REDUCED DENSITY MATRIX ESTIMATION FOR Fermi SUBSYSTEM USING PARTICULATE PARTICLE CONVERSION OF

Abstract

A computing system including a classical computing device includes a processor that generates a Haar-random unitary matrix. The processor further calculates a single-particle-based Fermi rotation based at least in part on the Haar-random unitary matrix, and outputs the single-particle-based Fermi rotation to a quantum computing device. A quantum computing device receives a specification of a Fermi wave function and also receives a single particle based Fermi rotation. The quantum computing device also applies a single-particle-based Fermi rotation to the Fermi wavelet function. The quantum computing device also measures the rotated Fermi wavelet function to obtain a classical shadow measurement. The processor of the classic computing device also receives classic shadow measurements. The processor also estimates k-reduced density matrix (k-RDM) elements of a k-RDM of the Fermi wavelet function based at least in part on the classical shadow measurements and the Haar-random unitary matrix. The processor also outputs the k-RDM element.

Description

Background Art

[0001] Fermions are particles with wave functions that cannot occupy the same quantum state at the same time. For example, electrons, protons, and neutrons are fermions. In quantum chemistry, for systems that include fermions, it is often desirable to estimate the value of an observable phenomenon, or simply an observable quantity. The observable quantity can be, for example, energy or a dipole moment. When getting closer to the behavior of a quantum chemical system, for example when estimating the amount of energy absorbed or released by a chemical reaction, the value of the observable quantity can be used. Summary of the invention

[0002] A computing system including a classical computing device is provided. The classical computing device includes a processor that generates a Haar-random unitary matrix, the processor calculates a single-particle basis fermion rotation based at least in part on the Haar-random unitary matrix, and outputs the single-particle basis fermion rotation to a quantum computing device. The quantum computing device receives a fermion single-particle basis fermion rotation and a specification of a fermion wave function. The fermion wave function over a plurality of fermions is standardized, and the plurality of fermions are in a fermion system in which the number of particles occupying a plurality of modes is conserved. The quantum computing device also applies the single-particle basis fermion rotation to the fermion wave function to obtain a rotated fermion wave function. The quantum computing device measures the rotated fermion wave function to obtain a classical shadow measurement result, which encodes an estimate of which mode in the mode is occupied by the fermion. The processor of the classical computing device also receives a classical shadow measurement result. The processor estimates a k-RDM element of a k-reduced density matrix (k-RDM) of a fermion wave function based at least in part on the classical shadow measurement result and the Haar-random unitary matrix. The processor outputs the k-RDM element to an additional computation process.

[0003] This summary is intended to introduce a selection of concepts in a simplified form, which are described in detail below. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. Furthermore, the claimed subject matter is not limited to implementations that solve any or all of the disadvantages mentioned in any part of this disclosure. BRIEF DESCRIPTION OF THE DRAWINGS

[0004] Figure 1 A computing system including a quantum computing device and a classical computing device according to an example embodiment is schematically illustrated.

[0005] Figure 2 In some examples, Figure 1 The data flow that occurs when estimates of observables are computed in a computing system.

[0006] Figure 3A It shows that it can be Figure 1A flowchart of an example method executed at a computing system of FIG.

[0007] Figure 3B shows that can be performed in some examples Figure 3A Additional steps of the method.

[0008] Figure 4 shows which can be instantiated Figure 1 A schematic diagram of an example computing environment for a computing system of FIG. DETAILED DESCRIPTION

[0009] Methods for determining properties of fermionic wave functions often have high computational complexity. For example, in the most general representation of fermionic wave functions as a sum of Slater determinants, the complexity grows exponentially with the number of modes. In some fermionic systems, such as those in which strong correlations are present, the complexity of computing properties of the system can be high even when using theoretically optimal algorithms. Although dedicated compact representations such as tensor network states and electron density can facilitate tractable parameter counting and often allow sufficient accuracy, these methods have lower accuracy for some fermionic systems.

[0010] Quantum computing devices allow the use of algorithms that provide significantly reduced computational complexity in many quantum chemical problems. For example, quantum computing devices can be used to implement variational quantum eigensolvers on which the ground state energy of the Hamiltonian is computed. In many quantum chemical applications of quantum computing devices, the estimation of multiple k-reduced density matrix (k-RDM) elements In the above expression, k is the number of ladder operator pairs used to compute the matrix elements. The k-reduced density matrix elements are then summed to compute estimates of the observables of interest (such as energy or dipole moment). For some special cases, such as 1-RDM or certain groups of k-RDMs, these methods for computing k-RDM elements are generally not applicable.

[0011] In recent years, quantum algorithms exploiting classical shadowing have been considered as a potential solution to the problem of efficiently computing k-RDM elements for fermionic systems. Using classical shadowing, an exponentially large number of observables that are local or low-rank can be predicted within a constant error threshold using polynomially many measurements. These observables can be computed on a random single-qubit basis or on a global Clifford basis chosen a priori. These algorithms exploiting classical shadowing can be extended to fermions via fermion-to-qubit mappings, such as the Jordan-Wigner mapping or the Bravyi-Kitaev mapping. However, observables that are naturally expressed in terms of fermionic operators are not easily converted to observables expressed in terms of Pauli operators. Expressing fermionic observables in terms of Pauli operators incurs considerable overhead, making it necessary to estimate the k-RDM on n modes. A quantum bit shadow.

[0012] In order to reduce the number of classical shadows used in the estimation of k-RDM elements, the classical shadows can be tailored to the fermion wave function of interest. In one previous approach, random fermion rotations are applied to the fermion wave function when computing the classical shadows. However, even when the fermion rotations are randomized in the Majorana basis, the classical shadows to estimate the k-RDM on n modes. In addition, in this previous approach, randomization of a constant number of fermion arrangements requires autonomous sampling of classical shadows on a random single qubit basis, and therefore requires a specific choice of the mapping of fermions to qubits to achieve tomographic completeness. The use of this bootstrap method further increases the computational complexity. In the Majorana representation, since the k-RDM decomposes into Majorana operators, so the sample complexity is optimal. “Sample complexity” here refers to the number of sample values ​​of the k-RDM required to achieve an error level below a specified threshold.

[0013] The devices and methods described herein overcome previous expectations that the sample complexity of classical shadows for fermion systems has generally not been improved. As discussed in further detail below, the random basis of the classical shadow can be adjusted based on observables that more succinctly describe the problem of calculating k-RDM elements than applying random rotations on the Majorana basis. For example, many electronic structure Hamiltonians are more easily represented using the number-conserving k-RDM than using the Majorana representation. In addition, many electronic structure Hamiltonians exhibit particle number symmetry. Therefore, a random basis can be selected that respects this symmetry. The basis can also be selected so that the public observable is of low rank. Specifically, a random basis measurement is selected for a monomer number-conserving fermion rotation, which is given by the following formula:

[0014] In the above formula, is the creation operator, a q is the annihilation operator, is a rotation space, ~ means "sampled from...", and Haar() means Haar integral. In addition, in the above formula, ln(u) pq represents the entry at row p and column q in the matrix logarithm ln(u).

[0015] Using the above expression for rotation can lead to significant computational savings. In any qubit representation of fermions, only The monomer number-conserving fermionic rotation is achieved with a single quantum gate. In addition, unlike the rotations used in previous methods that use classical shadowing to calculate k-RDM elements, the monomer number-conserving fermionic rotation is tomographically complete.

[0016] Although the number-conserving fermion rotation can reduce the sample complexity of computing k-RDM elements, the above formula for the rotation is not t-designed except in the single-particle subspace. In addition, the Haar integral in the random basis is a polynomial of order t = 3η, where η is the number of fermions. Since the degree of the polynomial can be very large, the Haar integral can be computationally expensive.

[0017] Compared to using a number-conserving fermion rotation basis, computing the k-RDM elements using number-conserving fermion rotations can be significantly more sample efficient. Multiple k-RDMs can be estimated in parallel using multiple samples given by:

[0018] As before The sample complexity given by the above formula is a super-exponential improvement over the scaling method, since n can be arbitrarily larger than η. Even for the worst-case scaling in the half-filled example, where η = n / 2, the sample complexity is given by: When k<<n, its ratio is 4 k The coefficient of is smaller, as is usually the case. In addition, the estimation algorithm is computationally efficient for any fixed k.

[0019] Using the devices and methods discussed in this paper, entangled fermionic properties can be efficiently computed for frequently applicable fermionic wave functions and classes of systems. Additionally, classical shadows can be encoded using small amounts of data. The methods discussed in this paper converge quickly to accurate estimates of common observables of interest and can be applied to any fermionic wave function.

[0020] Now turn to Figure 1 , schematically illustrates a computing system 10 according to an example embodiment. Figure 1 The computing system 10 depicted in FIG. 1 includes a quantum computing device 12 communicatively coupled to a classical computing device 20. At the quantum computing device 12, a plurality of qubits are instantiated at a qubit encoding surface. The quantum computing device 12 may include a plurality of logical qubits, each logical qubit being formed by a respective plurality of physical qubits. The plurality of physical qubits forming each logical qubit may be used to encode the state of the logical qubit and perform quantum error correction. At the quantum computing device 12, gates and measurements are performed on the logical qubits to perform quantum computations. For example, the quantum computing device 12 may be a topological quantum computing device.

[0021] Figure 1 The illustrated classical computing device 20 includes a processor 22 . Figure 1 The classical computing device 20 also includes a memory 24 communicatively coupled to the processor 22. The classical computing device 20 is configured to receive measurements from the quantum computing device 12 and perform further computations on those measurements at the processor 22. In some examples, the quantum computing device 12 may also receive control inputs from the classical computing device 20.

[0022] like Figure 1 As shown, the processor 22 of the classical computing device 20 generates a Haar-random unitary matrix u having a dimension of n×n. As discussed in further detail below, the Haar-random unitary matrix u can be randomized with respect to the orbital rotation U(u). In addition, the processor 22 calculates the single-particle basis fermion rotation U(u) based at least in part on the Haar-random unitary matrix u. The processor 22 also outputs the single-particle basis fermion rotation U(u) to the quantum computing device 12.

[0023] like Figure 1 As shown in the example of , the quantum computing device 12 receives specifications of fermion wave functions ρ on n fermions 14 in a population-conserving fermion system 36 occupying n modes 16. For example, the fermions 14 can be electrons and atomic nuclei contained in molecules, which can simulate the fermion wave function ρ. The specifications of the fermion wave function ρ can be received, for example, as eigenstates of the Hamiltonian.

[0024] The quantum computing device 12 receives the single-particle-based fermion rotation U(u) from the classical computing device 20. The quantum computing device 12 then applies the single-particle-based fermion rotation U(u) to the fermion wave function to obtain a rotated fermion wave function

[0025] The quantum computing device 12 also measures the rotated fermion wave function To obtain the classic shadow measurement results Classic shadow measurement results It is a classic shadow The classical shadow measurement result encodes an estimate of which of the modes is occupied by the fermions. The quantum computing device 12 converts the classical shadow measurement result The classical shadow measurement results are sent to the classical computing device 20, which receives and further processes the classical shadow measurement results.

[0026] When the classical shadow of the fermion wave function ρ changes randomly with the U basis By When sampling from a given probability distribution, we can The result of this approximation process is given by the following measurement channel:

[0027] Inverting the measurement channel produces an unbiased estimate of the density matrix:

[0028] Inverting the measurement channel can be done without explicitly calculating Estimate the expected value of any observable under the condition of . The expected value of an observable can be estimated as:

[0029] Given a finite number N of classical shadows and the corresponding Haar-random unitary matrix An estimate of O is given by:

[0030] Therefore, computing system 10 can estimate O without directly computing because The dimension of may be large, so the above process for estimating O may allow for saving processing time and memory usage.

[0031] Now offering about Classic Shadow The classical shadow of the quantum state ρ is the result of the measurement in a random basis indexed by u. The associated bit string Measuring a specific value The probability of is given by:

[0032] The above formula for probability indicates the reversal state is an estimate of ρ. The quantum channel corresponding to the measurement of the inversion state is given by the equation discussed above. In addition, as discussed above, an unbiased estimate of the fermion wave function ρ can be obtained by inverting the measurement channel and can be applied to the unbiased estimate by calculation As mentioned above, using classical shadowing to compute the estimated value of the observable O can result in savings in processing time and memory. It is difficult to express explicitly due to its large dimensionality, but can be avoided when the measurement channel is effectively reversible

[0033] The variance of the estimate of O is given by:

[0034] The upper bound of the variance of the estimate of O is the shadow norm

[0035] Measurement channel and the shadow norm can be expressed in the form of a t-design. The equations for the measurement channel and the shadow norm can be expressed in terms of the rotation operator Write as follows:

[0036] In the above formula, as well as is the dimension of ρ.

[0037] The following discussion provides a formalism that can be used to describe the rotations applied to the fermion wave function ρ. The following definitions are provided. The n-dimensional complex vector space V n The base element is defined as η-fermion space ∧ n V n The base element is in These basis elements of the η-fermion space are indexed by the following ordered set of integers:

[0038] symbol Also for is defined. In addition, The rank-1 projection on is defined as: where the digital operators are composed of ladder operator pairs Given. When , the rank-1 projection operator is given by:

[0039] In this example, It is rank In the layout unit The projection on the space occupied by it.

[0040] set up is the base element V n The unitary operation on ∧ n V n The expression U η (u) is a fermion rotation which then has the following action:

[0041] In the above formula, Used to represent the line x 1 ,x 2 ,… and column y 1 ,y 2 ,…to form the submatrix u=u (1,…,u),(1,…,n) .therefore, is the power of u. The fermion rotation is The homomorphism of has:

[0042] Fermion rotations can also be described as single-particle basis rotations, since each creation operator is rotated into a linear combination of the other creation operators, as follows:

[0043] In addition to the rotation of the creation operator caused by the fermion rotation, the product of the creation and annihilation operators can also be rotated, as shown above. Let k-RDM be represented as:

[0044] The rotated k-RDM is:

[0045] For the coefficient matrix The linear combination of the observables O, the rotated observables can be expressed as:

[0046] In the above equation for the rotated observable, the dimension The unitary matrix U ηThe conjugate of (u) is equivalent to the smaller dimension The unitary matrix U k (u). Since the order η of the outer product can be inferred from the context, U η The subscript of (u) is omitted hereinafter.

[0047] The form related to the Haar integral is given below. In this example, the Haar-random unitary matrix u is rotated with respect to the orbit is randomized. The corresponding rotation operator is a linear operator of the integral of the Wingaren basis. This linearity of the integral of the Wingaren basis allows the calculation of quantities such as the measurement channel and the shadow variance without the need for t-design results.

[0048] The rotation operator of degree t on The Haar measure on is: in and |k> are for and An orthogonal basis.

[0049] Weingarten integral basis of degree t on is defined as: in,

[0050] In the above definition, S q is a symmetry group of order q. is a Wingaren function which depends only on the conjugacy class of the permutation.

[0051] set up is a unique element group. Then is the set of all vectors whose elements are d elements. In the above formula, S n is the symmetry group over n elements.

[0052] The above measurement channel is reversible. Therefore, fermion rotations form a random measurement basis of the classical shadow. Since fermion rotations are a result of tomographic completeness, the measurement channel is reversible, which means that for any n-mode, the η-fermion density matrix ρ can be formed by fermion rotations via conjugate diagonal operators. In other words, any quantum state There is a definite number of η fermions, written in configuration basis, which can all be expressed as linear combinations. For some coefficient α j and using U(u j ) produces fermion rotation.

[0053] In addition to the fact that the fermion rotation is fully tomographic, the k-RDM is also diagonal with respect to the fermion rotation. The k-RDM It is diagonal with respect to the fermion rotation, so the k-RDM forms a complete basis for ρ when conjugated with the fermion rotation.

[0054] In pure state Upper given inversion measurement channel In the case of , any observable O of the fermion wave function ρ can be estimated from one or more classical shadows. Given one or more classical shadows The estimated value of the observable O is given by Given, where:

[0055] Since there is such an arrangement s z Make So you can replace and the above equation for the estimate of the observable is transformed by exchanging the fermion rotation through the measurement channel:

[0056] Therefore, the observable O can be evaluated by to estimate. The estimate of O can be calculated as:

[0057] In the above formula, the coefficient a d Given by:

[0058] Numeric operators The symmetric difference of is given by:

[0059] Using the above estimation and inverted measurement channels for O The style can be derived from one or more classic shadows The group of estimates the expectation of any given observable Q. The expression obtained is:

[0060] However, using the above formula The computational complexity is Since every operator in η has Dimension.

[0061] The calculations discussed below The method allows The calculation methods include Matrix multiplication, not matrix multiplication matrix. Therefore, using the techniques discussed below, The computational complexity can be Since k is usually much smaller than η, significant computational savings can be achieved.

[0062] return Figure 1 In the example of FIG. 1 , at the classical computing device 20 , the processor 22 receives the classical shadow measurement result from the quantum computing device 12 The processor 22 then uses the classical shadow measurement results at least in part to and the Haar-random unitary matrix u to estimate the k-RDM element 32 of the k-RDM 30 of the fermion wave function ρ. After calculating the k-RDM element 32, the processor 22 outputs the k-RDM element 32 to an additional calculation process 34.

[0063] Figure 2 4 shows a data flow 40 that occurs at computing system 10 in some examples. Figure 2 In the example of FIG. 5 , at the additional calculation process 34 , the processor calculates an estimate of the observable quantity O based at least in part on the k-RDM element 32 . Figure 2 As shown in the example of FIG. 1 , processor 22 calculates the estimated value of O using multiple k-RDM elements 32 included in k-RDM 30. However, in other examples, processor 22 may use a single k-RDM element 32 to calculate the estimated value of O.

[0064] In some examples, the processor 22 may calculate corresponding k-RDM elements 32 of the plurality of k-RDMs 30. When the processor 22 calculates the plurality of k-RDM elements 32, the quantum computing device 12 may determine a plurality of classical shadow measurement results associated with the corresponding plurality of Haar-random unitary matrices u In this example, the processor 22 of the classical computing device 20 determines the amount of the classical shadow measurement results based at least in part on the plurality of classical shadow measurement results. and the corresponding Haar-random unitary matrix to estimate the k-RDM element 32. In the example of calculating multiple k-RDM elements 32, the processor 22 can further convert the estimated value of the observable O is calculated as a linear combination of multiple k-RDM elements 32.

[0065] The following provides an estimate of the observable O Further details regarding the calculation of the linear combination of k-RDM elements 32. When the observable O is expressed as a linear combination of k-RDM elements 32, O has the following form: in is the coefficient matrix.

[0066] A single estimate of O computed using a k-RDM and the corresponding coefficient matrix o is as follows: Where D n,η,k is a diagonal estimation matrix. Each entry of the diagonal estimation matrix is ​​given by: in You can use complexity To calculate the diagonal estimate matrix All unique entries of .

[0067] exist is the special case of a single k-RDM 30, the single-shot estimate simplifies to:

[0068] Therefore, in this example, a single evaluation An estimate of O is provided simultaneously for each k-RDM 30. In some examples, processor 22 computes estimates of observable O in parallel using k-RDMs 30 as shown in the above equation.

[0069] Next, we discuss the estimated value of the observable O The following notation is used for expected values:

[0070] The variance of ρ with respect to ρ is state-dependent and is given by:

[0071] The upper bound on the state-dependent variance is given by the shadow norm:

[0072] As mentioned above, the average variance of estimating O on multiple k-RDMs 30 simultaneously is given by:

[0073] Therefore, since k is typically small (e.g., 1, 2, 3, or 4), in most examples the value of the observable O can be estimated with a low level of error.

[0074] Figure 3A A flow chart of a method 100 that can be used in a computing system including a quantum computing device and a classical computing device is shown. Figure 3A In the method 100 shown in the example of , steps 102, 104, 106, 116, 118, and 120 are performed at a classical computing device, and steps 108, 110, 112, and 114 are performed at a quantum computing device. For example, the quantum computing device may be a topological quantum computing device.

[0075] At step 102, method 100 includes: generating a Haar-random unitary matrix at a classical computing device. At step 106, method 100 also includes: computing a single-particle basis fermion rotation based at least in part on the Haar-random unitary matrix. Method 100 also includes: outputting the single-particle basis fermion rotation to the quantum computing device at step 106.

[0076] At step 108, method 100 also includes: receiving, at a quantum computing device, a specification of a fermion wave function over a plurality of fermions in a population-conserving fermion system occupying a plurality of modes. The specification of the fermion wave function is received at the quantum computing device from a classical computing device. In addition, the specification of the fermion wave function can be expressed as a Hamiltonian that characterizes the fermion system. For example, the fermion wave function can describe electrons and nuclei in a molecule simulated at the computing system. The Haar-random unitary matrix generated at step 102 has dimensions n×n, where n is the number of fermion modes that specify the fermion wave function.

[0077] At step 110, method 100 further includes: receiving a single-particle-based fermion rotation from a classical computing device. At step 112, method 100 further includes: applying the single-particle-based fermion rotation to the fermion wave function to obtain a rotated fermion wave function. Applying the single-particle-based fermion rotation to the fermion wave function may include calculating the product where U is the single-particle basis fermion rotation, ρ is the fermion wave function, and is the conjugate transpose of the single-particle basis fermion rotation.

[0078] At step 114, method 100 further includes: measuring the rotated fermion wave function to obtain a classical shadow measurement result, which encodes an estimate of which of the modes is occupied by the fermion. The classical shadow measurement result can be represented as a bit string. Therefore, the classical shadow measurement result can be generated in a form on which classical computing operations can be performed.

[0079] At step 116, method 110 also includes receiving classical shading measurements at the classical computing device. In some examples, the classical shading measurements and / or the Haar-random unitary matrix may be stored in a memory included in the classical computing device.

[0080] At step 118 , the method 110 further includes estimating k-RDM elements of the k-RDM of the fermion wave function based at least in part on the classical shadowmetry and the Haar-random unitary matrix.

[0081] Figure 3B Additional steps of method 100 are shown that may be performed in some examples. Figure 3B As shown, step 118A can be performed in an example where the classical shadow measurement result is included in a plurality of classical shadow measurement results, and the plurality of classical shadow measurement results are generated at a quantum computing device. In such an example, the classical shadow measurement result is generated based at least in part on a corresponding plurality of Haar-random unitary matrices, and the corresponding plurality of Haar-random unitary matrices include a Haar-random unitary matrix. At step 118A, step 118 may include: at the classical computing device, estimating the k-RDM element based at least in part on a plurality of classical shadow measurement results and a corresponding Haar-random unitary matrix. By utilizing a plurality of classical shadow measurement results and a corresponding Haar-random unitary matrix, a more accurate estimate of the k-RDM element may be obtained.

[0082] In some examples, such as Figure 3B As shown, at step 118B, step 118 may additionally or alternatively include computing a plurality of k-RDM elements. In some examples, the plurality of k-RDM elements may be included in the complete k-RDM approximated at the classical computing device.

[0083] return Figure 3A At step 120, method 100 may further include outputting the k-RDM elements to an additional computational process. For example, the additional computational process may be included in a quantum chemical simulation program. At step 120A, Figure 3B As shown, step 120 may include: computing an estimate of an observable based at least in part on the k-RDM elements. The observable may be, for example, energy, a dipole moment, or some other observable of a quantum mechanical system including a plurality of fermions. The estimate of the observable may be obtained by using The dimension of the operator is calculated, where n is the number of modes and k is the number of ladder operator pairs to calculate the k-RDM element. In addition, it can be calculated with The estimate of the k-RDM element is computed with a sample complexity of , where ∈ is the standard deviation of the estimate of the k-RDM element and η is the number of fermions specified by the fermion wave function.

[0084] In an example of performing step 118B, step 120 may further include: at step 120B, calculating the estimated value of the observable as a linear combination of the plurality of k-RDM elements. Additionally or alternatively, in an example of performing step 118B, step 120 may further include: calculating multiple estimated values ​​of the observable in parallel based at least in part on the plurality of k-RDM elements, the plurality of estimated values ​​including the estimated value of the observable. Thus, a sample of the plurality of values ​​of the observable may be estimated.

[0085] Using the above apparatus and methods, k-RDM elements can be estimated in a manner that results in a super-exponential speedup in sample complexity. Although previous methods for k-RDM estimation have been Scaling sample complexity, but the apparatus and methods discussed above allow for Scaling. Thus, the apparatus and methods discussed above may allow for significantly more efficient performance of computing systems when calculating k-RDM elements in quantum chemistry simulations.

[0086] As a further advantage of the devices and methods discussed above, since the classical shadow measurement results are expressed as classical bit strings when received at the classical computing device, the classical shadow measurement results can be stored in the memory of the classical computing device and processed at a subsequent time. In some examples, such calculations can be performed without concurrent operation of the quantum computing device, thereby allowing greater flexibility in scheduling computing tasks at the computing system.

[0087] In some embodiments, the methods and processes described herein may be bound to a computing system of one or more computing devices. In particular, such methods and processes may be implemented as a computer application or service, an application programming interface (API), a library, and / or other computer program products.

[0088] Figure 4 A non-limiting embodiment of a computing system 200 that can perform one or more of the above methods and processes is schematically illustrated. The computing system 200 is shown in simplified form. The computing system 200 can embody the above and Figure 1Components of computing system 200 may take the form of one or more personal computers, server computers, tablet computers, home entertainment computers, network computing devices, video gaming devices, mobile computing devices, mobile communication devices (e.g., smart phones), and / or other computing devices, as well as wearable computing devices (such as smart watches and head-mounted augmented reality devices).

[0089] The computing system 200 includes a logic processor 202, a volatile memory 204, and a non-volatile storage device 206. The computing system 200 may optionally include a display subsystem 208, an input subsystem 210, a communication subsystem 212, and / or Figure 2 Other components not shown.

[0090] Logical processor 202 includes one or more physical devices configured to execute instructions. For example, a logical processor may be configured to execute instructions that are part of one or more applications, programs, routines, libraries, objects, components, data structures, or other logical constructs. Such instructions may be implemented to perform a task, implement a data type, transform the state of one or more components, implement a technical effect, or otherwise achieve a desired result.

[0091] The logical processor may include one or more physical processors (hardware) configured to execute software instructions. Additionally or alternatively, the logical processor may include one or more hardware logic circuits or firmware devices configured to execute hardware-implemented logic or firmware instructions. The processor of the logical processor 202 may be single-core or multi-core, and the instructions executed thereon may be configured for sequential, parallel and / or distributed processing. The various components of the logical processor may optionally be distributed between two or more separate devices that may be remotely located and / or configured for coordinated processing. Various aspects of the logical processor may be virtualized and executed by a remotely accessible networked computing device configured in a cloud computing configuration. In this case, it is worth noting that these virtualized aspects run on different physical logical processors of various different machines.

[0092] The non-volatile storage device 206 includes one or more physical devices configured to store instructions executable by a logical processor to implement the methods and processes described herein. When implementing such methods and processes, the state of the non-volatile storage device 206 may be transformed, for example, to hold different data.

[0093] The non-volatile storage device 206 may include a removable and / or built-in physical device. The non-volatile storage device 206 may include an optical memory, a semiconductor memory, and / or a magnetic memory or other mass storage device technology. The non-volatile storage device 206 may include a non-volatile, dynamic, static, read / write, read-only, sequential access, location addressable, file addressable, and / or content addressable device. It is worth noting that the non-volatile storage device 206 is configured to retain instructions even when power is supplied to the non-volatile storage device 206.

[0094] The volatile memory 204 may include physical devices that include random access memory. The volatile memory 204 is typically used by the logical processor 202 to temporarily store information during the processing of software instructions. It is worth noting that when power is supplied to the volatile memory 204, the volatile memory 204 typically does not continue to store instructions.

[0095] Aspects of the logic processor 202, volatile memory 204, and non-volatile storage device 206 may be integrated together into one or more hardware logic components. For example, such hardware logic components may include field programmable gate arrays (FPGAs), programmable and application specific integrated circuits (PASIC / ASIC), program and application specific standard products (PSSP / ASSP), systems on chips (SOCs), and complex programmable logic devices (CPLDs). In addition, as described above, the computing system 200 may include a quantum computing device.

[0096] The terms "module", "program", and "engine" may be used to describe aspects of a computing system 200 that is typically implemented in software by a processor to perform a specific function using a portion of volatile memory, which involves a transformation process that specifically configures the processor to perform the function. Thus, a module, program, or engine may be instantiated using a portion of volatile memory 204, via a logical processor 202 that executes instructions held by a non-volatile storage device 206. It is noteworthy that different modules, programs, and / or engines may be instantiated from the same application, service, code block, object, library, routine, API, function, etc. Likewise, the same module, program, and / or engine may be instantiated by different applications, services, code blocks, objects, routines, APIs, functions, etc. The terms "module", "program", and "engine" may cover individuals or groups of executable files, data files, libraries, drivers, scripts, database records, and the like.

[0097] When included, the display subsystem 208 can be used to present a visual representation of the data held by the non-volatile storage device 206. The visual representation can take the form of a graphical user interface (GUI). As the methods and processes described herein change the data held by the non-volatile storage device, and thus convert the state of the non-volatile storage device, the state of the display subsystem 208 can also be transformed to visually represent the changes in the underlying data. The display subsystem 208 can include one or more display devices utilizing almost any type of technology. Such a display device can be combined with the logical processor 202, the volatile memory 204, and / or the non-volatile storage device 206 in a shared housing, or such a display device can be a peripheral display device.

[0098] When included, the input subsystem 210 may include or interface with one or more user input devices, such as a keyboard, mouse, touch screen, or game controller. In some embodiments, the input subsystem may include or interface with selected natural user input (NUI) components. Such components may be integrated or peripheral, and the conversion and / or processing of input actions may be handled on-site or off-board. Example NUI components may include microphones for voice and / or speech recognition; infrared, color, stereo, and / or depth cameras for machine vision and / or gesture recognition; head trackers, eye trackers, accelerometers, and / or gyroscopes for motion detection and / or intent recognition; and electric field sensing components for assessing brain activity; and / or any other suitable sensors.

[0099] When included, the communication subsystem 212 can be configured to communicatively couple the various computing devices described herein to each other and to communicatively couple to other devices. The communication subsystem 212 can include wired and / or wireless communication devices compatible with one or more different communication protocols. As a non-limiting example, the communication subsystem can be configured to communicate via a wireless telephone network, or a wired or wireless local area network or wide area network. In some embodiments, the communication subsystem can allow the computing system 200 to receive messages to and / or from other devices via a network such as the Internet.

[0100] The following paragraphs discuss several aspects of the present disclosure. According to one aspect of the present disclosure, a computing system is provided, including a classical computing device. The classical computing device includes a processor that generates a Haar-random unitary matrix. The processor also calculates a single-particle basis fermion rotation based at least in part on the Haar-random unitary matrix. The processor also outputs the single-particle basis fermion rotation to the quantum computing device. The quantum computing device receives a specification of a fermion wave function over a plurality of fermions in a fermion system in which the number of particles is conserved occupying a plurality of modes. The quantum computing device also receives a single-particle basis fermion rotation. The quantum computing device also applies the single-particle basis fermion rotation to the fermion wave function to obtain a rotated fermion wave function. The quantum computing device also measures the rotated fermion wave function to obtain a classical shadow measurement result, which encodes an estimate of which mode in the mode is occupied by the fermion. The processor of the classical computing device also receives a classical shadow measurement result. The processor of the classical computing device also estimates the k-RDM element of the k-RDM of the fermion wave function based at least in part on the classical shadow measurement results and the Haar-random unitary matrix. The processor of the classical computing device also outputs the k-RDM element to an additional computing process. The above features can have a technical effect of estimating the k-RDM element with low sample complexity. The above features can also have a technical effect of allowing the classical shadow measurement results to be processed asynchronously with the operation of the quantum computing device.

[0101] According to this aspect, the classical shadow measurement results can be included in a plurality of classical shadow measurement results, which are generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix. The processor of the classical computing device can estimate the k-RDM element based at least in part on the plurality of classical shadow measurement results and the corresponding Haar-random unitary matrix. The above features can have a technical effect of allowing the DM-RDM element to be estimated with higher accuracy.

[0102] According to this aspect, at the additional computational process, the processor may compute an estimate of the observable based at least in part on the k-RDM element. This feature may have a technical effect of computing a quantity of interest associated with a fermionic system.

[0103] According to this aspect, the classical shadow measurement results can be included in a plurality of classical shadow measurement results, which are generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, which include Haar-random unitary matrices. The processor of the classical computing device can also calculate a plurality of k-RDM elements. The processor can also calculate an estimate of the observable as a linear combination of the plurality of k-RDM elements. The above features can have a technical effect of allowing the processor to estimate the observable with higher accuracy.

[0104] According to this aspect, the classical shadow measurement result can be included in a plurality of classical shadow measurement results, which are generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, which include the Haar-random unitary matrix. The processor of the classical computing device can also calculate a plurality of k-RDM elements. The processor can also calculate a plurality of estimates of observables in parallel based at least in part on a plurality of k-RDM elements, the plurality of estimates including the estimated value of the observable. The above features can have the technical effect of efficiently calculating observables.

[0105] According to this aspect, the processor may calculate having An estimate of a k-RDM element with a sample complexity of , where ∈ is the standard deviation of the estimate of the k-RDM element, η is the number of fermions specified by the fermion wave function, and k is the number of ladder operator pairs that the processor utilizes to compute the k-RDM element. The above features may have a technical effect of estimating k-RDM elements with low sample complexity.

[0106] According to this aspect, the processor may calculate having The estimated value of the operator-dimensional observable of , where n is the number of modes and k is the number of ladder operator pairs that the processor utilizes to compute the k-RDM elements. The above features may have a technical effect of reducing the size of matrices that are multiplied when computing the k-RDM elements.

[0107] According to this aspect, the Haar-random unitary matrix may have dimensions n x n, where n is the number of modes specified by the fermion wave function.The above features may have the technical effect of allowing a processor to efficiently compute single-particle-based fermion rotations.

[0108] According to this aspect, the quantum computing device may be a topological quantum computing device. This feature may have the technical effect of allowing the quantum computing device to compute rotated fermion wave functions via topological braiding operations performed on fermion wave functions.

[0109] According to this aspect, the additional computation process may include storing the k-RDM elements in a memory. This feature may have the technical effect of allowing the processor to asynchronously compute one or more additional quantities from the operation of the quantum computing device using the k-RDM elements.

[0110] According to another aspect of the present disclosure, a method for use with a computing system is provided. The method includes generating a Haar-random unitary matrix at a classical computing device. The method also includes: at the classical computing device, computing a single-particle basis fermion rotation based at least in part on the Haar-random unitary matrix. The method also includes: at the classical computing device, outputting the single-particle basis fermion rotation to the quantum computing device. The method also includes: at the quantum computing device, receiving a specification of a fermion wave function over a plurality of fermions in a population-conserving fermion system occupying a plurality of modes. The method also includes: receiving a single-particle basis fermion rotation at the quantum computing device. The method also includes: at the quantum computing device, applying the fermion single-particle basis fermion rotation to the fermion wave function to obtain a rotated fermion wave function. The method also includes: at the quantum computing device, measuring the rotated fermion wave function to obtain a classical shadow measurement result, the result encoding an estimate of which mode in the mode is occupied by the fermion. The method also includes: receiving a classical shadow measurement result at the classical computing device. The method also includes: estimating, at a classical computing device, a k-RDM element of a k-RDM of a fermion wave function based at least in part on the classical shadow measurement result and the Haar-random unitary matrix. The method also includes: outputting the k-RDM element to an additional computing process at the classical computing device. The above features can have a technical effect of estimating k-RDM elements with low sample complexity. The above features can also have a technical effect of allowing classical shadow measurement results to be processed asynchronously with the operation of a quantum computing device.

[0111] According to this aspect, the classical shadow measurement results can be included in a plurality of classical shadow measurement results, which are generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix. The method may also include: at the classical computing device, estimating the k-RDM element based at least in part on the plurality of classical shadow measurement results and the corresponding Haar-random unitary matrix. The above features may have a technical effect of allowing the DM-RDM element to be estimated with higher accuracy.

[0112] According to this aspect, the method may further include: at an additional computational process, computing an estimate of an observable based at least in part on the k-RDM element. This feature may have a technical effect of computing a quantity of interest associated with a fermionic system.

[0113] According to this aspect, the classical shadow measurement results can be included in a plurality of classical shadow measurement results, which are generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, which include Haar-random unitary matrices. The method can also include calculating a plurality of k-RDM elements at the classical computing device and calculating an estimate of the observable as a linear combination of the plurality of k-RDM elements. The above features can have a technical effect of allowing the observable to be estimated with greater accuracy.

[0114] According to this aspect, the classical shadow measurement results can be included in a plurality of classical shadow measurement results, which are generated at a quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, which include Haar-random unitary matrices. The method may also include calculating a plurality of k-RDM elements at the classical computing device. The method may also include: at the classical computing device, calculating in parallel a plurality of estimates of observable quantities based at least in part on a plurality of k-RDM elements, the plurality of estimates including estimates of the observable quantities. The above features may have the technical effect of efficiently calculating observable quantities.

[0115] According to this aspect, it is possible to calculate The estimate of the k-RDM element with a sample complexity of , where ∈ is the standard deviation of the estimate of the k-RDM element, η is the number of fermions specified by the fermion wave function, and k is the number of ladder operator pairs that the processor utilizes to calculate the k-RDM element. The above features may have a technical effect of estimating the k-RDM element with low sample complexity.

[0116] According to this aspect, it is possible to calculate The estimated value of the operator-dimensional observable of , where n is the number of modes and k is the number of ladder operator pairs that the processor utilizes to compute the k-RDM elements. The above features may have a technical effect of reducing the size of matrices that are multiplied when computing the k-RDM elements.

[0117] According to this aspect, the Haar-random unitary matrix may have dimensions n x n, where n is the number of modes specified by the fermion wavefunction.The above features may have the technical effect of allowing a processor to efficiently compute single-particle-based fermion rotations.

[0118] According to this aspect, the quantum computing device may be a topological quantum computing device. This feature may have the technical effect of allowing the quantum computing device to compute a rotated fermion wave function via a topological braiding operation performed on the fermion wave function.

[0119] According to another aspect of the present disclosure, a computing system is provided, including a classical computing device. The classical computing device may be a processor that generates a plurality of Haar-random unitary matrices. The processor also calculates a plurality of single-particle basis fermion rotations based at least in part on the Haar-random unitary matrix. The processor also outputs a single-particle basis fermion rotation to the quantum computing device. The quantum computing device receives a specification of a fermion wave function over a plurality of fermions that occupy a plurality of modes of a fermion system in which the number of particles is conserved. The quantum computing device also receives a plurality of single-particle basis fermion rotations. The quantum computing device also applies each single-particle basis fermion rotation to the fermion wave function to obtain a plurality of rotated fermion wave functions. The quantum computing device also measures the rotated fermion wave function to obtain a plurality of classical shadow measurement results that encode estimates of which mode in the mode is occupied by the fermion. The processor of the classical computing device also receives the classical shadow measurement results and the corresponding Haar-random unitary matrix. The processor also estimates multiple k-reduced density matrices (k-RDMs) of fermion wave functions in parallel based at least in part on classical shadow measurement results and Haar-random unitary matrices. The processor also calculates estimates of observable quantities based at least in part on multiple k-RDMs. The above features can have a technical effect of estimating observable quantities with low sample complexity. The above features can also have a technical effect of allowing classical shadow measurement results to be processed asynchronously with the operation of the quantum computing device.

[0120] As used herein, "and / or" is defined to include OR∨, as specified by the following truth table: A B A∨B real real real real Fake real Fake real real Fake Fake Fake

[0121] It is noteworthy that the configuration and / or method described herein are exemplary in nature, and these specific embodiments or examples should not be considered as restrictive, because many variations are possible. The specific routine or method described herein can represent one or more of any number of processing strategies. Therefore, the various actions shown and / or described can be performed in the order shown and / or described, in other orders, in parallel or omitted. Similarly, the order of the above process can also be changed.

[0122] The subject matter of the present disclosure includes all novel and nonobvious combinations and subcombinations of the various processes, systems and configurations, and other features, functions, acts, and / or properties disclosed herein, and any and all equivalents thereof.

Claims

1. A computing system comprising: A classical computing device, comprising a processor, wherein: Generate Haar-random unitary matrix; computing single-particle basis fermion rotations based at least in part on the Haar-random unitary matrix; and The single-particle-based fermion rotation is output to a quantum computing device, wherein: The quantum computing device: receiving a specification of a fermion wave function over a plurality of fermions in a population-conserving fermion system occupying a plurality of modes; receiving the single-particle-based fermion rotation; applying the single-particle-based fermion rotation to the fermion wave function to obtain a rotated fermion wave function; and measuring the rotated fermion wavefunction to obtain a classical shadowmetry that encodes an estimate of which of the modes is occupied by the fermion; and The processor of the classical computing device also: Receiving the classical shadow measurement result; estimating a k-reduced density matrix (k-RDM) element of the fermion wave function based at least in part on the classical shadowmetry and the Haar-random unitary matrix; as well as The k-RDM elements are output to an additional computation process.

2. The computing system of claim 1, wherein: The classical shadow measurement result is included in a plurality of classical shadow measurements, the plurality of classical shadow measurements being generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix; and The processor of the classical computing device estimates the k-RDM elements based at least in part on the plurality of classical shadow measurements and the corresponding Haar-random unitary matrices.

3. The computing system of claim 1, wherein during the additional computation process, the processor computes an estimate of an observable based at least in part on the k-RDM elements.

4. The computing system of claim 3, wherein: The classical shadow measurement result is included in a plurality of classical shadow measurements, the plurality of classical shadow measurements being generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix; and The processor of the classical computing device also: calculating a plurality of k-RDM elements; and The estimate of the observable is calculated as a linear combination of the plurality of k-RDM elements.

5. The computing system of claim 3, wherein: The classical shadow measurement result is included in a plurality of classical shadow measurements, the plurality of classical shadow measurements being generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix; and The processor of the classical computing device also: calculating a plurality of k-RDM elements; and A plurality of estimates of the observable are computed in parallel based at least in part on the plurality of k-RDM elements, the plurality of estimates including the estimate of the observable.

6. The computing system of claim 5, wherein the processor computes The processor calculates the k-RDM element with a sample complexity of , where ∈ is the standard deviation of the estimate of the k-RDM element, η is the number of fermions specified by the fermion wave function, and k is the number of ladder operator pairs used by the processor to calculate the k-RDM element.

7. The computing system of claim 3, wherein the processor computes The processor calculates the estimated value of the observable of the operator dimension of , where n is the number of modes and k is the number of ladder operator pairs, and the processor calculates the k-RDM elements using the ladder operator pairs.

8. The computing system of claim 1, wherein the Haar-random unitary matrix has dimensions n x n, where n is the number of modes specified by the fermion wave function.

9. The computing system of claim 1, wherein the quantum computing device is a topological quantum computing device.

10. The computing system of claim 1, wherein the additional computing process includes storing the k-RDM elements in a memory.

11. A method for use with a computing system, the method comprising: On a classical computing device: Generate Haar-random unitary matrix; computing single-particle basis fermion rotations based at least in part on the Haar-random unitary matrix; and outputting the single-particle-based fermion rotation to a quantum computing device; At a quantum computing device: receiving a specification of a fermion wave function over a plurality of fermions in a population-conserving fermion system occupying a plurality of modes; receiving the single-particle-based fermion rotation; applying the single-particle-based fermion rotation to the fermion wave function to obtain a rotated fermion wave function; and measuring the rotated fermion wave function to obtain a classical shadowmetry result encoding an estimate of which of the modes is occupied by the fermion; as well as At the classical computing device: Receiving the classical shadow measurement result; estimating a k-reduced density matrix (k-RDM) element of the fermion wave function based at least in part on the classical shadowmetry and the Haar-random unitary matrix; as well as The k-RDM elements are output to an additional computation process.

12. The method according to claim 11, wherein: The classical shadow measurement result is included in a plurality of classical shadow measurements, the plurality of classical shadow measurements being generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix; and The method also includes, at the classical computing device, estimating the k-RDM elements based at least in part on the plurality of classical shadow measurements and the corresponding Haar-random unitary matrices.

13. The method of claim 11, further comprising, during the additional computation, computing an estimate of an observable based at least in part on the k-RDM elements.

14. The method according to claim 13, wherein: The classical shadow measurement result is included in a plurality of classical shadow measurements, the plurality of classical shadow measurements being generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix; and The method further comprises, at the classical computing device: Calculate multiple k-RDM elements; as well as The estimate of the observable is calculated as a linear combination of the plurality of k-RDM elements.

15. The method of claim 13, wherein: The classical shadow measurement result is included in a plurality of classical shadow measurements, the plurality of classical shadow measurements being generated at the quantum computing device at least in part based on a corresponding plurality of Haar-random unitary matrices, the corresponding plurality of Haar-random unitary matrices including the Haar-random unitary matrix; and The method further comprises, at the classical computing device: calculating a plurality of k-RDM elements; and A plurality of estimates of the observable are computed in parallel based at least in part on the plurality of k-RDM elements, the plurality of estimates including the estimate of the observable.

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