exist Figure 2 2A to 2B, waveform 208 is depicted as a curve that can oscillate continuously around a center point 210 over time 212. Figure 2E depicts an example generalized oscillator 102, generalized spring 104, and generalized wall 106 at five discrete points in time associated with waveform 208. Figure 2 At the first time point depicted in A, the generalized vibrator 102 and the generalized spring 204 can have zero displacement relative to the generalized rest position 210 of the generalized spring 204, and the generalized vibrator 102 can have a positive generalized velocity, where the generalized vibrator 102 can move toward the wall. In some cases, the generalized rest position 210 can correspond to the center point 210 of the waveform 208. Figure 2 A and Figure 2 During the time period between 2B and 2C, the generalized spring 204 can decelerate the generalized oscillating mass 102, where the generalized deceleration force can be proportional to the displacement (e.g., compression) of the generalized spring 104 relative to the rest position 210. At time 2B, the generalized velocity of the generalized oscillating mass 102 can be zero relative to the generalized wall 106, and the generalized spring 204 can continue to accelerate the generalized mass 102 in a negative direction away from the generalized wall 106. At time 2C, the generalized velocity of the generalized oscillating mass can have a similar (e.g., the same) amplitude and opposite sign relative to time 2A. In other aspects, some physical properties of the harmonic approximation depicted at time 2C (e.g., generalized displacement, generalized force, or acceleration, etc.) can be similar (e.g., the same) as at time 2A. At time 2D, some properties of the harmonic approximation (e.g., generalized displacement, generalized force, generalized acceleration) can have a similar (e.g., the same) amplitude and opposite sign relative to time 2B. In other aspects, some physical properties of the harmonic approximation depicted at time 2D (eg, zero generalized velocity, etc.) may be similar (eg, the same) as compared to time 2A. Figure 2 E depicts the state of the generalized oscillating mass 102, the generalized spring 104, and the generalized wall 106, which in some cases can be compared to Figure 2 The state depicted in A is the same (eg, when times 2A and 2E are exactly one cycle apart).

[0036] In some cases, waveform 208 may be a generalized waveform 208 that represents an approximation of a different (e.g., non-wave-based) classical physical system. In some cases, the harmonic approximation may be reversible. For example, in some cases, waveform 208 may be approximated by one or more oscillators, and a system of one or more oscillators may be approximated by generalized waveform 208.

[0037] although Figure 2 A relatively simple example is depicted, comprising a sinusoidal waveform corresponding to a single generalized oscillating mass 102, but more complex waves (e.g., multi-dimensional waves, waveforms with multiple higher-order harmonics, etc.) can be modeled in a similar (e.g., identical) manner.

[0038] Example quantum simulation

[0039] In general, a quantum simulation of a classical physical system may include initializing one or more qubits with a quantum state that encodes one or more properties of the physical system. In some cases, the property of the classical physical system may be a generalized property of a harmonic approximation of the classical physical system, include a generalized property of a harmonic approximation of the classical physical system, or be associated with a generalized property of a harmonic approximation of the classical physical system. The quantum simulation may further include simulating the classical physical system using the one or more qubits by the quantum computing system. In some cases, the simulation may include simulating the time evolution of a Hamiltonian. The quantum simulation may further include, for example, measuring one or more observables associated with the one or more qubits. In some cases, after simulating the time evolution of the Hamiltonian, the observables may be associated with the final state of the one or more qubits.

[0040] Example quantum states encoding classical physical properties

[0041] The quantum simulation of a classical physical system may include initializing one or more qubits with a quantum state that encodes one or more properties of the classical physical system (e.g., properties of a harmonic approximation of the classical physical system). In some cases, the one or more properties may include generalized momentum or generalized velocity and generalized displacement or generalized position associated with one or more generalized oscillating masses 102.

[0042] In some cases, the initial quantum state encoding the generalized momentum or generalized velocity and the generalized displacement or generalized position can be described by the following equations

[0043] ,

[0044] Wherein E may be a constant greater than zero; i may be the square root of -1; M may be an NxN diagonal matrix of generalized mass values ​​associated with the N generalized oscillating masses 102; It can be the matrix square root of M; can be the vector of N generalized velocities of N generalized oscillating masses 102 at time t; can be a vector with N(N+1) / 2 real-valued entries, where each entry can be written as or , where k > j, is the generalized spring constant of the generalized spring 104 between the j-th and k-th generalized oscillators 102, and is the generalized spring constant of the generalized spring 104 between the jth generalized oscillator 102 and the generalized wall 106. In some cases, the time t associated with the initial quantum state may be zero.

[0045] In some cases, E can be equal to K(t) + U(t), where K(t) can be the generalized kinetic energy associated with the classical physical system at time t (e.g., the generalized kinetic energy of the plurality of generalized oscillating masses 102), and U(t) can be the generalized potential energy associated with the classical physical system at time t. For example, in this way, E can be the generalized total energy of the classical physical system, which can be constant over time.

[0046] In some cases, instead of or in addition to generalized momentum and displacement, the initial quantum state can encode other properties of the classical physical system. For example, the generalized kinetic energy K(t) at time t can be written as , which can correspond to mv 2 The sum of all generalized oscillating masses 102 of , where v is the generalized velocity of the corresponding generalized oscillating mass 102 and m is the generalized mass value of the corresponding generalized oscillating mass 102. For example, in this way, the initial state The generalized kinetic energy K(0) of a classical physical system can be encoded. Similarly, the potential energy U(t) at time t can be written, for example, as For example, in this way, the initial state The generalized potential energy of the harmonic approximation of classical physical systems can be encoded.

[0047] In some cases, the initial quantum state encoding the generalized momentum and generalized displacement The encoding may be performed in an amount of time that is logarithmic with respect to the number of generalized oscillators 102 being encoded. For example, in some cases, one or more compact representations of K and M may enable access to any term of K or M in a time that is sublinear (e.g., constant) with respect to the number N of generalized oscillating masses 102 associated with the classical physical system. In some cases, K may be the generalized spring constant associated with the generalized spring 104 connecting the jth generalized oscillating mass 102 to the generalized wall 106. and the spring constant associated with the generalized spring 104 connecting the jth generalized oscillating mass 102 to the kth generalized oscillating mass 102 In some cases, a compact representation of M may include receiving an oscillator index j as input and generating M entries m j As a function of the output, where m j can correspond to the mass of the jth generalized oscillating mass 102 associated with the classical physical system. In some cases, such a function can be implemented in a quantum circuit (e.g., including one or more quantum gates) to provide for any value of m given an oscillator index j. jIn some cases, such access may be referred to as an "oracle access." Similarly, in some cases (e.g., when K is d-sparse), a compact representation of K may include receiving as input an oscillator index j and returning one or more non-zero generalized spring constants associated with the jth generalized oscillating mass 102. and function.

[0048] Similarly, in some cases, and One or more compact representations of may enable access to the time N in a sublinear (e.g., constant or logarithmic) time with respect to the number N of generalized oscillating masses 102 associated with the classical physical system. or Any item of 、 , K and M, the initial state can be initialized efficiently , which encodes the generalized momentum and displacement of the classical physical system at time zero (e.g., in a time logarithmic with respect to the size of the classical physical system). For example, in some cases, a unitary operator S can be provided that computes the mass m with respect to the input j j and the non-zero terms of K with respect to input (j, k), i.e., κ jk , and their positions. For example, one or more unitary operators S can be provided to perform the mapping

[0049] ,

[0050] ,

[0051] ,

[0052] where j and k can be any numbers between 1 and N; l can be any number between 1 and d, where K is d-sparse; and a(j, l) can be the column index of the lth nonzero entry in the jth row of K.

[0053] In some cases, efficient preparation can be provided and The unitary operator U of . Given U and S, the initial state can be efficiently prepared by calling U, S, their inverses and other gates .

[0054] For example, in some cases, the initial state The matrix can be applied via a quantum random walk and Come prepare, where B †can be the Hermitian adjoint of B, and B can be ,in and (where 1 ≤ j < k < N), where can be a typical vector with 1 in the jth position and 0 in all other positions.

[0055] In some cases, the unitary operator U can perform the mapping

[0056] ,

[0057] ,

[0058] in

[0059] ,

[0060]

[0061] can be normalized states encoding the initial state of the generalized oscillating mass 102 with their amplitudes, and where α > 0 and β > 0 can also be known. In this case, close to

[0062]

[0063] The initial state is - can be generated by calling U, S, their inverses and two-qubit gates.

[0064] For example, in some cases, it is possible to generate a signal close to The initial state is The first preparation action can prepare the state by calling the unitary operator U and an O(1) two-qubit gate. , as shown below:

[0065] .

[0066] The second ready action can be transformed , making its amplitude and The amplitude of the oscillation can be written as

[0067] .

[0068] The second preparatory action can be performed, for example, by constructing the unitary operator

[0069] ,

[0070] in And use the unitary operator V In some cases, the unitary operator V can be constructed according to known methods based on block coding and quantum random walks. Additional example implementation details are further described in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0071] In some cases, alternative encodings can be used to encode one or more properties of a classical physical system in the quantum state of one or more qubits. Example alternative encodings are described further below, after discussing an example Hamiltonian. (The example alternative encodings described below are based in part on the matrices discussed below with respect to the example Hamiltonian and are therefore easier to understand in terms of the example Hamiltonian.)

[0072] Example Hamiltonian for simulating time evolution

[0073] Quantum simulation of a classical physical system can include using one or more qubits to simulate the time evolution of a Hamiltonian. In some cases, the Hamiltonian can be configured to simulate the time evolution of a classical physical system.

[0074] In some cases, the appropriate Hamiltonian can be efficiently determined from K and M. For example, the Newtonian equations for classical harmonic oscillator dynamics can be written in matrix form as

[0075] ,

[0076] where F is an NxN matrix whose diagonal and off-diagonal entries are and The above equation can also be written as

[0077] ,

[0078] in This can also be written as

[0079] ,

[0080] This is equivalent to the Hamiltonian Thus, a Hamiltonian comprising the matrix square root of A can be used to simulate the time evolution of a harmonic approximation comprising one or more generalized oscillating masses 102 and one or more generalized springs 104 .

[0081] In some cases, a Hamiltonian involving the square root of a matrix A can be obtained from a matrix satisfying BB † = A is determined by the NxM matrix B, where B † is the Hermitian adjoint matrix of B. For example, the Hamiltonian H can be written as

[0082] .

[0083] In this way, for example, H can act on the space space on it, and can have a square H 2 , the first piece of the square is A. In this way, for example, a Hamiltonian H whose square includes A can act as a Hamiltonian including the "square root" of A. The Schrödinger equation due to the Hamiltonian H can be written as ,in is the state of the quantum system at time t.

[0084] In some cases, the two subspaces of the block according to H can be considered separately, and the Schrödinger equation due to the Hamiltonian H can be rewritten as

[0085] ,

[0086] in and yes The two components of .

[0087] In some cases, the initial state can be configured , so that for a certain , In such a case, for a , evolutionary state Can satisfy In this case, the Schrödinger equation due to the Hamiltonian H can be rewritten as

[0088] .

[0089] For example, in this way, a quantum computing system can simulate the dynamics of a classical oscillator by simulating a quantum system evolving under H and Perform appropriate initialization.

[0090] In some cases, B can be ,in

[0091] and

[0092]

[0093] where 1 ≤ j< k < N, where can be a typical vector with 1 in the jth position and 0 in all other positions. In some cases, B can be an NxM matrix, where M can be an integer greater than or equal to one. In some cases (e.g., the choices of B described in this paragraph), each product Can be In some cases, the vector Can have equal or , which can correspond to the example quantum state encoding described above,

[0094] .

[0095] In some cases, a suitable Hamiltonian (e.g., Hamiltonian H) can be constructed efficiently in a time that is logarithmic with respect to the size of the classical physical system. For example, in some cases, one or more compact representations of K and M can enable access to any term of K or M in a time that is sublinear (e.g., constant) with respect to the number N of generalized oscillating masses 102 associated with the classical physical system. In some cases, a compact representation of M can include receiving an oscillator index j as input and generating an M term m j As a function of the output, where m j can correspond to the mass of the jth generalized oscillating mass 102 associated with the classical physical system. In some cases, such a function can be implemented in a quantum circuit (e.g., including one or more quantum gates) to provide for any value of m given an oscillator index j. j In some cases, such access may be referred to as an "oracle access." Similarly, in some cases (e.g., when K is d-sparse), a compact representation of K may include receiving as input an oscillator index j and returning all non-zero generalized spring constants associated with the jth generalized oscillating mass 102. and In some cases (e.g., when K is d-sparse), a compact representation of K may include receiving as input an oscillator index j and returning one or more non-zero generalized spring constants associated with the jth generalized oscillating mass 102 and In the case where K is d-sparse, the Hamiltonian H can also be d-sparse. In some cases, it can be determined from a circuit that provides efficient access to these values ​​(e.g., "oracle access"). and m j To calculate the term of Hamiltonian H , and then calculated according to the standard method In some cases where the unitary operator S provides a(j, l), b(j, l) can be easily computed from a(j, l), where b(j, l) can be the column index of the lth nonzero entry in the jth row of H.

[0096] The time evolution of the Hamiltonian H can be simulated using existing methods. For example, in some cases, the exponential In some cases, such methods can be used to approximate . , d, t and ||H|| max Nearly optimal scaling is achieved in (the absolute maximum term of H). Additional example implementation details are further described in US Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0097] Example quantum observables

[0098] A quantum simulation may include, for example, measuring one or more observables associated with the final state of one or more qubits (e.g., the final state after simulating the time evolution of the Hamiltonian). In some cases, the observables may be observables that encode one or more properties of a classical physical system. In some cases, the one or more properties may be global properties of the entire classical physical system, or aggregate properties associated with multiple components of the classical physical system (e.g., multiple generalized oscillating masses 102). In some cases, the global or aggregate properties may be properties that cannot be efficiently determined using classical methods (e.g., in sublinear time relative to the size of the classical physical system).

[0099] For example, in some cases, the quantum simulation may include measuring an observable that encodes the generalized kinetic energy of the plurality of generalized oscillating masses 102. In some cases, the plurality of generalized oscillating masses 102 may include all of the generalized oscillating masses 102 of a harmonic approximation of a classical physical system. In some cases, the plurality of generalized oscillating masses 102 may include a subset (e.g., an exact subset) of the harmonic approximation of the classical physical system.

[0100] In some cases, a unitary operator v can be provided that labels all generalized oscillatory masses 102 of the subset of interest V by performing the following mapping

[0101] ,

[0102] where if the jth generalized oscillating mass 102 is a member of the subset of interest, then v j = -1, and if the jth generalized oscillating mass 102 is not a member of the subset of interest, then v j = 1. In such cases, the generalized kinetic energy of the subset of interest can be estimated efficiently using a quantum algorithm that uses O( ) Preparation and and their inverse and controlled versions of quantum circuits, where δ is the error probability and is the additive error associated with the estimate. For example, the generalized kinetic energy of the subsets can be written as

[0103] ,

[0104] in is the projection operator onto V. Therefore, by estimating and the error probability δ , we can determine the generalized kinetic energy of the subset V of interest, which is just the unitary operator The expected value of . Such observables can be efficiently estimated according to known methods such as high confidence amplitude estimation. In this way, for example, the provided method can output the generalized kinetic energy of the subset of interest in a time that is logarithmic with respect to the number N of generalized oscillating masses 102 of the classical physical system in the presence of additive error. Estimation within E. Additional example implementation details are further described in US Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0105] Example alternative quantum states encoding classical physical properties

[0106] Alternative encodings can be used to encode properties of classical physical systems and simulate them using quantum algorithms (e.g., the time evolution of the Hamiltonian).

[0107] In an example encoding, the initial state Can be initialized based on the normalized state

[0108] ,

[0109] Where X>0 can be a constant; can be the Moore-Penrose pseudoinverse of B; and P can be The components are projected to correspond to A (or B † ). In other words, P can be a projection operator onto a subspace orthogonal to the null space of A.

[0110] In some cases, the choice of quantum state encoding may be related to a computational complexity trade-off, and the best encoding choice may depend on the specific use case. For example, encoding using Moore-Penrose pseudoinverse may increase the cost of preparing the initial state. The cost of initial state preparation may be reduced, but may provide more direct access to one or more generalized displacements of one or more generalized oscillating masses 102. In some cases (e.g., simulating wave equations where A corresponds to a discretized Laplace operator), this increased initial state preparation cost may dominate the overall cost of performing quantum computations according to the provided methods. Additional example implementation details are further described in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0111] Example mapping for simulating universal quantum circuits

[0112] In some cases, any quantum circuit can be mapped to the quantum simulation of the present disclosure. In some cases, this universal mapping can prove that the provided systems and methods are BQP complete. In some cases, the provided quantum simulation can be mapped to a classical physical system (or its harmonic approximation) by applying the mapping described herein in the opposite direction. For example, in this way, any quantum circuit can be mapped to a classical physical system of a harmonic oscillator. In some cases, a classical physical system can be simulated (e.g., according to classical methods), and the resulting classical physical state can be mapped to a provided quantum state, and then mapped to the final quantum state of an arbitrary quantum circuit. For example, in this way, any quantum circuit can be simulated using classical methods for simulating harmonic systems.

[0113] In some cases, an arbitrary quantum circuit can be mapped to a quantum simulation of the present disclosure by mapping the arbitrary quantum circuit to a plurality of gates from a universal set and mapping the gates from the universal set to the quantum simulation of the present disclosure. The universal set of quantum gates can be, for example, {H, T}, where H can be a single-qubit Hadamard gate and T can be a three-qubit Toffoli gate. In some cases, a plurality of L gates U operating on n qubits L ...U1 can be mapped to a coupled oscillator system, where the number of oscillators N can be equal to (L+1)2 n+1 ; The generalized mass of each oscillator can be 1, so that the generalized mass matrix M can be equal to 𝟙 N , i.e., an NxN diagonal matrix with entries equal to 1; and as described above with respect to constructing an appropriate Hamiltonian, the matrices A and F can be expressed as

[0114] ,in

[0115] If U l If it is the Toffoli Gate,

[0116] ,

[0117] And if U l is a Hadamard gate, then W can be obtained through the following mapping l

[0118] .

[0119] For example, in this way, the off-diagonal terms of A can be , and the diagonal entries can be 4. In this case, the corresponding off-diagonal entries of the matrix K can be This corresponds to a 5-sparse system of coupled oscillators, where the spring constant is non-negative and can be efficiently accessed based on the following equation

[0120] .

[0121] In some cases, the initial state of such a 5-sparse system can be 、 and , where j > 2, so that the total energy E is equal to 1. In this case, the time evolution of the Hamiltonian in time t can be simulated so that the proportionality constant can be In this case, a first generalized kinetic energy of the first plurality of generalized oscillating masses 102 may be obtained, and a second generalized kinetic energy of the second plurality of generalized oscillating masses 102 may be obtained. In some cases, by labeling each oscillator with (l, r) ,in And r = binary ( ) n ...r0; defining the first plurality of generalized oscillating masses as the set of generalized oscillating masses 102, where l = L + 1, r1 = 0 and r0 = 0; and defining the first plurality of generalized oscillating masses as the set of generalized oscillating masses 102, where l = L + 1, r1 = 1 and r0 = 0, the first and second pluralities of generalized oscillating masses 102 can be determined. In this case, the difference between the first generalized kinetic energy and the second generalized kinetic energy can be the sum of ... and single-qubit expectations within an error probability of 1 / 3. This mapping can, for example, prove that the presented systems and methods are BQP complete.

[0122] In addition, the 5 sparsely coupled oscillator systems mapped in this way can be mapped to classical physical systems (or their harmonic approximations) by applying one or more mappings described herein in the opposite direction. For example, in this way, any quantum circuit can be mapped to the classical physical system of the harmonic oscillator. In some cases, a classical physical system can be simulated (for example, according to a classical method), and the resulting classical physical state can be mapped to a provided quantum state, and then mapped to the final quantum state of any quantum circuit. For example, in this way, any quantum circuit can be simulated using a classical method for simulating a harmonic system. Additional example implementation details are further described in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0123] Example quantum computing system

[0124] Figure 3An example quantum computing system 300 is depicted. Example system 300 is an example of a system in which the systems, components, and techniques described below can be implemented on one or more classical computers or quantum computing devices in one or more locations. One of ordinary skill in the art, using the disclosure provided herein, will understand that other quantum computing structures or systems can be used without departing from the scope of this disclosure.

[0125] System 300 includes quantum hardware 302 in data communication with one or more classical processors 304. Quantum hardware 302 includes components for performing quantum computations. For example, quantum hardware 302 includes a quantum system 310, a control device 312, and a readout device 314 (e.g., a readout resonator). Quantum system 310 may include one or more multi-level quantum subsystems, such as registers of qubits. In some implementations, the multi-level quantum subsystems may include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, and the like.

[0126] The type of multi-level quantum subsystem used by system 300 may vary. For example, in some cases, it may be convenient to include one or more readout devices 314 attached to one or more superconducting qubits (e.g., transmon qubits, flux qubits, gmon qubits, xmon qubits, or other qubits). In other cases, ion traps, photonic devices, or superconducting cavities may be used (e.g., utilizing them to avoid the need for qubits to prepare states). Further examples of implementations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0127] Quantum circuits can be constructed and applied to qubit registers included in quantum system 310 via a plurality of control lines coupled to one or more control devices 312. Example control devices 312 that operate on qubit registers can be used to implement quantum gates or quantum circuits having multiple quantum gates, such as Pauli gates, Hadamard gates, controlled NOT (CNOT) gates, controlled phase gates, T-gates, multi-qubit quantum gates, coupler quantum gates, and the like. One or more control devices 312 can be configured to operate quantum system 310 using one or more corresponding control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystem can be a superconducting qubit, and control device 312 can be configured to provide control pulses to the control lines to generate a magnetic field to adjust the frequency of the qubit.

[0128] The quantum hardware 302 may further include a readout device 314 (e.g., a readout resonator). Measurement results 308 obtained via the measurement device may be provided to the classical processor 304 for processing and analysis. In some implementations, the quantum hardware 302 may include quantum circuits, and the control device 312 and the readout device 314 may implement one or more quantum logic gates that operate the quantum hardware 302 via physical control parameters (e.g., microwave pulses) sent via wires included in the quantum hardware 302. Another example of a control device includes an arbitrary waveform generator, in which a DAC (digital-to-analog converter) creates a signal.

[0129] Readout device 314 can be configured to perform quantum measurements on quantum system 310 and send measurement results 308 to classical processor 304. In addition, quantum hardware 302 can be configured to receive data specifying physical control qubit parameter values ​​306 from classical processor 304. Quantum hardware 302 can use the received physical control qubit parameter values ​​306 to update the actions of control device 312 and readout device 314 on quantum system 310. For example, quantum hardware 302 can receive data specifying new values ​​representing the voltage strength of one or more DACs included in control device 312, and the quantum hardware can update the actions of the DACs on quantum system 310 accordingly. Classical processor 304 can be configured to initialize quantum system 310 with an initial quantum state, for example, by sending data specifying an initial parameter set 306 to quantum hardware 302.

[0130] The readout device 314 can utilize the quantum system elements (such as quantum bits) and The state of an element (e.g., a quantum bit) is measured by the impedance difference of the state. For example, due to the nonlinearity of the quantum bit, when the quantum bit is in the state or status The resonant frequency of the readout resonator can assume different values ​​when the qubit is in the same state. Therefore, the microwave pulse reflected from the readout device 314 carries an amplitude and phase shift that depends on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device 314 to block the propagation of microwaves at the qubit frequency.

[0131] In some implementations, quantum system 310 may include, for example, a plurality of qubits 320 arranged in a two-dimensional grid 322. For clarity, Figure 1The two-dimensional grid 322 depicted in FIG includes 16 qubits arranged in a square form, however, in some implementations, the system 310 may include fewer or more qubits. In some embodiments, the plurality of qubits 320 may interact with each other through a plurality of qubit couplers (e.g., qubit coupler 324). The qubit coupler may define the nearest neighbor interaction between the plurality of qubits 320. In some implementations, the strength of the plurality of qubit couplers is an adjustable parameter. In some cases, the plurality of qubit couplers included in the quantum computing system 300 may be couplers with a fixed coupling strength. In some implementations, the plurality of qubits 320 may include data qubits (such as qubit 326) and measurement qubits (such as qubit 328). A data qubit is a qubit that participates in the calculation performed by the system 300. A measurement qubit is a qubit that can be used to determine the result of the calculation performed by the data qubit. That is, during the calculation process, the unknown state of the data qubit is transferred to the measurement qubit using appropriate physical operations and is measured via appropriate measurement operations performed on the measurement qubit.

[0132] In some implementations, each qubit in the plurality of qubits 320 can operate using a corresponding operating frequency, such as an idle frequency and / or an interaction frequency and / or a readout frequency and / or a reset frequency. The operating frequencies of different qubits may be different. For example, each qubit may be idle at a different operating frequency. The operating frequency of the qubit 320 can be selected before the calibration system performs the calculation. Some operating frequencies are better than other operating frequencies. One metric for evaluating the pros and cons of a particular operating frequency for a particular qubit is the energy relaxation time (T1) of the qubit at that frequency. A lower energy relaxation time may result in larger quantum computing errors.

[0133] In various implementations, the example system 300 can be implemented as a client device, a server device, or both. The example system 300 can be implemented as part of a distributed computing system. The example system 300 can be implemented with other example systems that may be the same or different. The example system 300 can be implemented in a server farm or other facility that operates multiple computing systems to provide computing services to or on behalf of multiple client systems. Advantageously, the technology according to example aspects of the present disclosure can provide improved calibration and maintenance of computing facilities, increase service uptime, reduce failure rates, and the like.

[0134] Example Method

[0135] Figure 4 Depicted is a flow chart of an example method for simulating a classical physical system according to an example embodiment of the present disclosure. Figure 4 The steps of the exemplary method 400 are described in a specific order for the purpose of illustration and discussion, but the method of the present disclosure is not limited to the specific order or arrangement shown. The steps of the exemplary method 400 may be omitted, rearranged, combined and / or adapted in various ways without departing from the scope of the present disclosure.

[0136] At 402, example method 400 may include encoding one or more first properties of a classical physical system in a state of one or more qubits. In some cases, the first property may be, include, correspond to, or otherwise be associated with a generalized property of generalized oscillating mass 102 or generalized spring 104. In some cases, the first property may be or include generalized momentum, generalized displacement, generalized mass, generalized spring constant, or generalized velocity. In some cases, encoding the first property may include modeling the classical physical system as a harmonic approximation and encoding the property of the harmonic approximation in the state of one or more qubits. In some cases, example method 400 may include, at 402, using one or more systems or performing operations related to Figures 1 to 3 Describes one or more activities.

[0137] At 404, the example method 400 may include simulating a time evolution of a Hamiltonian, wherein the Hamiltonian is configured such that the time evolution of the Hamiltonian corresponds to a time evolution of one or more first properties of a classical physical system. In some cases, a quadrant of the square of the Hamiltonian may include a matrix encoding one or more second properties of the classical physical system. In some cases, the matrix encoding the one or more second properties may include a matrix product of a first matrix encoding one or more masses or generalized masses associated with the classical physical system and a second matrix encoding one or more spring constants or generalized spring constants associated with the classical physical system. In some cases, the Hamiltonian may be, include, or consist of the Hamiltonian H described above. In some cases, simulating a classical physical system may include executing a quantum algorithm whose complexity is logarithmic with respect to the size of the classical physical system or the size of the harmonic approximation. In some cases, the example method 400 may include, at 404, using one or more systems or executing a method regarding Figures 1 to 3 Describes one or more activities.

[0138] At 406, example method 400 may include measuring an observable associated with one or more qubits to generate one or more measurement values. In some cases, the observable may be, include, encode, or otherwise correspond to, kinetic energy associated with a classical physical system. In some cases, the observable may be, include, encode, or otherwise correspond to, generalized kinetic energy associated with a harmonic approximation of a classical physical system. In some cases, the observable may be, include, encode, or otherwise correspond to, generalized kinetic energy of an interesting subset of a plurality of generalized oscillating masses 102 associated with a harmonic approximation of a classical physical system. In some cases, example method 400 may include, at 406, using one or more systems or performing a method for ... Figures 1 to 3 Describes one or more activities.

[0139] At 408, the example method 400 may include estimating one or more third properties of the classical physical system based at least in part on the one or more measurements. In some cases, the estimated third property may be, include, or consist of a high confidence amplitude estimate. In some cases, the example method 400 may include using one or more systems or performing a method for determining the properties of the classical physical system. Figures 1 to 3 Describes one or more activities.

[0140] Figure 5 An example method 500 for performing quantum computation using a quantum circuit according to example aspects of the present disclosure is depicted. For example, in some cases, a quantum circuit may include, be included in, or be implemented by a quantum system 310. Figure 5 For purposes of illustration and discussion, the steps are depicted as being performed in a particular order, but the methods of the present disclosure are not limited to the particular order or arrangement shown. Individual steps of method 500 may be omitted, rearranged, combined, and / or adapted in various ways without departing from the scope of the present disclosure. Method 700 may be implemented by any suitable computing system, such as a quantum computing system including quantum hardware in communication with one or more quantum control devices, such as Figure 3 Quantum computing system 300.

[0141] At 502, example method 500 may include obtaining data indicative of a quantum circuit. Obtaining data may include, for example, receiving data from a computing device (e.g., a user device, a server device); receiving data from a user (e.g., via an input / output device); reading data from one or more non-transitory computer-readable media; generating data (e.g., using an algorithm); and the like. The data indicative of a quantum circuit may include, for example, a circuit design, a circuit diagram, one or more unitary matrices, software code (e.g., quantum software code in a quantum computing language), and the like.

[0142] At 504, example method 500 may include preparing one or more qubits in a known quantum state. Preparing one or more qubits in a known quantum state may include, for example, preparing one or more qubits in a known basis state (e.g., by manipulating a plurality of qubits such that a qubit characterized by a particular basis state, such as or , can be separated (e.g., physically separated, individually identified, etc.) from qubits not characterized by the basis state. Preparing one or more qubits in a known quantum state can include, for example, performing quantum gating using control device 312 to generate a known multi-qubit basis state. Preparing one or more qubits can include generating a known multi-qubit basis state with respect to Figure 3 The described approach uses the control device 312 .

[0143] At 506, the example method 500 may include applying one or more quantum gates to one or more qubits to perform a quantum algorithm. For example, in some cases, the control device 312 may be configured to Figure 3 The described method realizes a quantum gate or a quantum circuit with multiple quantum gates, such as a Pauli gate, a Hadamard gate, a controlled NOT (CNOT) gate, a controlled phase gate, a T gate, a multi-qubit quantum gate, a coupler quantum gate, etc.

[0144] At 508, the example method 500 may include measuring the state of at least one of the one or more qubits using a readout device. The readout device may be, for example, readout device 314, and step 506 may in some cases be performed with respect to Figure 3 Execute as described.

[0145] Figure 6 A block diagram of an example computing system 5 is depicted that can perform aspects of example embodiments of the present disclosure. System 5 includes a computing device 50, a server computing system 60, and a third-party system 70 communicatively coupled via a network 49. System 5 also includes a quantum computing system 80 communicatively coupled to the server computing system.

[0146] The computing device 50 can be any type of computing device (e.g., a classic computing device), such as, for example, a mobile computing device (e.g., a smartphone or tablet), a personal computing device (e.g., a laptop or desktop computer), a workstation, a cluster, a game console or controller, a wearable computing device, an embedded computing device, or any other type of computing device. In some embodiments, the computing device 50 can be a client computing device or a server computing device. The computing device 50 can include one or more processors 51 and a memory 52. ​​The one or more processors 51 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.), and can be a single processor or multiple processors operatively connected. The memory 52 can include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 52 can store data 53 and instructions 54, which are executed by the processor 51 to cause the user computing device 50 to perform operations as described herein.

[0147] The computing device 50 may also include one or more input components for receiving user input. For example, the user input component may be a touch-sensitive component (e.g., a touch-sensitive display or touchpad) that is sensitive to the touch of a user input object (e.g., a finger or a stylus). The touch-sensitive component may be used to implement a virtual keyboard. Other example user input components include a microphone, a traditional keyboard, or other devices by which a user can provide user input.

[0148] The quantum computing system 80 may include one or more processors 81 (e.g., classical processor 304) and memory 82. The one or more processors 81 may be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.), and may be a single processor or multiple processors operatively connected. The memory 82 may include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 82 may store data 83 and instructions 84 that are executed by the processor 81 to cause the quantum computing system 80 to perform operations as described herein.

[0149] Quantum computing system 80 may also include a quantum system 85 for performing quantum computations. In some cases, quantum system 85 may be, include, or consist of quantum hardware 302, as described above with reference to Figure 3 described.

[0150] In some implementations, a quantum computing system 80 may include or otherwise be implemented by one or more server computing systems 60. Where the quantum computing system 80 includes multiple server computing devices, such server computing devices may operate according to a sequential computing architecture, a parallel computing architecture, or some combination thereof.

[0151] The third-party system 70 may include one or more processors 71 and a memory 72. The one or more processors 71 may be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.), and may be a single processor or a plurality of processors operatively connected. The memory 72 may include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, a flash memory device, a disk, etc., and combinations thereof. The memory 72 may store data 73 and instructions 74, which are executed by the processor 71 to cause the third-party system 70 to perform operations. In some implementations, the third-party system 70 includes one or more server computing devices or is otherwise implemented by one or more server computing devices.

[0152] The server computing system 60 may include one or more processors 61 and memory 62. The one or more processors 61 may be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, an FPGA, a controller, a microcontroller, etc.), and may be a single processor or multiple processors operatively connected. The memory 62 may include one or more non-transitory computer-readable storage media, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 62 may store data 63 and instructions 64, which are executed by the processor 61 to cause the server computing system 60 to perform operations. In some implementations, the server computing system 60 includes one or more server computing devices or is otherwise implemented by one or more server computing devices.

[0153] The network 49 can be any type of communication network (e.g., classical or quantum), such as a local area network (e.g., an intranet), a wide area network (e.g., the Internet), or some combination thereof, and can include any number of wired or wireless links. In general, communications over the network 49 can be carried via any type of wired or wireless connection using various communication protocols (e.g., TCP / IP, HTTP, SMTP, FTP), encodings or formats (e.g., HTML, XML), or protection schemes (e.g., VPN, secure HTTP, SSL).

[0154] Figure 6An example computing system that can be used to implement the present disclosure is shown. Other computing systems can also be used. For example, in some implementations, quantum computing system 80 can include server computing system 60, or vice versa. In some implementations, quantum computing system 80 can be communicatively coupled to computing device 50, third-party system 70, or server computing system 60 via network 49.

[0155] The digital, classical and / or quantum subject matter and implementations of digital function operations and quantum operations described in this specification may be implemented in digital electronic circuit systems, suitable quantum circuit systems or more generally quantum computing systems, in tangibly implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more thereof. The term "quantum computing system" may include, but is not limited to, a quantum computer / computing system, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0156] The implementation of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs (e.g., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or to control the operation of the data processing device). The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more quantum bits / qubit structures, or a combination of one or more thereof. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, the artificially generated propagated signal being generated to encode the digital and / or quantum information for transmission to an appropriate receiver device for execution by the data processing device.

[0157] The terms quantum information and quantum data refer to information or data carried by, stored in, or stored in a quantum system, wherein the smallest non-trivial system is a qubit (i.e., a system that defines a unit of quantum information). It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, e.g., having two or more levels. For example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational base state is identified by a base state and a first excited state, however, it should be understood that other arrangements in which computational states are identified by higher-level excited states (e.g., qubits) are also possible.

[0158] The term "data processing device" refers to digital and / or quantum data processing hardware and includes all types of devices, apparatuses, and machines for processing digital and / or quantum data, including, for example, a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The device may also be or further include specialized logic circuitry, such as an FPGA (field programmable gate array) or an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system. Specifically, a quantum simulator is a specialized quantum computer that lacks the capability to perform general-purpose quantum computations. In addition to the hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these.

[0159] A digital or classical computer program, which may also be referred to or described as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages; and it may be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or may be written in a quantum programming language such as QCL, Quipper, Cirq, etc.

[0160] A digital and / or quantum computer program may, but does not necessarily, correspond to a file in a file system. A program may be stored in a portion of a file that stores other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordination files (e.g., files storing one or more modules, subroutines, or code portions). A digital and / or quantum computer program may be deployed to execute on a digital computer or a quantum computer or on multiple digital and / or quantum computers, which are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system (e.g., qubit). Generally, a digital data communication network cannot transmit quantum data, while a quantum data communication network can transmit quantum data and digital data simultaneously.

[0161] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers operating in conjunction with one or more digital and / or quantum processors, where appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by dedicated logic circuitry (e.g., FPGAs or ASICs or quantum simulators), and devices can also be implemented as such dedicated logic circuitry, or by a combination of dedicated logic circuitry or quantum simulators with one or more programmable digital and / or quantum computers.

[0162] For a system of one or more digital and / or quantum computers or processors to be "configured to" or "operable to" perform specific operations or actions, it is meant that the system has installed thereon software, firmware, hardware, or a combination thereof that, in operation, causes the system to perform those operations or actions. For one or more digital and / or quantum computer programs to be configured to perform specific operations or actions, it is meant that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform those operations or actions. A quantum computer can receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform the operations or actions.

[0163] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general-purpose or special-purpose digital and / or quantum microprocessor, or both, or any other kind of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0164] Some example elements of a digital and / or quantum computer are a central processing unit for executing or performing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory may be supplemented by or incorporated into a dedicated logic circuit or quantum simulator. Generally, a digital and / or quantum computer will also include one or more mass storage devices for storing digital and / or quantum data, such as magnetic, magneto-optical, or optical disks, or quantum systems suitable for storing quantum information, or be operatively coupled to receive digital and / or quantum data from or transmit digital and / or quantum data to the mass storage devices, or both. However, a digital and / or quantum computer need not have such devices.

[0165] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media, and memory devices, including, for example, semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks; as well as quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data with high fidelity and efficiency for a long period of time, such as a light-matter interface that uses light for transmission, uses matter for storage, and preserves quantum characteristics such as superposition or quantum coherence of quantum data.

[0166] The control of the various systems or portions thereof described in this specification may be implemented in a digital and / or quantum computer program product comprising instructions stored on one or more tangible, non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification may be implemented as an apparatus, method, or electronic system, respectively, which may include one or more digital and / or quantum processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0167] Although this specification contains many specific implementation details, these details should not be interpreted as limitations on the scope of what may be claimed, but rather as descriptions of features that are unique to a particular implementation. Certain features described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented in multiple implementations, either individually or in any suitable subcombination. Furthermore, although features are described above as functioning in certain combinations and even initially claimed as such, one or more features from a claimed combination may, in some cases, be separated from that combination, and a claimed combination may be directed to a subcombination or a variation of a subcombination.

[0168] Similarly, although operations are depicted in a particular order in the accompanying drawings, this should not be construed as requiring that the operations be performed in the particular order shown or in a continuous order, or that all of the illustrated operations be performed, to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the implementations described above should not be construed as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged in multiple software products.

[0169] Specific implementations of the present subject matter have been described. Other implementations are within the scope of the appended claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order or sequential order shown to achieve the desired results. In some cases, multitasking and parallel processing can be advantageous.

[0170] Various aspects of the present disclosure have been described with respect to their illustrative implementations. Those of ordinary skill in the art may, after carefully reading this disclosure, conceive of numerous other implementations, modifications, and changes within the scope and spirit of the appended claims. Any and all features in the following claims may be combined or rearranged in any possible manner. Therefore, the scope of the present disclosure is by way of illustration and not limitation, and the present disclosure does not exclude such modifications, changes, and / or additions to the subject matter that would be readily understood by those of ordinary skill in the art. In addition, this document uses a list of example elements connected by conjunctions such as "and," "or," and "but" to describe terms. It should be understood that such conjunctions are provided only for the purpose of explanation. For example, a list connected by a specific conjunction such as "or" may refer to "at least one" or "any combination" of the example elements listed therein, wherein, unless otherwise indicated, "or" should be understood as "and / or." In addition, terms such as "based on" should be understood as "at least partially based on."

[0171] A person of ordinary skill in the art will understand, using the disclosure provided herein, that the elements of any claim, operation, or process discussed herein may be adjusted, rearranged, expanded, omitted, combined, or modified in various ways without departing from the scope of this disclosure. Some claims are described with letter references to claim elements for illustrative purposes and are not intended to be limiting. Letter references do not imply a specific order of operations. For example, letter identifiers such as (a), (b), (c) ..., (i), (ii), (iii) ..., etc. may be used to illustrate operations. Such identifiers are provided for the convenience of the reader and do not indicate a specific order of steps or operations. An operation shown by a list identifier (a), (i), etc. may be performed before, after, or in parallel with another operation shown by a list identifier (b), (ii), etc.

Claims

1. A method for modeling a classical physical system using a quantum computing system, comprising: encoding one or more first properties of a classical physical system comprising a network of oscillators in a state of one or more qubits; as well as The one or more qubits are used by one or more quantum computing devices to simulate the classical physical system.

2. The method of claim 1 , wherein the one or more first properties of the classical physical system include at least one of: a generalized momentum associated with at least one oscillator in the network of oscillators; a generalized velocity associated with at least one oscillator in the network of oscillators; a generalized displacement associated with at least one element in the network of elements; and A generalized location associated with at least one element in the network of elements.

3. The method of claim 1, wherein: simulating the classical physical system comprises performing a quantum computation; and The complexity of the quantum computation is logarithmic with respect to the size of the classical physical system.

4. The method of claim 1, wherein simulating the classical physical system comprises simulating the time evolution of a Hamiltonian.

5. The method of claim 4, wherein the Hamiltonian is configured such that a time evolution of the Hamiltonian corresponds to a time evolution of the one or more first properties of the classical physical system.

6. The method of claim 4, wherein the square of the Hamiltonian comprises a matrix encoding one or more second properties of the classical physical system.

7. The method of claim 6, wherein the matrix encoding the one or more second attributes comprises: A matrix product of a first matrix encoding one or more generalized masses associated with the classical physical system and a second matrix encoding one or more generalized spring constants associated with the classical physical system.

8. The method of claim 1, further comprising: measuring an observable associated with the one or more qubits to generate one or more measurement values; as well as One or more third properties of the classical physical system are estimated based at least in part on the one or more measurements.

9. The method of claim 8, wherein the one or more third properties include generalized kinetic energy associated with the classical physical system.

10. The method of claim 1, wherein: The classical physical system is a first classical physical system; The first classical physical system is a harmonic approximation of the second classical physical system; and The method further comprises: measuring an observable associated with the one or more qubits to generate one or more measurement values; as well as One or more third properties of the second classical physical system are estimated based at least in part on the one or more measurements. The method of claim 10 , wherein the harmonic approximation of the one or more third properties corresponds to the generalized kinetic energy of the first classical physical system.

12. A quantum computing system configured to perform operations comprising: encoding one or more first properties of a classical physical system comprising a network of oscillators in a state of one or more qubits; as well as The one or more qubits are used by one or more quantum computing devices to simulate the classical physical system.

13. The quantum computing system of claim 12, wherein the classical physical system is a harmonic approximation of a second classical physical system.

14. The quantum computing system of claim 12, wherein the one or more first properties of the classical physical system include at least one of: a generalized momentum associated with at least one oscillator in the network of oscillators; a generalized velocity associated with at least one oscillator in the network of oscillators; a generalized displacement associated with at least one element in the network of elements; and A generalized location associated with at least one element in the network of elements.

15. The quantum computing system of claim 12, wherein: simulating the classical physical system comprises performing a quantum computation; and The complexity of the quantum computation is logarithmic with respect to the size of the classical physical system.

16. The quantum computing system of claim 12, wherein simulating the classical physical system comprises simulating the time evolution of a Hamiltonian.

17. The quantum computing system of claim 16, wherein the Hamiltonian is configured such that a time evolution of the Hamiltonian corresponds to a time evolution of the one or more first properties of the classical physical system.

18. The quantum computing system of claim 12, further comprising: measuring an observable associated with the one or more qubits to generate one or more measurement values; as well as One or more third properties of the classical physical system are estimated based at least in part on the one or more measurements.

19. The quantum computing system of claim 18, wherein the one or more third properties include generalized kinetic energy associated with the classical physical system.

20. A method for modeling a quantum computing system using a classical computing system, comprising: mapping the quantum circuit to a classical physical system by one or more classical computing devices, the classical physical system comprising an oscillator network; simulating the classical physical system by the one or more classical computing devices; as well as A quantum computation result associated with the quantum circuit is determined by the one or more classical computing devices based on the simulation.

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