Efficient shadow tomography of pauli operators and local fermionic operators with two-copy measurements

Abstract

Methods, systems, and apparatus for estimating expectation values of a quantum state with respect to a set of observables and to a target precision. In one aspect a method includes computing, using multiple physical copies of the quantum state and for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable; generating, using the estimated magnitudes, an updated set of observables; computing, using the updated set of observables, a classical description of a mimicking quantum state; and performing Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state to obtain, for each observable in the updated set of observables, an estimate of a sign of the expectation value of the quantum state with respect to the observable.

Description

[0001] EFFICIENT SHADOW TOMOGRAPHY OF PAULI OPERATORS AND LOCAL FERMIONIC OPERATORS WITH TWO-COPY MEASUREMENTS

[0002] BACKGROUND

[0003] This disclosure relates to quantum computing.

[0004] Shadow tomography is a process that can be used to estimate 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 p denote a quantum state and assume access to multiple copies of the quantum state p is possible. Let {£} represent a collection of M observables. The task of shadow tomography is to estimate the expectation values Tr(jjS) up to some additive error e for each operator 5 in the collection of observables.

[0005] SUMMARY

[0006] This disclosure describes techniques for efficient shadow tomography of Pauli operators and local fermionic operators with two-copy measurements.

[0007] In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for estimating expectation values of a quantum state with respect to a set of observables and to a target precision, the method comprising: computing, using multiple physical copies of the quantum state and for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable to within a first predetermined threshold range that is based on the target precision; generating, using the estimated magnitudes, an updated set of observables, comprising removing observables from the set of observables that correspond to an expectation value with a magnitude that is smaller than a second predetermined threshold value that is based on the target precision; computing, using the updated set of observables, a classical description of a mimicking quantum state, wherein the mimicking quantum state has a same dimension as the quantum state, and expectation values of the mimicking quantum state with respect to the updated set of observables have respective magnitudes that are the same as respective magnitudes of the expectation values of the quantum state with respect to the updated set of observables; and performing Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state to obtain, for each observable in the updated set of observables, an estimate of a sign of the expectation value of the quantum state with respect to the observable.

[0008] Other implementations of these aspects includes corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.

[0009] The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the set of observables comprises a complete set of n-qubit Pauli operators, wherein n represents the size of the quantum state.

[0010] In some implementations computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing two-copy measurements of the quantum state.

[0011] In some implementations performing two-copy measurements of the quantum state comprises performing 0 (log(|S |) e-4) two-copy measurements, where|5| represents the size of the set of observables and e represents the target precision.

[0012] In some implementations computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing Bell sampling on a tensor product of two physical copies of the quantum state.

[0013] In some implementations expectation values of the mimicking quantum state with respect to the updated set of observables comprise a same or different sign as corresponding expectation values of the quantum state with respect to the updated set of observables.

[0014] In some implementations performing Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state comprises preparing multiple physical copies of the mimicking quantum state.

[0015] In some implementations performing Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state to obtain, for each observable in the updated set of observables, an estimate of a sign of the expectation value of the quantum state with respect to the observable comprises, for each observable in the set of updated observables: obtaining a sample for the observable, wherein the sample comprises an expectation value of a tensor product of the observable with respect to the tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state; computing an expectation value of the observable with respect to the mimicking quantum state; and dividing the sample for the observable by the expectation value of the observable with respect to the mimicking quantum state to obtain an estimate of the sign of the expectation value of the quantum state with respect to the observable.

[0016] In some implementations performing Bell sampling comprises obtaining

[0017] 0 (log(|S | ) e-4) Bell samples, where|S| represents the size of the set of observables and e represents the target precision.

[0018] In some implementations computing the expectation value of the observable with respect to the mimicking quantum state comprises exactly computing the expectation value of the observable with respect to the mimicking quantum state using the classical description of the mimicking quantum state.

[0019] In some implementations computing the classical description of the mimicking quantum state comprises performing a classical search algorithm.

[0020] In general, another innovative aspect of the subject matter described in this specification can be implemented in a method for estimating expectation values of a quantum state with respect to a set of observables and to a target precision, the method comprising: computing, using multiple physical copies of the quantum state and for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable to within a first predetermined threshold range that is based on the target precision; generating, using the estimated magnitudes, an updated set of observables, comprising removing observables from the set of observables that correspond to an expectation value with a magnitude that is smaller than a second predetermined threshold value that is based on the target precision; obtaining a frustration graph for the updated set of observables, wherein each vertex in the frustration graph represents a respective observable in the updated set of observables and pairs of vertices that represent anti-commutating observables in the updated set of observ ables are connected by respective edges; classifying vertices of the frustration graph using multiple classifications, wherein adjacent vertices in the frustration graph are classified as different classifications; and for each classification of the multiple classifications: measuring observables that correspond to vertices that are classified in the classification, comprising performing Clifford measurements of single physical copies of the quantum state.

[0021] Other implementations of these aspects includes corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.

[0022] The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the set of observables comprises a set of k-local fermionic operators.

[0023] In some implementations the method further comprises using the estimated expectation values of the quantum state with respect to the set of k-local Pauli operators to reconstruct k-qubit reduced density matrices of the quantum state.

[0024] In some implementations classifying vertices of the frustration graph using multiple classifications comprises coloring vertices of the frustration graph using multiple colors, wherein the colouring comprises fractional colouring.

[0025] In some implementations multiple colors comprise at most a number of colors that is polynomial in e-2, where e represents the target precision.

[0026] In some implementations computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing two-copy measurements of the quantum state.

[0027] In some implementations performing two-copy measurements of the quantum state comprises performing O(log(|S|) ε-4) two-copy measurements, where |S| represents the size of the set of observables and ε represents the target precision.

[0028] In some implementations computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing Bell sampling on a tensor product of two physical copies of the quantum state. In some implementations the observables that correspond to vertices that are classified in the classification are measured approximately simultaneously.

[0029] In some implementations performing Clifford measurements of single physical copies of the quantum state comprise using 0 physical copies of the quantum state,

[0030]

[0031] where represents the number of classifications used to classify the vertices of the frustration graph, | S| represents the size of the set of observables, and e represents the target precision.

[0032] In some implementations the size of a largest clique of vertices included in the frustration graph is at most 9ε-2, where ε represents the target precision.

[0033] The subject matter described in this specification can be implemented in particular embodiments so as to realize one or more of the following advantages.

[0034] The present disclosure describes shadow tomography techniques that minimize several computational resources, including sample complexity (e.g., the total number of physical copies of a quantum state required by the shadow tomography process), time complexity (e.g., the computational cost or size of quantum circuits used to implement the shadow tomography process), and the amount of quantum memory or parallel qubit registers required (e.g., the number of physical copies of the quantum state used at a time).

[0035] Known shadow tomography processes, i.e., different to those described in the present 5 disclosure, include processes that require a sample complexity that scales as O log — ) and have efficient time complexify. Whilst these are both satisfactory complexities, even the best known shadow tomography process uses a lot of quantum memory - it requires storing 5

[0036] O(log — ) physical copies of the quantum state in memory. This means that such shadow tomography processes would need a quantum computer that is O(log — ) times bigger (e.g., in terms of number of qubits) than the quantum computer needed to prepare just one physical copy of the quantum state. However, if only one physical copy of the quantum state is used at a time, the memory requirement is reduced but the shadow tomography process then requires too many physical copies of the quantum state.

[0037] The present disclosure describes various shadow tomography techniques that improve on known shadow tomography processes in several ways. For example, the presently described shadow tomography techniques use only two physical copies of the quantum state at a time (i.e., require only 2-copy measurements). One technique for learning a k-body reduced densify matrix of a fermionic state requires only two physical copies of the quantum state at a time, achieves efficient time complexity, and has a sample complexity that is polynomial in log S' and log l / e. Another technique for learning the set of all Pauli observables (of which there are 4") requires only n0^ physical copies of the quantum state using 2-copy measurements (despite there being exponentially many Pauli observables). Another technique for learning the set of all Pauli observables uses fewer physical copies of l

[0038] the quantum state - O(—) using 2-copy measurements (with a running time that is exponential in n). Each technique only requires a quantum computer that is double the minimum possible size (i.e., the number of qubits needed to prepare just one physical copy of the quantum state.)

[0039] The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below; Other features, aspects, and advantages of the subject matter will become apparent from the description, the draw ings, and the claims.

[0040] BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram of an example system efficient shadow tomography of Pauli operators and local fermionic operators with two-copy measurements.

[0041] FIG. 2 is a flow' diagram of a first process for estimating expectation values of an unknown quantum state with respect to a set of observables and to a target precision.

[0042] FIG. 3 is a flow diagram of a process for computing a classical description of a mimicking quantum state.

[0043] FIG. 4 is a flow' diagram of a second process for estimating expectation values of an unknown quantum state with respect to a set of observables and to a target precision.

[0044] FIG. 5 depicts an example classical / quantum computer.

[0045] DESCRIPTION

[0046] This specification describes shadow tomography processes for estimating properties, e.g., expectation values, of a quantum state with respect to a set of observables, where the set of observables is either a subset of the full set of Pauli operators on n qubits (that is 4nPauli operators) or a set of observables that arise from the task of learning the k-body reduced density matrix (k-RDM) of a fermionic state. The processes require at most two physical copies of the quantum state at a time, and are both sample efficient and computationally efficient. FIG. 1 is a block diagram of an example system 100 for efficient shadow tomography of Pauli operators and local fermionic operators with two-copy measurements. 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 can be implemented.

[0047] The example system 100 includes a quantum computing device 102 in data communication with a classical processor 104. For illustrative purposes, the quantum computing device 102 and classical processor 104 shown in FIG. 1 are illustrated as separate entities, however in some implementations the classical processor 104 may be included in quantum computing device 102. For example, in some implementations the quantum computing device 102 can be directly connected to the classical processor 104. In other implementations, the quantum computing device 102 can be connected to the classical processor 104 through a network, e.g., a local area network (LAN), wide area network (WAN), the Internet, or a combination thereof.

[0048] The classical processor 104 includes components for performing classical computation and can 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, or to control the operation of. a data processing apparatus. The computer storage medium can 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.

[0049] The quantum computing device 102 includes components for performing quantum computation. For example, the quantum computing device 102 can include a qubit array comprising a plurality of physical qubits, quantum circuitry, and control devices configured to operate physical qubits in the qubit array and apply quantum circuits to the qubits. In some implementations the quantum computing device 102 can be a noisy device, e g., a noisy intermediate-scale quantum device (NISQ device), a trapped ion quantum computer, or a neutral atom quantum computer. An example quantum computing device is described in the more detail below with reference to FIG. 5.

[0050] The example system 100 is configured to perform shadow' tomography of an unknown quantum state using at most tw o physical copies of the quantum state at a time. In one example, the system 100 performs shadow tomography for sets of observables that arise from the problem of learning the k-body reduced density matrix (k-RDM) of a fermionic state. In another example, the system 100 performs shadow tomography for the set of all Pauli observables (of which there are 4").

[0051] In both examples, the absolute values of the expectation values to be learned are computed using a quantum computing protocol, e.g., Bell sampling. Bell sampling is a process of measuring pairs of qubits in a tensor product state in the Bell basis (which consists of the four maximally entangled two-qubit states called Bell states) to determine which entangled Bell state the pair of qubits occupies. The measurement outcomes relate to expectation vales of observables on a quantum state in the tensor product state. For example, the sign of Tr(pO) where p is the quantum state and O is an observable can be determined based on the distribution of observed Bell states across corresponding qubit pairs in the tensor product. One example of Bell sampling is provided described by Montanaro, “Learning stabilizer states by bell sampling,” arXiv: 1707.04012, 2017.

[0052] The quantum computing hardware 102 can be programmed to prepare or access (from quantum memory' 120) multiple physical copies of the unknown quantum state p and perform Bell sampling to measure a tensor product of two physical copies of the quantum state p®p in the Clifford basis that diagonalizes commuting observables P®P for all observables P in the set observables. The classical processor 104 is then configured to filter the set of observables to include only those that have at least a minimum absolute value, e.g., are larger than a threshold that is dependent on a target precision of the shadow tomography process.

[0053] In some implementations, e.g., those where the system 100 performs shadow tomography for the set of all Pauli observables, the classical processor 104 is configured to compute a classical description of a quantum state that mimics the unknown quantum state. The quantum computing hardware 102 can then be programmed to perform Bell sampling on a tensor product of the unknown quantum state and the mimicking quantum state to estimate the signs of the absolute values of the expectation values to be learned.

[0054] In other implementations, e.g., those where the system 100 performs shadow tomography for sets of observables that arise from the problem of learning the k- RDM of a fermionic state, the classical processor 104 is configured to construct a frustration graph for the set of observables. Properties of the fractional chromatic number of the graph are used to determine a measurement protocol that is implemented by the quantum computing hardware 104 to estimate the signs of the absolute values of the expectation values to be learned.

[0055] Example operations performed by the example system 100 are described in more detail below with reference to FIGS. 2-4. FIG. 2 is a flow diagram of a process 200 for estimating expectation values of an unknown quantum state with respect to a set of observables and to a target precision. For example, the set of observables can be a complete set of n-qubit Pauli operators, wherein n represents the size of the unknown quantum state. For convenience, the 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, the system 100 of FIG. 1, appropriately programmed, can perform example process 200.

[0056] The system uses multiple physical copies of the quantum state to compute, for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable (step 202). The estimates of the magnitudes of the expectation values are computed to within a first predetermined threshold that is based on the target precision, e.g., the threshold e / 3 where e represents the target precision.

[0057] Computing the estimates of the magnitudes of the expectation values includes performing two-copy measurements of the unknown quantum state, e.g., using Bell sampling. For example, the system can prepare or access (from quantum memory) multiple physical copies of the unknown quantum state p and perform Bell sampling to measure a tensor product of two physical copies of the quantum state p p in the Clifford basis that diagonalizes commuting Pauli observables P®P for all Pauli operators P in the set of n-qubit Pauli operators. In some implementations 0(Zo^(|S|)e-4) two-copy measurements can suffice to compute estimates {uP}Pesthat satisfy

[0058] |uP- \Tr(pP~)\ | < e / 3

[0059] for all P E S (with high probability), where S represents the set of observables, |S| represents the size of the set of observables, e represents the target precision, uPrepresents an estimate of the magnitude of the expectation value of the quantum state p with respect to the observable P, and e / 3 represents the first predetermined threshold.

[0060] The system uses the magnitudes estimated at step 202 to generate an updated set of observables (step 204). The system removes observables from the set of observables that have negligible expectation values, e.g., observables that have expectation values with a magnitude that is smaller than a second predetermined threshold that is based on the target precision. In more detail, if an estimate uPcomputed at step 202 is less than the second predetermined threshold, e.g., 2e / 3. then the system uses 0 is an e-approximation to the expected value Tr(pP)' and removes the corresponding observable P from the set of observables. The updated set of observables is therefore equal to

[0061] Se= {P E S: |uP> y}.

[0062]

[0063] This set is a random variable determined by the output of Bell sampling, and implies that S£Q {P E S:

[0064]

[0065] \Tr(pP') | > The set S£can be referred to as a set of e-supported observables. To complete the learning task of example process 200, it therefore suffices to compute the sign of Tr pP') for all observables P E S£.

[0066] The system uses the updated set of observables to compute a classical description of a mimicking quantum state (step 206). The mimicking quantum state mimics the unknown quantum state in the sense that it has a same dimension as the quantum state and expectation values of the mimicking quantum state with respect to the updated set of observables have a same magnitude as expectation values of the quantum state with respect to the updated set of observables (however, the expectation values of the mimicking quantum state with respect to the updated set of observables can have different signs). That is, the mimicking quantum state o can be computed to satisfy

[0067] |Tr(crP)| > e / 6 for all P E S£

[0068] It can be shown that there always exists at least one mimicking state o for the unknown quantum state p (for example, o = p is a valid mimicking state). Further, given the updated set of observables

[0069]

[0070] a mimicking state can be found without using any additional physical copies of the quantum state p, e.g., by a classical search algorithm. That is, step 206 of example process 200 can be performed without using any additional physical copies of the quantum state p. An example classical search algorithm that can be used to compute a classical description of a mimicking quantum state is described below with reference to FIG.

[0071] 3.

[0072] The system performs Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state (e.g., p o' ) to obtain, for each observable in the updated set of observables, an estimate of a sign of the expectation value of the quantum state with respect to the observable (step 208). To perform the Bell sampling, the system generates multiple physical copies of the mimicking state o, e.g., using known state preparation techniques. The system can then use the resulting samples to estimate the mean values

[0073] T

[0074]

[0075] r^P 0 P(p 0 cr)] = Tr(Pp)Tr(P<j), P 6 Se.

[0076] Since o is a mimicking quantum state as defined above, each of these mean values has magnitude at least e2and all of them can be computed up to e2 / 2 error using only O log |S| / e4) Bell samples. Since the mimicking quantum state o is known, the system can compute the quantity Tr(aP) exactly (including the sign). The system can then use this know ledge and the estimates of the mean values Tr^P 0 P p 0 o’)) to infer the sign of each mean value Tr(pP) for P E Se.

[0077] That is. for each observable in the set of updated observables, the system computes an expectation value of a tensor product of the observable with respect to the tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state, e.g., computes Tr(P 0 P(p 0 cr)). The system then computes an expectation value of the observable with respect to the mimicking quantum state, e.g., computes Tr(Po’), and divides Tr(P 0 P(p 0 <7))by Tr (P P) to obtain an estimate of the sign of the expectation value of the quantum state with respect to the observable. In some implementations the system can perform the Bell sampling using 0 (log(| S |) e-4) Bell samples, where | S' | represents the size of the set of observables and e represents the target precision.

[0078] The system outputs the magnitudes computed at step 202 and corresponding signs computed at step 208 as estimated expectation values of the unknown quantum state with respect to the set of observables.

[0079] FIG. 3 is a flow' diagram of a process 300 for computing a classical description of a mimicking quantum state. For convenience, the 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, the system 100 of FIG. 1, appropriately programmed, can perform example process 300.

[0080] The system obtains an input (step 302). The input includes multiple physical copies of the quantum state p that the mimicking quantum state is to mimic, e.g., the quantum state p described above with reference to FIG. 2. For example, the system can prepare or access (from quantum memory) the multiple physical copies of the quantum state. The input also includes the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables included in the set of observables, as computed at step 202 of example process 200 described above. That is, the system obtains the set of magnitudes {uP}, where each uP> 0 and satisfies \rPuP— Tr Pp\ < e / for all observables in the set of observables for some unknown choices of signs rP6 {+1, -1}. By convention, in some implementations the system can set rP= 0 in the case that uPis negligible, e.g., uP< 2e / 3, where e represents the target precision defined above with reference to example process 200.

[0081] The system sets an initial classical description of the mimicking quantum state (step 304). For example, the system can set the initial classical description of the mimicking quantum state ro(o)as a classical description of a maximally mixed quantum state. That is, the system sets a0-1= I / 2nwhere 1 represents the identity matrix and n represents the size of the quantum state (and mimicking quantum state). The system also sets the quantities T ■ = n / e2and / ?: (n / T), where T represents a total number of iterations and e represents the target precision.

[0082] For each iteration of the total number of iterations, e.g., for 0, 1,..., T — 1, the system performs the following steps. The system determines whether the set of observables includes an observable that generates an expectation value with respect to the classical description of the mimicking quantum state for the iteration that is within a predetermined distance from a corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable (step 306). The system determines whether a magnitude of the difference between an expectation value of the classical description of the mimicking quantum state for the iteration with respect to an observable and a corresponding magnitude of the expectation value of the quantum state with respect to the observable is greater than the target precision and whether a magnitude of the sum of the expectation value of the classical description of the mimicking quantum state for the iteration with respect to the observable and the corresponding magnitude of the expectation value of the quantum state with respect to the observable is greater than the target precision. That is, the system searches for an observable P in the set of observables such that \Tr (Pa>^) — uP| > e and |

[0083]

[0084] Tr (Pio^)) + up\ > e, where to represents the classical description of the mimicking quantum state for the iteration t.

[0085] In response to determining that the set of observables does not include such an observable, the system outputs the classical description of the mimicking quantum state for the iteration as a final classical description of the mimicking quantum state (step 308). That is, the system outputs

[0086]

[0087] a =

[0088] In response to determining that the set of observables includes such an observable, the system updates the classical description of the mimicking quantum state for the iteration. The system determines whether the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable is larger than a third predetermined threshold (step 310). For example, the system determines whether uP> 2e / 3.

[0089] In response to determining that the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable is not larger than the third predetermined threshold, the system sets a sign rPof the expectation value of the quantum state with respect to the observable as equal to zero (step 312). That is, the system sets rP— 0.

[0090] In response to determining that the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable is larger than the third predetermined threshold, the system uses multiple physical copies of the quantum state to measure the sign rPe +1, —1} of the expectation value of the quantum state with respect to the observable (step 314). The number of physical copies can include

[0091]

[0092] 2) copies.

[0093] The system then updates the classical description of the mimicking quantum state for the iteration using the set or measured sign (step 316). The system computes a product of the observable with the sign of: the difference between the expectation value of the classical description of the mimicking quantum state for the iteration with respect to the observable and the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable multiplied by the set or measured sign. That is, the system computes M(t):= sign{Tr P j(t>— rPuP] ■ P. The system then sets the updated classical description of the mimicking quantum state for the iteration as a normalized exponential of the computed product, e g., sets

[0094] ,=>

[0095] Tr (exp(— XT=I M^))

[0096]

[0097] If the iteration is not a final iteration, the system provides the updated classical description of the mimicking quantum state as input for a subsequent iteration (step 318). If the iteration is a final iteration, the system outputs o = a1"* (step 320).

[0098] FIG. 4 is a flow diagram of a process 400 for estimating expectation values of an unknown quantum state with respect to a set of observables and to a target precision. For example, the set of observables can be a set of k-local fermionic operators. For convenience, the 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, the system 100 of FIG. 1, appropriately programmed, can perform example process 400.

[0099] The system uses multiple physical copies of the quantum state to compute, for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable (step 402). Step 402 of example process 400 is similar to step 202 of example process 200. For brevity, details are not repeated.

[0100] The system uses the estimated magnitudes to generate an updated set of observables (step 404). Step 404 of example process 400 is similar to step 204 of example process 200. For brevity7, details are not repeated.

[0101] The system obtains, e.g., receives or generates data that represents, a frustration graph for the updated set of observables. Each vertex in the frustration graph represents a respective observable in the updated set of observables and pairs of vertices that represent anti-commutating observables in the updated set of observables are connected by respective edges. By construction, the size of a largest clique of vertices included in the frustration graph is at most 9ε-2, where ε represents the target precision. This property of the frustration graph enables an efficiently computable fractional coloring of the frustration graph, as described below.

[0102] The system classifies, e.g., colors, vertices of the frustration graph using multiple classifications, e.g., colors (step 406). For example, the system can use a known graph coloring algorithm to color the graph. In some implementations the coloring can be a fractional coloring. In some implementations the multiple colors can include at most a number of colors that is polynomial in e-2, where e represents the target precision. The runtime of the coloring can be polynomial in the size n of the quantum state.

[0103] For each classification (color) of the multiple classifications (colors), the system measures observables that correspond to vertices that are colored in the color (step 408). The system measures the observables using Clifford measurements of single physical copies of the quantum state. Observables that correspond to vertices that are colored in the color are measured approximately simultaneously, e.g., within limitations of the hardware performing the measurements. In some implementations

[0104]

[0105] physical copies of the quantum state are used to perform the measurements, where represents the number of colors used to color the vertices of the frustration graph, | S | represents the size of the set of observables, and e represents the target precision.

[0106] The system outputs the magnitudes computed at step 202 and measurements obtained at step 408 as estimated expectation values of the unknow n quantum state with respect to the set of observables. In some implementations the system can use the estimated expectation values of the quantum state with respect to the set of k-local Pauli operators to reconstruct k-qubit reduced density matrices of the quantum state, e.g., for quantum chemistry applications.

[0107] FIG. 5 depicts an example classical / quantum computer 500 for performing some or all of the classical and quantum operations described in this specification. The example classical / quantum computer 500 includes an example quantum computing device 502. The quantum computing device 502 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 do not limit implementations of the inventions described and / or claimed in this document.

[0108] The example quantum computing device 502 includes a qubit assembly 552 and a control and measurement system 504. The qubit assembly includes multiple qubits, e.g., qubit 506, that are used to perform algorithmic operations or quantum computations. While the qubits shown in FIG. 5 are arranged in a rectangular array, this is a schematic depiction and is not intended to be limiting. The qubits can also be arrange in other forms, e.g., in a line. The qubit assembly 552 also includes adjustable coupling elements, e.g., coupler 508, that allows for interactions betw een coupled qubits. In the schematic depiction of FIG. 5, each qubit is adjustably coupled to each of its four adjacent qubits by means of respective coupling elements. However, this is an example arrangement of qubits and couplers and other arrangements are possible, including arrangements that are non -rectangular, arrangements that allow" for coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits. Each qubit can be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The specific physical realization of the multiple qubits and how they interact with one another is dependent on a variety of factors including the type of the quantum computing device 502 included in the example computer 500 or the type of quantum computations that the quantum computing device is performing. For example, in an atomic quantum computer the qubits may be realized via atomic, molecular or solid-state quantum systems, e.g.. hyperfine atomic states. As another example, in a superconducting quantum computer the qubits may be realized via superconducting qubits or semi-conducting qubits, e.g., superconducting transmon states. As another example, in a NMR quantum computer the qubits may be realized via nuclear spin states.

[0109] In some implementations a quantum computation can proceed by loading qubits, e.g., from a quantum memory, and applying a sequence of unitary operators to the qubits.

[0110] Applying a unitary operator to the qubits can include applying a corresponding sequence of quantum logic gates to the qubits, e.g., to implement a quantum algorithm such as a quantum principle component algorithm. Example quantum logic gates include single-qubit gates, e.g., Pauli-X. Pauli-Y. Pauli-Z (also referred to as X, Y. Z). Hadamard gates. S gates, rotations, two-qubit gates, e.g., controlled-X, controlled-Y, controlled-Z (also referred to as CX, CY, CZ), controlled NOT gates (also referred to as CNOT) controlled swap gates (also referred to as CSWAP), and gates involving three or more qubits, e.g., Toffoli gates. The quantum logic gates can be implemented by applying control signals 510 generated by the control and measurement system 504 to the qubits and to the couplers.

[0111] For example, in some implementations the qubits in the qubit assembly 552 can be frequency tunable. In these examples, each qubit can have associated operating frequencies that can be adjusted through application of voltage pulses via one or more drive-lines coupled to the qubit. Example operating frequencies include qubit idling frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idling frequency may put the qubit into a state where it does not strongly interact with other qubits, and where it may be used to perform single-qubit gates. As another example, in cases where qubits interact via couplers with fixed coupling, qubits can be configured to interact with one another by setting their respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. In other cases, e.g., when the qubits interact via tunable couplers, qubits can be configured to interact with one another by setting the parameters of their respective couplers to enable interactions between the qubits and then by setting the qubit’s respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed in order to perform multi-qubit gates.

[0112] The type of control signals 510 used depends on the physical realizations of the qubits. For example, the control signals may include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system.

[0113] A quantum computation can be completed by measuring the states of the qubits, e.g., using a quantum observable such as X or Z, using respective control signals 510. The measurements cause readout signals 512 representing measurement results to be communicated back to the measurement and control system 504. The readout signals 512 may include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and / or the qubits. For convenience, the control signals 510 and readout signals 512 show n in FIG. 5 are depicted as addressing only selected elements of the qubit assembly (i.e. the top and bottom rows), but during operation the control signals 510 and readout signals 512 can address each element in the qubit assembly 552.

[0114] The control and measurement system 504 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 552, as described above, as well as other classical subroutines or computations. The control and measurement system 504 includes one or more classical processors, e.g., classical processor 514, one or more memories, e.g., memory 516, and one or more I / O units, e g., I / O unit 518, connected by one or more data buses. The control and measurement system 504 can be programmed to send sequences of control signals 510 to the qubit assembly, e.g. to carry out a selected series of quantum gate operations, and to receive sequences of readout signals 512 from the qubit assembly, e.g. as part of performing measurement operations.

[0115] The processor 514 is configured to process instructions for execution within the control and measurement system 504. In some implementations, the processor 514 is a single-threaded processor. In other implementations, the processor 514 is a multi -threaded processor. The processor 514 is capable of processing instructions stored in the memory 516.

[0116] The memory 516 stores information within the control and measurement system 504. In some implementations, the memory' 516 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 516 can include storage devices capable of providing mass storage for the system 504, e.g. a hard disk device, an optical disk device, a storage device that is shared over a network by multiple computing devices (e.g.. a cloud storage device), and / or some other large capacity storage device.

[0117] The input / output device 518 provides input / output operations for the control and measurement system 504. The input / output device 518 can include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers, whereby to send control signals 510 to and receive readout signals 512 from the qubit assembly, as appropriate for the physical scheme for the quantum computer. In some implementations, the input / output device 518 can also include one or more network interface devices, e.g., an Ethernet card, a serial communication device, e.g., an RS-232 port, and / or a wireless interface device, e.g., an 802.11 card. In some implementations, the input / output device 518 can include driver devices configured to receive input data and send output data to other external devices, e.g., keyboard, printer and display devices.

[0118] Although an example control and measurement system 504 has been depicted in FIG.

[0119] 5, implementations of the subject matter and the functional operations described in this specification can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them.

[0120] The example system 500 also includes an example classical processor 550. The classical processor 550 can be used to perform classical computation operations described in this specification according to some implementations.

[0121] Implementations of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term ‘‘quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0122] Implementations of the subject matter described in this specification can 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, or to control the operation of, data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0123] The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term '’qubit" encompasses all quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible.

[0124] The term “data processing apparatus7’ refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by w ay of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardw are, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0125] A digital computer program, which may also be referred to or described as a program, softw are, a softw are application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can 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, a software application, a module, a software module, a script, or code, can 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 can be written in a quantum programming language, e.g., QCL or Quipper.

[0126] A computer program may. but need not. correspond to a file in a file system. A program can 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, subprograms, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0127] The processes and logic flows described in this specification can 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 can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0128] For a system of one or more computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For example, a quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions. Computers suitable for the execution of a computer program can be based on general or special purpose processors, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read-only memory, a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0129] The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices.

[0130] Quantum circuit elements (also referred to as quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of 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 represent and operate on information in more than one state simultaneously. 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-SQUID or DC-SQUID), among others.

[0131] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operations on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors.

[0132] In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, e.g., superconducting quantum and / or classical circuit elements.

[0133] Fabrication of the superconducting circuit elements can entail the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography).

[0134] During operation of a quantum computational system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature.

[0135] In certain implementations, control signals for the quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0136] 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 by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory’ devices; magnetic disks, e.g., internal hard disks or removable disks; magnetooptical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence. Control of the various systems described in this specification, or portions of them, can be implemented in a computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more processing devices and memory to store executable instructions to perform the operations described in this specification.

[0137] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0138] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations 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 into multiple software products.

[0139] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

[0140] What is claimed is:

Claims

CLAIMS1. A method for estimating expectation values of a quantum state with respect to a set of observables and to a target precision, the method comprising:computing, using multiple physical copies of the quantum state and for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable to within a first predetermined threshold range that is based on the target precision;generating, using the estimated magnitudes, an updated set of observables, comprising removing observables from the set of observables that correspond to an expectation value with a magnitude that is smaller than a second predetermined threshold value that is based on the target precision;computing, using the updated set of observables, a classical description of a mimicking quantum state, wherein the mimicking quantum state has a same dimension as the quantum state, and expectation values of the mimicking quantum state with respect to the updated set of observables have respective magnitudes that are the same as respective magnitudes of the expectation values of the quantum state with respect to the updated set of observables; andperforming Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state to obtain, for each observable in the updated set of observables, an estimate of a sign of the expectation value of the quantum state with respect to the observable.

2. The method of claim 1, wherein the set of observables comprises a complete set of n-qubit Pauli operators, wherein n represents the size of the quantum state.

3. The method of claim 1 or claim 2, wherein computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing two-copy measurements of the quantum state.

4. The method of claim 3, wherein performing two-copy measurements of the quantum state comprises performing O(log(|S|) ε-4) two-copy measurements, where |S| represents the size of the set of observables and ε represents the target precision.

5. The method of any preceding claim, wherein computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing Bell sampling on a tensor product of two physical copies of the quantum state.

6. The method of any preceding claim, wherein expectation values of the mimicking quantum state with respect to the updated set of observables comprise a same or different sign as corresponding expectation values of the quantum state with respect to the updated set of observables.

7. The method of any preceding claim, wherein performing Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state comprises preparing multiple physical copies of the mimicking quantum state.

8. The method of any preceding claim, wherein performing Bell sampling on a tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state to obtain, for each observable in the updated set of observables, an estimate of a sign of the expectation value of the quantum state with respect to the observable comprises, for each observable in the set of updated observables:obtaining a sample for the observable, wherein the sample comprises an expectation value of a tensor product of the observable with respect to the tensor product of a physical copy of the quantum state and a physical copy of the mimicking quantum state;computing an expectation value of the observable with respect to the mimicking quantum state; anddividing the sample for the observable by the expectation value of the observable with respect to the mimicking quantum state to obtain an estimate of the sign of the expectation value of the quantum state with respect to the observable.

9. The method of claim 8, wherein performing Bell sampling comprises obtaining O(log(|S|) ε-4) Bell samples, where |S| represents the size of the set of observables and ε represents the target precision.

10. The method of any preceding claim, wherein computing the expectation value of the observable with respect to the mimicking quantum state comprises exactly computing theexpectation value of the observable with respect to the mimicking quantum state using the classical description of the mimicking quantum state.

11. The method of any preceding claim, wherein computing the classical description of the mimicking quantum state comprises performing a classical search algorithm.

12. The method of any preceding claim, wherein computing the classical description of the mimicking quantum state comprises:setting an initial classical description of the mimicking quantum state;for each iteration in a sequence of a predetermined number of iterations:determining whether the set of observables includes an observable that generates an expectation value with respect to the classical description of the mimicking quantum state for the iteration that is within a predetermined distance from a corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable;in response to determining that the set of observables does not include such an observable, outputting the classical description of the mimicking quantum state for the iteration as a final classical description of the mimicking quantum state; in response to determining that the set of observables includes such an observable, updating the classical description of the mimicking quantum state for the iteration, comprising:determining whether the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable is larger than a third predetermined threshold;in response to determining that the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable is not larger than the third predetermined threshold, setting a sign of the expectation value of the quantum state with respect to the observable as equal to zero;in response to determining that the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable is larger than the third predetermined threshold, using multiple physical copies of the quantum state to measure the sign of the expectation value of the quantum state with respect to the observable; andupdating the classical description of the mimicking quantum state for the iteration using the set or measured sign; andproviding the updated classical description of the mimicking quantum state as input for a subsequent iteration.

13. The method of claim 12, wherein the predetermined number of steps is dependent on the target precision.

14. The method of any of claims 12 or 13. wherein setting the initial classical description of the mimicking quantum state comprises setting the initial classical description of the mimicking quantum state as a maximally mixed state.

15. The method of any of claims 12 to 14, wherein determining whether the set of observables includes an observable that generates an expectation value with respect to the classical description of the mimicking quantum state for the iteration that is within the predetermined distance from a corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable comprises:determining whether a magnitude of the difference between an expectation value of the classical description of the mimicking quantum state for the iteration with respect to an observable and a corresponding magnitude of the expectation value of the quantum state with respect to the observable is greater than the target precision; anddetermining whether a magnitude of the sum of the expectation value of the classical description of the mimicking quantum state for the iteration with respect to the observable and the corresponding magnitude of the expectation value of the quantum state with respect to the observable is greater than the target precision.

16. The method of claims 1 to 11, wherein updating the classical description of the mimicking quantum state for the iteration using the set or measured sign comprises:computing a product of the observable with the sign of the difference between the expectation value of the classical description of the mimicking quantum state for the iteration with respect to the observable and the corresponding estimate of the magnitude of the expectation value of the quantum state with respect to the observable multiplied by the set or measured sign; andsetting an updated classical description of the mimicking quantum state for the iteration as a normalized exponential of the computed product.

17. A method for estimating expectation values of a quantum state with respect to a set of observables and to a target precision, the method comprising:computing, using multiple physical copies of the quantum state and for each observable in the set of observables, an estimate of a magnitude of the expectation value of the quantum state with respect to the observable to within a first predetermined threshold range that is based on the target precision;generating, using the estimated magnitudes, an updated set of observables, comprising removing observables from the set of observables that correspond to an expectation value with a magnitude that is smaller than a second predetermined threshold value that is based on the target precision;obtaining a frustration graph for the updated set of observables, wherein each vertex in the frustration graph represents a respective observable in the updated set of observables and pairs of vertices that represent anti-commutating observables in the updated set of observables are connected by respective edges;classifying vertices of the frustration graph using multiple classifications, wherein adjacent vertices in the frustration graph are classified as different classifications; and for each classification of the multiple classifications:measuring observables that correspond to vertices that are classified in the classification, comprising performing Clifford measurements of single physical copies of the quantum state.

18. The method of claim 17. wherein the set of observables comprises a set of k-local fermionic operators.

19. The method of claim 18, further comprising using the estimated expectation values of the quantum state with respect to the set of k-local Pauli operators to reconstruct k-qubit reduced density matrices of the quantum state.

20. The method of any of claims 17 to 19, wherein classifying vertices of the frustration graph using multiple classifications comprises coloring vertices of the frustration graph using multiple colors, wherein the colouring comprises fractional colouring.

21. The method of claim 20, wherein multiple colors comprise at most a number of colors that is polynomial in ε-2, where ε represents the target precision.

22. The method of any of claims 17 to 21, wherein computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing two-copy measurements of the quantum state.

23. The method of claim 22, wherein performing two-copy measurements of the quantum state comprises performing O(log(|S|) ε-4) two-copy measurements, where |S| represents the size of the set of observables and ε represents the target precision.

24. The method of any of claims 17 to 23, wherein computing the estimates of the magnitudes of the expectation values of the quantum state with respect to the observables in the set of observables comprises performing Bell sampling on a tensor product of two physical copies of the quantum state.

25. The method of any of claims 17 to 24. wherein the observables that correspond to vertices that are classified in the classification are measured approximately simultaneously.

26. The method of any of claims 17 to 25, wherein performing Clifford measurements of single physical copies of the quantum state comprise using O(x|S|10ε-5) physical copies of the quantum state, where x represents the number of classifications used to classify the vertices of the frustration graph, |S| represents the size of the set of observables, and ε represents the target precision.

27. The method of any of claims 17 to 26, wherein the size of a largest clique of vertices included in the frustration graph is at most 9ε-2, where ε represents the target precision.

28. A system comprising:a quantum computer; anda classical computer coupled to the quantum computer, the classical computer comprising:one or more data processing apparatuses; andnon- transitory computer readable storage media in data communication with the one or more data processing apparatuses and storing instructions executable by the data processing apparatuses;wherein the system is configured to perform operations according to the method of any one of claims 1 to 27.

Images