Devices, computer implementations, and quantum mechanical devices for optimizing, monitoring, and controlling actual physical systems.

Abstract

Methods and apparatuses are provided for performing at least one of optimizing, monitoring, and controlling an actual physical system. Data representing the physical system is acquired and used to generate one or more Hamiltonians. These Hamiltonians are used in the quantum signal processing circuits of a quantum computer to simulate the imaginary time evolution of a Thermal Pure Quantum (TPQ) state of a set of qubits representing the Gibbs state of the system. Optionally, a Bayesian machine can be trained based on the evolution of the TPQ state to predict the operation of the physical system over time. Classical shadow tomography is then used to provide a classical representation of the TPQ state to a classical computer to facilitate classical optimization or control of the actual physical system.

Description

Technical Field

[0001] <Cross - Reference to Related Applications> This application claims the benefit of the filing dates of UK Patent Application No. GB2208524.5 filed on June 10, 2022 and US Patent Application No. 18 / 201,410 filed on May 24, 2023. The entire contents of these applications are incorporated herein by reference.

[0002] The present invention relates to an apparatus and a method for performing at least one of optimizing, monitoring, and controlling an actual physical system. Optionally, the apparatus and method are implemented using quantum computing resources. Further, the present invention relates to a software product executable on an apparatus for performing the aforementioned method.

Background Art

[0003] It is known to use a mathematical model of an actual physical system for at least one of monitoring, optimizing, and controlling the actual physical system. In many cases, since an actual physical system can only be approximately defined by studying its components, it is necessary to collect data representing the operation of the actual physical system in order to more accurately define the operating characteristics of the actual physical system. Further, systems often have various operating modes, and their operating characteristics vary depending on the operating mode in use. Therefore, in control systems of actual physical systems such as aircraft, chemical production facilities, manufacturing facilities, nuclear power plants, and financial systems, it is necessary to represent the actual physical system by a mathematical model before accurately monitoring, optimizing, and controlling the actual physical system. However, collecting a large amount of data from an actual physical system is often costly and time - consuming and is infeasible in many situations.

[0004] Various types of mathematical models are feasible and can be generated, for example, using a Bayesian machine. However, there are various feasible implementations of Bayesian machines, such as Boltzmann machines, Born machines, and Ising machines. However, deriving the parameters of a mathematical model from data collected from the operation of an actual physical system is not an easy computational task, especially when the actual physical systems are very complex in their respective operating methods.

[0005] The present disclosure aims to provide an improved apparatus for generating one or more mathematical models that describe a given actual physical system from data representing the operation of the given actual physical system. Here, the one or more mathematical models can be used for at least one of monitoring, optimizing, and controlling a given actual physical system.

Summary of the Invention

Means for Solving the Problems

[0006] According to a first aspect, there is provided an apparatus for performing at least one of optimizing, monitoring, and controlling an actual physical system. The apparatus comprises a hybrid computing configuration including one or more classical computers connected to one or more quantum computers, and the one or more quantum computers are configured to execute one or more quantum circuits configured using the one or more classical computers. The apparatus is configured to perform the following operations in use: (i) Obtain data representing the operation of the actual physical system, (ii) Generate one or more Hamiltonians using the data, define at least one quantum circuit executable on one or more of the quantum computers, and the observed value (M) obtained from the execution of at least one of the quantum circuits represents Gibbs states. The generation of the observed value (M) includes the use of a combination of Thermal Pure Quantum (TPQ) states and classical shadow tomography. (iii) Train a computational machine based on the observed value (M) representing the Gibbs state. (iv) Generate a mathematical model representing the actual physical system including the computational machine. (v) Apply the mathematical model to the data obtained from the actual physical system to (a) generate a monitoring output representing the operating state or operating conditions of the actual physical system, and (b) one or more inputs to an optimization function applied to the mathematical model to generate values of parameters used to operate the actual physical system in a more optimized manner, and (c) one or more inputs to a control function applied to the mathematical model to generate values of parameters used to control the operation of the actual physical system. Generate at least one of the above.

[0007] The implementation of the devices and methods described herein has the advantage that, by using a combination of Thermal Pure Quantum (TPQ) states and classical shadow tomography, observables (M) can be obtained from at least one quantum circuit generated from Hamiltonians that describe an actual physical system, where the observables (M), i.e., the expected values, generated from the execution of at least one quantum circuit represent theoretical Gibbs states. By using Thermal Pure Quantum (TPQ) states and classical shadow tomography, at least one quantum circuit uses fewer qubits, fewer observables (M), and becomes shallower when providing computational results for generating a mathematical model. Due to such advantages, the device can train, for example, a computational machine implemented as a quantum Boltzmann machine in a much more computationally efficient way.

[0008] Since observable values are measurable properties of a physical system, for example, when measuring free energy, pressure, and temperature, M = 3.

[0009] Optionally, the device is configured to generate the Thermal Pure Quantum (TPQ) state using imaginary time evolution of an n - qubit random state |φ> using TIFF2025522402000002.tif722.

[0010] Optionally, the device is configured to construct an efficient classical representation of these TPQ states from the results of a randomized subset of the observables (M) using the classical shadow tomography.

[0011] Optionally, the device is the observable of a single prepared TPQ state It is configured to estimate M Gibbs state expectation values using TIFF2025522402000003.tif620.

[0012] According to a second aspect, a computer-implemented method is provided that uses an apparatus for performing at least one of optimizing, monitoring, and controlling an actual physical system. The apparatus comprises a hybrid computing configuration including one or more classical computers connected to one or more quantum computers, and one or more of the quantum computers are configured to execute one or more quantum circuits configured using one or more of the classical computers. The computer-implemented method includes the following operations: (i) Obtaining data representing the operation of the actual physical system, (ii) Using the data to generate one or more Hamiltonians, defining at least one quantum circuit executable on one or more of the quantum computers, and observations (M) obtained from the execution of at least one of the quantum circuits represent Gibbs states (i.e., statistical distributions). The generation of the observations (M) includes the use of a combination of Thermal Pure Quantum (TPQ) states and classical shadow tomography, (iii) Training a computational machine based on the observations (M) representing the Gibbs states, (iv) Generating a mathematical model representing the actual physical system including the computational machine, (v) Applying the mathematical model to data obtained from the actual physical system to (a) generate a monitoring output representing the operating state or operating conditions of the actual physical system, and (b) generate one or more inputs to an optimization function applied to the mathematical model to generate values of parameters used to operate the actual physical system in a more optimized manner, (c) one or more inputs to the control function applied to the mathematical model to generate the values of the parameters used to control the operation of the actual physical system, generate at least one of.

[0013] Optionally, the method uses the apparatus to perform an imaginary time evolution of the random state |φ> of n qubits using TIFF2025522402000004.tif722 to generate the thermal pure quantum (TPQ) state.

[0014] Optionally, the method includes using the classical shadow tomography to construct an efficient classical representation of these TPQ states from the results of a randomized subset (M) of the observables.

[0015] Optionally, the method estimates M Gibbs state expectation values using the observables of a single prepared TPQ state using TIFF2025522402000005.tif620.

[0016] According to a third aspect, there is provided a software product executable on the computing system of the apparatus of the first aspect for implementing the method of the second aspect.

[0017] According to a fourth aspect, there is provided an actual physical system coupled to the apparatus of the first aspect, the apparatus being configured to use the method of the second aspect to perform at least one of monitoring, optimizing, and controlling the operation of the actual physical system.

[0018] According to a fifth aspect, there is provided a quantum circuit configured to calculate one or more observables (i.e., expected values) representing a Gibbs state, the quantum circuit being generated from one or more Hamiltonians representing an actual physical system, the observables being arranged to train a quantum Boltzmann machine for use in at least one of monitoring, controlling, and optimizing the actual physical system, and the observables being generated using a combination of a thermal pure quantum (TPQ) state implemented in the quantum circuit and classical shadow tomography.

[0019] According to a sixth aspect, there is provided a quantum mechanical apparatus comprising one or more classical computers coupled to an actual physical system and one or more quantum computers in data communication with the one or more classical computers. The one or more quantum computers comprise a plurality of qubits configured to execute a quantum circuit, the quantum circuit including: (a) a first Clifford circuit configured to randomize a thermal pure quantum (TPQ) state |φ> of the plurality of qubits; (b) a quantum signal processing circuit configured to approximate the imaginary time evolution of |φ> according to one or more Hamiltonians representing the actual physical system; and (c) a second Clifford circuit configured to perform a randomized measurement of |φ> and thereby calculate one or more observables representing Gibbs states. The one or more classical computers are configured to control the actual physical system based on the one or more calculated observables.

[0020] Optionally, the one or more quantum computers further comprise a quantum Bayesian machine coupled to the quantum circuit for training by the one or more calculated observables.

[0021] Optionally, the quantum Bayesian machine includes a Boltzmann machine, a Born machine, or an Ising machine.

[0022] Optionally, one or more of the quantum computers are configured to provide a classical representation of the state of the actual physical system to one or more classical computers by applying classical shadow tomography to the quantum Bayesian machine.

[0023] It is understood that the concepts, techniques, and structures disclosed herein can be implemented in ways other than those summarized above. Therefore, this summary should not be construed as limiting, but rather as exemplary.

Brief Description of the Drawings

[0024] The methods of making, using, and processes of the disclosed apparatus and methods can be understood by referring to the figures of the accompanying drawings. It should be understood that the components and structures shown in the figures are not necessarily to scale, and emphasis has been placed on showing the principles of the concepts described herein. The same reference numerals indicate corresponding parts throughout different figures. Further, the implementation of the apparatus and methods is illustrated in the figures and is not limiting.

[0025]

Figure 1

[0026]

Figure 2

[0027]

Figure 3

[0028]

Figure 4

[0029]

Figure 5

[0030] In the accompanying drawings, underlined numbers are used to represent the item in which the underlined number is located, or the item adjacent to the underlined number. Non-underlined numbers are related to the item identified by the line linking the non-underlined number and the item. When the number is non-underlined and there is an associated arrow, the non-underlined number is used to identify the general item to which the arrow points.

DETAILED DESCRIPTION OF THE INVENTION

[0031] In practice, since actual physical systems are often too complex to be accurately modeled, instead, various measurable characteristics ("observables") are modeled according to a probability distribution that changes over time. The equilibrium probability distribution (i.e., the stationary distribution) that remains invariant with respect to future changes in the system is called the "Gibbs state" of the system. For example, the air temperature in a room can be measured with a thermometer and remains invariant over an appropriate time scale in the equilibrium state. Other observables include, for example, the hardness of a material, the engine temperature of a vehicle, and even the price of a commodity. Even in actual physical systems that are complex enough to avoid accurate modeling for predicting future behavior, observables are thought to exist. One aspect of the concepts, techniques, and structures disclosed herein is to determine appropriate mathematical models for those observable things, use those models to predict results, and in some cases, optimize and control the operation of those actual physical systems through a feedback loop.

[0032] Referring to FIG. 1, a flowchart generally indicated at 10 is shown. Generally, the flowchart 10 includes steps 20 to 50 that need to be executed to efficiently implement a mathematical model of a particular actual physical system.

[0033] In the first step 20, data representing the operation of a given actual physical system, such as measurements of operation parameters as a function of time and operation modes, is collected.

[0034] In the second step 30, using Hamiltonians via corresponding quantum circuits on a quantum computer, and using the quantum computer, observables (M) of the Hamiltonian at finite temperature are generated by using pure thermal states and thermal shadows (details will be described later). The observables (M) provide observables representing probability distributions such as the theoretical Gibbs state. The observables (M) correspond to the operation of the actual physical system.

[0035] In the third step 40, a computing machine (e.g., but not limited to, a quantum Boltzmann machine) representing a given actual physical system is trained using an observed value (M) of a Hamiltonian representing a theoretical Gibbs state. The computing machine is incorporated into a mathematical model to represent a given actual physical system.

[0036] In the fourth step 50, the mathematical model is used to monitor, optimize, or control a specific actual physical system. For example, the mathematical model is used to simulate how a specific actual physical system operates under various operating conditions and find the optimal way for the system to operate from the simulation. Then, the optimal way of operation is applied to the specific actual physical system, for example, by controlling the operating settings and parameters to make the system function more efficiently or productively (e.g., improving energy efficiency, improving the efficiency of input materials, reducing waste generation, etc.).

[0037] According to the present disclosure, the implementation of the described apparatus and method generates an observed value (M) representing the value of a theoretical Gibbs state from a quantum circuit derived from a Hamiltonian using a pure thermal state and a thermal shadow. Directly calculating the Gibbs state using a quantum computer is computationally very difficult and may be intractable in some cases.

[0038] Referring to FIG. 2, a quantum computing device generally designated 100 is shown. The quantum computing device 100 includes one or more classical binary computers 110 connected to one or more quantum computers 120. The one or more classical binary computers 110 include devices such as reduced instruction set computers (RISC), array processors, graphical processors unit (GPU), etc., which are conventionally based on silicon semiconductor technology. The one or more quantum computers 120 include devices such as cryogenic-cooled Josephson junction (superconducting) quantum processors, trapped ion quantum processors, photon-based quantum computers, etc. These quantum computers 120 are configured to perform operations on qubits, for example, in the range of 2 to 100 qubits, more optionally in the range of 2 to 1000 qubits, or even more optionally in the range of 2 to 1 million qubits, which is recently envisioned. The one or more classical binary computers 110 each provide an input port 130 and an output port 140 for data input and output.

[0039] During operation, one or more computational tasks are input into the one or more classical binary computers 110 via the input port 130. Among other tasks, during operation, the one or more classical binary computers 110 configure the one or more quantum computers 120 to perform at least a portion of one or more computational tasks that are optimal to execute using quantum computing. Such a configuration includes devising one or more Hamiltonians used to generate one or more corresponding quantum circuits (e.g., including qubits and quantum logic gates) configured using the one or more quantum computers 120.

[0040] Although the related processes executed in the embodiments in relation to FIG. 1 and the apparatuses capable of executing such processes in relation to FIG. 2 have been described, the programming of the apparatuses for executing these processes will be described in more detail with reference to FIG. 3. In particular, according to the concepts, techniques, and structures disclosed herein, the expected value representing the Gibbs state of an actual physical system, i.e., the observed value (M), is not calculated directly from the Gibbs state itself, but is calculated from the thermal pure quantum (TPQ) state of one or more quantum circuits. To configure a quantum circuit with one or more quantum computers 120, it should be understood that it may include changing the physical characteristics of one or more qubits, thereby changing how those qubits react to each other's presence and environment according to the principles of quantum mechanics.

[0041] The TPQ state is relatively easy to calculate using a quantum computer, but it is very difficult, and in many cases technically impossible or computationally intractable, to directly calculate the Gibbs state using a quantum computer. Advantageously, the embodiments are configured to calculate a "very general observed value" representing a given Gibbs state, and the observed value (M) is generated from the Hamiltonian at a finite temperature (e.g., non-zero temperature).

[0042] In a scenario where it is desired to predict a total of M Gibbs state expected values, the quantum circuits of the order of log(M), i.e., O(log(M)), and / or the observed values from experiments on a particular actual physical system are sufficient input parameters (element 30 in FIG. 1) for training a computational engine such as a quantum Boltzmann machine using the expected values with additive errors, and it is not necessary to directly calculate the Gibbs state (element 40 in FIG. 1). In particular, by using thermal shadowing when generating the observed value (M), the number of required observed values is reduced, and that number is of the order of log(M).

[0043] Such relationships used in implementations of the present disclosure demonstrate a reduction in the number of required observables (M) compared to known previous approaches. This reduction represents an advantage provided by embodiments of the concepts, techniques, and structures disclosed herein. For example, the number of qubits required to be used in a quantum computer to perform a particular calculation is reduced, and the depth of the quantum circuits used is also reduced. Accordingly, a method of using a TPQ state that does not require the preparation of a purified Gibbs state is provided, which means a reduction in the number of qubits required for one or more quantum computers 120 compared to other approaches. Another advantage results from the fact that, unlike known methods which require many auxiliary qubits and deeper circuits, it is not necessary to generate a Gibbs state on one or more quantum computers 120. Thus, by using these thermal pure states (without shadows), the number of qubits required and the depth of the quantum circuits are saved.

[0044] It is also shown that it is sufficient to simply initialize the algorithm with a quantum 2 - design, which means that the depth of the quantum circuit of the first subroutine of the algorithm (i.e., the preparation of a random pure state) can be reduced from exponential (Haar random) to polynomial (Clifford random). Reducing the depth of the quantum circuit in this way reduces the computational noise, and thus improves the accuracy of the observables (M) representing the Gibbs state used in the training of the corresponding computational machine, such as the training of a quantum Boltzmann machine. These random pure states are then evolved in imaginary time and randomly measured to construct shadows. The method described above is called utilizing "pure thermal shadows".

[0045] Accordingly, embodiments may include a quantum circuit that implements a pure thermal shadow by quantum signal processing and a random Clifford circuit for use in the implementation. Such an approach is not plagued by barren plateaus, can in principle process any Hamiltonian, and can trade off between the depth of the quantum circuit and the accuracy of the quantum computation. These all have advantages compared to other known methods such as variational quantum algorithms. It will be understood that one or more quantum computers 120 are contemplated to be fault-tolerant quantum computers. Finally, numerical simulation results of the efficiency of a quantum circuit of a 10-spin 1 / 2 XXZ Heisenberg model are demonstrated in the implementation of this disclosure.

[0046] Referring to FIG. 3, the Gibbs state is a mixed quantum state of the form TIFF2025522402000006.tif734. Here, H is the system Hamiltonian, β is the inverse temperature, and TIFF2025522402000007.tif630 is the partition function. These theoretical Gibbs states describe quantum systems in thermodynamic equilibrium with the environment at finite temperature and play a central role in quantum statistical mechanics. These properties are important in a wide range of applications, such as the design of complex quantum materials in condensed matter physics and quantum chemistry, optimization by quantum semidefinite programming, and machine learning by quantum Boltzmann machines. An exemplary Gibbs state of a particular quantum system in thermal equilibrium with the environment is shown on the left side of FIG. 3 (upper).

[0047] Preparing the Gibbs state and calculating the Gibbs state expectation value are very non-trivial tasks. Existing known algorithms can be quite complex to implement and may only be applicable to a limited set of systems. Classical algorithms are troubled by the intractability of dealing with the partition function due to the exponentially growing Hilbert space or may encounter sign problems in some fermionic systems. Fault-tolerant quantum algorithms have better asymptotic scaling but are currently limited by hardware constraints and error processes. Variational quantum algorithms can prepare the Gibbs state and address some hardware limitations, but many experimental measurements are required for each optimization step and may be troubled by unproductive plateaus regarding convergence during the calculation. Therefore, using variational quantum algorithms is not a desirable approach. Other quantum approaches based on minimally entangled typical thermal states (METTS) set up Markov chains with potentially long thermalization times. Therefore, a more efficient method for calculating the expectation value representing the Gibbs state, i.e., the observable (M), is needed.

[0048] The present disclosure provides a more efficient quantum algorithm for estimating observables (M) representing a number of Gibbs state expectation values without preparing and measuring the corresponding true Gibbs state. Calculating the true Gibbs state is computationally complex and, in some cases, intractable.

[0049] As shown on the right side of the upper part of FIG. 3, in an embodiment of the present disclosure, by combining a thermally pure quantum state and classical shadow tomography, the estimation of observables (M) corresponding to a number of Gibbs state expectation values is achieved according to the embodiment. One way to generate a thermally pure quantum (TPQ) state is the imaginary time evolution of a random state |φ> of n qubits It is to use TIFF2025522402000008.tif721. When the probability distribution on the initial state |φ> forms at least a quantum 2-design, the imaginary-time evolution state can approximate the expected value of TIFF2025522402000009.tif57 with an error that decreases exponentially with the system size n. Next, classical shadow tomography is used to construct an efficient classical representation of these TPQ states from the results of randomized observations. For a sufficiently large system, the number M of expected values of the Gibbs state can be estimated using the observations, i.e., measurements, of a single prepared TPQ state TIFF2025522402000010.tif517. This is a remarkable result because the number of measurements required is similar to the case when the Gibbs state is actually prepared (e.g., via purification using 2n qubits). In embodiments of the present disclosure, the actual implementation of the algorithm is described below.

[0050] There are three steps in the appropriate algorithm for configuring one or more quantum computers 120 to calculate a pure thermal shadow. In the first step of the algorithm, an initial state is prepared using a Clifford quantum circuit of polynomial depth. This preparation is sufficient to generate a quantum 3-design.

[0051] The second step of the algorithm involves approximating the imaginary-time evolution using quantum signal processing (QSP). This approach is very general (i.e., applicable in principle to any quantum Hamiltonian H) and provides excellent flexibility because the depth of the quantum circuit and the accuracy of the quantum calculation can be systematically traded off.

[0052] In the third step, another Clifford circuit is used for randomized measurement. The complete circuit including the Clifford circuit, the QSP circuit, and the randomized measurement is shown in Fig. 3 (lower part). This circuit has a total number of qubits proportional to the system size n. Optionally, in an implementation of the present disclosure, this quantum circuit can be numerically simulated for a quantum system of 10 spin-1 / 2 particles of the XXZ-Heisenberg model.

[0053] The TPQ state |φ> is any pure state that can estimate a fixed set of properties (expectation values) of a sufficiently large mixed thermal state specified by a particular statistical ensemble. In the case of a thermodynamic (standard) Gibbs ensemble, this is randomly drawn and for all O j within a predefined set of Hermitian operators {O j} is defined as any pure state that satisfies TIFF2025522402000011.tif5112. In the case of thermal pure quantum states, the observable O j has the advantage of having an operator norm that is at most polynomial in the system size.

[0054] The pure state is TIFF2025522402000012.tif17133, where U ∈ Cl(2 n ) is a random unitary drawn from the n-qubit Clifford group and satisfies equation (1). These TPQ states are different from those known so far. It is known to use a particular form of Haar random states for U|0>, but an exponential circuit depth is required in a quantum computer. However, in an implementation of the present disclosure, choosing U ~ Cl(2 n ) gives a unitary 3 design and reproduces Haar integrals up to the third moment. This is sufficient for the purposes of the implementation of the present disclosure, TIFF2025522402000013.tif has the advantage of requiring only 525 quantum gates.

[0055] Random pure state The expected value of any Hermitian operator O at TIFF2025522402000014.tif710 is, on average, TIFF2025522402000015.tif9153. Here, TIFF2025522402000016.tif57 represents the ensemble average U ∈ Cl(2 n ) with respect to the Clifford group of n qubits. This approximation is obtained from the Taylor expansion up to first order at TIFF2025522402000017.tif411, which is to deal with the appearance of U at the normalization constant TIFF2025522402000018.tif621. Since TIFF2025522402000019.tif411 is exponentially small with respect to n, this expansion is justified.

[0056] A similar expansion and the calculation of the variance of the expected value with respect to U are TIFF2025522402000020.tif26132. The bias of Equation (3) and the variance of Equation (4) are both proportional to the purity of the Gibbs state TIFF2025522402000021.tif711. The Gibbs state TIFF2025522402000022.tif57 minimizes the Helmholtz free energy TIFF2025522402000023.tif529, so it can be written as follows. TIFF2025522402000024.tif14130 The last equality is derived from the properties of the free energy, namely extensivity TIFF2025522402000025.tif516 and monotonicity with respect to the inverse temperature TIFF2025522402000026.tif519.

[0057] The term multiplying the purities of Equation (3) and Equation (4) has the form of an expected value and is bounded by the spectral norm TIFF2025522402000027.tif58. This is a polynomially large operator TIFF2025522402000028.tif58, in which case the variance decays exponentially with n. After applying Markov's inequality, the following can be seen. TIFF2025522402000029.tif521 satisfies TIFF2025522402000030.tif527 and TIFF2025522402000031.tif740 in Equation (1).

[0058] Thus, according to the embodiments of the concepts, techniques, and structures disclosed herein, the expected value of O for a random pure state TIFF2025522402000032.tif69 can be used as an estimator of the Gibbs state expected value of an operator O of polynomial size for a sufficiently large system and finite β. The maximum error between the exact Gibbs state expected value and the expected value estimated according to the embodiment is shown in Figure 4(a) as a function of the degree of the polynomial.

[0059] Importantly, TIFF2025522402000033.tif415, i.e., when the Gibbs state approaches the ground state of H, the Gibbs state becomes pure and TIFF2025522402000034.tif618, and the error remains finite for any system size n. For all other β, the exponential decay rate, i.e., the required size of n, is TIFF2025522402000035.tif620. Thus, the exact error depends on the specific system Hamiltonian H, the inverse temperature β, and the spectral norm of the observable It varies depending on TIFF2025522402000036.tif57. In certain cases, this means that using only a single TPQ state is sufficient (for example, when used in some implementations of the present disclosure). In contrast, in known previous approaches such as algorithms based on METTS where the dispersion does not disappear, it is necessary to calculate multiple pure states, which can be computationally cumbersome.

[0060] To implement a thermal shadow algorithm in a quantum device, several different elements are required as follows. First, a method of uniformly sampling from the Clifford group of n qubits to generate a random Clifford circuit U is used. There is an efficient polynomial-time algorithm for this, which is implemented in the algorithms of quantum computing libraries. This algorithm is used for both the generation of a random pure state at the beginning of the algorithm and the randomized measurement at the end of the algorithm.

[0061] Next, a routine for approximating the non-unitary operator in Equation (2) TIFF2025522402000037.tif615 is used. For this purpose, quantum signal processing (QSP), that is, a framework for performing matrix operations on a quantum computer, is used. Intuitively speaking, QSP applies a target polynomial to the eigenvalues of a block-encoded matrix. Assume that the Hamiltonian is given in the following form. TIFF2025522402000038.tif631 However, TIFF2025522402000039.tif516 and Pk is a Pauli operator of n qubits. First, a preprocessing procedure for rescaling the spectrum of H to the interval [0,1] is used. This allows the rescaled Hamiltonian to be block-encoded into a larger unitary matrix. Beneficially, minimum-maximum rescaling TIFF2025522402000040.tif662 can be used. However, λ min (λ max ) is the minimum (maximum respectively) eigenvalue. By this rescaling, the eigenvalues can be pushed into intervals much smaller than [0,1], preventing the approximation error from becoming large.

[0062] In practice, the extreme eigenvalues are unknown, and it is necessary to rely on the lower bound of λ min and the upper bound of λ max . Imaginary time TIFF2025522402000041.tif542 is defined as follows. TIFF2025522402000042.tif662 Therefore, TIFF2025522402000043.tif65 over time TIFF2025522402000044.tif34 it is possible to evolve and obtain the desired non-unitary operator up to a constant multiple. This factor is irrelevant because it is canceled in equation (2). Next, TIFF2025522402000045.tif65's spectrum is converted to an even function TIFF2025522402000046.tif817 using polynomial approximation. Since the polynomial is even, the absolute value simplifies the polynomial approximation.

[0063] For example, the Python pyqsp library can be used to find the polynomial and construct the appropriate QSP circuit. QSP is used in the TIFF2025522402000047.tif766 block-encoding circuit. The circuit diagram for generating the TPQ state of equation (2) using QSP is shown at the bottom of Figure 3. The success probability depends on the initial random state and averages to TIFF2025522402000048.tif648 value. It can be confirmed that as β increases, the expected success probability decreases. This is expected from the intuition that low-temperature sampling is difficult.

[0064] To increase this probability, fixed-point amplitude amplification is used. This is because there is no consistent algorithm for imaginary-time evolution. Amplitude amplification When TIFF2025522402000049.tif is repeated 865 times, the protocol is expected to succeed with probability TIFF2025522402000050.tif 611. This can be implemented via yet another layer of QSP. This protocol functions for any Hamiltonian and temperature as long as an appropriate block-encoding circuit can be found. Note, however, that for specific choices of H and β, the circuit depth is expected to be polynomial in n. In Fig. 4(b), the errors between the shadow directly constructed from the true Gibbs state TIFF2025522402000051.tif 57, the shadow constructed from the exact TPQ state, and the TPQ state generated by QSP are shown as a function of the number of shadows.

[0065] In summary, the present invention provides a method for generating and using a mathematical model in at least one of monitoring, optimizing, and controlling a given physical system. This method includes the following steps, as shown in Fig. 5.

[0066] Step 1: Obtain data representing the operation of the physical system.

[0067] Step 2: Generate at least one Hamiltonian and use it to generate a quantum circuit using a thermal pure quantum (TPQ) state and a thermal shadow. Here, M observables obtained by executing the quantum circuit on a quantum computer represent the Gibbs state. The M observables representing the Gibbs state involve using a combination of a thermal pure quantum (TPQ) state and classical shadow tomography.

[0068] Step 3: Use the M observables to train a computational machine such as a quantum Boltzmann machine.

[0069] Step 4: Incorporate the computing machine into the mathematical model that describes the operation of the actual physical system.

[0070] Step 5: Apply the mathematical model to the actual physical system to perform at least one of monitoring, optimizing, and controlling the actual physical system. Optionally, such optimization can be realized using a variational quantum eigensolver (VQE) method.

[0071] Optionally, the thermal pure quantum (TPQ) state is the imaginary-time evolution of an n-qubit random state |φ> generated using TIFF2025522402000052.tif825. When the probability distribution over the initial state |φ> forms at least a quantum 2-design, the imaginary-time evolved state can approximate the expectation value of TIFF2025522402000053.tif57 up to an error that decays exponentially with the system size n. It is shown that TIFF2025522402000053.tif57 can be approximated by the expectation value of TIFF2025522402000053.tif57.

[0072] Optionally, use classical shadow tomography to construct an efficient classical representation of these TPQ states from the results of randomized measurements. Optionally, for a sufficiently large system, the measurements of TIFF2025522402000054.tif517 of one prepared TPQ state can be used to estimate the expectation values of M Gibbs states. Since the number of required observables is similar to the case where the Gibbs states are actually prepared (e.g., via purification using 2n qubits), this is a remarkable result.

[0073] It will be understood that the quantum Boltzmann machine is a natural application of the thermal shadow algorithm according to the present disclosure. Such a machine is an example showing how the algorithm can be used to solve a specific set of problems that are problems of Hamiltonian learning and generative modeling in the implementation of the present disclosure.

[0074] A quantum Boltzmann machine learns the parameters of a Hamiltonian that generates samples matching an input data distribution at a given temperature. The quantum Boltzmann machine is trained in an iterative process that generally includes at least two steps. The first step involves collecting samples for fixed Hamiltonian parameters and evaluating how well the model Hamiltonian fits the behavior of an actual physical system, which is represented by input data obtained, for example, from experimental measurements performed on the actual physical system. The second step involves updating the Hamiltonian parameters according to an update rule. This update can be efficiently implemented using the shadow of a pure thermal state, and to update the Hamiltonian parameters, it is necessary to calculate certain observables of the Gibbs state. However, it will be understood that the use of the quantum Boltzmann machine is merely an example.

[0075] Importantly, the present disclosure includes the use of a method for providing solutions to problems having several requirements. The first requirement is that a given quantum system is in thermodynamical equilibrium with an environment, where the environment is defined by a Hamiltonian and a finite temperature. The second requirement is that a given set of observables is provided from the given quantum system, from which the expected values for a previously specified system need to be calculated. The third requirement is that the calculations need to be performed accurately and efficiently in three ways.

[0076] The first computational requirement is to reduce the number of samples compared to an exponential function of the system size and a linear function of the number of observables. The second computational requirement is to reduce the number of qubits required compared to a linear or worse overhead of quantum circuit ancillary qubits as a function of the system size. The third computational requirement is to reduce the depth of the quantum circuit compared to an exponential function of the system size.

[0077] The solutions to the aforementioned problems include, as described above, calculating M observables using a classical shadow of a thermal pure quantum (TPQ) state. The M observables are used, for example, to train a quantum Boltzmann machine as described above.

[0078] As a result, it will be understood that pure thermal states can generally be prepared using quantum signal processing. Here, the details of a particular implementation depend on the characteristics of the actual physical system. Further, in another usage application of the implementation of the present disclosure, it will be understood that it is necessary to calculate some observables of the Gibbs state, such as in Quantum Semi Definite Programming (QSDP). Such programming is conceptually separated.

[0079] Using the aforementioned mathematical model including the calculation engine of step 5, at least one feedback control loop is provided in a particular actual physical system to control the particular actual physical system. One or more parameters of the control loop are generated by the mathematical model. The input parameters for the output parameters of the control loop are monitored as a function of time, and the time drift of the parameters can be used to determine the state of the actual physical system. The state of the actual physical system is, for example, suboptimal operating conditions of the actual physical system, an impending failure of components of the actual physical system, the need for maintenance of the actual physical system (to improve reliability), etc.

[0080] Changes to the foregoing disclosed embodiments are possible without departing from the scope of the disclosure as defined by the appended claims. Expressions such as "comprising," "including," "incorporating," "having," "being," etc., used to describe and claim the disclosure are intended to be construed non-exclusively, that is, there may also be items, components, or elements not explicitly described. References to the singular are also construed to relate to the plural. The term "exemplary" is used herein to mean "serving as an example, instance, or illustration." Embodiments described as "exemplary" should not necessarily be construed as preferred or advantageous over other embodiments, nor do they preclude the incorporation of features from other embodiments. The term "optionally" is used herein to mean "provided in some embodiments and not provided in other embodiments." For clarity, it is understood that specific features of the disclosure described in the context of separate embodiments may be provided in combination in a single embodiment. Conversely, for brevity, various features of embodiments described in the context of a single embodiment may be provided separately, or in any suitable combination, or in other described embodiments of the present disclosure as appropriate.

[0081] Although specific materials are referred to herein, it is understood that other materials having similar functional and / or structural properties may be substituted where appropriate, and that those skilled in the art will be able to select such materials and understand how to incorporate them into embodiments of the concepts, techniques, and structures described herein without departing from the scope of their teachings.

[0082] Various embodiments of concepts, systems, devices, structures, and techniques for which protection is sought are described herein with reference to the accompanying drawings. Alternative embodiments can be devised without departing from the scope of the concepts, systems, devices, structures, and techniques described herein. Note that in the following description and drawings, various connection and positional relationships (e.g., above, below, adjacent, etc.) between elements are shown. These connections and / or positional relationships can be direct or indirect, unless otherwise specified, and the concepts, systems, devices, structures, and techniques described are not intended to be limited in this regard. Thus, the coupling of entities can refer to either a direct or an indirect coupling, and the positional relationship between entities can be a direct or an indirect positional relationship.

[0083] As an example of an indirect positional relationship, a reference herein to forming layer "A" above layer "B" includes situations where one or more intermediate layers (e.g., layer "C") are between layer "A" and layer "B", provided that the relevant characteristics and functions of layer "A" and layer "B" are not substantially changed by the intermediate layer. The following definitions and abbreviations are used in the interpretation of the claims and the specification. Here, the terms "comprise", "comprising", "include", "including", "have", "having", or "with", or other variations thereof, are intended to cover non-exclusive inclusion. For example, a composition, mixture, process, method, article, or device that includes a list of elements is not necessarily limited to only those elements, and can include other elements not explicitly listed or inherent in such composition, mixture, process, method, article, or device.

[0084] Furthermore, as used herein, the term "exemplary" is used in the sense of "serving as an example, instance, or illustration". Embodiments or designs described herein as "exemplary" are not necessarily to be construed as being preferred or advantageous over other embodiments or designs. The terms "one or more" and "at least one" are understood to include any integer greater than or equal to one, i.e., 1, 2, 3, 4, etc. The term "a plurality" is understood to include any integer greater than or equal to two, i.e., 2, 3, 4, 5, etc. The term "connected" includes both indirect "connections" and direct "connections".

[0085] References herein to "one embodiment", "an embodiment", "an example of an embodiment", etc., indicate that the described embodiment may include a particular feature, structure, or characteristic, but not every embodiment may include that particular feature, structure, or characteristic. Moreover, such phrases do not necessarily refer to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is within the knowledge of those skilled in the art to affect such feature, structure, or characteristic in connection with other embodiments, whether or not explicitly described.

[0086] In the following description, the terms "above", "below", "right", "left", "vertical", "horizontal", "top", "bottom", and derivatives thereof shall relate to the structures and methods described and shown in the drawings. The terms "above", "uppermost", "at the uppermost", "disposed above", or "disposed at the uppermost" mean that a first element, such as a first structure, is present above a second element, such as a second structure, and there may be intervening elements, such as an interface structure, between the first element and the second element. The term "in direct contact" means that a first element, such as a first structure, and a second element, such as a second structure, are connected without an intermediate element.

[0087] To change a claim element, the use of ordinal numbers such as "first", "second", "third", etc. within a claim does not, in itself, mean that one claim element has precedence, priority, or order over another claim element, or that it implies the chronological order in which the acts of a method are performed. Instead, it is simply used as a label to distinguish one claim element with a particular name from another element with the same name (other than by the use of the ordinal number), thereby differentiating the claim elements.

[0088] The terms "about" and "approximately" may be used in some embodiments to mean within ±20% of a target value, within ±10% (percent) of a target value in some embodiments, within ±5% of a target value in some embodiments, and within ±2% of a target value in still some embodiments. The terms "about" and "approximately" may include the target value. The term "substantially equal" may be used in some embodiments to refer to values that are within ±20% of each other, within ±10% of each other in some embodiments, within ±5% of each other in some embodiments, and within ±2% of each other in still some embodiments.

[0089] The term "substantially" may be used in some embodiments to refer to values that are within ±20% of a comparative measurement value, within ±10% in some embodiments, within ±5% in some embodiments, and within ±2% in still some embodiments. For example, a first direction that is "substantially" perpendicular to a second direction may, in some embodiments, refer to a first direction that is within ±20% of an angle of 90° with the second direction, within ±10% of an angle of 90° with the second direction in some embodiments, within ±5% of an angle of 90° with the second direction in some embodiments, and within ±2% of an angle of 90° with the second direction in still some embodiments.

[0090] It should be understood that the disclosed subject matter is not limited, in its application, to the details of construction and arrangement of components set forth in the following description or illustrated in the drawings. The disclosed subject matter is capable of other embodiments and of being practiced and carried out in various ways. It should also be understood that the expressions and terminology used herein are for the purpose of description and should not be regarded as limiting. Accordingly, those skilled in the art will understand that the concepts upon which this disclosure is based can readily be utilized as a basis for designing other structures, methods, and systems for carrying out some of the purposes of the disclosed subject matter. Therefore, the claims should be regarded as including such equivalent constructions insofar as they do not depart from the spirit and scope of the disclosed subject matter.

[0091] The disclosed subject matter has been described and illustrated in the foregoing exemplary embodiments, but the present disclosure has been made by way of example only, and it is understood that numerous changes in the details of the implementation of the disclosed subject matter are possible without departing from the spirit and scope of the disclosed subject matter.< / oj> < / oj>

Claims

1. A device for optimizing, monitoring, and controlling an actual physical system, the device comprising a hybrid computing configuration including one or more classical computers connected to one or more quantum computers, the one or more quantum computers configured to execute one or more quantum circuits configured using one or more classical computers, and the device configured to perform the following operations when in use: (i) Obtain data representing the operation of the actual physical system, (ii) Using the data, generate one or more Hamiltonians, define at least one quantum circuit executable on one or more quantum computers, where the observed values ​​(M) obtained from the execution of at least one of the quantum circuits represent Gibbs states, and the generation of the observed values ​​(M) involves the use of a combination of thermal pure quantum (TPQ) states and classical shadow tomography, (iii) Train a computational machine based on the observed value (M) representing the cast state, (iv) Generate a mathematical model representing the actual physical system including the computing machine, (v) Applying the mathematical model to the data obtained from the actual physical system, (a) A monitoring output representing the actual operating state or operating conditions of the physical system, (b) One or more inputs to an optimization function applied to the mathematical model to generate parameter values ​​used to operate the actual physical system in a more optimized manner, (c) One or more inputs to a control function applied to the mathematical model to generate parameter values ​​used to control the operation of the actual physical system, To generate at least one of the following: Device.

2. The aforementioned device performs imaginary time evolution of the random state |φ> of n qubits. The system is configured to generate the thermal pure quantum (TPQ) state using The apparatus according to claim 1.

3. The apparatus is configured to use classical shadow tomography to construct an efficient classical representation of the TPQ state from the results of a randomized subset of (M) of the observed values. The apparatus according to claim 1.

4. The device has prepared the observed values ​​of a single TPQ state. It is configured to estimate M Gibbs state expectation values ​​using The apparatus according to claim 1.

5. A computer implementation method using a device that optimizes, monitors, and controls an actual physical system, wherein the device comprises a hybrid computing configuration including one or more classical computers connected to one or more quantum computers, the one or more quantum computers being configured to execute one or more quantum circuits configured using one or more classical computers, and the computer implementation method includes the following operations: (i) Obtain data representing the operation of the actual physical system, (ii) Using the data, generate one or more Hamiltonians, define at least one quantum circuit executable on one or more quantum computers, where the observed values ​​(M) obtained from the execution of at least one of the quantum circuits represent Gibbs states, and the generation of the observed values ​​(M) involves the use of a combination of thermal pure quantum (TPQ) states and classical shadow tomography, (iii) Train a computational machine based on the observed value (M) representing the cast state, (iv) Generate a mathematical model representing the actual physical system including the computing machine, (v) Applying the mathematical model to the data obtained from the actual physical system, (a) A monitoring output representing the actual operating state or operating conditions of the physical system, (b) One or more inputs to an optimization function applied to the mathematical model to generate parameter values ​​used to operate the actual physical system in a more optimized manner, (c) One or more inputs to a control function applied to the mathematical model to generate parameter values ​​used to control the operation of the actual physical system, To generate at least one of the following: Computer implementation method.

6. The above method uses the above apparatus to perform the imaginary time evolution of a random state |φ> of n qubits. Using the above, the process includes generating the thermal pure quantum (TPQ) state, The computer implementation method according to claim 5.

7. The method includes constructing an efficient classical representation of the TPQ state from the results of a randomized subset of (M) observations using classical shadow tomography. The computer implementation method according to claim 5.

8. The method described above uses the observed values ​​of a single prepared TPQ state. This involves estimating the expected values ​​of M Gibbs states using The computer implementation method according to claim 5.

9. One or more classical computers coupled to an actual physical system, One or more quantum computers that communicate data with one or more classical computers, Equipped with, One or more of the quantum computers comprises a plurality of qubits configured to execute a quantum circuit, and the quantum circuit is (a) A first Clifford circuit configured to randomize the thermal pure quantum (TPQ) states |φ> of multiple qubits, (b) A quantum signal processing circuit configured to approximate the imaginary time evolution of |φ> according to one or more Hamiltonians representing the actual physical system, (c) A second Clifford circuit configured to perform a randomized measurement of |φ> and thereby compute one or more observations representing Gibbs states, Includes, One or more of the classical computers are configured to control the actual physical system based on one or more of the calculated observed values. Quantum mechanics device.

10. One or more of the quantum computers further comprises a quantum Bayesian machine coupled to the quantum circuit for training with one or more of the calculated observations. The quantum mechanical apparatus according to claim 9.

11. The aforementioned quantum Bayesian machine includes a Boltzmann machine, a Born machine, or an Ising machine. The quantum mechanical apparatus according to claim 10.

12. One or more of the quantum computers are configured to provide one or more of the classical computers with a classical representation of the state of the actual physical system by applying classical shadow tomography to the quantum Bayesian machine. The quantum mechanical apparatus according to claim 10.

Images