Simultaneous measurement of commuting operators

Abstract

A computer-implemented method for determining a measurement value for each operator of a plurality of operators. The method includes grouping the plurality of operators into one or more sets of operators that commute with each other, each set including one or more of the plurality of operators. For each set of operators, determine a transformed subset of operators based on the set of operators, such that the set of operators is equal to a product of the transformed subsets of operators, and a mapping circuit based on the transformed subset of operators, the mapping circuit including an arrangement of quantum gates configured to operate on a plurality of qubits in a quantum computer. Performing a measurement routine for each set of operators, the measurement routine including preparing a trial state using a first arrangement of quantum gates using a plurality of qubits in the quantum computer, operating the mapping circuit on the plurality of qubits in the trial state, and performing a measurement on each qubit of the plurality of qubits to obtain a qubit measurement for each qubit.

Description

[Background technology]

[0001] The present disclosure relates to determining energy levels. In particular, the present disclosure relates to methods for determining the energy levels of a physical system. Furthermore, the present disclosure relates to determining measurement results of commuting operators using a quantum computer.

[0002] Being able to determine the possible eigenstates and energies of a physical system, such as a molecule or atom, is very useful in many technical fields. By determining how the energy may change when the system is perturbed, it is possible to derive many molecular properties. For example, by solving the Schrödinger equation associated with the molecular electronic structure Hamiltonian for a number of atomic nuclear geometries, it is possible to construct the potential energy surface (PES) of a molecular system. Knowledge of the PES is particularly important in chemistry, as it allows scientists to determine, among other things, reaction rates.

[0003] Determining excited states is necessary to determine the light spectrum, as well as other charge and energy transfer processes, in photovoltaic materials. Characterization of excited states can also provide a better understanding of many chemical reactions, such as those involving photolysis. Furthermore, classical methods such as density functional theory often do not allow the determination of excited states, even for materials for which ground-state energy calculations are possible.

[0004] Many current methods of obtaining information about the eigenstates and energies of physical systems rely on classical computers that use complex algorithms to simulate the physical system. However, such methods often require prohibitive amounts of computing resources or do not return solutions with sufficient accuracy. Quantum computers are capable of simulating systems much more efficiently than is possible with classical computers, and experimental development of quantum computers is progressing using a variety of architectures. Miniature devices based on trapped ions or superconducting systems are now available with a clear roadmap to large-scale implementations.

[0005] There are known methods to find the energy levels of a physical system using quantum computers. For example, variational quantum eigensolver (VQE) methods can be used to estimate the energy levels of a physical system to a specified accuracy, given knowledge of the system's Hamiltonian.

[0006] Known methods such as VQE require many iterations of preparing trial states and many iterations of measuring the trial states in order to run on a quantum computer, which means that it can take a long time and a lot of processing power to obtain useful results. It is therefore desirable to be able to reduce the number of iterations of preparing and measuring trial states in determining the energy levels of a physical system.

[0007] The present invention seeks to address these and other disadvantages encountered in the prior art by providing an improved method for determining the energy level of a physical system in which measurements of multiple operators used to determine the energy level can be obtained simultaneously. [Mode for carrying out the invention]

[0008] According to one aspect of the disclosure, a method for determining a measurement result value for each operator of a plurality of operators is provided, the method including: grouping the plurality of operators into one or more sets, each set including one or more of the plurality of operators; determining, for each set of operators, a subset of transformed operators based on the set of operators, a mapping circuit based on the subset of transformed operators, and a post-measurement processing routine based on the subset of transformed operators; and performing the measurement routine for each set of operators, the measurement routine including: preparing a trial state using a first arrangement of quantum gates using a plurality of qubits on a quantum computer; operating the mapping circuit on the plurality of qubits in the trial state; performing a measurement on each qubit of the plurality of qubits to obtain a qubit measurement for each qubit; and applying the post-measurement processing routine to the qubit measurement values ​​to convert the qubit measurements into operator measurements for each of the operators in the set of operators.

[0009] An example use of the method includes determining the energy level of a physical system. Thus, disclosed herein is a method for determining an estimate of an expected energy value of a physical system using a quantum computer. The expected energy value is described by a sum of expected values ​​of a plurality of operators, the method including determining a measurement value for each operator of the plurality of operators, the determining including grouping the plurality of operators into one or more sets, each set including one or more of the plurality of operators, and for the set of operators, determining a subset of transformed operators based on the set of operators, a mapping circuit based on the subset of transformed operators, and a post-measurement processing routine based on the subset of transformed operators. Determining a measurement result for each operator of the plurality of operators further includes running a measurement routine for each set of operators, the measurement routine including preparing a trial state using a plurality of qubits on the quantum computer using a first arrangement of quantum gates, operating a mapping circuit on the plurality of qubits in the trial state, performing a measurement on each qubit of the plurality of qubits to obtain a qubit measurement for each qubit, and applying a post-measurement processing routine to the qubit measurement values ​​to convert the qubit measurements to operator measurements for each of the operators in the set of operators. The method further includes determining an estimate of an expected energy value of the physical system based on at least the determined operator measurement values ​​for each operator in each set.

[0010] Optionally, determining the subset of transformed operators, determining the mapping circuit, and determining the post-measurement processing routine are performed using a classical computer, which further performs the step of applying the post-measurement processing routine to the qubit measurements to convert the qubit measurements into operator measurements for each of the operators in the set of operators. This is advantageous over conventional methods because it allows the process of determining the operator measurements from the qubit measurements to be determined classically, instead of using additional gates in the mapping circuit. This therefore reduces the computational requirements on the quantum computer, allowing simpler mapping circuits with as few qubit gates as possible to be used.

[0011] The steps of preparing the trial states, operating the mapping circuitry, and performing a measurement on each qubit may be performed using a quantum computer.

[0012] Optionally, the measurement routine is executed multiple times for each set to obtain a corresponding plurality of operator measurements for each operator in each set. The method may further include determining a predicted value for each operator in each set based on an average of the corresponding plurality of operator measurements. Optionally, determining the estimate of the energy predicted value includes a sum of the predicted values ​​for each operator in each set.

[0013] Optionally, the mapping circuit includes at least one multi-qubit gate configured to act on at least two of the plurality of qubits. This allows for grouping of sets of operators into groups of operators that commonly commute, which is advantageous over conventional methods as it allows for groups with a large number of operators, and therefore multiple operator measurements can be obtained simultaneously using the methods disclosed herein.

[0014] Optionally, the mapping circuit includes one or more multi-qubit gates, the number of multi-qubit gates being proportional to the number of the plurality of qubits, with the proportionality having an upper limit on the number of the plurality of qubits multiplied by the number of independent operators in the set of operators, each operator in the set of operators can be constructed from one or more independent operators.

[0015] In some embodiments, the mapping circuit includes one or more single-qubit gates configured to apply a rotation to each qubit of the plurality of qubits.

[0016] In some embodiments, each operator in a set generally commutes with every other operator in the set. This is advantageous over conventional methods because it allows for larger groupings of operators, which means that many more operator measurements can be taken simultaneously using the methods disclosed herein.

[0017] Optionally, determining the subset of transformed operators includes determining one or more independent operators of the set of operators, where each operator in the set of operators can be constructed from one or more independent operators, and transforming the one or more independent operators into the subset of transformed operators.

[0018] Optionally, transforming the one or more independent operators into the subset of transformed operators includes determining whether the number of independent operators matches the number of the plurality of qubits, and in response to determining that the number of independent operators is less than the number of the plurality of qubits, constructing one or more new transformed operators that are added to the subset of transformed operators such that the number of transformed operators matches the number of qubits.

[0019] Optionally, the qubit measurements represent measurements of a subset of the transformed operators. A post-measurement processing routine is then used to convert the transformed operator measurements into operator measurements. This is advantageous over traditional methods because it allows the process of determining the operator measurements from the qubit measurements to be determined classically, instead of using additional gates in the mapping circuit. This therefore reduces the computational requirements on the quantum computer and allows simpler mapping circuits with as few qubit gates as possible to be used.

[0020] According to a further aspect of the present disclosure, a computer-readable medium is provided that includes instructions that, when executed by a processor, cause the processor to perform any one of the disclosed methods.

[0021] According to another aspect of the present disclosure, an apparatus is provided that includes a classical computer and a quantum computer configured to perform any one of the disclosed methods. [Brief description of the drawings]

[0022] Specific embodiments are described below, by way of example only, and with reference to the accompanying drawings. [Figure 1] We present a state-of-the-art variational quantum eigensolver (VQE) method for determining the energy levels of physical systems. [Figure 2a] 1 shows a measurement routine for determining the measurement results of multiple Pauli operators according to known methods. [Figure 2b] 13 illustrates a measurement routine for determining a measurement result of multiple Pauli operators, according to some embodiments. [Diagram 3] We show how embodiments of the present disclosure can be incorporated into a VQE framework to determine the energy levels of a physical system. [Figure 4] 1 is an illustration of a mapping circuit according to a particular embodiment; [Diagram 5] 1 is a flow chart illustrating a method according to an embodiment of the present disclosure. [Figure 6] 1 illustrates a block diagram of one implementation of a computing device according to some embodiments. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] Reference is made herein to equations numbered (1) through (31), which are provided in Appendix A at the end of the description.

[0024] In hybrid quantum-classical algorithms, such as variational quantum eigensolvers (VQE), the problem of determining the energy levels of a physical system is specified by a problem Hamiltonian H. This Hamiltonian is specific to a physical system, such as an atom or molecule, and describes the energy levels of that physical system, as explained below. The Hamiltonian of the problem is divided into a sum of so-called Pauli operators according to equation (1). The coefficients a i is calculated by a classical computer, and the Pauli term P i have the property that their predictions for any given trial state can be estimated on a quantum computer. <h>is the Pauli operator P i It is estimated by measuring the expected values ​​of in turn and calculating their coefficient-weighted sum on a classical computer.

[0025] Reference is now made to Fig. 1 which illustrates a variational quantum eigensolver (VQE) method for determining energy levels of physical systems according to the state of the art. Dashed box 102 illustrates the portion of the method performed using a quantum computer, using quantum circuits. Dashed box 104 illustrates the portion of the method performed using a classical computer, using classical circuits. The arrow between dashed boxes 102 and 104 illustrates the interface between the quantum and classical computers.

[0026] As will be appreciated by those skilled in the art, the energy state of a physical system can be described using a Hamiltonian operator, including the sum of multiple Pauli operators. Using standard VQE methods, a quantum expectation value estimation routine (indicated by box 108) can be used together with a classical optimizer 112 to determine the energy levels of the Hamiltonian H of the physical system. The classical optimizer adjusts the trial state wave function |ψ(λ)> according to a parameter λ. For a given normalized |ψ(λ)>, it is possible to evaluate the energy. E(λ)≡<ψ(λ)|H|ψ(λ)>=Σa i <ψ(λ)|P i |ψ(λ)>

[0027] To describe standard VQE in more detail, we first consider the Hamiltonian operator H as a finite sum H=Σa i P i In the formula, a i is a complex coefficient, and P i is the Pauli operator. Each a i P i can be described as an augend. The number of augends m is assumed to be a polynomial in the size of the system, as in the case of the electronic Hamiltonian in quantum chemistry.

[0028] To evaluate the energy state of a physical system, knowledge of the Hamiltonian is used to determine a hypothetical trial state |ψ(λ)>, which can be prepared using multiple qubits on a quantum computer. The hypothetical trial state has energy E(λ) that depends on a parameter λ. The trial state is prepared in a quantum processor, and an arrangement of quantum gates, also referred to as a quantum circuit, is used to determine the expected values ​​of each summand, one at a time. Given the expected value estimates, a classical computer 104 is used to calculate the corresponding complex coefficients a of each Pauli operator. i A weighted sum is determined based on E(λ)=E(λ)+λ ... R(λ):|0>→|ψ(λ)> where |0> is the starting state of the qubit. The variational principle (VP) justifies the whole VQE procedure when finding the ground state, and for the eigenvalues ​​of the ground state of H, E min and VP is such that E(λ) ≥ E if and only if |ψ(λ)> is a ground state. min states that E(λ) is equal to the energy level of a physical system. Similarly, the local minima of the E(λ) curve represent other energy levels / states of the physical system. Therefore, it is possible to use the VQE method to determine estimates of the energy levels of a physical system described by a Hamiltonian.

[0029] A typical VQE process prepares an initial trial state |ψ(λ)〉 using a preparation circuit R contained within a quantum computer. The preparation of the initial trial state is shown in box 106 of Figure 1. The preparation circuit R is a particular arrangement of quantum gates determined by the parameter λ that is used to prepare the trial state on multiple qubits in the quantum computer.

[0030] The expected value of each Pauli operator term in the Hamiltonian can then be estimated for a given trial state. This determination is illustrated in block 108 of FIG. 2. In other words, to determine the energy eigenvalues ​​of a Hamiltonian with m summands, the quantum computing device calculates the energy eigenvalues ​​of the trial states P1, P2, ...P m Measurements are taken.

[0031] These measurements are then communicated to a classical computing device, represented by dashed box 104 in FIG. 1, which computes the expected values ​​<ψ(λ)|P1|ψ(λ)>, <ψ(λ)|P2|ψ(λ)>, …<ψ(λ)|P m Calculate |ψ(λ)>. As one skilled in the art will appreciate, each measurement P1 can be obtained directly using a simple circuit with a circuit depth D=O(1). To determine the expected value of each Pauli operator, multiple measurements are performed on a trial state to obtain multiple measurement results of that Pauli operator. In other words, the same quantum circuit is applied multiple times to a qubit in a given trial state, and then the qubit is measured to provide a measurement result value. The measurement result values ​​form a statistical distribution from which the expected value of the Pauli operator can be obtained, for example by averaging over the multiple measurement result values.

[0032] After calculating the expected value of each Pauli operator, the classical computing device calculates the corresponding complex coefficient a i The weighted sum of the expected values ​​of each Pauli operator, weighted by , is determined to find the energy value of the Hamiltonian of the initial trial state. Based on this eigenvalue, the classical computer 104 updates the parameter λ in box 112, which allows the construction of a new trial state. The quantum computer is instructed to create a new trial state, and the entire process is repeated until the optimization procedure satisfies that the desired energy levels are determined to a specified accuracy. The measurement process is repeated N=O(1 / ε) for each Pauli operator to achieve accuracy within ε of the expectation. 2 ) times. The number of iterations therefore scales polynomially with the required precision. For each iteration, a trial state |ψ(λ)> needs to be prepared on the quantum computer using the preparation circuit R, since the qubits in the trial state are measured each time, thus causing the state of the qubits to collapse. Thus, in known state-of-the-art VQE methods, N state preparations are required to determine the expected value of a single Pauli operator. Thus, for a Hamiltonian with m Pauli operator terms, known methods require N×m state preparations on the quantum computer to determine the expected value of the energy of the trial state |ψ(λ)>, thus requiring a large number of state preparation and measurement operations. Thus, known methods are limited in that a large number of calculations, and therefore longer processing times and longer quantum computer operations, are required to obtain useful results.

[0033] However, new superconducting, trapped ion, and proposed networked quantum computer architectures allow for parallel measurement and readout of qubits. The disclosed method exploits the ability to perform measurements on two or more Pauli operators simultaneously to reduce the total number of state preparations and measurements (and therefore execution time) when determining the energy expectation of a trial state.

[0034] As discussed in more detail below, the terms of the Hamiltonian are grouped according to certain properties, and measurements can then be taken simultaneously for all terms in a group. After the usual state preparation circuitry to prepare a trial state |ψ(λ)〉, the disclosed method performs a novel rotation or mapping circuitry built specifically for the group, so that after a single measurement of all n qubits, the measurement results for all terms in the group can be determined from measurements on the n qubits.

[0035] Figure 3 illustrates how the method of the present disclosure may be incorporated into a VQE framework to determine the energy level of a physical system. Specifically, Figure 3 illustrates method steps corresponding to boxes 106, 108, and 110 of the VQE method illustrated in Figure 1, where measurements of multiple operators in a group may be obtained simultaneously, rather than one at a time. The operators are grouped together into sets of operators based on particular properties of the operators, as discussed in more detail below.

[0036] In step 310, the number of measurements N per group or set is determined. As above, N represents the number of repeated state preparations and corresponding measurements that are performed to obtain a distribution of measurements from which the expected values ​​of all operators in the group can be obtained. In other words, the number of measurements N per group represents the number of repetitions of the steps in the dashed box 320.

[0037] Dashed box 320 includes method steps of the present disclosure that are used to simultaneously obtain measurements of all operators in the group. In step 312, a trial state is prepared using multiple qubits on a quantum computer, similar to step 106 of FIG. 1. In step 314, a rotation or mapping circuit is applied to the multiple qubits prepared in the quantum state. Rotation circuits are discussed in more detail below and are constructed using a specific arrangement of quantum gates depending on the group of operators. In step 316, each qubit prepared in the quantum state and to which the rotation circuit has been applied is measured to obtain a measurement. The measurement will be +1 or -1. Thus, in step 316, multiple qubit measurements are obtained. The qubit measurements are then input to a post-measurement routine (otherwise referred to as classical post-processing) in step 318, which converts the qubit measurements into measurement results for each operator in the set. The post-measurement routine includes determining one or more products of any of one or more of the qubit measurements to determine a measurement for each operator in the set. The particular product of a particular qubit measurement is determined based on the set of operators itself, which is discussed in more detail below.

[0038] The process identified in dashed box 320 is repeated N times for each group / set (N is determined in step 310, which may be the same for each set of operators or may be specific to a given set of operators) to obtain N measurements for each operator in each set. From the N measurements of each operator, a predicted value for the operator can be determined by taking the average of the N measurements. Thus, predicted values ​​for all operators in a set can be obtained from N repetitions of box 320 for that set.

[0039] In step 330, a predicted value of the Hamiltonian is determined by summing the predicted values ​​of each term in the Hamiltonian (each weighted operator predicted value), where the predicted values ​​of the operators that make up the Hamiltonian are determined in step 320. In other words, an estimate of the predicted value of the energy of the physical system represented by the Hamiltonian is determined based on the predicted values ​​of the operators. Furthermore, an estimate of the error associated with the predicted value of the Hamiltonian can also be determined. Furthermore, the output of step 330 can be input to a classical optimizer, as in step 112 of FIG. 1, to update the trial state in a broader VQE framework.

[0040] A high level illustration of the disclosed method and a comparison with known techniques is shown in Figures 2a and 2b to illustrate the advantages it provides over known methods. Figure 2a shows the measurement routine of box 108 of Figure 1 according to known methods. The measurement routine is used to determine the measurement result of the Pauli operator of the Hamiltonian of a physical system. The measurement routine is used within the VQE (as box 108) to determine the energy levels of the physical system.

[0041] FIG. 2a shows a measurement routine for determining the measurement results of multiple Pauli operators according to a known method. FIG. 2a shows four different measurement routines 210, 220, 230, and 240 for determining the measurement results of each of P1, P2, P3, and P4, respectively. In this known method, each of 210, 220, 230, and 240 is executed separately to determine each measurement result one at a time. This is because each measurement routine involves state preparation (212, 222, 232, 242) on qubits in a quantum computer, which are then manipulated using quantum gates (214, 224, 234, 244), after which the qubits are measured (216, 226, 236, 246). The measurements of all the qubits can be used to obtain a measurement of a single Pauli operator. This process must be repeated for each of the Pauli operators P1, P2, P3, and P4 using the same state preparation and using quantum gates constructed based on the particular Pauli operator.

[0042] In the particular example shown in Figures 2a and 2b, the process requires four qubits (as indicated by the four qubit wires and four measurement results, one measurement per qubit), however, one of skill in the art will appreciate that in some examples any number of qubits required for the associated Pauli operators may be used.

[0043] FIG. 2b illustrates a measurement routine for determining measurement results of multiple Pauli operators according to the method of the present disclosure. In stark contrast to the method illustrated in FIG. 2a, FIG. 2b allows for the measurement results of two or more operators to be determined simultaneously using a new rotation circuit C. FIG. 2b illustrates state preparation 252, which prepares qubits in a quantum computer into a trial state, similar to steps 212, 222, 232, and 242 of FIG. 2a. A new circuit C is then used to operate qubits in the trial state at 254. The new circuit C includes an arrangement of quantum gates, including multi-qubit gates, and is discussed in more detail below. Following operation of circuit C on the qubits in the trial state, the qubits are measured at 256. The measurement results of the qubits are then processed using a new post-measurement routine P (258) to simultaneously determine measurements of each of the Pauli operators P1, P2, P3, and P4. A discussion of the hardware used for state preparation and operation of the quantum circuit on the qubits is discussed below. Thus, in stark contrast to the known method of FIG. 2a, the disclosed method makes it possible to obtain measurements of two or more Pauli operators using a single state preparation, and a single set of circuitry and measurement operations.

[0044] Thus, the new mapping circuitry and new post-measurement routines allow for the preparation of a single trial state and a set of qubit measurements to simultaneously obtain information about two or more Pauli operators. More specifically, the disclosed method allows for the simultaneous measurement of all Pauli operators in a group, each Pauli operator in a group having specific properties, as discussed in more detail below.

[0045] Methods of the present disclosure, such as method 250 shown in Figure 2b, can be used within the framework of VQE, but can determine energy estimates in significantly less time than known VQE methods. Specifically, the method of Figure 2b can be used to replace box 108 in Figure 1 (of known VQE methods) to obtain estimates of expected values ​​of multiple Pauli operators simultaneously. Importantly, the methods of the present disclosure can determine measurements of multiple Pauli operators in a Hamiltonian simultaneously, rather than running a measurement routine for each Pauli operator individually.

[0046] Grouping Operators As discussed above, the methods of the present disclosure can be used to simultaneously determine the measurement results of each operator in a group or set of operators. The operators are grouped according to certain properties, and the operators in a group commute with each other. For any group of operators that commute with each other, measurements of each can be obtained simultaneously by applying a mapping circuit to the quantum computer and performing some classical post-processing. The mapping circuit and classical post-processing are described in more detail below.

[0047] General commutation operator For a problem defined with n qubits, there are 4 possible Hamiltonians (excluding the identity term) n -1 possible Pauli operator. Each Pauli operator is a function of the other two 2n-1 - Swap with 2. The maximum number of operators that can be swapped is 2. n -1, but only n of these will be independent (the rest can be constructed from products of operators in the independent set). The number of qubits on which the problem is defined represents the number of qubits on which the Pauli operators can operate, which may correspond to the number of terms that make up the Pauli operators. The number of qubits may be dictated, at least in part, by the particular physical system described by the Hamiltonian, and may also be dictated, at least in part, by how the problem is represented on a quantum computer.

[0048] Typically, chemical Hamiltonians have O(n 4 ) terms. One way to sort these terms into groups of operators that commute with each other is to take each operator in turn, see if it can be placed within an existing group, and if not, start a new group. Doing this for all the operators in the Hamiltonian, one can place all the Pauli operators in a group, and every Pauli operator in a group commutes with every other Pauli operator in the same group.

[0049] More specifically, for a problem defined on n qubits, each operator in the Hamiltonian can have up to n subterms, where each subterm can be a Pauli matrix X, Y, or Z, or alternatively an identity matrix I. The methods of the present disclosure apply generally to the group of Pauli operators that commute.

[0050] As examples of operators that generally commute, consider the following two Pauli operators: P1=X1Z2I3Y4 P2=Y1Z2I3Y4 If P1 and P2 commute, then the products P1P2 and P2P1 are equal.

[0051] As can be appreciated by one skilled in the art, the product P1P2 should be: P1P2=X1Y1×Z2Z2×I3I3×Y4X4=iZ1×1×1×-iZ4=-i×i×Z1Z4=Z1Z4 As should be known, the square of any Pauli matrix is ​​1, so XY = -YX = iZ.

[0052] As can be appreciated by one skilled in the art, the product P2P1 should be: P2P1=Y1X1×Z2Z2×I3I3×X4Y4=-iZ1×1×1×iZ4=-i×i×Z1Z4=Z1Z4 Thus, even though corresponding terms in the same position within each operator may not commute (e.g., the first X1 term and Y1, and the fourth terms Y4 and X4 do not commute with each other so that XY does not equal YX), P1P2=P2P1, and therefore P1 and P2 commute.

[0053] Those skilled in the art will appreciate that a clear distinction is being made here with locally commuting Pauli operators, where each term in an operator must commute with a corresponding term at the same position in another operator. In other words, locally commuting operators are more restricted in that the operator must have the same Pauli matrix or identity matrix at each position in the string of subterms. It is clear that the above example of generally commuting operators P1 and P2 does not commute locally because the first and fourth terms do not commute (they are different Pauli matrices).

[0054] It will be appreciated that groups of locally exchanging operators are more restricted than groups of commonly exchanging operators. The properties of commonly exchanging allow for a large number of operators to be grouped together. The disclosed method applies a novel mapping circuit based on groups of commonly exchanging operators to the qubits of the trial state to determine the measurement results of each operator in the group. Thus, by grouping operators into groups of commonly exchanging operators, it is possible to have larger groups and therefore more simultaneous measurements can be taken at once.

[0055] Expected value of standard form operator For a group of operators of a certain form, the mapping circuitry required to make measurements simultaneously on all operators is known and no classical post-processing is required. This form is: For n Pauli operators defined on n qubits, these operators are given by, for 1 ≤ i ≤ n,

number

[0056] It can be shown that a conjecture for a group of operators of this form can be provided by equation (2) in Appendix A. U is the same for all i, and U is O ij = Z, followed by a Hadamard gate for every qubit. Thus, the operator

number

[0057] An exemplary set of operators defined on four qubits in this standard form is provided in equation (3) in Appendix A. A specific rotation or "mapping" circuit for this particular group of generally commuting operators applied to the qubits is provided in equation (4), where H i is the Hadamard gate applied to qubit i, and cZ ij is the control Z gate applied to qubits i and j.

[0058] Further one-qubit rotations can be performed to yield a transformed subset of operators based on the set of original operators. The transformed operators are in standard form, and there may be an additional single-qubit rotation applied to one or more of the qubits. This further single-qubit rotation, as well as measurements on the qubits after applying the control Z and Hadamard gates described above, provide measurements of the transformed subset of operators. Since the original operators can be obtained from the product of the transformed operators, measurements of the original operators can be obtained using a post-measurement processing routine that determines measurements of the original operators from the product of qubit measurements (equivalent to the product of the transformed operators). The methods of the present disclosure are used to manipulate or convert a general group of operators that commonly exchange into the specific form (hereinafter referred to as the "standard form") described above. Such manipulations or transformations consist of one-qubit rotations, described in more detail below, and taking products of the original operators, which corresponds to classical post-processing, otherwise referred to as post-measurement processing. In other words, the method of the present disclosure involves transforming a set of operators into a subset of transformed operators, which is a subset of the standard form operators, optionally with an additional single-qubit rotation applied to each qubit. In that case, the original set of operators that generally commute is equal to the product of the transformed operators. Thus, a measurement of the original set of operators can be obtained from a product of measurements of the transformed operators.

[0059] The rotation or mapping circuitry to be applied is determined based on the transformation between the transformed operators and the standard form operators, and the form of the standard form operators. In other words, the mapping circuitry includes a single-qubit rotation representing the transformation between a subset of the transformed operators and the standard form operators, and further includes the two-qubit controlled Z gates and Hadamard gates described above. The number and operation of the controlled Z gates depends on the form of the resulting transformed operators (or, equivalently, the form of the standard form operators). A post-measurement routine that converts the qubit measurements back to measurements of the original set of operators that typically commute is also determined.

[0060] Thus, in some embodiments of the present disclosure, the method includes manipulating a group of commonly commuting operators into a subset of transformed operators, which is based on a group of operators in a standard form and may have an additional single-qubit rotation applied to each qubit. However, those skilled in the art will appreciate that the step of converting the group of commonly commuting operators into a particular form may be omitted, for example, if the group of commonly commuting operators is already in the particular form required by the method.

[0061] Moreover, it will be appreciated that the particular forms of the operators and corresponding mapping circuits may not be exactly as described above, but may instead require other particular characteristics of the group of operators. The corresponding mapping circuits may equally require the application of any other suitable multi-qubit gates to particular pairs or sets of qubits.

[0062] Mapping circuit calculation As described above, a group of commonly commuting operators may be manipulated into a transformed subset of operators of a particular form, and a corresponding mapping circuit may be determined based on the transformed subset of operators. The following discussion provides one particular method, adding a second alternative to some of the manipulations, according to the present disclosure for manipulating operators and determining a particular mapping circuit based on the transformed operators, and is not intended to be limiting.

[0063] We note that to simplify the method and the rotation circuitry, one can treat the qubits that all operators locally exchange individually. In what follows, we assume that such qubits have been removed from the system.

[0064] Binary Framework The method of the present disclosure uses a binary framework for representing the Pauli matrices of the Pauli operators of a Hamiltonian, in which the Pauli matrices are represented using the following notation:

number

[0065] The Pauli operator for n qubits is the 2n-dimensional binary vector

number

number

[0066] Based on the particular form of the operators as discussed above, the method of the present disclosure includes finding a matrix S in the form of equation (5) in Appendix A. In this form, Q -1 is a 2n × 2n matrix, S' is a 2n × n matrix that represents the group of operators in the standard form above, and R -1 is an n × M matrix. The -1 superscript is purely notational; in particular, R -1 is not necessarily reversible.

[0067] The matrix S′ representing the group of operators of the standard form considered above has the structure of a matrix as provided in equation (7), where A is an n×n symmetric matrix with diagonal elements equal to 0, and I is an n×n identity matrix. -1 contains information about the one-qubit rotation required to convert between the transformed group of operators and the standard form operators. The matrix R -1 contains information on how to construct the measurements of the original operators from the measurements of the transformed operators. In other words, the matrix R -1 denotes a post-measurement routine that allows the measurement of the original operator to be determined from the product of qubit measurements (or equivalently, from the product of measurements of the transformed operator).

[0068] In some embodiments of the present disclosure, the method may include finding K independent Pauli operators from which all other M operators in the group may be constructed. In some embodiments, this may include performing Gauss-Jordan elimination to convert S into a reduced row-echelon form, where the matrix S may then be written in the form provided in equation (8), where:

number

number

[0069] Next,

Number

Number

Number

Number

[0070] As considered above, the matrix

Number

Number

Number

[0071] In both cases (K = n and K < n), Gauss elimination is used to find its pivot row

Number

Number

Number

Number

Number

[0072] and evaluate. In this case

Number

Number

[0073]

Number

Number

number

number

number

[0074] Using equation (8), as provided in equation (12),

number

[0075] From this we can see that we have a matrix Q that contains information about the one-qubit rotations required to convert between the transformed operators and the standard form operators. -1 It can be seen that is provided as in equation (13) in Appendix A (

number

number

[0076] When K < n (i.e., when the number of independent Pauli operators within a mutually commuting group of Pauli operators is less than the number of qubits for which the problem is defined),

Number

Number

[0077] When K < n, the method involves constructing n - K additional independent commuting Pauli operators, whereby the entire set of operators can be transformed into the "standard form" as considered above. Additional necessary operators can be easily constructed from the form in which the operators are currently contained. The entire system including the additional operators is arranged within the new matrix S フル inside. The i-th operator within the existing K has either an X or a Y on the qubit corresponding to the i-th row within the identity matrix. The remaining operators can only have Z's in these same locations. There is no restriction on the Pauli matrices of the qubits not corresponding to the rows of the identity matrix. The additional operators need to each have one X on one of the rows not within the identity matrix. Each X can commute or anti-commute with the terms in the same position as the existing operators. For each operator with which it anti-commutes, a Z is placed in the same position as the X or Y of the operator. Here, since it is known that there are no non-Z operators among the others, the new operators now commute with all of the existing operators.

[0078] Thus, a new matrix is obtained that includes all of the independent Pauli operators, including the additionally constructed ones. This is through equation (15) provided in Appendix A

Number

number

number

[0079] Using equations (8) and (15), as provided in equation (17),

number

number

number

[0080] Construction of the mapping circuit The above discussion describes how a group of commuting Pauli operators can be transformed into a "standard form" using various matrix operations depending on whether the number of independent terms in the group is less than or equal to the number of qubits for which the problem is defined. The method disclosed herein converts the original group of commuting Pauli operators into a transformed operator (Q -1 The method involves determining a subset of operators in standard form (S') using a matrix Q that provides information on how to convert between the transformed subset of operators and the operators in standard form (S'). -1 and a matrix R that gives information on how to obtain the measurements of the original group of operators from the measurements of the transformed group of operators. -1 It is decided that:

[0081] Measurements of the transformed subset of operators are obtained using a quantum computer and a mapping circuit built in quantum computer hardware, for example, using a quantum computer and quantum gates as described below. The mapping circuit comprises an arrangement of quantum gates that operate on one or more qubits in the quantum computer. The mapping circuit is configured to map a matrix Q -1 , where Z is the control of the standard form operator. In other words, the mapping circuit comprises single-qubit quantum gates operating on the qubits that represent the transformation between the transformed operators and the standard form operators. The mapping circuit further comprises two-qubit quantum gates, specifically the control Z in this example, that operate on specific qubits depending on the exact form of the standard form operators. Specifically, the mapping circuit applies a single-qubit rotation to the qubits that represents the single-qubit transformation described by ij A control Z gate operates on qubits i, j, where =Z.

[0082] As another example, with some modifications to the post-measurement processing routine, a block of control-X or CNOT gates can be used. More specifically, the upper-left K×K submatrix of S′, represented by E, is symmetric. Thus, E is

number

number

[0083] The following considerations are based on the Q obtained above. -1 It provides details on how matrices are transformed into quantum circuits (mapping circuits) that contain arrangements of quantum gates. -1 The matrices describe how the transformed operators relate to the standard-form operators, and corresponding circuitry applies single-qubit rotations that convert between the transformed operators and the standard-form operators.

[0084] It will be appreciated that the particular quantum gates used within a desired mapping circuit will depend on the particular quantum hardware used to obtain the measurements of the transformed operators. In this particular non-limiting example, a controlled Z gate and three types of one-qubit rotation gates are used.

[0085] matrix Q -1 There are two possible types of single-qubit rotation given in 1. Map X to Z, Y to itself, and Z to X,

number

number

[0086] When applying gates to perform the above rotations, the first two rotations cannot be performed exactly as described, and a -1 coefficient is required before one of the terms. Since there is no Y term in the final group of operators (due to the particular "standard form" of the operators considered above), a -1 coefficient can be included in this term. The gates needed to perform the desired rotations are as follows: 1.

number

number

[0087] Next, whenever the standard form operator has an off-diagonal Z, a control Z gate is applied. Finally, to measure in the X basis,

number

[0088] As can be appreciated by those skilled in the art, any multiqubit gate can be implemented by a series of one-qubit and two-qubit gates, and conversely, a series of one-qubit and two-qubit gates can be written in terms of multiqubit gates. Thus, the terms multiqubit and two-qubit gates should be understood to refer to the property that the effect of a gate cannot be calculated by looking at the effect of the gate on a single qubit in isolation.

[0089] Building measurements As discussed above, the matrix R -1 contains information on how the original Pauli operator measurement can be constructed from the Z measurements of each qubit by applying the above mapping circuit. After applying the above mapping circuit to each qubit, we measure the qubit. The measurement result of each qubit is a value of either +1 or -1, which represents the measurement of the transformed operator. Thus, the matrix R -1 contains information about how the measurements of the transformed operator can be transformed into the measurements of the original operator. -1 The jth column of contains the information needed for the jth Pauli operator. The measurement of the jth Pauli operator is -1 ij This can be done by taking the product of the measurements of each qubit i where x = 1. As can be appreciated by one skilled in the art, the process of converting measurements of the transformed operator to measurements of the original operator involves a classical transformation. Specifically, in this particular example, this process involves the multiplication of two or more real numbers. It will therefore be appreciated that this transformation routine is a classical routine that is executed on a classical computer, rather than a quantum computer.

[0090] It will be appreciated by those skilled in the art that the binary framework does not allow one to preserve information about the phase of the Pauli operators. To resolve them, one applies Q to S' to "undo" the rotation of one qubit, and then obtains the product given by the R matrix while keeping track of the phase factor. The product of two Pauli operators is provided by equations (20) to (25) in Appendix A.

[0091] Working Example An example of the above method is provided in Appendix B below. Specifically, the example provides one particular, non-limiting example of a method of transforming a group of mutually commuting operators into a subset of transformed operators, where the transformed operators are in a "standard form" to which one-qubit rotations as described above may be added; determining a mapping circuit built on a quantum computer corresponding to transforming the transformed operators into computation-based measurements; and determining a post-measurement routine for transforming measurements of the transformed operators into measurements of the original operators. FIG. 4 is a schematic diagram of the mapping circuit obtained in this particular example. Item 410 represents a single qubit gate representing the transformation between the transformed operators and the standard form operators. Item 420 represents a two-qubit controlled Z gate applied between qubits (q1, q4) and qubits (q2, q3). Item 430 represents a Hadamard gate equivalent applied to all qubits to measure the qubits in the X basis. The Hadamard gate equivalent in 430 is

number

[0092] method FIG. 5 is a flow chart illustrating a method according to the present disclosure. The illustrated method is suitable for simultaneously determining measurement results (also referred to as operator measurements) of multiple operators using a quantum computer. In other words, the method is suitable for simultaneously obtaining measurements of multiple operators. This is done by preparing a trial state using multiple qubits, applying a mapping circuit to the qubits in the trial state, and subsequently measuring the qubits to obtain qubit measurements. The method further includes performing a post-measurement processing routine for each of the multiple operators to convert the qubit measurements into measurement results or operator measurements. Optionally, the method further includes determining an expected value for each of the multiple operators by repeating the routine on the quantum computer to obtain multiple measurement results for each operator. The expected value for each operator can then be determined based on the multiple measurement results for that operator. Specific steps of the method are described in more detail below.

[0093] In step 500, the plurality of operators are grouped into distinct sets of operators. Each set includes one or more of the plurality of operators, and the operators are grouped such that each operator in a given set commutes with all other operators in the same set. More specifically, the operators are grouped such that the operators in a given set commute with each other and generally with each other. In an embodiment, the act of grouping the operators into sets of generally commuting operators includes using one of several possible classification algorithms. The classification algorithm may be executed on a classical computer, such as the classical computer 1150 shown in FIG. 6. More specifically, the grouping of the operators in step 500 may be executed by a processor 1152 on the classical computer. The resulting grouping may be stored in a main memory 1154 or a static memory 1156 on the classical computer.

[0094] After the operators have been grouped into sets of commonly commuting operators, the method proceeds to subroutine 510. Subroutine 510 includes steps 512-526 that are performed for each of the sets determined in step 500. Subroutine 510 is used to simultaneously determine a measurement result, and optionally, an expected value, for each operator in a given set. The subroutine is repeated for each set to determine a measurement result, and optionally, an expected value, for each operator of the plurality of operators. Steps 512-526 are discussed in detail below with respect to one particular set, although the steps may be repeated in the same manner for each set.

[0095] Subroutine 510 starts from step 512. In step 512, based on the original set of generally commuting operators, a transformed subset of operators is determined. The step of determining the transformed subset of operators may include obtaining independent operators within the set. In particular, in a set of m operators, there may be K independent operators (K < m) from which the remaining m - K operators can be constructed. Thus, the measurement values of the K independent operators can be obtained, and the measurement values of the remaining m - K operators within the set can be constructed from the measurement values of the K independent operators. Once the independent operators within the set are obtained, the group of operators in the "standard form" described above with reference to Equation (2) is determined, and based on the group of operators in the standard form, the transformed subset of operators is determined. Determining the transformed subset of operators includes determining a transformation that uses single-qubit rotations to transform the group of operators in the standard form into the transformed subset of operators. The transformation may include mathematical techniques such as the matrix operations described above (see the "binary framework" section above). Further, if K < n (where n is the number of qubits in the system), the step of determining the transformed operators may further include constructing n - K additional operators that also commute with all other operators. This step may be executed on a classical computer such as the classical computer 1150 shown in FIG. 6. More specifically, determining the transformed subset of operators may be executed by a processor 1152 on a classical computer (i.e., the mathematical operations may be executed using the processor 1152). The resulting transformed subset of operators is stored in the main memory 1154 or the static memory 1156 on the classical computer.

[0096] In step 514, a mapping circuit to be prepared on the quantum computer is determined based on the subset of transformed operators determined in step 512. This may include determining the form and structure of the mapping circuit. The mapping circuit is determined based on the transformation technique used to transform the set of transformed operators into computation-based measurements that can be generated on the quantum computer (see the "Construction of the Mapping Circuit" section above). For example, the mapping circuit comprises single-qubit rotations that transform between the transformed operators and standard-form operators, and two-qubit gates, such as control Z, that operate on specific qubits depending on the standard-form operators. In other words, the specific form of the mapping circuit is based on the subset of transformed operators. This step can be determined on a classical computer, for example using the processor 1152. In particular, a classical computer can be used to determine the specific types and arrangements of single-qubit and two-qubit gates in the mapping circuit. This information can then be stored in main memory 1154 or static memory 1156 before being sent to the quantum computer 1110, which constructs the mapping circuit. In particular, the information may be sent to a control means within the quantum computer that controls the quantum processor to prepare the mapping circuit using physical quantum gates. Those skilled in the art will know how a variety of single-qubit and two-qubit quantum gates can be physically implemented using a variety of quantum computer architectures.

[0097] In step 516, a post-measurement routine for converting the qubit measurements into measurements of the original set of operators is determined based on the subset of transformed operators determined in step 512. Steps 514 and 516 may be performed simultaneously or sequentially. Specifically, either step 514 or 516 may be performed first, or both steps may be performed simultaneously. The post-measurement routine includes classical mathematical operations that convert the qubit measurements into operator measurements. In particular, the post-measurement routine may include taking a product of two or more of the qubit measurements to determine a measurement of each of the original operators in the set. The particular product is determined based on the subset of transformed operators. Specific details on how the post-measurement routine is determined are provided in the "Constructing Measurements" section above. Determining the post-measurement routine may be performed by a classical processor 1152 on the classical computer 1150. Instructions for the routine may then be stored in a main memory or static memory on the classical computer 1150.

[0098] In step 518, a trial state is prepared on the quantum computer. The trial state may be based on a particular parameter (λ) that is iteratively updated in the VQE framework. Alternatively, the trial state may be prepared based on knowledge of the physical system whose energy levels are to be determined. The trial state is prepared on a plurality of qubits. Specifically, the trial state is prepared with a number of qubits that matches the number of qubits for which the problem is defined (i.e., the number of qubits that matches the maximum number of terms in the Pauli operator), and the trial state is prepared on the qubits using an arrangement of quantum gates, such as single-qubit or two-qubit gates, or other multi-qubit gates. The particular type and arrangement of quantum gates used to prepare the trial state will depend on the trial state itself. One skilled in the art will know how to prepare a particular trial state on a plurality of qubits using an arrangement of quantum gates.

[0099] In step 520, the mapping circuit determined in step 514 and built on the quantum computer is applied to the qubits prepared in the trial state in step 518. Specifically, the single-qubit and two-qubit gates in the mapping circuit are applied to the qubits in the trial state in the quantum processor. An example of a mapping circuit for a particular subset of transformed operators is shown in FIG.

[0100] In step 522, after applying the mapping circuit to each qubit in the trial state, each qubit is measured, for example, using a measurement means 1104 on the quantum computer. This gives a qubit measurement for each qubit, each of which has a value of +1 or -1. The qubit measurement indicates the measurement result of a subset of the transformed operators. Those skilled in the art will know how to measure qubits on a quantum computer using any suitable measurement means available to those skilled in the art. The qubit measurements can then be sent to a classical computer for readout or further processing, as described below.

[0101] After measuring the qubits in step 522, the post-measurement routine determined in step 516 is applied to the qubit measurements to convert the qubit measurements from the measurement results of the transformed subset of operators to the measurement results of each of the operators in the original set of operators. As discussed above, the post-measurement processing routine may include taking a product of two or more of the qubit measurements to determine a measurement of the original operator in the set. Alternatively, for one or more of the operators in the set, the measurement of that operator may be given by a single qubit measurement, i.e., there is a one-to-one mapping from qubit measurements to operator measurement operators. An example of a post-measurement processing routine is given in the example in Appendix B. In particular, the matrix provided in (49) provides details on how the original operator measurements are constructed from the qubit measurements for this particular example. It can be seen that some operator measurements are determined from products of multiple qubit measurements, while other operator measurements are simply readouts of single qubit measurements. It will be appreciated that the post-measurement processing routine may be executed on a classical computer since it involves multiplication of real numbers. Thus, classical processor 1152 can be used to apply post-measurement processing routines to the qubit measurements to determine corresponding operator measurements, which can then be stored in the memory of the classical computer.

[0102] The resulting operator measurements can then be read in a classical computer, for example, by displaying the results on a display 1158. At this point, the method can end for the particular set for which steps 512-524 were performed. Steps 512-514 can then be repeated for all other sets of operators to determine measurement results for all operators of the plurality of operators.

[0103] Alternatively, instead of ending the method at step 524 for each set, the method may include multiple repetitions of steps 518-524 to obtain multiple measurements of each operator in the set. The method may then proceed to step 526, where an expected value for each operator in the set is determined. In some implementations of the method, the expected value for each operator is determined by taking an average of the measurements for that operator, which may be determined using a classical computer. Steps 512-526 may then be repeated for all sets to obtain expected values ​​for all operators of the plurality of operators. An expected energy value for the physical system may then be determined by taking a weighted sum of the expected values ​​for all operators. The expected values ​​may be determined using a classical processor 1152 on the classical computer 1150.

[0104] The design of the disclosed method is motivated by technical considerations of the internal functioning of quantum computers. In particular, taking into account the constraints of modern quantum computers, such as the time and required computational cost of preparing trial states for a large number of qubits, and manipulating and measuring those qubits, the disclosed method can exploit the quantum properties of qubits and quantum gates built on quantum computers to simultaneously obtain measurements of Pauli operators. It will be appreciated that in the context of determining the energy levels of a physical system, the final number of state preparations, operations, and measurements can be reduced by performing measurements simultaneously on a large number of different Pauli operators in a Hamiltonian. Thus, considering the capabilities and constraints of quantum computers, it is clear that the present method offers a distinct advantage over known methods in which operators are measured one at a time. Appendix C, "Minimizing Errors," provides further information on how to reduce the number of state preparations by grouping operators into groups of commonly commuting operators, compared to not grouping the operators and determining measurements individually.

[0105] Further advantages over known methods are provided by the use of at least a post-measurement processing routine to convert qubit measurements into operator measurements. In particular, using a classical computer to perform the post-measurement processing routine can reduce the requirements of the quantum computer by reducing the number of quantum gates in the mapping circuit. The inventors have thought carefully about how best to reduce the number of quantum gates in the mapping circuit without making the required quantum circuit too complex and therefore too difficult to implement. The disclosed method uses a specific technique to obtain transformed operators that relate in a specific way to operators of a particular "standard" form. In theory, additional gates could be used to achieve what the method achieves, but the functionality has been pushed to the classical computer performing the post-measurement processing routine, thus reducing the requirements on the quantum computer and maximizing the processing power of both the quantum computer and the classical computer. More specifically, the post-measurement processing routine performs the function of converting qubit measurements into operator measurements. Since the qubit measurements represent measurements of transformed operators, the post-processing routine converts the measurements of the transformed operators into measurements of the original operators. Although this function could theoretically be performed using additional quantum gates in the mapping circuit, by specifically considering the internal functions of quantum and classical computers when designing the method, greater efficiency and processing speed can be achieved by reducing the number of quantum gates and by leveraging classical processing power for this particular function. Thus, in view of the difficulties and constraints in implementing large-scale quantum circuits on a quantum computer, it is clear that the disclosed method provides a relatively simpler way to obtain measurements of operators that are much easier to implement on a quantum computer.

[0106] Resource requirements The method described here requires many matrix operations, all of which are performed on a classical computer, the most complex of which is Gauss-Jordan elimination. For a square matrix of size D, this process takes time in the order of O(D 3 ) The largest matrix that performs such an elimination is of size 2n × M max where M max is the maximum number of terms in a group. The complexity of performing Gauss-Jordan elimination on such a matrix is

number

[0107] Considering the number of two-qubit gates in the mapping circuit, to measure a group of operators of the "standard form" format, two-qubit gates, specifically controlled Z gates or equivalent implementations, are needed for every off-diagonal Z matrix present. Thus, the maximum number is 1 / 2n(n-1), which is O(n 2 ). However, if K ≠ n, then the maximum number of two-qubit gates needed is actually 1 / 2 n(n-1)-1 / 2 (nK)(nK-1)=nK-1 / 2 K(K+1), which is O(nK), which is a much smaller solution than the similarly sized but faster solution of O(n 2 ) because the additional operator is フル Because of the way that nK is constructed to have full rank, i.e., how the −1 / 2(nK)(nK-1) terms arise, there is never an off-diagonal Z between two of the additional operators.

[0108] Device 6 shows a block diagram of one implementation of a computing device 1100 on which a set of instructions may be executed to cause the computing device to perform any one or more of the techniques of the present disclosure. Although only a single computing device is shown, the term "computing device" is also taken to include any collection of machines (e.g., computers) that individually or collectively execute a set (or sets) of instructions to perform any one or more of the techniques discussed herein. The computing device 1100 comprises a quantum computing system 1110 and a classical computing system 1150. The quantum computing system 1110 is in communication with the classical computing system 1150. The classical computing system is arranged to instruct the quantum computing system to create quantum states and perform measurements on those quantum states according to instructions stored in a memory.

[0109] The quantum computing system 1110 comprises a quantum processor 1102, which in turn comprises at least two qubits and at least one coupler capable of coupling the qubits. The qubits may be physically implemented, for example, using photons, trapped ions, electrons, one or more atomic nuclei, superconducting circuits, and / or quantum dots. In other words, the qubits may be physically implemented in a variety of ways, including the polarization state of a single photon, the spatial light path of a single photon, two different energy states of an atom or ion, the spin orientation of one or more particles, such as a nucleus. The quantum computer also comprises means for storing the qubits and maintaining the qubits in a suitable environment to enable quantum computation, for example, means for supercooling the qubits. The qubits may be operated by one or more quantum circuits formed by suitable quantum gate arrays.

[0110] Quantum gates operate on a number of qubits and can be thought of as quantum analogues of basic low-level instructions in classical circuits, such as NOT or AND gates. Typically, quantum circuits are decomposed into a series of single-qubit and two-qubit gates obtained from a universal gate set, along with state creation and qubit measurement or readout. However, it is also possible to build quantum circuits using quantum gates that operate on more than two qubits, i.e. "multi-qubit" gates. The measurement results are classical data that are then processed by a classical computer. Many quantum computers based on superconducting circuits and trapped ions have already demonstrated, on a small scale, all the functionality required for large-scale quantum computing devices.

[0111] Here, we describe possible implementations and methods of operation of qubits in a quantum computer. These implementations are merely examples, and one of ordinary skill in the art will be familiar with other methods of implementing quantum computers. Birefringent waveplates can be used to manipulate the polarization state of a single photon, for example, to cause linear or horizontal polarization of the photon to represent two different states of the photon. Qubits can also be implemented using beam splitters. For example, the presence or absence of a photon along a particular optical path can be implemented using a beam splitter that splits the photon beam into two separate paths. The presence of a photon in either path represents two different states of the photon. Alternatively, or additionally, two separate electronic energy states of an atom or ion can represent two separate and distinct states of a qubit. For example, the transition energies between these levels may correspond to the energy of electromagnetic radiation of a certain frequency, so that the separate energy states of an atom or ion can be addressed using a radiation source such as a laser or microwave emitter. Alternatively or additionally, two distinct spin states (spin "up" and spin "down") of a particle or particles, e.g., a nucleus, can represent two distinct states of a qubit. Manipulation of nuclear spins may be implemented using magnetic fields, using methods known to those skilled in the art.

[0112] Alternatively or additionally, superconducting electronic circuits may be used to generate qubits. These systems use Josephson junctions, which are nonlinear inductors supercooled below 100K and allow the creation of anharmonic oscillators. Anharmonic oscillators (unlike harmonic oscillators) do not have equally spaced energy levels, and therefore the two states can be controlled separately and used to store the qubits. The qubits are connected to a microwave cavity, and single and two-qubit gates can be performed using microwave signals.

[0113] The quantum computing device 1110 also comprises a measurement means 1104 and a control means 1106. The control means 1106 may comprise control hardware and / or a control device. The control means 1106 is configured to receive instructions from the classical computer 1150, which may instruct the control means 1106 to create a particular state in the quantum processor using a particular quantum gate arrangement. Furthermore, the control means may be configured to receive instructions to build a quantum circuit in the quantum processor. The measurement means 1104 may comprise measurement hardware and / or a measurement device. The measurement means comprises hardware configured to make measurements from states created by the control means 1106 in the quantum processor 1102.

[0114] An exemplary classical computing device 1150 includes a processor 1152, a main memory 1154 (e.g., read only memory (ROM), flash memory, dynamic random access memory (DRAM) such as synchronous DRAM (SDRAM) or Rambus DRAM (RDRAM), etc.), a static memory 1156 (e.g., flash memory, static random access memory (SRAM), etc.), and a secondary memory (e.g., a data storage device), which communicate with each other via a bus.

[0115] The processing device 1152 represents one or more general-purpose processors, such as a microprocessor, a central processing unit, etc. More specifically, the processing device 1152 may be a complex instruction set computing (CISC) microprocessor, a reduced instruction set computing (RISC) microprocessor, a very long instruction word (VLIW) microprocessor, a processor implementing other instruction sets, or a processor implementing a combination of instruction sets. The processing device 1152 may also be one or more special-purpose processing devices, such as an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a digital signal processor (DSP), a network processor, etc. The processing device 1152 is configured to execute processing logic for performing the operations and steps discussed herein.

[0116] The data storage device may include one or more machine-readable storage media (or, more specifically, one or more non-transitory computer-readable storage media) on which one or more sets of instructions embodying any one or more of the techniques or functions described herein are stored. The instructions may also reside, completely or at least partially, within the main memory 1154 and / or within the processing device 1152 during execution thereof by the computer system, with the main memory 1154 and the processing device 1152 also constituting computer-readable storage media.

[0117] In general, classical computer 1150 instructs control means 1106 of quantum computer 1110 to create a particular state in quantum processor 1102. Control means 1106 operates qubits in quantum processor 1102 based on the instructions. Once the qubits have been manipulated to establish a desired state in quantum processor 1102, measurement means 1104 takes a measurement from that state. Quantum computer 1110 then communicates the measurement result to the classical computer.

[0118] The various methods described herein may be implemented by a computer program. The computer program may include computer code arranged to instruct a computer to perform the functions of one or more of the various methods described above. The computer program and / or code for performing such methods may be provided to an apparatus, such as a computer, on one or more computer readable media, or more generally on a computer program product. The computer readable medium may be transitory or non-transitory. The one or more computer readable media may be, for example, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, or a propagation medium for data transmission, for example, for downloading code via the Internet. Alternatively, the one or more computer readable media may take the form of one or more physical computer readable media, such as semiconductor or solid state memory, magnetic tape, removable computer diskettes, random access memory (RAM), read only memory (ROM), rigid magnetic disks, and optical disks, such as CD-ROM, CD-R / W, DVD, etc.

[0119] In one implementation, the modules, components, and other features described herein may be implemented as individual components or integrated into the functionality of a hardware component such as an ASICS, FPGA, DSP, or similar device.

[0120] In addition, the modules and components may be implemented as firmware or functional circuitry within a hardware device. Further, the modules and components may be implemented in any combination of hardware devices and software components, or in software alone (e.g., code stored or otherwise embodied in a machine-readable medium or transmission medium).

[0121] Unless otherwise indicated, and as will be apparent from the discussion that follows, discussions utilizing terms such as "receive," "determine," "compare," "enable," "maintain," "identify," and the like throughout the specification are understood to refer to operations and processes of a computer system or similar electronic computing device that manipulate and transform data represented as physical (electronic) quantities in the computer system's registers and memory into other data similarly represented as physical quantities in the computer system's memory or registers or other such information storage, transmission, or display device.

[0122] Herein, we refer to the energy level of a physical system. The physical system may be either an atom, a molecule, a collection of atoms, an enzyme or a part thereof, a chemical substance, or a material such as a potential superconductor. In addition, many other problems can be solved by mapping to a Hamiltonian and solving by finding an energy level such as a ground state. For example, various optimization problems such as scheduling tasks or searching for faults in a circuit can be effectively solved by this method. As will be understood by those skilled in the art, the energy level of a physical system refers to the eigenvalue of the corresponding Hamiltonian.

[0123] To give an example of the many industrial applications of the present method, the quest for more efficient means of producing fertilizer is an example of a technological problem that could be aided by a better understanding of the energy levels of the reactants. The production of ammonia via the Haber-Bosch process is essential for the production of fertilizer, but requires both high pressure and high temperature and is therefore a very energy intensive process. In contrast, nitrogenase is an enzyme that accomplishes the same task at room temperature and standard pressure, and therefore there is a strong interest in understanding the nitrogenase enzyme. It is known that increased knowledge of the energy levels of the iron-molybdenum cofactor (FeMo-co) within the MoFe protein contained in the nitrogenase enzyme would significantly advance the discovery of more efficient methods to generate ammonia.

[0124] The techniques described herein may be embodied on a computer-readable medium, which may be a non-transitory computer-readable medium carrying computer-readable instructions arranged to execute on a processor to cause the processor to perform any or all of the methods described herein.

[0125] The term "computer-readable medium" as used herein refers to any medium that stores data and / or instructions to cause a processor to operate in a specific manner. Such storage media may include non-volatile media and / or volatile media. Non-volatile media may include, for example, optical or magnetic disks. Volatile media may include dynamic memory. Exemplary forms of storage media include floppy disks, flexible disks, hard disks, solid-state drives, magnetic tapes, or any other magnetic data storage media, CD-ROMs, any other optical data storage media, any physical media with one or more patterns of holes, RAM, PROM, EPROM, FLASH-EPROM, NVRAM, and any other memory chip or cartridge.

[0126] It will be understood that the above description of specific embodiments is by way of example only and is not intended to limit the scope of the present disclosure. Many modifications of the described embodiments are contemplated and are intended to be within the scope of the present disclosure.

[0127] The above implementations have been described by way of example only, and the described implementations and arrangements should be considered in all respects only as illustrative and not restrictive, It will be understood that variations in the described implementations and arrangements may be made without departing from the scope of the invention.

[0128] Appendix A: List of Expressions 1 Introduction

number

number

number

[0129] Appendix B: Working Examples There are six Pauli operators.

number

[0130] We note that not all operators commute locally on any qubit, and therefore consider the four-qubit system as a whole. Our method is used to perform one measurement instead of six. Writing the operators in binary form, the matrix S becomes:

number

[0131] Performing Gauss-Jordan elimination to convert S to reduced row echelon form gives

number

[0132] The pivot columns are numbers 1, 2, and 4, so P1, P2, and P4 are three independent Pauli operators from which the remaining operators in the group can be constructed. Thus, we can write

number

[0133] next,

number

number

number

number

number

[0134] Then the lower half of this matrix has rank 3, and if we run Gaussian elimination again, the first three rows become the inverse submatrix

number

number

[0135] Then we have the following:

number

[0136] Now, change the fourth column to

number

number

[0137] S フル Since the lower half of is full rank, taking the inverse can give

number

[0138] Finally, rotate qubits 1 and 2

number

number

number

[0139] Then we have

number

number

[0140] Q -1 The transformed operator, given by S', is:

number

[0141] It is these transformed operators that we obtain measurements by applying the rotation circuit and then measuring each qubit in the Pauli Z basis. Then, the measurements of the original operators are calculated using the -1 It is obtained by taking the product of the measurements shown in the matrix.

[0142] The rotation circuit is shown in Figure 4. Using R, S', and Q, we find that the phase of the six original operators is (+1+1+1+1+1+1). Therefore, we can construct the measurement of the original Pauli ring as follows: P1 from the product of measurements of qubits 3 and 4, P2 from the product of measurements of qubits 1, 2, 3, and 4, P3 from the product of measurements of qubits 1 and 2, P4 from the measurement of qubit 2, P5 from the measurement of qubit 1, P6 from the product of measurements of qubits 1, 3, and 4.

[0143] Appendix C: Error Minimization We can now consider the expected error impact of simultaneous measurements on a group of exchanging operators. We have a Hamiltonian of the form

number

number

number

number

number

[0144] It turns out that it is useful to define

number

number

[0145] Fixed total number of state preparations N u Then the error is minimized when

number

number

[0146] Instead, we simultaneously measure all the operators in the group to be exchanged, and from these measurements, we obtain k When considering constructing a measure of H k The variance of must be taken into account. The error is given by

number

number

number

number

[0147] Therefore, to estimate the Hamiltonian expectation with the same level of accuracy, the number of state preparations required in the grouped case compared to the ungrouped case turns out to be:

number

[0148] The following numbered paragraphs are also disclosed:

[0149] 1. A method for determining an estimate of an energy level of a physical system using a quantum computer, the energy level being described by a sum of expected values ​​of a number of operators, the method comprising: determining a measurement value for each operator of the plurality of operators, the determining comprising: grouping the plurality of operators into one or more sets, each set including one or more of the plurality of operators; For each set of operators, A subset of transformed operators based on a set of operators, a mapping circuit based on a subset of the transformed operators; and determining a post-measurement processing routine based on the transformed subset of operators; Determining a measurement result for each operator of the plurality of operators further includes running a measurement routine for each set of operators, the measurement routine comprising: preparing a trial state using a first configuration of quantum gates using a plurality of qubits on a quantum computer; operating a mapping circuit on a plurality of qubits in a trial state; performing a measurement on each qubit of the plurality of qubits to obtain a qubit measurement for each qubit; applying a post-measurement processing routine to the qubit measurements to convert the qubit measurements into operator measurements for each of the operators in the set of operators; The method further includes determining an estimate of an energy level of the physical system based on at least the determined operator measurements for each operator in each set.

[0150] 2. The method of paragraph 1, wherein determining the subset of transformed operators, determining the mapping circuit, and determining the post-measurement processing routine are performed using a classical computer, and the classical computer further performs the step of applying the post-measurement processing routine to the qubit measurements to transform the qubit measurements into operator measurements for each of the operators in the set of operators.

[0151] 3. The method of paragraph 1 or 2, wherein preparing the trial state, operating the mapping circuit, and performing the measurement on each qubit are performed using a quantum computer.

[0152] 4. The method of paragraph 1, wherein the measurement routine is executed multiple times for each set to obtain a corresponding plurality of operator measurements for each operator in each set, and the method further includes determining an expected value for each operator in each set based on an average of the corresponding plurality of operator measurements.

[0153] 5. The method of paragraph 4, wherein determining an estimate of the energy level includes a sum of expected values ​​for each operator in each set.

[0154] 6. The method of any preceding paragraph, wherein the mapping circuitry includes at least one multi-qubit gate configured to act on at least two of the plurality of qubits.

[0155] 7. The method of paragraph 6, wherein the mapping circuit includes one or more multiqubit gates, the number of multiqubit gates being proportional to the number of the plurality of qubits, the proportionality having an upper limit on the number of the plurality of qubits multiplied by the number of independent operators in the set of operators, and each operator in the set of operators can be constructed from one or more independent operators.

[0156] 8. The method of any preceding paragraph, wherein the mapping circuitry includes one or more single-qubit gates configured to apply a rotation to each qubit of the plurality of qubits.

[0157] 9. Determining a subset of transformed operators determining one or more independent operators of a set of operators, each operator in the set of operators being capable of being constructed from one or more independent operators; and transforming one or more independent operators into a subset of transformed operators.

[0158] 10. Transforming one or more independent operators into a subset of transformed operators determining whether a number of independent operators matches a number of the plurality of qubits; In response to determining that the number of independent operators is less than the number of the plurality of qubits, and constructing one or more new transformed operators that are added to the subset of transformed operators such that the number of transformed operators matches the number of qubits.

[0159] 11. The method of any preceding paragraph, in which the qubit measurements represent measurements of a subset of transformed operators.

[0160] 12. A method for determining a measurement value for each operator of a plurality of operators, comprising: grouping the plurality of operators into one or more sets, each set including one or more of the plurality of operators; For each set of operators, A subset of transformed operators based on a set of operators, a mapping circuit based on a subset of the transformed operators; determining a post-measurement processing routine based on the subset of transformed operators; executing a metrology routine for each set of operators, the metrology routine comprising: preparing a trial state using a first configuration of quantum gates using a plurality of qubits on a quantum computer; operating a mapping circuit on a plurality of qubits in a trial state; performing a measurement on each qubit of the plurality of qubits to obtain a qubit measurement for each qubit; and applying a post-measurement processing routine to the qubit measurements to convert the qubit measurements into operator measurements for each of the operators in the set of operators.

[0161] 13. A computer-readable medium comprising instructions that, when executed by a processor, cause the processor to perform the method of any preceding paragraph.

[0162] 14. An apparatus comprising a classical computer and a quantum computer configured to perform the method of any one of paragraphs 1 to 12.< / h>

Claims

1. 1. A computer-implemented method for enabling determination of a measurement value for each operator of a plurality of operators, comprising: grouping the plurality of operators into one or more sets of mutually commutating operators, each set including one or more of the plurality of operators; For each set of operators, a subset of transformed operators based on the set of operators, such that the set of operators is equal to a product of the subsets of transformed operators; a mapping circuit based on the subset of transformed operators, the mapping circuit comprising an arrangement of quantum gates configured to operate on a plurality of qubits in a quantum computer; and determining a post-measurement processing routine for transforming qubit measurements into operator measurements for each of the operators in the set of operators based on the transformed subset of operators.

2. and executing a metrology routine for each set of operators, the metrology routine comprising: preparing a trial state using a first configuration of quantum gates using the plurality of qubits in the quantum computer; operating the mapping circuit on the plurality of qubits in the trial state; performing a measurement on each qubit of the plurality of qubits to obtain a qubit measurement for each qubit; and applying the post-measurement processing routine to the qubit measurements to convert the qubit measurements to the operator measurements for each of the operators in the set of operators.

3. 3. The method of claim 2, wherein the method is configured to use the quantum computer to estimate an energy level of a physical system, the energy level being described by a sum of expected values ​​of the plurality of operators, the method further comprising determining an estimate of the energy level of the physical system based on the determined operator measurements for at least each operator in each set.

4. 4. The method of claim 2 or claim 3, wherein determining the subset of transformed operators, determining the mapping circuit, and determining the post-measurement processing routine are performed using a classical computer, the classical computer further performing the step of applying the post-measurement processing routine to the qubit measurements to convert the qubit measurements into operator measurements for each of the operators in the set of operators.

5. 5. The method of claim 2, wherein the preparing the trial states, operating the mapping circuit, and performing a measurement on each qubit are performed using the quantum computer.

6. 4. The method of claim 3, wherein the measurement routine is executed multiple times for each set to obtain a corresponding plurality of operator measurements for each operator in each set, the method further comprising determining an expected value for each operator in each set based on an average of the corresponding plurality of operator measurements.

7. The method of claim 6 , wherein determining the estimate of the energy level comprises a sum of the expected values ​​of each operator in each set.

8. 8. A method according to claim 1, wherein the mapping circuitry comprises at least one multi-qubit gate configured to act on at least two of the plurality of qubits.

9. 9. The method of claim 8, wherein the mapping circuit comprises one or more multi-qubit gates, the number of multi-qubit gates being proportional to the number of the plurality of qubits, the proportionality having an upper limit on the number of the plurality of qubits multiplied by the number of independent operators in the set of operators, and each operator in the set of operators can be constructed from the one or more independent operators.

10. 10. A method according to any preceding claim, wherein the mapping circuitry comprises one or more single-qubit gates configured to apply a rotation to each qubit of the plurality of qubits.

11. Determining the subset of transformed operators determining one or more independent operators of the set of operators, each operator in the set of operators being capable of being constructed from the one or more independent operators; Transforming the one or more independent operators into a subset of the transformed operators.

12. Transforming the one or more independent operators into a subset of the transformed operators, determining whether a number of the independent operators matches the number of the plurality of qubits; in response to determining that a number of the independent operators is less than the number of the plurality of qubits; and constructing one or more new transformed operators that are added to the subset of transformed operators such that the number of transformed operators matches the number of qubits.

13. 13. A method according to any preceding claim, wherein the qubit measurements represent measurements of a subset of the transformed operators.

14. A non-transitory computer readable storage medium comprising instructions that, when executed by a processor, cause the processor to perform the method of any one of claims 1 to 13.

15. 14. An apparatus comprising a classical computer and a quantum computer configured to carry out the method of any one of claims 1 to 13.

Images