Enhanced classical shadow using matched gate quantum circuits

Abstract

Methods, systems, and devices for enhancing classical shadows using matched gate quantum circuits. In one aspect, a method for calculating a classic shadow of an n-qubit quantum state includes: resampling, by a classic computer, unitary operators from a random unitary set, where the random unitary set includes a group of generalized match gates; and for each sampled unitary operator: applying, by a quantum computer, a quantum circuit to the n-qubit quantum state to obtain an evolved quantum state, wherein the quantum circuit implements the sampled unitary operator; measuring, by the quantum computer, the evolved quantum state to obtain a corresponding bit string; and storing, by the classic computer, a record of the respective bit string and the sampled unitary operator; and providing the record as a classical shadow of the quantum state.

Description

Background Art

[0001] The present disclosure relates to quantum computing.

[0002] Shadow tomography is a procedure that can be used to estimate the properties of an unknown quantum state without resorting to full-state tomography (which is prohibitively expensive in terms of the number of measurements required). Let ρ denote a quantum state, and assume that multiple copies of this quantum state ρ are accessible. Let {O i} represents a set of M observables. The task of shadow tomography is to estimate the value of each operator O in the set of observables. i The expected value Tr(ρO i ), until a certain additive error ∈ is reached. This can be efficiently achieved by randomly selecting measurement operators from the complete set of tomograms.

[0003] To specify the protocol, choose the set of unitary You can set Then, randomly sample a unitary from the set and measure the state in the computational basis To obtain the ground state |b k > k |. Now, consider the state It is expected that the mapping from ρ to this state defines the quantum channel Requires quantum channel is invertible if and only if by drawing the unitary This is true only if the set of measurement operators defined for measurement in the computational basis is tomographically complete. Assuming this is true, we can Applied to the definition of quantum channel, we get gather is the classical shadow of the quantum state ρ. Summary of the invention

[0004] This disclosure describes techniques for enhancing classical shadowing using matched-gate quantum circuits.

[0005] ​Generally speaking, one innovative aspect of the subject matter described in this specification can be implemented in a method for computing a classical shadow of an n-qubit quantum state, the method comprising: repeatedly sampling, by a classical computer, a unitary operator from a random unitary set, wherein the random unitary set comprises a generalized match gate group, and for each sampled unitary operator: applying, by the quantum computer, a quantum circuit to a quantum state p to obtain an evolved quantum state, wherein the quantum circuit implements the sampled unitary operator; measuring, by the quantum computer, the evolved quantum state on a computational basis to obtain a corresponding bit string; and storing, by the classical computer, a record of the corresponding bit string and the sampled unitary operator; and providing the stored record as the classical shadow of the quantum state.

[0006] Other implementations of these aspects include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each computer program being configured to perform the actions of the method. One or more systems of classical and quantum computers may be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which software, firmware, hardware, or a combination thereof causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform actions.

[0007] The foregoing and other implementations may each optionally include one or more of the following features, alone or in combination. In some implementations, the generator of the generalized matching gate group includes an operator X j X j+1 , X j Y j+1 , Y j X j+1 or Y j Y j+1 The unitary operator generated by the action on an array of n qubits, where 1≤j≤n, X j represents the Pauli-X operator applied to qubit j, and Y j represents the Pauli-Y operator applied to qubit n; and the unitary operator generated by the action of the Pauli-X operator on qubit n.

[0008] In some implementations, the group of generalized matching gates has a one-to-one correspondence with the group of 2n×2n orthogonal matrices O(2n).

[0009] In some implementations, for each element R in the group O(2n), there exists a unique unitary operator in the group of generalized matching gates that satisfies

[0010] In some implementations, sampling a unitary operator from the random unitary set includes sampling from a group O(2n) according to a Haar measure and constructing a corresponding generalized matching gate unitary operator.

[0011] In some implementations, classical shading is given by

[0012]

[0013] where J represents the number of repetitions performed to sample the unitary operator, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a super-operator that projects the input onto a set of Majorana operators of degree 2l, where Equivalent to a quantum channel.

[0014] In some implementations, the method further includes using the classical shadow of the quantum state to perform one or more operations, the operations comprising one or more of: predicting an expectation value of an observable relative to the quantum state, performing direct fidelity estimation, performing entanglement verification, estimating a correlation function, or predicting entanglement entropy.

[0015] In general, another innovative aspect of the subject matter described in this specification can be implemented in a method for computing an expectation value of a projection operator, the method comprising: obtaining a classical shadow of an n-qubit quantum state, wherein the classical shadow comprises a quantum channel of unitary operators sampled from a random unitary set and a measured bit string; generating an updated unitary operator comprising multiplying a unitary operator defining the projection operator with i) a unitary operator sampled from the random unitary set and ii) an operator that prepares the measured bit string from a vacuum state; and computing an expectation value of the quantum channel with respect to the vacuum state, the computing comprising evaluating a derivative of a polynomial comprising: a Pfaffian of a matrix comprising the updated unitary operator.

[0016] Other implementations of these aspects include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each computer program being configured to perform the actions of the method. A system of one or more classical computers may be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which software, firmware, hardware, or a combination thereof causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform an action.

[0017] The foregoing and other implementations may each optionally include one or more of the following features, alone or in combination: In some implementations, the projection operator includes an operator that projects the quantum state into a pure fermionic Gaussian state.

[0018] In some implementations, a pure fermion Gaussian state includes a unitary operator from a random unitary set applied to the vacuum state.

[0019] In some implementations, the projection operator is given by is given by , where |0> represents the vacuum state, and is a unitary operator that defines the projection operator, where In a random unitary set.

[0020] In some implementations, the random unitary set includes a generalized matching gate group.

[0021] In some implementations, classical shading is given by

[0022]

[0023] where J represents the number of repetitions performed to sample the unitary operator, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a superoperator that projects the input onto a set of Majorana operators of degree 2l, where Equivalent to a quantum channel.

[0024] In some implementations, generating an updated unitary operator includes redefining the unitary operator U j Absorbing unitary operator and prepare |b from |0> j >.

[0025] In some implementations, computing the expectation value of the quantum channel with respect to the vacuum state includes computing the expectation value of a superoperator with respect to the vacuum state, where the superoperator projects an updated unitary operator applied to the vacuum state onto a set of Majorana operators of degree 2l.

[0026] In some implementations, calculating the expectation value of the superoperator relative to the vacuum state includes calculating

[0027] In some implementations, the matrix including the updated unitary operator is M0+zQM0Q T Given, where z is the independent variable of the polynomial, and Q represents the element in SO(2n) corresponding to the updated unitary operator.

[0028] In some implementations, evaluating the derivative of the polynomial includes performing numerical differentiation or polynomial interpolation techniques.

[0029] In general, another innovative aspect of the subject matter described in this specification can be implemented in a method for computing an expectation value of an operator, the method comprising: obtaining a classical shadow of an n-qubit quantum state, wherein the classical shadow comprises a quantum channel of unitary operators sampled from a random unitary set and a measured bit string; generating an updated unitary operator comprising multiplying a unitary operator defining the operator with the unitary operator sampled from the random unitary set; and computing an expectation value of the quantum channel with respect to an outer product of a computational ground state included in the operator and a vacuum state, the computing comprising evaluating a derivative of a polynomial comprising: a Pfaffian of a matrix comprising the updated unitary operator.

[0030] Other implementations of these aspects include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each computer program being configured to perform the actions of the method. A system of one or more classical computers may be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which software, firmware, hardware, or a combination thereof causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform an action.

[0031] The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations, the operator includes a Slater determinant and a vacuum state.

[0032] In some implementations, the Slater determinant includes a unitary operator from a set of random unitaries that preserves an excitation number applied to a computational basis state having an excitation number.

[0033] In some implementations, the operator is composed of Given, where |0> represents the vacuum state, U φ is the unitary operator that defines this operator, where U φ In a random unitary set, and |x> represents the computational ground state.

[0034] In some implementations, the random unitary set includes a generalized matching gate group.

[0035] In some implementations, classical shading is given by

[0036]

[0037] where J represents the number of repetitions performed to sample the unitary operator, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a superoperator that projects the input onto a set of Majorana operators of degree 2l, where Equivalent to a quantum channel.

[0038] In some implementations, generating an updated unitary operator includes redefining the unitary operator U j By absorbing unitary operator U φ .

[0039] In some implementations, computing an expectation value of the quantum channel relative to an outer product of a computational ground state included in the operator and a vacuum state includes computing an expectation value of a superoperator relative to an outer product of the computational ground state included in the operator and a vacuum state, wherein the superoperator projects an updated unitary operator applied to the bit string onto a set of Majorana operators of degree 2l.

[0040] In some implementations, calculating the expectation value of the superoperator relative to the outer product of the calculated ground state and the vacuum state included in the operator includes calculating

[0041] In some implementations, the matrix including the updated unitary operator is given by Given, where for x∈{0,1}, z is the independent variable of the polynomial, Q represents the element of O(2n) corresponding to the updated unitary operator, in S x =[2n]{2j-1:x j =1} and M[S] represents the constraints of the matrix S on the rows and columns with indices in S.

[0042] In some implementations, evaluating the derivative of the polynomial includes performing numerical differentiation or polynomial interpolation techniques.

[0043] The subject matter described in this specification can be implemented in specific embodiments to realize one or more of the following advantages.

[0044] Systems implementing the presently described techniques use a new set of unitaries in the classical shadow framework—the generalized matching gate group. Using this continuous family of unitaries provides several technical advantages when using shadow tomography, especially on noisy quantum computing devices. For example, it is often difficult to implement specific gates accurately. Therefore, protocols built on continuous rotations rather than discrete rotations may be particularly advantageous because such protocols may be less sensitive to over-rotation and under-rotation errors caused by averaging a series of continuous rotations, especially if the errors can be fully or partially characterized (where "averaging" in this case refers to the fact that random unitaries are sampled and the estimates of the expected value are averaged over the different samples, which occurs when the data is post-processed to extract one or more expected values.)

[0045] Additionally, the techniques described so far can be used in the context of analogical quantum simulations, where it may be practical to implement the evolution of random generalized matched gate unitaries using the correspondence between random generalized matched gate unitaries and the evolution of free fermions. This allows the use of shadow tomography in analogical simulators based on neutral atoms or ion traps. In analogical simulators, only coarse-grained control over the system is possible. In particular, it may be difficult to implement a particular gate. However, the techniques described so far do not require specific gates, but only implementations of random gates sampled from a specific family of gates. In other words, the random matched gates are generated by the evolution under a randomly distributed free fermion Hamiltonian (with a suitable distribution). This can be used in analogical simulators where there is enough control to evolve under a random free fermion Hamiltonian, but not enough control to implement a specific gate exactly.

[0046] Additionally, computing the fidelity of an experimentally prepared state to some target state is a standard experimental task. Using previously known methods, an exponential amount of classical postprocessing is required to estimate the expectation values ​​of an operator using classical shadowing, such as a projection operator onto an arbitrary fermionic Gaussian state. The presently described technique reduces the amount of classical postprocessing required to evaluate these expectation values. Specifically, when using classical shadowing based on random matching gates, the amount of classical postprocessing scales cubically with the system size. Furthermore, numerical experiments confirm that the sampling efficiency (e.g., the number of physical measurements required) of the presently described technique is also improved and scales efficiently with the system size.

[0047] Furthermore, the ability to efficiently evaluate the expected value of the projection operator (the fidelity) using matched-gate classical shadowing opens up new possibilities for benchmarking device and algorithm performance. This is particularly true in the field of quantum simulation in chemistry and materials science where fermionic Gaussian states play a central role. For example, the classical shadowing technique described presently enables efficient evaluation of the fidelity of physically prepared states with arbitrary fermionic Gaussian states. If a circuit is executed that should (in the absence of noise) have a certain fidelity with fermionic Gaussian states, this can be verified using the classical shadowing technique described presently. Additionally, states typically associated with quantum simulations typically support a small number of fermionic Gaussian states, and the technique described presently can provide an efficient way to verify the execution of some medium-sized circuits.

[0048] In addition, the techniques described so far further expand the set of operators that can be efficiently estimated using shadow tomography. As with the evaluation of projection operators onto fermionic Gaussian states, the techniques for evaluating matrix elements described herein will help benchmark quantum computers and quantum algorithms. A generalization of the proposed exact techniques can also calculate other off-diagonal matrix elements. This in turn allows for efficient evaluation of the fidelity of a wave function that is a superposition of (polynomially many) Gaussian states.

[0049] The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is a block diagram of an example system using a matched-gate quantum circuit to enhance classical shadowing.

[0051] Figure 2 is a flow chart of the process for computing the classical shadow of an n-qubit quantum state.

[0052] Figure 3 is a flow chart of an example process for computing the expectation value of a projection operator that projects a quantum state onto a purely fermionic Gaussian state.

[0053] Figure 4 is a flow chart of an example process for computing the expected value of an operator.

[0054] Figure 5 Two example graphs are shown that illustrate the sampling efficiency of the presently described techniques.

[0055] Figure 6 Depict an example classical / quantum computer. DETAILED DESCRIPTION

[0056] This specification describes techniques for computing classical shadows and evaluating expressions (e.g., expected values) using classical shadows. The classical shadows are computed using a new set of unitary states, a generalized matching gate group. The quantum channel generated by this new set of unitary states is reversible. Furthermore, the classical post-processing required to use the resulting quantum channel is efficient. Specifically, the classical shadows generated using the currently described quantum channel can be used to evaluate expected values ​​at a cost that scales cubically with the number of qubits. This provides a technical improvement over the prior art, where the current technology evaluates expected values ​​at a cost that scales exponentially with the number of qubits.

[0057] In this disclosure, the term "set" is understood to refer to both a set (eg, a set of matching gate units) and a probability distribution (eg, distributed according to the Haar measure). A "sample from a set" means a random draw from the set according to the distribution.

[0058] Figure 1 is a block diagram of an example system 100 for enhancing classical shadowing using matched-gate quantum circuits. The example system 100 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices in one or more locations in which the systems, components, and techniques described herein may be implemented.

[0059] The example system 100 includes a quantum computing device 102 in data communication with a classical processor 104. For purposes of illustration, Figure 1 The quantum computing device 102 and the classical processor 104 shown in FIG. 1 are shown as separate entities, however, in some implementations, the classical processor 104 may be included in the quantum computing device 102. For example, in some implementations, the quantum computing device 102 may be directly connected to the classical processor 104. In other implementations, the quantum computing device 102 may be connected to the classical processor 104 through a network (e.g., a local area network (LAN), a wide area network (WAN), the Internet, or a combination thereof).

[0060] The classical processor 104 includes components for performing classical computations and may be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible, non-transitory storage medium, for execution by a data processing device or to control the operation of the data processing device. The computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of them.

[0061] The quantum computing device 102 includes components for performing quantum computations. For example, the quantum computing device 102 may include a qubit array, a quantum circuit system, and a control device configured to operate physical qubits in the qubit array and apply quantum circuits to the qubits. In some implementations, the quantum computing device 102 may be a noisy device, such as a noisy intermediate-scale quantum device (NISQ device), a trapped ion quantum computer, or a neutral atom quantum computer. Figure 6 An example quantum computing device is described in further detail.

[0062] The classical processor 104 is configured to receive a request 106 to evaluate the expected value of an operator (e.g., a projection operator or a matrix element between a Slater determinant and a vacuum state) with respect to a quantum state. The classical processor 104 is then configured to perform shadow tomography using the quantum computing device 102 and compute a classical shadow of the quantum state. The classical processor 104 repeatedly randomly samples a unitary operator from a set of unitary operators. The set of unitary operators includes a unitary operator 108 from a generalized matching gate group, as described below with reference to Figure 2 The example process 200 is described in more detail.

[0063] The classical processor 104 is configured to provide data 110 to the quantum computing device 102, which represents the randomly sampled unitary operator and the quantum state specified in the request 106. For each randomly sampled unitary operator, the quantum computing device 102 is configured to prepare an array of physical qubits 112 in the quantum state and apply a quantum circuit 114 to the quantum state. The application of the quantum circuit evolves the quantum state under the randomly sampled unitary operator. Then, the quantum computing device 102 is configured to measure 118 the evolved quantum state in the computational basis to obtain a corresponding bit string.

[0064] exist Figure 1 In the example shown, the quantum computing device 102 is configured to store a record of each measured bit string and the randomly sampled unitary operator corresponding to the measured bit string in the storage device 120. After the quantum computing device 102 has obtained one or more bit strings of each randomly sampled unitary operator 110, the quantum computing device 102 is configured to provide the stored record 120 to the classical processor 104 as a classical shadow 122 of the quantum state. In other implementations, the classical processor 104 can be configured to store a record of each bit string measured by the quantum computing device 102. Figure 2 Example operations performed by the quantum computing device 102 and the classical processor 104 to compute the classical shadow of a quantum state using generalized matching gates are described in more detail.

[0065] The classical processor 104 is configured to evaluate 124 a requested expected value of an operator with respect to the quantum state using the classical shadow 122 of the quantum state and provide as output an evaluated expected value 126. Due to the particular structure of the classical shadow 122, the classical processor 104 can evaluate this expected value efficiently, e.g., with a cost that scales at most cubically with the number of qubits used by the quantum computing device 102 to compute the classical shadow. Figure 3 and Figure 4 Example operations performed by the classical processor 104 for evaluating the expectation value of an operator with respect to a quantum state are described in more detail.

[0066] Figure 2 is a flow chart of a process 200 for computing the classical shadow of an n-qubit quantum state. For convenience, process 200 will be described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, a suitably programmed Figure 1 The system 100 may perform the example process 200 .

[0067] The system repeatedly samples unitary operators from a random unitary set by a classical processor (step 202). The random unitary set is a group of generalized matching gates, where each unitary operator in the set of unitary operators generates a corresponding generalized matching gate quantum circuit. The generalized matching gate quantum circuit is a quantum circuit that can be applied to n qubits with nearest neighbor interactions, for example, a one-dimensional chain of n qubits, where the quantum circuit includes two qubit gates that implement a rotation e of a certain angle θ. -iθXX (For example, by the operator The quantum gate generated by the rotation is given below in matrix form, where X represents the Pauli-X matrix. An example two-qubit gate:

[0068]

[0069] The quantum circuit also includes a single-qubit X gate on the last qubit in the qubit line, and optionally one or more single-qubit Z(θ) quantum gates. In some implementations, the single-qubit X gate can also be parameterized, for example depending on θ.

[0070] In other words, the generator of the generalized matching gate group consists of the operator X j X j+1 , X j Y j+1 , Y j X j+1 or Y j Y j+1 The unitary operator generated by the action on an array of n qubits, where 1≤j≤n1≤j≤n, X jrepresents the Pauli-X operator applied to qubit j, and Y j denotes the Pauli-Y operator applied to qubit j; and the unitary operator generated by the action of the Pauli-X operator on qubit n.

[0071] In some implementations, a matching gate is defined as a class of two-qubit gates consisting of the form The two-qubit X rotation of the form and A matching gate circuit can be defined as a nearest neighbor matching gate (for example, where n qubits are placed in a one-dimensional chain) and a PauliX operator X on the last qubit of the one-dimensional chain. n Another way to look at such a matched gate circuit is to view it as a qubit representation of a fermionic Gaussian unitary under the Jordan-Wigner transformation, where the Gaussian unitary is a unitary operator that transforms between valid sets of Majorana operators (i.e., equivalent to applying the matrix Q∈O(2n) to a set of 2n Majorana operators).

[0072] In some implementations, the system can randomly sample unitary operators from the random unitary set by sampling from the group O(2n) according to the Haar measure. The group of generalized matching gates has a one-to-one correspondence with the group of 2n×2n orthogonal matrices O(2n). For each element R in the group O(2n), there is a unique unitary operator in the group of generalized matching gates that satisfies Therefore, the system can sample random unitary operators from the group O(2n) and construct the corresponding generalized matching gate unitary operators from the sampled unitary operators using the one-to-one correspondence described above.

[0073] For each unitary operator randomly sampled at step 202, the system prepares an n-qubit quantum state using a quantum computing device, and applies a quantum circuit to the quantum state to obtain an evolved quantum state (step 204). Applying the quantum circuit to the n-qubit quantum state enables the evolution of the n-qubit quantum state under the sampled unitary operator. That is, the quantum circuit is a matching gate quantum circuit generated by the unitary operator. The system then measures the evolved quantum state, for example in a computational basis, to obtain a corresponding bit string (step 206). The system stores a record of the bit string and the random unitary operator used to generate the bit string (step 208).

[0074] After the system performs steps 204-208 one or more times for each randomly sampled unitary operator, the system provides a stored record as a classical shadow of the quantum state (step 210).

[0075] For example, classical shading is given by equation (1) below.

[0076]

[0077] In equation (1), J represents the number of unitary operators sampled at step 202, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a superoperator that projects its input onto a set of Majorana operators of degree 2l.

[0078] In some implementations, the system can use the classical shadow of the quantum state to perform one or more operations. For example, the system can use the classical shadow to predict the expected value of an operator with respect to the quantum state, as described below with reference to Figure 2 and Figure 3 Alternatively, the system can use or provide classical shadows for use in subsequent computations, for example, to perform direct fidelity estimation, entanglement verification, correlation function estimation, or entanglement entropy prediction.

[0079] Numerical experiments have demonstrated that the sampling efficiency of the example process 200, for example the number of physical measurements required to compute classical shadows, can scale efficiently with the system size n (e.g. Figure 5 shown).

[0080] Figure 3 is a flow chart of an example process 300 for computing the expected value of a projection operator. The projection operator can be an operator that projects a quantum state ρ onto a pure fermion Gaussian state |φ>. A quantum state is a pure fermion Gaussian state if it can be generated by applying a unitary operator from a generalized matching gate group to a vacuum state, e.g., |φ>=U|O>, where U is a unitary operator from a generalized matching gate group and |0) represents the vacuum state (which is, for example, a zero-particle state that maps to an all-zero qubit state under the Jordan-Wigner transformation). That is, the projection operator can be given by For convenience, process 300 will be described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, a suitably programmed Figure 1 The system 100 may perform the example process 300 .

[0081] The system obtains a classical shadow of an n-qubit quantum state (step 302). The classical shadow is generated using a quantum channel of unitary operators sampled from a set of random unitary operators and the measured bit string. Figure 2 The example process 200 describes a technique to prepare classical shadows. For example, the random unitary set can be a generalized matching gate group.

[0082] The system generates an updated unitary operator by multiplying the unitary operator defining the projection operator with i) a unitary operator sampled from a random unitary set and ii) an operator that prepares the measured bit string from the vacuum state (step 304). That is, the system redefines the unitary operator U to absorb the unitary operator and prepare |b from |0> j >, as shown below.

[0083] To effectively assess expected value (in represents the projection operator, and The classical shadow representing the quantum state ρ, e.g. as defined in equation (1) above), for any integer l∈[0, .., n], n qubits The unitary operator U in the generalized matching gate group and the computation of the ground state |b>, it is sufficient to evaluate the expression given by equation (2)

[0084]

[0085] Because n qubits Elements and super operators of the generalized matching gate group Commutative, so the unitary operator of the projective operator is defined as Can be used for To redefine the unitary operator U, thereby absorbing the unitary operator and prepare the calculation of the ground state |b> from the vacuum state.

[0086] The system then uses the redefined unitary operator to evaluate the expectation value of the projection operator relative to the classical shadow of the quantum state (step 306). Specifically, the system evaluates the expectation value of the quantum channel relative to the vacuum state. That is, the system evaluates the quantity given by equation (3) below,

[0087]

[0088] where U now represents the redefined unitary operator. It can be shown that the expression given by equation (3) above is equal to

[0089]

[0090] in is a polynomial, where Pf denotes the Pfaffian of the matrix, Q is an element of the special orthogonal group SO(2n) corresponding to the updated unitary operator U, and M0 is given by Thus, to evaluate the expectation value of the projection operator with respect to the classical shadow of the quantum state, the system can compute the derivative of the polynomial that includes the Pfaffian of the matrix, which in turn includes the updated unitary operator.

[0091] The system can use standard methods, such as numerical differentiation or polynomial interpolation techniques, to compute the matrix M0+zQM0Q for a particular value of z at a cost that scales at most cubically with n. T Thus, due to the structure of the classical shadow obtained at step 302, only the system implementing the example process 300 is required to perform multiple physical measurements that scale with the system size n (as described above with reference to Figure 2 ) and a certain amount of classical post-processing with n-cubic scaling.

[0092] Figure 4 is a flow chart of an example process 400 for computing the expected value of an operator. The operator may be the outer product of a Slater determinant |φ> and a vacuum state 0> (which is, for example, a zero-particle state having a mapping to an all-zero qubit state under a Jordan-Wigner transformation). The Slater determinant |φ> may be obtained by applying a unitary operator U in a generalized matching gate group. φ to generate, the generalized matching gate group saves the excitation number (and serves as the identifier on the vacuum state |0)) to the state with the excitation number (e.g., |φ>=U φ |x>). For example, the operator can be used For convenience, process 400 will be described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, a suitably programmed Figure 1 The system 100 may perform the example process 400 .

[0093] The system obtains a classical shadow of an n-qubit quantum state (step 402). The classical shadow is generated using a quantum channel of unitary operators sampled from a set of random unitary operators and the measured bit string. Figure 2 The example process 200 describes a technique to prepare classical shadows. For example, the random unitary set can be a generalized matching gate group.

[0094] The system generates an updated unitary operator by multiplying the unitary operator defining the operator with the unitary operator sampled from the random unitary set (step 304). That is, the system redefines the unitary operator U to absorb the unitary operator U φ , as shown below.

[0095] To effectively assess expected value (where |φ>=U φ|x> represents the Slater determinant, |0> represents the vacuum state, and The classical shadow representing the quantum state ρ, as defined in equation (1) above), for any integer l∈[0, .., n], n qubits The unitary operator U in the generalized matching gate group and the calculated ground state |b>, it is sufficient to evaluate the expression given in equation (5)

[0096]

[0097] Because n qubits Elements and super operators of the generalized matching gate group Commutative, so define the unitary operator U of the Slater determinant φ Can be used for To redefine the unitary operator U, thereby absorbing the unitary operator U φ .

[0098] The system then uses the redefined unitary operator to evaluate the expectation value of the operator relative to the classical shadow of the quantum state (step 406). Specifically, the system evaluates the expectation value of the quantum channel relative to the outer product of the computational ground state included in the operator and the vacuum state. That is, the system evaluates the quantity given by equation (6) below,

[0099]

[0100] where U now represents the redefined unitary operator. It can be shown that the expression given by equation (6) above is equal to

[0101]

[0102] in is a polynomial where Pf denotes the Pfaffian of the matrix, Q is an element of the special orthogonal group SO(2n) corresponding to the updated unitary operator U, M[S] denotes the constraints of the matrix S on the rows and columns with indices in S, and for x∈{0,1}n, in S x =[2n]{2j-1:x j =1}. Thus, to evaluate the expectation value of the operator with respect to the classical shadow of the quantum state, the system can compute the derivative of a polynomial that includes the Pfaffian of the matrix that in turn includes the updated unitary operator.

[0103] The system can use standard methods, such as numerical differentiation or polynomial interpolation techniques, to compute the Pfaffian of the matrix for a particular z value at a cost that scales at most cubically with n. Thus, due to the structure of the classical shadow obtained at step 402, only the system implementing the example process 400 is required to perform multiple physical measurements that scale with the system size n (as described above with reference to Figure 2 ) and a certain amount of classical post-processing with n-cubic scaling.

[0104] Figure 5 Two example graphs are shown that illustrate the sampling efficiency of the presently described techniques. Specifically, Figure 5 includes b(n,ζ) (for classical shadow estimation using the matching gate described currently |φ>< 0 Variance bounds on the expectation value of ||φ>, where |φ> is any n-mode, ζ-fermion Slater determinant) versus n Linear and log-log plots 500, 550, where ξ∈{0, 2, 10, 50, 100, 200, 500} (evaluated and connected at integer values ​​of n). The quantity y=√n ln(n) is also plotted for comparison. As described herein, estimating the expectation value allows estimating the overlap between any pure state and ||φ>. The quantity b(n,0) is also equivalent to the variance bound for estimating the expectation value of an arbitrary Gaussian density operator.

[0105] As shown in the figure, the variance is calculated for various parameters, and in each case, the variance is small, and as n gets larger, the variance is always below This means that the number of measurements with fixed precision is better than this.

[0106] Figure 6 An example classical / quantum computer 600 is depicted for performing some or all of the classical and quantum operations described in this specification. The example classical / quantum computer 600 includes an example quantum computing device 602. Quantum computing device 602 is intended to represent various forms of quantum computing devices. The components shown here, their connections and relationships, and their functions are exemplary only and are not intended to limit implementations of the inventions described and / or claimed in this document.

[0107] The example quantum computing device 602 includes a qubit assembly 652 and a control and measurement system 604. The qubit assembly includes a plurality of qubits, such as qubit 606, which are used to perform an algorithmic operation or quantum computation. Figure 6 The qubits are shown arranged in a rectangular array, but this is a schematic depiction and is not intended to be limiting. The qubits may also be arranged in other forms, such as in a line. Qubit assembly 652 also includes adjustable coupling elements, such as coupler 608, which allow interaction between coupled qubits. Figure 6 In the schematic depiction of , each qubit is adjustably coupled to each of its four neighboring qubits through a corresponding coupling element. However, this is an example arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that allow coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits.

[0108] Each qubit can be a physical two-level quantum system or device with levels representing logical values ​​0 and 1. The specific physical implementation of multiple qubits and how they interact with each other depends on various factors, including the type of quantum computing device 602 included in the example computer 600, or the type of quantum computing that the quantum computing device is performing. For example, in an atomic quantum computer, a qubit can be implemented via an atom, a molecule, or a solid-state quantum system (e.g., a hyperfine atomic state). As another example, in a superconducting quantum computer, a qubit can be implemented via a superconducting qubit or a semiconductor qubit (e.g., a superconducting transmon state). As another example, in an NMR quantum computer, a qubit can be implemented via a nuclear spin state.

[0109] In some implementations, quantum computations may be performed by loading qubits from a quantum memory and applying a series of unitary operators to the qubits. Applying unitary operators to the qubits may include applying a corresponding sequence of quantum logic gates to the qubits, for example to implement a quantum algorithm such as a quantum principal component algorithm. Example quantum logic gates include single-qubit gates such as Pauli-X, Pauli-Y, Pauli-Z (also referred to as X, Y, Z), Hadamard gates, S gates, rotations; two-qubit gates such as controlled X gates, controlled Y gates, controlled Z gates (also referred to as CX, CY, CZ), controlled NOT gates (also referred to as CNOT), controlled exchange gates (also referred to as CSWAP); and gates involving three or more qubits, such as Toffoli gates. Quantum logic gates may be implemented by applying control signals 610 generated by a control and measurement system 604 to the qubits and couplers.

[0110] For example, in some implementations, the qubits in qubit assembly 652 may be frequency tunable. In these examples, each qubit may have an associated operating frequency that may be adjusted by applying a voltage pulse via one or more drive lines coupled to the qubit. Example operating frequencies include qubit idle frequency, qubit interaction frequency, and qubit readout frequency. Different frequencies correspond to different operations that a qubit may perform. For example, setting the operating frequency to a corresponding idle frequency may place the qubit in a state in which it does not strongly interact with other qubits, but may be used to perform a single-qubit gate. As another example, in the case where qubits interact via a coupler with a fixed coupling, the qubits may be configured to interact with each other by setting their respective operating frequencies to a certain gate-related frequency that is detuned from their common interaction frequency. In other cases, for example, when qubits interact via a tunable coupler, the qubits may be configured to interact with each other by setting the parameters of their respective couplers to enable interaction between qubits, and then by setting the respective operating frequencies of the qubits to a certain gate-related frequency that is detuned from their common interaction frequency. Such interactions may be performed to perform multi-qubit gates.

[0111] The type of control signal 610 used depends on the physical implementation of the qubit. For example, the control signal may include RF or microwave pulses in NMR or superconducting quantum computer systems, or light pulses in atomic quantum computer systems.

[0112] Quantum computations can be accomplished by measuring the state of a qubit (e.g., using a quantum observable, such as X or Z) using a corresponding control signal 610. The measurement causes a readout signal 612 representing the measurement result to be transmitted back to the measurement and control system 604. The readout signal 612 may include an RF, microwave, or optical signal, depending on the physical scheme of the quantum computing device and / or qubit. For convenience, Figure 6 The illustrated control signals 610 and readout signals 612 are depicted as addressing only selected elements of the qubit assembly (ie, the top and bottom rows), but during operation, the control signals 610 and readout signals 612 may address every element in the qubit assembly 652.

[0113] The control and measurement system 604 is an example of a classical computer system that can be used to perform the various operations on the qubit components 652 described above, as well as other classical subroutines or computations. The control and measurement system 604 includes one or more classical processors (e.g., classical processor 614), one or more memories (e.g., memory 616), and one or more I / O units (e.g., I / O unit 618) connected via one or more data buses. The control and measurement system 604 can be programmed to send a sequence of control signals 610 to the qubit components, for example, to perform a series of selected quantum gate operations, and to receive a sequence of readout signals 612 from the qubit components, for example, as part of performing a measurement operation.

[0114] The processor 614 is configured to process instructions for execution within the control and measurement system 604. In some implementations, the processor 614 is a single-threaded processor. In other implementations, the processor 614 is a multi-threaded processor. The processor 614 is capable of processing instructions stored in the memory 616.

[0115] The memory 616 stores information within the control and measurement system 604. In some implementations, the memory 616 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 616 may include a storage device capable of providing mass storage for the system 604, such as a hard disk device, an optical disk device, a storage device shared by multiple computing devices on a network (e.g., a cloud storage device), and / or some other mass storage device.

[0116] Input / output devices 618 provide input / output operations for control and measurement system 604. Input / output devices 618 may include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers, thereby sending control signals 610 to qubit components and receiving readout signals 612 from qubit components, as appropriate for the physical scheme of a quantum computer. In some implementations, input / output devices 618 may also include one or more network interface devices (e.g., Ethernet cards), serial communication devices (e.g., RS-232 ports), and / or wireless interface devices (e.g., 802.11 cards). In some implementations, input / output devices 618 may include driver devices configured to receive input data and send output data to other external devices (e.g., keyboards, printers, and display devices).

[0117] Although the example control and measurement system 604 has been Figure 6However, the subject matter and functional operations described in this specification may be implemented in other types of digital electronic circuit systems or in computer software, firmware or hardware, including the structures disclosed in this specification and their structural equivalents or a combination of one or more of them.

[0118] The example system 600 includes an example classical processor 650. The classical processor 650 can be used to perform the classical computing operations described in this specification according to some implementations.

[0119] The subject matter and implementation of the operations described in this specification may be implemented in digital electronic circuit systems, analog electronic circuit systems, suitable quantum circuit systems, or more generally, quantum computing systems, in tangibly embodied software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or a combination of one or more of them. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptographic system, or a quantum simulator.

[0120] Implementations of the subject matter described in this specification may be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or to control the operation of the data processing device. The computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagated signal capable of encoding digital and / or quantum information, such as a machine-generated electrical, optical, or electromagnetic signal, which is generated to encode digital and / or quantum information for transmission to an appropriate receiver device for execution by the data processing device.

[0121] The terms quantum information and quantum data refer to information or data carried, held or stored by a quantum system, wherein the smallest non-trivial system is a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., having two or more energy levels. For example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational ground state is identified by the ground state and the first excited state, however it should be understood that other arrangements in which computational states are identified by higher-order excited states are possible.

[0122] The term "data processing device" refers to digital and / or quantum data processing hardware, and covers all types of devices, apparatuses and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may also be or further include a dedicated logic circuit system, such as an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system. Specifically, a quantum simulator is a dedicated quantum computer that does not have the ability to perform general quantum computations. In addition to hardware, the device may also include code that creates an execution environment for a digital and / or quantum computer program, for example, code that constitutes a processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0123] A digital computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it may be deployed in any form, including as a stand-alone program or as a module, component, subroutine or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or may be written in a quantum programming language such as QCL or Quipper.

[0124] A computer program may, but need not, correspond to a file in a file system. A program may be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, subroutines, or code portions). A computer program may be deployed to execute on one computer or on multiple 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 send quantum data using a quantum system, such as a qubit. Generally speaking, a digital data communication network cannot send quantum data, however, a quantum data communication network can send both quantum data and digital data.

[0125] The processes and logic flows described in this specification may be performed by one or more programmable computers operating with one or more processors as appropriate, executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows may also be performed by a dedicated logic circuit system, and the device may also be implemented as a dedicated logic circuit system, such as an FPGA or ASIC or a quantum simulator, or by a combination of a dedicated logic circuit system or a quantum simulator and one or more programmed digital and / or quantum computers.

[0126] For a system of one or more computers to be "configured to" perform a particular operation or action, it is meant that the system has installed thereon software, firmware, hardware, or a combination thereof that, in operation, causes the system to perform the operation or action. For one or more computer programs to be configured to perform a particular operation or action, it is meant that the one or more programs include instructions that, when executed by a data processing device, cause the device to perform the operation or action. For example, 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.

[0127] Computers suitable for executing computer programs may be based on general or special purpose processors or any other kind of central processing unit. Typically, the central processing unit will receive instructions and data from a read-only memory, a random access memory, or a quantum system suitable for sending quantum data (e.g. photons), or a combination thereof.

[0128] Elements of a computer include a central processing unit for executing or implementing instructions and one or more memory devices for storing instructions and digital, analog and / or quantum data. The central processing unit and the memory may be supplemented by or incorporated into a dedicated logic circuit system or a quantum simulator. In general, a computer will also include one or more mass storage devices for storing data, such as magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information, or be operatively coupled to receive data from or transmit data to or both of the one or more mass storage devices. However, a computer need not have such devices.

[0129] Quantum circuit elements (also called quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, quantum circuit elements are configured to utilize quantum mechanical phenomena such as superposition and entanglement to perform operations on data in a non-deterministic manner. Certain quantum circuit elements such as qubits can be configured to simultaneously represent information in more than one state and operate on the information. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUIDs or DC-SQUIDs), etc.

[0130] In contrast, classical circuit elements typically process data in a deterministic manner. Classical circuit elements can be configured to collectively execute instructions of a computer program by performing basic arithmetic, logic, and / or input / output operations on data, where the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to send data to and / or receive data from quantum circuit elements via electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuit systems, fast single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices, and ERSFQ devices, where ERSFQ devices are energy-saving versions of RSFQ that do not use bias resistors.

[0131] In some cases, some or all of the quantum and / or classical circuit elements may be implemented using, for example, superconducting quantum and / or classical circuit elements. The manufacture of superconducting circuit elements may require the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering) or epitaxial techniques and other deposition processes may be used to deposit these materials. The process for manufacturing circuit elements described herein may require the removal of one or more materials from the device during manufacture. Depending on the material to be removed, the removal process may include, for example, wet etching techniques, dry etching techniques or stripping processes. The materials forming the circuit elements described herein may be patterned using known photolithography techniques (e.g., photolithography or electron beam lithography).

[0132] During operation of a quantum computing system using superconducting quantum circuit elements and / or superconducting classical circuit elements (such as the circuit elements described herein), the superconducting circuit elements are cooled in a cryostat to a temperature that allows the superconducting material to exhibit superconducting properties. Superconductor (or superconducting) materials can be understood as materials that exhibit superconducting properties at or below the superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature is 1.2 Kelvin) and niobium (superconducting critical temperature is 9.3 Kelvin). Therefore, superconducting structures such as superconducting traces and superconducting ground planes are formed of materials that exhibit superconducting properties at or below the superconducting critical temperature.

[0133] In certain implementations, classical circuit elements electrically and / or electromagnetically coupled to quantum circuit elements (e.g., qubits and qubit couplers) may be used to provide control signals for the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0134] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including, for example, semiconductor memory devices, such as EPROM, EEPROM and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data for a long time with high fidelity and efficiency, such as light-matter interfaces that use light for transmission, use matter for storage, and preserve quantum characteristics such as superposition or quantum coherence of quantum data.

[0135] The control of the various systems or parts thereof described in this specification may be implemented in a computer program product, which includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more processing devices. The systems or parts thereof described in this specification may each be implemented as an apparatus, method, or system, which may include one or more processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0136] Although this specification contains many implementation details, these details should not be interpreted as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to a particular implementation. Certain features described in this specification in the context of separate implementations may also be implemented in a single implementation in combination. Conversely, individual features described in the context of a single implementation may also be implemented in multiple implementations, either individually or in any suitable subcombination. In addition, although features are described above as acting in certain combinations and even initially claimed, one or more features from a claimed combination may be separated from the combination in some cases, and a claimed combination may be directed to a subcombination or a change in a subcombination.

[0137] Similarly, although operations are depicted in a particular order in the accompanying drawings, this should not be understood as requiring that the operations be performed in the particular order shown or in a continuous order, or that all of the operations shown be performed, to achieve the desired result. In some contexts, multitasking and parallel processing can be advantageous. In addition, the separation of the various system modules and components in the implementation described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged in multiple software products.

[0138] Specific implementations of the subject matter have been described. Other implementations are within the scope of the appended claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired results. As an example, the processes depicted in the accompanying drawings do not necessarily require the particular order or ordered sequence shown to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for computing a classical shadow of an n-qubit quantum state, the method comprising: Repeatedly sampling unitary operators from a random unitary set by a classical computer, wherein the random unitary set includes a generalized matching gate group; For each sampled unitary operator: applying, by a quantum computer, a quantum circuit to the n-qubit quantum state to obtain an evolving quantum state, wherein the quantum circuit implements the sampled unitary operator, The quantum computer measures the evolving quantum state to obtain a corresponding bit string, and storing, by the classical computer, a record of the corresponding bit string and the sampled unitary operator; as well as The record is provided by the classical computer as a classical shadow of the quantum state.

2. The method of claim 1, wherein the generator of the generalized matching gate group comprises: By operator X j X j+1 , X j Y j+1 , Y j X j+1 or Y j Y j+1 The unitary operator generated by the action on an array of n qubits, where 1≤j≤n, X j represents the Pauli-X operator applied to qubit j, and Y j represents the Pauli-Y operator applied to qubit j; and Unitary operator generated by the action of the Pauli-X operator on qubit n.

3. The method of claim 1 or claim 2, wherein The generalized matching gate group has a one-to-one correspondence with the 2n×2n orthogonal matrix group O(2n); and For each element R in the group O(2n), there exists a unique unitary operator in the generalized matching gate group that satisfies 4. The method of claim 3, wherein sampling the unitary operator from the random unitary set comprises sampling from the group O(2n) according to the Haar measure and constructing a corresponding generalized matching gate unitary operator using the one-to-one correspondence.

5. A method according to any one of claims 1 to 4, wherein the classical shadow is given by where J represents the number of repetitions performed to sample the unitary operator, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a superoperator that projects its input onto a set of Majorana operators of degree 2l.

6. The method of any one of claims 1 to 5, further comprising using the classical shadow of the quantum state to perform one or more operations, the operations comprising one or more of the following: Predict an expected value of an observable relative to the quantum state, perform direct fidelity estimation, perform entanglement verification, estimate a correlation function, or predict entanglement entropy.

7. A computer-implemented method for computing an expected value of a projection operator, the method comprising: Obtaining a classical shadow of an n-qubit quantum state, wherein the classical shadow comprises a quantum channel of unitary operators sampled from a random unitary set and the measured bit string; generating an updated unitary operator comprising multiplying a unitary operator defining the projection operator with i) the unitary operator sampled from the random unitary set and ii) an operator that prepares the measured bit string from a vacuum state; as well as An expectation value of the quantum channel with respect to the vacuum state is calculated, the calculation comprising evaluating a derivative of a polynomial comprising: a Pfaffian of a matrix including the updated unitary operator.

8. The method of claim 7, wherein the projection operator comprises an operator that projects the quantum state onto a pure fermion Gaussian state.

9. The method of claim 8, wherein the pure fermion Gaussian state comprises unitary operators from the random unitary set applied to the vacuum state.

10. The method of claim 9, wherein the projection operator is is given by , where |0> represents the vacuum state, and is the unitary operator defining the projection operator, where in the random unitary set.

11. The method of any one of claims 7 to 10, wherein the random unitary set comprises a generalized matching gate group.

12. The method of claim 11, wherein the classical shadow is given by where J represents the number of repetitions performed to sample the unitary operator, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a superoperator that projects the input onto a set of Majorana operators of degree 2l, where Equivalent to the quantum channel.

13. The method of claim 12, wherein generating the updated unitary operator comprises redefining the unitary operator U j To absorb the unitary operator and prepare |b from |0) j >.

14. The method of claim 13, wherein calculating the expectation value of the quantum channel relative to the vacuum state comprises calculating the expectation value of the superoperator relative to the vacuum state, wherein the superoperator projects the updated unitary operator applied to the vacuum state onto a set of Majorana operators of degree 2l.

15. The method of claim 14, wherein calculating the expectation value of the superoperator relative to the vacuum state comprises calculating 16. The method of any one of claims 7 to 15, wherein the matrix including the updated unitary operator is composed of M0+zQM0Q T Given, where z is the independent variable of the polynomial, and Q represents the element of SO(2n) corresponding to the updated unitary operator.

17. The method of any one of claims 7 to 16, wherein evaluating the derivative of the polynomial comprises performing numerical differentiation or polynomial interpolation techniques.

18. A method for calculating an expected value of an operator, the method comprising: Obtaining a classical shadow of an n-qubit quantum state, wherein the classical shadow comprises a quantum channel of unitary operators sampled from a random unitary set and the measured bit string; generating an updated unitary operator, comprising multiplying a unitary operator defining said operator with said unitary operator sampled from said random unitary set; and An expectation value of the quantum channel with respect to an outer product of a computational ground state and a vacuum state included in the operator is calculated, the calculation comprising evaluating a derivative of a polynomial comprising: a Pfaffian of a matrix including the updated unitary operator.

19. The method of claim 18, wherein the operators include Slater determinants and vacuum states.

20. The method of claim 19, wherein the Slater determinant comprises a unitary operator in the random unitary set, the unitary operator preserving the excitation number applied to a calculated ground state having an excitation number.

21. The method of claim 20, wherein the operator is Given, where |0) represents the vacuum state, U φ is the unitary operator defining the operator, where U φ In the random unitary set, and |x> represents the computational ground state.

22. The method of any one of claims 18 to 21, wherein the random unitary set comprises a generalized matching gate group.

23. The method of claim 22, wherein the classical shadow is given by where J represents the number of repetitions performed to sample the unitary operator, and U j represents the unitary operator sampled at repetition j, b j represents the bit string stored at repetition j, and represents a superoperator that projects the input onto a set of Majorana operators of degree 2l, where Equivalent to the quantum channel.

24. The method of claim 23, wherein generating the updated unitary operator comprises redefining the unitary operator U j To absorb the unitary operator U φ .

25. The method of claim 23, wherein calculating the expectation value of the quantum channel relative to the outer product of the computational ground state and the vacuum state included in the operator comprises: An expectation value of the superoperator relative to the outer product of a computational ground state included in the operator and the vacuum state is calculated, wherein the superoperator projects the updated unitary operator applied to the bit string onto a set of Majorana operators of degree 2l.

26. The method of claim 25, wherein calculating the expectation value of the superoperator relative to the outer product of the calculated ground state included in the operator and the vacuum state comprises calculating 27. The method of any one of claims 18 to 26, wherein the matrix including the updated unitary operator is composed of Given, where for x∈{0,1}, z is the independent variable of the polynomial, Q represents the element of O(2n) corresponding to the updated unitary operator, ,in S x =[2n]{2j-1:x j =1} and M[S] represents the constraints of the matrix S on the rows and columns with indices in S.

28. The method of any one of claims 18 to 27, wherein evaluating the derivative of the polynomial comprises performing numerical differentiation or polynomial interpolation techniques.

29. A system, comprising: Quantum computers; as well as A classical computer coupled to the quantum computer, the classical computer comprising: one or more data processing devices; and a non-transitory computer-readable storage medium in data communication with the one or more data processing devices and storing instructions executable by the data processing devices; The system is configured to perform the operations of the method according to any one of claims 1 to 6.

30. The system of claim 29, wherein the quantum computer comprises a noisy quantum computing device.

31. The system of claim 29, wherein the quantum computer comprises a superconducting quantum computer.

32. The system of claim 29, wherein the quantum computer comprises an analog simulator based on neutral atoms or ion traps.

33. A system, comprising: one or more data processing devices; as well as A non-transitory computer-readable storage medium that communicates data with the one or more data processing devices and stores instructions that, when executed by the data processing devices, cause the one or more data processing devices to perform the operations of the method according to any one of claims 7 to 28.

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