Modeling exponentially large classical physical systems using quantum computing

Abstract

According to embodiments, systems and methods are provided for simulating a classical physical system. In one example, the method may include initializing one or more qubits with an initial quantum state that encodes one or more physical properties of a classical physical system that includes an oscillator network. An exemplary method may include simulating the classical physical system with one or more quantum computing devices using the one or more qubits.

Description

[Technical Field]

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

[0002] CROSS-REFERENCE TO RELATED APPLICATIONS This application is based on and claims priority to U.S. Provisional Patent Application No. 63 / 486,537, filed February 23, 2023, the disclosure of which is incorporated herein by reference in its entirety for all purposes.

[0003] Quantum computing is a computing method that exploits quantum effects such as basis state superposition and quantum entanglement to perform certain calculations more efficiently than classical digital computers. In contrast to digital computers, which store and manipulate information in the form of bits (e.g., "1" or "0"), quantum computing systems can manipulate information using quantum bits ("qubits"). A qubit can refer to a quantum device that allows for the superposition of multiple states (e.g., data in both "0" and "1" states) and / or the superposition of multiple states of data itself. In conventional terminology, the superposition of "0" and "1" states in a quantum system can be expressed, for example, as a|0〉 + b|1〉. The "0" and "1" states of a digital computer are analogous to the |0〉 and |1〉 basis states of the qubit, respectively. Summary of the Invention

[0004] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the description that follows, or may be learned from the description, or may be learned by practice of the embodiments.

[0005] An exemplary aspect of the present disclosure provides an exemplary method. In some implementations, the exemplary method may include encoding one or more first properties of a classical physical system in the state of one or more qubits. In the exemplary method, the classical physical system may include an oscillator network. The exemplary method may include simulating the classical physical system by one or more quantum computing devices using the one or more qubits.

[0006] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description, drawings, and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate exemplary embodiments of the present disclosure and, together with the description, explain associated principles.

[0007] Detailed descriptions of embodiments directed to those skilled in the art are set forth herein with reference to the accompanying drawings. [Brief explanation of the drawings]

[0008] [Figure 1] 1 illustrates an exemplary system of a generalized harmonic oscillator, according to an exemplary embodiment of the present disclosure. [Figure 2] 1 illustrates an exemplary generalized waveform according to an exemplary aspect of the present disclosure. [Figure 3] 1 illustrates an example of a quantum computing system according to an exemplary aspect of the present disclosure. [Figure 4] FIG. 1 shows a flowchart diagram of an exemplary quantum computing method according to the present disclosure. [Figure 5] FIG. 1 shows a flowchart diagram of an exemplary quantum computing method according to an exemplary aspect of the present disclosure. [Figure 6] 1 illustrates a block diagram of an exemplary computing system according to an exemplary aspect of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0009] overview Exemplary embodiments according to some aspects of the present disclosure are directed to systems and methods for efficiently simulating classical physical systems using quantum computing. More specifically, systems and methods according to examples of the present disclosure can simulate a wide variety of classical physical systems (e.g., electromagnetic waves, elastic waves, molecular vibrations, etc.) that can be modeled using harmonic approximations. Harmonic approximations can include, for example, approximating a classical physical system as a system of harmonic oscillators, which can be mathematically similar to a system of interconnected masses and spring oscillators. In some examples, the systems and methods of the present disclosure can calculate some properties of a classical physical system in a time that is logarithmic with respect to the size of the classical physical system. In this way, for example, methods of the present disclosure can model some classical physical systems that are exponentially large with respect to the complexity of the quantum computation used to model the system.

[0010] An exemplary method may include, for example, initializing a plurality of qubits with an initial quantum state that encodes a physical property of a classical physical system at a first time. An exemplary method may include simulating the time evolution of a Hamiltonian to generate a second quantum state that encodes a physical property of the classical physical system at a second time. An exemplary method may include, for example, measuring an observable associated with the second quantum state, the observable corresponding to a property of an object associated with the classical physical system at the second time.

[0011] The initial quantum state can encode, for example, physical properties associated with one or more generalized momentum and generalized displacement (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 equivalent) to corresponding properties of a spring-and-mass oscillating system that corresponds 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. A spring of the spring-and-mass approximation can have, for example, a generalized spring constant. As a non-limiting illustrative example, an electrical circuit such as a series resistor-inductor-capacitor (RLC) circuit can generate an output waveform that corresponds to a harmonic oscillator, and the output waveform can be mathematically similar to a spring-and-mass oscillator. In such a circuit, charge can correspond to the generalized position of the oscillating mass, current can correspond to the generalized velocity, inductance can correspond to the generalized mass, etc.

[0012] In some examples, methods for encoding an initial quantum state may have complexity that is logarithmic with respect to the size of the classical physical system being encoded. For example, in some examples, the classical physical system or harmonic approximation may be characterized by sparse connections between generalized oscillatory masses. For example, each oscillatory mass may be connected to only d other oscillatory masses, where d may be a constant. Such a system may be referred to as a "d-sparse" system. In such examples, various efficient coding implementations are possible. Some exemplary encoding implementations are further described below and in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0013] In some examples, 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 examples, the Hamiltonian can be constructed in time that is logarithmic to the size of the classical physical system, and its evolution can be simulated in time that is logarithmic to the size of the classical physical system. For example, in some examples where the classical physical system (or harmonic approximation) is d-sparse, a unitary can be provided that receives an index j indicating a particular seismic mass and efficiently returns one or more of the following: a generalized mass of the seismic mass; one or more of d nonzero spring constants associated with the seismic mass; and one or more indices k associated with each seismic mass connected to the jth seismic mass by each spring having a nonzero spring constant. In such examples, a Hamiltonian can be efficiently constructed based on the output of the provided unitary, and its evolution can be efficiently simulated according to known methods. In some examples, the complexity of the simulation can be sublinear with respect to d. Details of exemplary embodiments are further described below and in US Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0014] In some examples, the measured property may be a global property of the entire classical physical system, and in some examples, may be measured in a time that is logarithmic with respect to the size of the classical physical system. For example, in some examples, the provided methods may efficiently estimate the generalized kinetic energy associated with the entire classical physical system. In some examples, the physical property of a subset of the classical physical system may be measured. For example, a subset of the generalized vibrational mass of a harmonic approximation may be identified, and the generalized kinetic energy of that subset may be efficiently measured. Other exemplary properties (e.g., potential energy, etc.) may also be measured.

[0015] In some examples, the classical physical system can be simulated multiple times to generate multiple measurements. In some examples, the number of times the classical physical system is simulated can vary depending on the desired accuracy.

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[0016] In some examples, systems and methods according to examples of the present disclosure may be BQP-complete, meaning that any problem in the class of bounded-error quantum polynomial-time (BQP) problems can be mapped to a harmonic approximation of a classical system of the present disclosure, and vice versa. Thus, for example, another BQP problem (e.g., another quantum algorithm) can be mapped to a harmonic approximation of a classical system and solved efficiently according to the provided systems and methods. Furthermore, the provided systems and methods may, in some examples, enable simulation of a BQP class of quantum algorithms using existing methods for simulating harmonic oscillator systems. For example, BQP quantum problems can be mapped to quantum computations of the present disclosure. The quantum computations of the present disclosure can be mapped to classical harmonic oscillator systems, which can then be simulated according to classical methods (e.g., on classical computing devices, etc.). While the complexity of such classical simulations may, in some examples, be exponentially larger than the complexity of the corresponding quantum computations, such classical simulations may still be useful in some examples (e.g., for small to moderate problem sizes, etc.). For example, classical computing systems may, in some instances, offer technical advantages over corresponding quantum computing systems, such as reduced noise, reduced computational cost (e.g., per bit or per qubit), etc. In such instances, classical simulation of BQP problems may be useful for some purposes (e.g., error rate benchmarking, etc.), even when classical simulation is associated with high computational complexity related to the size of the problem.

[0017] Exemplary embodiments according to certain aspects of the present disclosure may provide multiple technical effects and advantages, such as improvements to computing technologies (e.g., quantum computing technologies). For example, the systems and methods of the present disclosure may simulate classical physical systems more efficiently than alternative methods, such as classical computing simulations. For example, in some instances, the complexity associated with the systems and methods of the present disclosure may be logarithmic with respect to the size of the classical physical system being simulated. Thus, for example, the classical physical system may be exponentially larger than the complexity of the exemplary quantum computations of the present disclosure. In contrast, alternative methods (e.g., classical computing methods) may, in some instances, require at least linear complexity in relation to the size of the classical physical system. Thus, the alternative methods may be associated with exponentially larger complexity compared to the provided systems and methods.

[0018] In some examples, the exponential speedup associated with the systems and methods of the present disclosure may enable tasks that may be difficult and / or significantly nontrivial to perform in practice using classical computing systems. For example, a task with exponential complexity may be easy when the problem size is small, but become virtually impossible when the problem size becomes too large. For example, 256-bit RSA encryption can be decrypted ("cracked") in less than a minute using brute force calculations, while a problem only eight times larger (2048-bit RSA encryption) may take trillions or quadrillions of years to decrypt using current classical computers. In contrast, a system or method with non-exponential complexity may, in some examples, scale more efficiently to large problem sizes. As one illustrative example, a system or method may perform O(n 2) and can perform a small calculation in under a minute, the same system or method may perform eight times the calculation in about 64 minutes instead of quadrillions of years. Thus, the systems and methods of the present disclosure may, in some cases, make feasible classical computing that may be difficult to perform without quantum methods.

[0019] Although this disclosure describes some activities that can be performed in logarithmic time with respect to classical physical systems, systems and methods having non-logarithmic efficiency can be used without departing from the scope of this disclosure. For example, in some examples, quantum computations of the present disclosure can simulate classical physical systems having sizes that are polynomial to the complexity of the quantum computation without departing from the scope of this disclosure.

[0020] Referring now to the drawings, exemplary embodiments of the present disclosure will be described in further detail.

[0021] Harmonic approximation example FIG. 1 illustrates an example of a harmonic approximation, in which a classical physical system can be modeled as an oscillator network including multiple generalized seismic masses. The harmonic approximation can involve mapping the classical physical system to a corresponding system of harmonic oscillators, which can include, for example, a generalized seismic mass 102, a generalized spring 104, and a generalized wall 106. The generalized seismic 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 involving waves, etc.) can be modeled or approximated according to the described harmonic approximation. Further examples of harmonic approximations are described below with respect to FIG. 2.

[0022] The generalized seismic mass 102 may include, for example, a harmonic approximation of the seismic mass, where the generalized seismic mass 102 is configured to mathematically resemble (e.g., be equivalent to, be approximated by, etc.) components or properties of a classical physical system. The generalized seismic mass 102 may have multiple generalized properties, including, for example, without limitation, a generalized position, a generalized mass property, and a generalized momentum. Each generalized property may be configured, for example, to mathematically resemble (e.g., be equivalent to, be approximated by, etc.) a corresponding physical property of a classical physical system and to mathematically resemble (e.g., be equivalent to, etc.) a corresponding physical property of a seismic mass in a mass-and-spring harmonic oscillator system. For example, the generalized velocity may be the rate of change of the generalized position, the generalized momentum may be the product of the generalized mass and the generalized velocity, and the generalized mass of the seismic mass 102 may represent the amount of generalized force required to accelerate or decelerate the generalized seismic mass 102 at a particular velocity. As a non-limiting illustrative example, a series resistor-inductor-capacitor (RLC) circuit can generate an output waveform corresponding to a harmonic oscillator, where charge may correspond to a generalized position of a vibrating mass, current may correspond to a generalized velocity, inductance may correspond to a generalized mass, etc. As another example, parallel RLC circuits can generate different output waveforms corresponding to different harmonic oscillators, where magnetic flux linkage may correspond to a generalized position, voltage may correspond to a generalized velocity, capacitance may correspond to a generalized mass, and charge may correspond to a generalized momentum.

[0023] The generalized spring 104 may be, for example, a spring associated with a harmonic approximation, where the generalized spring 104 is configured to mathematically resemble (e.g., be equivalent to, be approximated by, etc.) components or properties of a classical physical system. The generalized spring 104 may have multiple generalized properties, including, for example, without limitation, 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 may correspond to the ratio of the force applied by the generalized spring 104 to the generalized displacement (e.g., a generalized distance compressed or stretched) of the generalized spring 104 relative to a generalized rest position. As a non-limiting illustrative example, a series resistor-inductor-capacitor (RLC) circuit can generate an output waveform corresponding to a harmonic oscillator, and the elastance may correspond to the generalized spring constant of the generalized spring 104, which directly corresponds to the RLC circuit.

[0024] The generalized wall 106 may correspond, for example, to a generalized immobile object (e.g., having a fixed generalized position) associated with a harmonic approximation, and one or more generalized seismic masses 102 may be attached to the generalized wall 106 via one or more generalized springs 104.

[0025] In general, the generalized seismic mass 102, the generalized spring 104, and the generalized wall 106 can possess any generalized physical properties (e.g., damping, driving force, etc.) that correspond to any physical properties that a corresponding classical oscillating system can possess. In some examples, the generalized physical properties of the generalized objects 102, 104, 106 can be derived from other generalized physical properties of the generalized objects 102, 104, 106 according to classical physical laws (e.g., Newton's laws). For example, the generalized kinetic energy of the generalized seismic mass 102 can be

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[0026] 2 illustrates an exemplary harmonic approximation of a waveform 208, where the generalized displacement of the waveform 208 with respect to time can be modeled or mapped as a system of harmonic oscillator(s). A variety of classical physical waveforms can be harmonically approximated in a similar (e.g., the same) manner as illustrated, including, but not limited to, sound waves, light waves, electromagnetic waves, etc.

[0027] In FIG. 2 , waveform 208 is depicted as a curve that can continuously oscillate over time 212 around center point 210. FIGS. 2A-2E illustrate an example generalized oscillator 102, generalized spring 104, and generalized wall 106 at five discrete time points associated with waveform 208. At an initial time point, as shown in FIG. 2A , generalized oscillator 102 and generalized spring 204 can have zero displacement relative to a generalized rest position 210 of generalized spring 204, and generalized oscillator 102 can have a positive generalized velocity, allowing generalized oscillator 102 to move toward the wall. In some examples, generalized rest position 210 can correspond to center point 210 of waveform 208. During the time period between FIG. 2A and FIG. 2B , generalized spring 204 can decelerate generalized seismic mass 102, and the generalized deceleration force can be proportional to the displacement (e.g., compression) of generalized spring 104 relative to rest position 210. At time 2B, the generalized velocity of the generalized seismic mass 102 may be zero relative to the generalized wall 106, and the generalized spring 204 may 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 seismic mass may have a similar (e.g., same) magnitude and opposite sign relative to time 2A. In other respects, some physical properties of the harmonic approximation (e.g., generalized displacement, generalized force, generalized acceleration, etc.) shown at time 2C may be similar (e.g., the same) compared to time 2A. At time 2D, some properties of the harmonic approximation (e.g., generalized displacement, generalized force, generalized acceleration, etc.) may have a similar (e.g., the same) magnitude and opposite sign relative to time 2B. In other respects, some physical properties of the harmonic approximation (e.g., zero generalized velocity, etc.) shown at time 2D may be similar (e.g., the same) compared to time 2B. FIG. 2E represents the state of the generalized oscillating mass 102, the generalized spring 104, and the generalized wall 106, which in some instances is identical to the state shown in FIG. 2A (e.g., the state when times 2A and 2E are exactly one period apart).

[0028] In some examples, the 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 examples, the harmonic approximation may be reversible. For example, in some examples, the waveform 208 may be approximated as one or more oscillators, and the system of one or more oscillators may be approximated as the generalized waveform 208.

[0029] While FIG. 2 shows a relatively simple example involving a sinusoidal waveform corresponding to a single generalized vibrating mass 102, more complex waves (e.g., multi-dimensional waves, waveforms with multiple higher harmonics, etc.) can be modeled in a similar (e.g., the same) manner.

[0030] Quantum Simulation Example In general, quantum simulation of a classical physical system may include initializing one or more qubits with quantum states that encode one or more properties of the physical system. In some examples, the properties of the classical physical system may be, include, or be associated with generalized properties of a harmonic approximation of the classical physical system. The quantum simulation may further include simulating the classical physical system with a quantum computing system using one or more qubits. In some examples, the simulating 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 examples, the observables may be associated with the final states of the one or more qubits after simulating the time evolution of the Hamiltonian.

[0031] Examples of quantum states that encode classical physical properties Quantum simulation of a classical physical system may include initializing one or more qubits with quantum states that encode one or more properties of the classical physical system (e.g., properties of a harmonic approximation of the classical physical system). In some examples, the one or more properties may include a generalized momentum or generalized velocity and a generalized displacement or generalized position associated with one or more generalized seismic masses 102.

[0032] In some examples, the initial quantum state encoding the generalized momentum or velocity and the generalized displacement or position can be described by the following equation:

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[0033] In some examples, E may be equal to K(t)+U(t), where K(t) may be the generalized kinetic energy associated with a classical physical system at time t (e.g., the generalized kinetic energy of the plurality of generalized seismic masses 102) and U(t) may be the generalized potential energy associated with the classical physical system at time t. Thus, for example, E may be the generalized total energy of a classical physical system, which may be constant over time.

[0034] In some instances, the initial quantum state can encode other properties of a classical physical system instead of, or in addition to, the generalized momentum and displacement. For example, the generalized kinetic energy K(t) at time t can be expressed as, for example,

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[0035] In some instances, the initial quantum state encodes the generalized momentum and the generalized displacement.

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[0036] Similarly, in some instances,

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[0037] In some instances,

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[0038] For example, in some examples, the initial state

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[0039] In some cases, the unitary U can be mapped as follows:

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[0040] For example, in some instances,

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[0041] The second preparatory action is to check whether the amplitude of |φ〉 is

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[0042] A second preparatory action can be performed, for example, by constructing a unitary using the following formula:

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[0043] In some examples, 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. Examples of alternative encodings are further described below after the description of the example Hamiltonian. (The example alternative encodings described below are based in part on the matrices described with respect to the example Hamiltonian below and are therefore more easily understood in view of the example Hamiltonian.)

[0044] Example Hamiltonian for simulating time evolution Quantum simulation of a classical physical system can include simulating the time evolution of a Hamiltonian using one or more qubits. In some examples, the Hamiltonian can be configured to simulate the time evolution of the classical physical system.

[0045] In some instances, an appropriate Hamiltonian can be efficiently determined from K and M. For example, Newton's equations for the dynamics of a classical harmonic oscillator can be written in matrix form as follows:

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[0046] In some examples, the Hamiltonian involving the matrix square root of A is BB † can be determined from the N × M matrix B that satisfies =A, where B †is a Hermitian adjoint of B. For example, the Hamiltonian H can be written as

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[0047] In some instances, we can consider two subspaces separately according to the blocks of H, and the Schrödinger equation induced by the Hamiltonian H can be rewritten as

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[0048] In some cases, the initial state

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[0049] Thus, for example, quantum computing systems

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[0050] In some instances, B

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[0051] In some examples, an appropriate Hamiltonian (e.g., Hamiltonian H) can be efficiently constructed in time logarithmic with respect to the size of the classical physical system. For example, in some examples, one or more compact representations of K and M can enable access to any component of K or M within a quasi-linear (e.g., constant) time with respect to the N generalized oscillating masses 1o2 associated with the classical physical system. In some examples, a compact representation of M can include a function that receives the oscillator index j as an input and generates the M components m j as an output, where m j ​may correspond to the mass of the jth generalized oscillatory mass 102 associated with the classical physical system. In some examples, such a function may be j In some examples, this access may be referred to as "oracle access." Similarly, in some cases (e.g., when K is d-sparse), a concise representation of K may be implemented as follows: jj and k jk In some examples (e.g., when K is d-sparse), a compact representation of K may take as input an oscillator index j and return one or more non-zero generalized spring constants k associated with the jth generalized seismic mass 102: jj and k jk If K is d-sparse, the Hamiltonian H may also be d-sparse. In some examples, the components of the Hamiltonian H

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[0052] The time evolution of the Hamiltonian H can be simulated according to existing methods. For example, in some cases, the exponent e -itH A method of applying an approximation of, for example, a Taylor series truncation, can be used. In some instances, such a method may be used to apply the parameter

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[0053] Examples of quantum observables Quantum simulation may include, for example, measuring one or more observables associated with a final state of one or more qubits (e.g., the final state after simulating the time evolution of a Hamiltonian). In some examples, the observables may be observables that encode one or more properties of a classical physical system. In some examples, 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 oscillatory masses 102). In some examples, the global or aggregate properties may be properties that cannot be efficiently determined using classical methods (e.g., in time that is sublinear compared to the size of the classical physical system).

[0054] For example, in some examples, the quantum simulation may include measuring an observable that encodes the generalized kinetic energy of the plurality of generalized oscillatory masses 102. In some examples, the plurality of generalized oscillatory masses 102 may include all of the generalized oscillatory masses 102 of a harmonic approximation of a classical physical system. In some examples, the plurality of generalized oscillatory masses 102 may include a subset (e.g., an exact subset) of the harmonic approximation of the classical physical system.

[0055] In some examples,

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[0056] Alternative examples of quantum states that encode classical physical properties Alternative encodings are possible for encoding properties of classical physical systems and simulating them using quantum algorithms (e.g., the time evolution of a Hamiltonian).

[0057] In one example encoding, the initial state

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[0058] The choice of quantum state encoding can in some instances be associated with a computational complexity trade-off, and the optimal choice of encoding may depend on the particular use case. For example, encoding using the Moore-Penrose pseudoinverse may be used to encode the initial state

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[0059] Example mapping for simulating general-purpose quantum circuits In some examples, any arbitrary quantum circuit can be mapped to a quantum simulation of the present disclosure. In some examples, this generic mapping can demonstrate that the provided systems and methods are BQP-complete. In some examples, the provided quantum simulation can be mapped to a classical physical system (or a harmonic approximation thereof) by applying the mapping described herein in reverse. Thus, for example, any arbitrary quantum circuit can be mapped to a classical physical system of harmonic oscillators. In some examples, the classical physical system can be simulated (e.g., according to classical methods), and the resulting classical physical state can be mapped to the provided quantum state, which can then be mapped to the final quantum state of the quantum circuit. In this way, for example, classical methods for simulating harmonic systems can be used to simulate any arbitrary quantum circuit.

[0060] In some examples, any quantum circuit can be mapped to the quantum simulations of the present disclosure by mapping the circuit from the universe to a number of gates, and then mapping the gates from the universe to the quantum simulations of the present disclosure. The universe of quantum gates can be, for example, {H,T}, where H can be a one-qubit Hadamar gate and T can be a three-qubit Toffoli gate. In some examples, a number of L gates U operating on n qubits can be mapped to the quantum simulations of the present disclosure. L ...U1 can be mapped to a system of coupled oscillators, where the number of oscillators, N, is (L+1)2 n+1 and each oscillator may have a generalized mass of 1, so that the matrix of generalized masses M is

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[0061] In some instances, the initial state of such a 5-sparse system is

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[0062] Furthermore, a sparse coupled oscillator system mapped in this manner can be mapped to a classical physical system (or a harmonic approximation thereof) by applying one or more of the mappings described herein in reverse. In this way, for example, any quantum circuit can be mapped to a classical physical system of harmonic oscillators. In some examples, 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, which can then be mapped to the final quantum state of any quantum circuit. In this way, for example, any quantum circuit can be simulated using classical methods for simulating harmonic systems. Additional exemplary implementation details are further described in U.S. Provisional Application No. 63 / 486,537, which is incorporated herein by reference.

[0063] Examples of quantum computing systems 3 illustrates an exemplary quantum computing system 300. The exemplary system 300 is an example of a system on one or more classical computers and / or quantum computing devices at one or more locations that can implement the systems, components, and techniques described below. Using the disclosure provided herein, one skilled in the art will understand that another quantum computing device or system can be used without departing from the scope of the present disclosure.

[0064] 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, control device(s) 312, and readout device(s) 314 (e.g., readout resonator(s)). Quantum system 310 may include one or more multi-level quantum subsystems, such as a register of qubits. In some implementations, the multi-level quantum subsystem may include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, etc.

[0065] The type of multi-level quantum subsystem utilized by system 300 may vary. For example, in some cases it may be convenient to include one or more readout device(s) 314 attached to one or more superconducting qubits, e.g., transmon, fluxon, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices, or superconducting cavities (e.g., which may prepare states without the need for qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0066] Quantum circuits may be constructed and applied to a register of qubits included in quantum system 310 via multiple control lines coupled to one or more control devices 312. Exemplary control devices 312 operating on a register of qubits may be used to implement a quantum circuit having a quantum gate or multiple quantum gates (e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled phase gates, T-gates, multi-qubit quantum gates, coupler quantum gates, etc.). One or more control devices 312 may be configured to operate on quantum system 310 with one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystem may be a superconducting qubit, and control device 312 may be configured to provide control pulses to the control lines to generate magnetic fields that tune the frequency of the qubit.

[0067] The quantum hardware 302 may further include a readout device 314 (e.g., a readout resonator). Measurements 308 obtained via the measurement device may be provided to a classical processor 304 for processing and analysis. In some implementations, the quantum hardware 302 may include quantum circuits, and the control device(s) 312 and readout device(s) 314 may implement one or more quantum logic gates that operate on the quantum system 302 using physical control parameters (e.g., microwave pulses) transmitted via wires included in the quantum hardware 302. Further examples of control devices include arbitrary waveform generators, in which a DAC (digital-to-analog converter) creates a signal.

[0068] The readout device(s) 314 may be configured to perform quantum measurements on the quantum system 310 and transmit the measurement results 308 to the classical processor 304. Additionally, the quantum hardware 302 may be configured to receive data from the classical processor 304 specifying physical control qubit parameter values ​​306. The quantum hardware 302 may use the received physical control qubit parameter values ​​306 to update the action of the control device(s) 312 and readout device(s) 314 on the quantum system 310. For example, the quantum hardware 302 may receive data specifying new values ​​representing voltage magnitudes of one or more DACs included in the control device 312 and may update the action of the DACs on the quantum system 310 accordingly. The classical processor 304 may be configured to initialize the quantum system 310 to an initial quantum state, for example, by transmitting data specifying a set of initial parameters 306 to the quantum hardware 302.

[0069] In some implementations, the readout device(s) 314 can measure the state of an element (e.g., a qubit) of a quantum system, such as a qubit, by utilizing the difference in impedance for the |0> and |1> states of the element. For example, due to the nonlinearity of the qubit, the resonant frequency of the readout resonator can be different when the qubit is in the |0> or |1> state. Thus, microwave pulses reflected from the readout device 314 convey amplitude and phase shifts that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 314 to prevent microwave propagation at the qubit frequency.

[0070] In some implementations, quantum system 310 may include a plurality of qubits 320 arranged, for example, in a two-dimensional grid 322. For clarity, two-dimensional grid 322 shown in FIG. 1 is square and includes 16 qubits, although in some implementations, system 310 may include fewer or more qubits. In some implementations, the plurality of qubits 320 may interact with one another through a plurality of qubit couplers, such as qubit coupler 324. The qubit coupler may define nearest-neighbor interactions between the plurality of qubits 320. In some implementations, the strength of the plurality of qubit couplers is a tunable parameter. In some cases, the plurality of qubit couplers included in quantum computing system 300 may be couplers with fixed coupling strengths. 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 a computation being performed by system 300. A measurement qubit is a qubit that may be used to determine the outcome of a computation performed by a data qubit. That is, during a computation, the unknown state of the data qubit is transferred to the measurement qubit using an appropriate physical operation and measured by an appropriate measurement operation performed on the measurement qubit.

[0071] In some embodiments, each qubit in the plurality of qubits 320 can operate using a respective operating frequency, such as an idle frequency, an interaction frequency, a readout frequency, and / or a reset frequency. The operating frequency can be different for each qubit. For example, each qubit can idle at a different operating frequency. The operating frequency of the qubits 320 can be chosen before calculations are performed by the calibration system. Some operating frequencies are better than others. One metric for assessing how good a particular operating frequency is for a particular qubit is the energy relaxation time (T1) of the qubit at that frequency. As the energy relaxation time decreases, quantum computation errors can increase.

[0072] In various implementations, the exemplary system 300 can be implemented as a client device, a server device, or both. The exemplary system 300 can be implemented as part of a distributed computing system. The exemplary system 300 can be implemented together with other exemplary systems, which can be the same or different. The exemplary system 300 can be implemented in a server farm or other facility that operates multiple computing systems to provide computational services to or on behalf of multiple client systems. Advantageously, techniques according to exemplary aspects of the present disclosure can provide improved calibration and maintenance of computing equipment, increased service uptime, reduced failure rates, etc.

[0073] Example of how to 4 illustrates a flowchart of an exemplary method for simulating a classical physical system, according to an exemplary embodiment of the present disclosure. While FIG. 4 depicts steps occurring in a particular order for purposes of illustration and explanation, the methods of the present disclosure are not limited to the particularly depicted order or arrangement. Various steps of method 400 may be omitted, rearranged, combined, and / or adapted in various ways without departing from the scope of the present disclosure.

[0074] At 402, the example method 400 may include encoding one or more first properties of a classical physical system in the states of one or more qubits. In some examples, the first property may be, include, correspond to, or otherwise be associated with a generalized property of the generalized seismic mass 102 or the generalized spring 104. In some cases, the first property may be or include a generalized momentum, a generalized displacement, a generalized mass, a generalized spring constant, or a generalized velocity. In some examples, encoding the first property may include modeling the classical physical system as a harmonic approximation and encoding the properties of the harmonic approximation in the states of one or more qubits. In some examples, the example method 400 may include, at 402, using one or more systems or performing one or more activities described with respect to FIGS. 1-3 .

[0075] At 404, the example method 400 may include simulating a time evolution of a Hamiltonian, where the Hamiltonian is configured such that the time evolution of the Hamiltonian corresponds to the time evolution of one or more first properties of the classical physical system. In some examples, a square quadrant of the Hamiltonian may include a matrix encoding one or more second properties of the classical physical system. In some examples, 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 examples, the Hamiltonian may be, may include, or may be included in the Hamiltonian H described above. In some examples, simulating the classical physical system may include performing a quantum algorithm having a complexity that is logarithmic with respect to the size of the classical physical system or the size of the harmonic approximation. In some examples, the example method 400 may include, at 404, using one or more systems or performing one or more activities described with respect to FIGS. 1-3.

[0076] At 406, the example method 400 may include measuring an observable associated with one or more qubits to generate one or more measurements. In some examples, the observable may be, include, encode, or otherwise correspond to kinetic energy associated with a classical physical system. In some examples, 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 examples, the observable may be, include, encode, or otherwise correspond to generalized kinetic energy of a subset of the plurality of generalized seismic masses 102 associated with a harmonic approximation of a classical physical system. In some examples, the example method 400 may include, at 406, using one or more systems or performing one or more activities described with respect to FIGS. 1-3 .

[0077] 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 examples, estimating the third property may be, may include, or may be included in a reliable amplitude estimation. In some examples, the example method 400 may include, at 408, using one or more systems or performing one or more activities described with respect to FIGS. 1-3.

[0078] FIG. 5 illustrates an exemplary method 500 for performing quantum computing using a quantum circuit according to exemplary aspects of the present disclosure. For example, the quantum circuit may, in some examples, include, be included in, or be implemented by quantum system 310. While FIG. 5 depicts steps occurring in a particular order for purposes of illustration and explanation, the methods of the present disclosure are not limited to the particularly depicted order or arrangement. Various steps of method 500 may be omitted, reordered, 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 quantum computing system 300 of FIG. 3.

[0079] 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), etc. Data indicative of a quantum circuit may include, for example, a circuit design, a circuit schematic, one or more unitary matrices, software code (e.g., quantum software code in a quantum computing language), etc.

[0080] At 504, the exemplary method 500 may include preparing one or more qubits in a known quantum state. Preparing the one or more qubits in a known quantum state may include, for example, preparing the one or more qubits in a known basis state (e.g., by manipulating multiple qubits such that qubits characterized by a particular basis state, such as |0〉 or |1〉, are separable (e.g., physically separated, separately identified, etc.) from qubits not characterized by that basis state). Preparing the one or more qubits in a known quantum state may include, for example, performing quantum gating using the control device 312 to generate the known multi-qubit basis state. Preparing the one or more qubits may include using the control device 312 in the manner described with respect to FIG. 3 .

[0081] 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 examples, the control device 312 can be used to implement a quantum gate or a quantum circuit having multiple quantum gates (e.g., 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.) in the manner described with respect to FIG.

[0082] At 508, example method 500 may include measuring the state of at least one of the one or more qubits using a readout apparatus. The readout apparatus may be, for example, readout device 314, and step 506 may, in some examples, be performed in the manner described with respect to FIG.

[0083] 6 shows a block diagram of an exemplary computing system 5 capable of performing aspects of an exemplary embodiment of the present disclosure. System 5 includes a computing device 50, a server computing system 60, and a third-party system 70, which are communicatively connected via a network 49. System 5 also includes a quantum computing system 80 that is communicatively coupled to the server computing system.

[0084] The computing device 50 may be any type of computing device (e.g., a classical computing device), such as a mobile computing device (e.g., a smartphone or tablet), a personal computing device (e.g., a laptop or desktop), 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 may be a client computing device or a server computing device. The computing device 50 may include one or more processors 51 and memory 52. ​​The one or more processors 51 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 52 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 52 may store data 53 and instructions 54, which are executed by the processor 51 to cause the user computing device 50 to perform the operations described herein.

[0085] Computing device 50 may also include one or more input components that receive user input. For example, the user input component may be a touch-sensitive component (e.g., a touch-sensitive display screen or touchpad) that senses the touch of a user input object (e.g., a finger or stylus). The touch-sensitive component may function to implement a virtual keyboard. Another exemplary user input component includes a microphone, a conventional keyboard, or another means by which a user can provide user input.

[0086] The quantum computing system 80 may include one or more processors 81 (e.g., classical processor(s) 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 operatively connected processors. 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, which are executed by the processor 81 to cause the quantum computing system 80 to perform the operations described herein.

[0087] Quantum computing system 80 may also include a quantum system 85 for performing quantum computations. In some examples, quantum system 85 may be, include, or be included in quantum hardware 302 described above with reference to FIG. 3.

[0088] In some implementations, quantum computing system 80 may include, or may otherwise be implemented by, one or more server computing devices 60. When server 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.

[0089] The third-party system 70 may include one or more processors 71 and 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 multiple operatively connected processors. The memory 72 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 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 embodiments, the third-party computing system 70 includes or is otherwise implemented by one or more server computing devices.

[0090] 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 operatively connected processors. 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 embodiments, the server computing system 60 includes or is otherwise implemented by one or more server computing devices.

[0091] Network 49 may be any type of communications network, such as a local area network (e.g., an intranet), a wide area network (e.g., the Internet), or any combination thereof, and may include any number of wired or wireless links. In general, communications over network 49 may occur over any type of wired or wireless connection, using a wide variety of communications protocols (e.g., TCP / IP, HTTP, SMTP, FTP), encodings or formats (e.g., HTML, XML), or protection schemes (e.g., VPN, Secure HTTP, SSL).

[0092] 6 illustrates one exemplary computing system that can be used to implement the present disclosure. Other computing systems can be used as well. 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.

[0093] The digital, classical, and / or quantum subject matter, and digital functional and quantum operational embodiments described herein may be implemented in digital electronic circuitry, suitable quantum circuitry, or, more generally, in a quantum computing system, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computing hardware including the structures disclosed herein and their equivalents, or in one or more combinations 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.

[0094] Digital and / or quantum implementations of the subject matter described herein can be implemented as one or more digital and / or quantum computer programs (e.g., as one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by or to control the operation of a data processing apparatus). 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 qubits / qubit structures, or a combination of one or more thereof. Alternatively or additionally, the program instructions can be encoded in an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information that is generated to encode the digital and / or quantum information for transmission to a suitable receiving device for execution by a data processing apparatus.

[0095] The terms "quantum information and quantum data" refer to information or data carried by, held, or stored in a quantum system, with the smallest nontrivial system being a qubit (i.e., a system defining a unit of quantum information). The term "qubit" is understood to encompass all quantum systems that can be suitably approximated as two-level systems in the corresponding context. Such quantum systems may include, for example, multi-level systems having two or more levels. By way of example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, a computational basis state is specified using a ground state and a first excited state, although it is understood that alternative setups are possible in which a computational state is specified using a higher-level excited state (e.g., a qubit).

[0096] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all types of apparatus, devices, and machines for processing digital and / or quantum data, including, by way of example, a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, as well as combinations thereof. An apparatus may also be or include special-purpose 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 apparatus designed to simulate or generate information about a specific quantum system. Specifically, a quantum simulator is a special-purpose quantum computer that does not have the capability to perform general-purpose quantum computation. An apparatus may also optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs (e.g., code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these).

[0097] A digital or classical computer program, which may also be referred to as or described as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including a compiled or interpreted language, or a declarative or procedural language, and may be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other apparatus suitable for use in a digital computing environment. A quantum computer program, which may also be referred to as or described as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including a compiled or interpreted language, or a declarative or procedural language, and may be converted to or written in a suitable quantum programming language (e.g., QCL, Quipper, Cirq, etc.).

[0098] A digital and / or quantum computer program may correspond to a file in a file system, but this is not necessarily the case. A program can be stored in part of a file holding another program 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 associated files (e.g., files storing one or more modules, subprograms, or portions of code). A digital and / or quantum computer program can be deployed to run on one digital and / or quantum computer, or on multiple digital and / or quantum computers, 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 quantum systems (e.g., qubits). Generally, digital data communication networks cannot transmit quantum data, but quantum data communication networks can transmit both quantum data and digital data.

[0099] The processes and logic flows described herein may be performed by one or more programmable digital and / or quantum computers operating with one or more digital and / or quantum processors, and, where appropriate, executing one or more digital and / or quantum computer programs to perform functions by performing operations on input digital and quantum data to generate outputs. The processes and logic flows may also be performed, and the apparatus may be implemented, as special purpose logic circuits (e.g., FPGAs or ASICs) or quantum simulators, or by a combination of special purpose logic circuits or quantum simulators with one or more programmed digital and / or quantum computers.

[0100] When one or more digital and / or quantum computers or processors are "configured" or "operable" to perform particular operations or actions, it means that, in operation, the system has installed thereon software, firmware, hardware, or a combination thereof that causes the operation or action to be performed. When one or more digital and / or quantum computer programs are configured to perform particular operations or actions, it means that the one or more programs contain instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer may receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform an operation or action.

[0101] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general-purpose or a special-purpose digital and / or quantum microprocessor, or both, or any other kind of central digital and / or quantum processing unit. Generally, a digital and / or quantum processing unit receives 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.

[0102] Some exemplary elements of a digital and / or quantum computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory can be supplemented by or incorporated into special purpose logic circuitry or a quantum simulator. Generally, a digital and / or quantum computer also includes one or more mass storage devices (e.g., magnetic, magneto-optical, or optical disks, or quantum systems suitable for storing quantum information) for storing digital and / or quantum data, or is operably coupled to receive digital and / or quantum data therefrom, transfer digital and / or quantum data thereto, or both. However, a digital and / or quantum computer need not have such devices.

[0103] 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 memories, media, and memory devices, including, by way of example, semiconductor memory devices (e.g., EPROM, EEPROM, and flash memory devices), magnetic disks (e.g., internal hard disks or removable disks), magneto-optical disks, CD-ROM and DVD-ROM disks, and quantum systems (e.g., trapped atoms or electrons). Quantum memory is understood to be a device capable of storing quantum data with high fidelity and efficiency for long periods of time, for example, a light-matter interface that uses light for transmission and matter for storage and preservation of the quantum characteristics of the quantum data, such as superposition or quantum coherence.

[0104] Control of the various systems described herein, or portions thereof, may be embodied in a digital and / or quantum computer program product stored on one or more tangible, non-transitory, machine-readable storage media and including instructions executable on one or more digital and / or quantum processing devices. The systems described herein, or portions thereof, may each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and a memory for storing executable instructions for performing the operations described herein.

[0105] While this specification contains details of many specific embodiments, these should not be construed as limiting the scope of what may be claimed, but rather as descriptions of features that may be inherent in particular embodiments. Certain features described herein in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Furthermore, while features may be described above as acting in a particular combination and may initially be claimed as such, one or more features from a claimed combination can, in some cases, be deleted from that combination, and the claimed combination may be directed to a subcombination or a variation of the subcombination.

[0106] Similarly, although operations are shown in the figures in a particular order, this should not be understood as requiring that such operations be performed in the particular order or sequence shown, or that all of the operations shown be performed, to achieve desirable results. In certain situations, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the above-described embodiments should not be understood as requiring such separation in all embodiments, and it should be understood that the program components and systems described generally can be integrated together in a single software product or packaged in multiple software products.

[0107] Specific implementations of the present subject matter have been described. Alternative implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

[0108] Aspects of the present disclosure have been described in terms of exemplary embodiments thereof. Those skilled in the art will be able to devise, from a consideration of this disclosure, numerous alternative embodiments, modifications, or variations that fall within the scope and spirit of the appended claims. Any features of the following claims can be combined or rearranged in any possible manner. Accordingly, the scope of the present disclosure is exemplary, not limiting, and disclosure of the present subject matter is not intended to exclude the inclusion of such modifications, variations, or additions to the subject matter as would be readily apparent to one of ordinary skill in the art. Furthermore, terms are described herein using lists of exemplary elements joined by conjunctions such as "and," "or," and "but." It should be understood that such conjunctions are provided for illustrative purposes only. A list joined by a particular conjunction, such as "or," can refer, for example, to "at least one of" or "any combination of" the exemplary elements listed therein, with "or" being understood as "and / or" unless otherwise indicated. Additionally, terms such as "based on" should be understood as "based at least in part on."

[0109] Those skilled in the art will understand, using the disclosure provided herein, that elements of any of the claims, operations, and methods described herein may be adapted, rearranged, extended, omitted, combined, or modified in various ways without departing from the scope of the present disclosure. Some of the claims are described using reference letters to elements of the claims for exemplary, illustrative purposes and not intended to be limiting. The reference letters do not imply a particular order of operations. For example, operations may be described using letter identifiers such as (a), (b), (c), ..., (i), (ii), (iii), .... Such identifiers are provided for the convenience of the reader and do not indicate a particular order of steps or operations. Operations indicated by list identifiers such as (a), (i), etc. may occur before, after, or in parallel with other operations indicated by list identifiers such as (b), (ii), etc.

Claims

1. 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 in the state of one or more qubits, the classical physical system comprising an oscillator network; simulating the classical physical system with one or more quantum computing devices using the one or more qubits; and A method comprising:

2. The one or more first properties of the classical physical system are: a generalized momentum associated with at least one oscillator of the oscillator network; a generalized velocity associated with at least one oscillator of the oscillator network; a generalized displacement associated with at least one oscillator of the oscillator network; a generalized position associated with at least one oscillator of the oscillator network; The method of claim 1 , comprising at least one of:

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

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 the time evolution of the Hamiltonian corresponds to the 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 that encodes one or more second properties of the classical physical system.

7. 7. The method of claim 6, wherein the matrix encoding the one or more second properties 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. measuring an observable associated with the one or more qubits to generate one or more measurements; estimating one or more third properties of the classical physical system based at least in part on the one or more measurements; and The method of claim 1 further comprising:

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

10. the classical physical system is a first classical physical system; the first classical physical system is a harmonic approximation of a second classical physical system; The method comprises: measuring an observable associated with the one or more qubits to generate one or more measurements; estimating one or more third properties of the second classical physical system based at least in part on the one or more measurements; and The method of claim 1 further comprising:

11. The method of claim 10 , wherein the harmonic approximation of the one or more third properties corresponds to a generalized kinetic energy of the first classical physical system.

12. 1. A quantum computing system configured to perform an operation, the operation comprising: encoding one or more first properties of a classical physical system in the state of one or more qubits, the classical physical system comprising an oscillator network; simulating the classical physical system with one or more quantum computing devices using the one or more qubits; and 1. A quantum computing system comprising:

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 one or more first properties of the classical physical system are: a generalized momentum associated with at least one oscillator of the oscillator network; a generalized velocity associated with at least one oscillator of the oscillator network; a generalized displacement associated with at least one oscillator of the oscillator network; a generalized position associated with at least one oscillator of the oscillator network; 13. The quantum computing system of claim 12, comprising at least one of:

15. simulating the classical physical system includes performing a quantum computation; the complexity of the quantum computation is logarithmic with respect to the size of the classical physical system; 13. The quantum computing system of claim 12.

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

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

18. measuring an observable associated with the one or more qubits to generate one or more measurements; estimating one or more third properties of the classical physical system based at least in part on the one or more measurements; and 13. The quantum computing system of claim 12, further comprising:

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

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

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