Quantum control sequence

Abstract

A method 200 for determining a control sequence for execution on a quantum computer and for determining a sequence of pulses of radiation includes receiving 202 a target quantum operation and determining 204 a circuit library or a pulse library comprising a plurality of quantum circuits or pulses, each of which approximate the target operation. A solution weight vector is determined 206 which synthesises the target operation when applied in a weighted summation over the circuit or pulse library and which satisfies a minimum condition. The minimum condition includes having a minimum of a weighted sum of a vector norm of a candidate weight vector and an error term for the candidate vector solution. A probability associated with each quantum circuit or pulse in the library is determined 208 and a plurality of selections of a quantum circuit or pulse from the library is performed 210 according to the probability distribution to form 212 a control sequence or sequence of pulses. The pulses may be pulses of electromagnetic radiation for application to one or more qubits in the quantum computer.

Description

[0001] TECHNICAL FIELD

[0002] The present disclosure relates to quantum computing. In particular, the present disclosure relates to methods and apparatus for determining a control sequence for execution on a quantum computer. BACKGROUND

[0003] A quantum computer is a device which utilises quantum mechanical effects to process quantum information. The basic unit of quantum information used in quantum computing is a qubit. A qubit has two basis states and is analogous to a bit used in classical computing. However, unlike a classic bit, which can only exist in one of its two states at any given time, a qubit can exist in a superposition of both of its states simultaneously. The overall superposition of states in a quantum computer scales as 2” where n is the number of qubits placed in a superposition of states. A quantum computer can use such a superposition of states, along with other quantum mechanical effects, such as quantum entanglement, to solve various computational problems. In particular, quantum computing has the potential to solve a range of computational problems which remain out of reach of even the world's largest classical supercomputers.

[0004] A practical quantum computer is configured to establish, maintain and manipulate a plurality of physical qubits. Physical qubits may be realised by maintaining and manipulating a two-state quantum mechanical physical system.

[0005] Quantum computations may be carried out by executing a state preparation circuit on a quantum computer comprising a plurality of qubits. A state preparation circuit may comprise one or more quantum operations (such as gate operations) executed on one or more qubits. After execution of a state preparation circuit a measurement may be performed of the quantum state of the qubits, where the measurement represents the result of a quantum computation.

[0006] As was explained above, a state preparation circuit may comprise a plurality of quantum operations such as gate operations to be executed on one or more qubits. Often a given target quantum operation to be carried out cannot be implemented directly in hardware as a native gate. In such situations a target quantum operation may be approximated or synthesised as one or more gates which can be implemented on the quantum hardware.

[0007] It is in this context that the subject matter contained in the present application has been devised. SUMMARY

[0008] According to a first aspect of the present disclosure there is provided a computer implemented method of determining a control sequence for execution on a quantum computer. The method comprises: receiving a target quantum operation for execution on the quantum computer; determining a circuit library comprising a plurality of quantum circuits, each of which approximate the target quantum operation, wherein the target quantum operation can be synthesised by a weighted summation over the plurality of circuits in the circuit library, the weighted summation being performed according to a weight vector comprising a weight coefficient corresponding to each circuit in the circuit library, wherein the plurality of different circuits in the circuit library are determined such that there are a plurality of candidate weight vectors which synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library; determining a solution weight vector from the plurality of candidate weight vectors, wherein the determined solution weight vector satisfies a minimum condition of all of the plurality of candidate weight vectors, wherein the minimum condition comprises having a minimum of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector, wherein the error term represents a difference between the target quantum operation and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the circuit library using the candidate weight vector; determining a probability associated with each quantum circuit in the circuit library, wherein each probability is proportional to the magnitude of the respective weight coefficient in the solution weight vector, wherein the probabilities associated with each quantum circuit in the circuit library form a probability distribution; performing a plurality of selections of a quantum circuit from the circuit library according to the probability distribution; and determining a control sequence of quantum circuits for execution on the quantum computer, wherein the control sequence comprises the plurality of selections of a quantum circuit from the circuit library.

[0009] The target quantum operation may comprise a single quantum gate. Additionally or alternatively, the target quantum operation may comprise a plurality of quantum gates which together form a quantum circuit. The target quantum operation may comprise a unitary gate. The target quantum operation may be represented as a unitary process matrix that acts on a vectorised density matrix representing the state of one or more qubits. The target quantum operation may act on a single qubit (e.g. a single qubit gate). The target quantum operation may act on a plurality of qubits (e.g. a multi-qubit gate such as a two or three qubit gate). The target quantum operation may represent any quantum operation which can be performed on a quantum computer.

[0010] The plurality of quantum circuits included in the circuit library may each comprise a quantum circuit which is implementable on the quantum computer. The plurality of quantum circuits included in the circuit library may each comprise a single quantum gate or a quantum circuit comprising a plurality of quantum gates. The plurality of quantum circuits included in the circuit library may be determined in dependence on one or more properties of the quantum computer on which the control sequence is to be executed. For example, execution of given quantum operations, circuits and / or gates on a given quantum computer may have a form of cost associated with them. The cost may, for example, comprise a measure of time (e.g. time needed to execute) and / or hardware complexity associated with execution of a given quantum operation, circuit and / or gate on a given quantum computer. Additionally or alternatively, the cost may, for example comprise a measure of impact on an accuracy of a computation carried out on the quantum computer. For example, the cost may comprise a measure of noise and / or error which may be introduced through execution of the given quantum operation, circuit and / or gate on the given quantum computer.

[0011] In general a given quantum computer may have some operations, gates and / or circuits which may be executed relatively easily and / or at relatively low cost and some operations, gates and / or circuits which are relatively difficult to execute and / or are executed at relatively high cost. For example, in the context of a fault-tolerant quantum computer, Clifford gates may be executed at relatively low cost (and / or may be executed relatively easily) whereas a T gate may be executed at relatively high cost (and / or may be relatively difficult to execute). In the context of realising a continuous set of parameterised gates (e.g. parameterised Pauli gates) parameterised by a rotation angle, a relatively small number of discrete rotation angles may be executed at relatively low cost (and / or may be executed relatively easily) whereas a relatively higher number of discrete rotation angles may be executed at relatively high cost (and / or may be relatively difficult to execute). In the context of control pulses (e.g. pulses of microwave radiation and / or laser pulses) used to directly control qubits, there may be some pulse shapes which may be executed at relatively low cost (and / or may be executed relatively easily) whereas there may other pulse shapes which may only be executed at relatively high cost (and / or may be relatively difficult to execute). The cost and / or complexity associated with different operations, gates and / or circuits may depend on the hardware of the quantum computer.

[0012] The plurality of quantum circuits included in the circuit library may each comprise a quantum circuit which is implementable on the quantum computer with relatively low cost and / or complexity. That is, the plurality of quantum circuits included in the circuit library may be determined to be quantum circuits which are implementable on the quantum computer with relatively low cost and / or complexity. Determining the plurality of quantum circuits to include in the circuit library may comprise determining quantum circuits subject to hardware cost criteria. For example, only quantum circuits which are sufficiently simple and / or have a sufficiently low cost associated with them may be included in the circuit library.

[0013] The target quantum operation may correspond with a quantum operation, circuit and / or gate which is difficult to execute on the quantum computer and / or may only be executed at a relatively high cost. In some examples, it may not be possible to directly implement the target quantum operation on the quantum computer. The target quantum operation can be synthesised by execution of at least some of the quantum circuits in the circuit library.

[0014] Each of the quantum circuits may not exactly match the target quantum operation. That is, each quantum circuit in the circuit library may represent an approximation of the target quantum operation and may differ from the target quantum operation. The quantum circuits in the circuit library may be determined as quantum circuits which closely approximate the target quantum operation but which are implementable (and / or are implementable relatively easily and / or at relatively low cost) on the quantum computer.

[0015] Whilst each quantum circuit in the circuit library differs from target quantum operation, the target quantum operation can be synthesised as a weighted summation of the quantum circuits included in the circuit library. The weighted summation comprises a weighted summation over the plurality of quantum circuits in the circuit library according to a weight vector. The weight vector comprises a weight coefficient corresponding to each quantum circuit in the circuit library. For example, for a circuit library comprising N quantum circuits, a corresponding weight vector will comprise N weight coefficients, wherein each of the N weight coefficients corresponds to a single quantum circuit in the circuit library. The weight coefficients may comprise real numbers. Each weight coefficient may comprise a positive number, zero, or a negative number. A weighted summation of the quantum circuits included in the circuit library comprises a summation of each quantum circuit multiplied by its respective weight coefficient in a weight vector.

[0016] The quantum circuits in the circuit library are determined such that there are a plurality of candidate weight vectors which synthesise the target quantum operation when applied in a weighted summation of the quantum circuits in the circuit library. That is, there are a plurality of solutions (each represented by a candidate weight vector) which lead to synthesis of the target quantum operation. This may be achieved by including a sufficiently large number of quantum circuits in the circuit library such that a plurality of solutions (candidate weight vectors) are available to synthesise the target quantum operation. The weighted summation of the quantum circuits (to synthesise the target quantum operation) is not executed directly in hardware. The weighted summation of the quantum circuits (to synthesise the target quantum operation) may instead be implemented on average through multiple executions of different quantum circuits in the circuit library.

[0017] The determining a solution weight vector from the plurality of candidate weight vectors may comprise determining each of the plurality of candidate weight vectors and selecting one of the candidate weight vectors as the solution weight vector. Alternatively, determining a solution weight vector from the plurality of candidate weight vectors may not comprise determining each and every candidate weight vector. For example, the solution weight vector may be determined using a search or optimisation process which does not require determination of all possible candidate weight vectors.

[0018] The minimum condition comprises determining a solution weight vector having a minimum (of the candidate weight vectors) of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector. Each candidate weight vector has an associated vector norm and an associated error term and thus a weighted sum of the vector norm and error term may be determined for each candidate weight vector. However, in practice the weighted sum of the vector norm and the error term may not be determined for each candidate weight vector since a minimum may be found using an optimisation process which does not require determination of all possible candidate weight vectors (or weighted sum). The solution weight vector is determined as the candidate weight vector having a minimum weighted sum of its vector norm and error term, from all of the candidate weight vectors. The weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector may comprise multiplying one or both of the vector norm and the error term by a respective weight factor and summing the terms. For example, the weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector may comprise a sum of the vector norm multiplied by a weight factor and the error term. Alternatively, the weighted sum may comprise a sum of the vector norm and the error term multiplied by a weight factor.

[0019] The vector norm may, for example, comprise the LI norm of a candidate weight vector. As will be described in further detail herein, the vector norm may represent a measurement overhead associated with synthesising the target quantum operation. In particular, a variance which is introduced into measurements taken after execution of the control sequence may depend (e.g., may be proportional to and / or may be limited by) the vector norm of the solution weight vector which is used. A candidate weight vector having a smaller vector norm may therefore produce results with smaller variance. As a result, fewer executions of quantum circuits may be required to synthesise the target quantum operation to a given accuracy.

[0020] The error term for a given candidate weight vector represents a difference between the target quantum operation and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the circuit library using the given candidate weight vector. The error term may therefore be considered to be a measure of the accuracy with which a given candidate weight vector synthesises the target quantum operation.

[0021] For example, some (or all) candidate weight vectors may in at least some examples, produce an exact synthesis of the target quantum operation. That is, a candidate weight vector when applied in a weighted summation of the quantum circuits in the circuit library may produce an exact synthesis of the target quantum operation. For such a candidate weight vector, the error term may be substantially zero. Additionally or alternatively, some (or all) candidate weight vectors may not produce an exact synthesis of the target quantum operation such that their error term is non-zero.

[0022] In some examples, all of the plurality of candidate weight vectors may represent an exact synthesis of the target quantum operation. The error term for each candidate weight vector may therefore be zero. In such examples, the weighted sum of a vector norm of a candidate weight vector and an error term for the candidate weight vector will therefore be dependent only on the vector norm of the candidate weight vector. In such examples, determining a solution weight vector from the plurality of candidate weight vectors may comprise determining a solution weight vector which has a vector norm which is a minimum of the vector norms of all of the plurality of candidate weight vectors.

[0023] In some examples, a weight factor applied in the weighted sum of the vector norm and error term may be chosen to prioritise a small vector norm of the solution weight vector, whilst allowing for inexact synthesis of the target quantum operation (i.e. a non-zero error term). For example, the weighted sum may comprise a sum of the vector norm multiplied by a weight factor and the error term, and the weight factor may be set to be relatively large. In such examples, the vector norm of candidate weight vectors will contribute more to the weighted sum (which is minimised to find the solution weight vector) such that a small vector norm is prioritised in determining the solution weight vector.

[0024] In some examples, a weight factor applied in the weighted sum of the vector norm and error term may be chosen to prioritise the accuracy of the synthesis of the target quantum operation. For example, the weighted sum may comprise a sum of the vector norm multiplied by a weight factor and the error term and the weight factor may be set to be relatively smaller (than when a small vector norm is prioritised). As the weight factor is reduced, the contribution of the error term to the weighted sum (which is minimised to find the solution weight vector) increases. Accordingly, a smaller error term and a more accurate synthesis is prioritised in determining the solution weight vector.

[0025] As was explained above, the weight coefficients in the candidate weight vectors may take on negative or positive values. Allowing solutions with positive and negative weight coefficients may provide greater flexibility in determining a solution weight vector which has the desired combination of a small vector norm and accurate synthesis of the target quantum operation.

[0026] The determining a probability associated with each quantum circuit in the circuit library may comprise determining a probability in dependence on the magnitude of the respective weight coefficient in the solution weight vector. As was explained above, a weight vector includes a weight coefficient for each quantum circuit in the circuit library. The probability associated with each quantum circuit may be determined based on the weight coefficient (in the solution weight vector) which corresponds to that quantum circuit. The probability associated with each quantum circuit may be determined as a normalised magnitude of the weight coefficient (in the solution weight vector) which corresponds to that quantum circuit. The normalised magnitude of a weight coefficient may comprise the magnitude of the weight coefficient divided by a normalisation factor. The normalisation factor may comprise a vector norm (e.g. the LI norm) of the solution weight vector.

[0027] The performing a plurality of selections of a quantum circuit from the circuit library according to the probability distribution may comprise selecting quantum circuits from the circuit library such that each quantum circuit has a probability of being selected which corresponds to its associated probability in the probability distribution. That is, quantum circuits with a relatively higher associated probability in the probability distribution are more likely to be selected in each selection than quantum circuits with a relatively lower associated probability. Accordingly, when many selections are performed a quantum circuit with a higher associated probability will be selected on average more times than a quantum circuit with a lower associated probability.

[0028] In some examples, the performing a plurality of selections of a quantum circuit from the circuit library according to the probability distribution may comprise performing a plurality of independent selections of a quantum circuit from the circuit library according to the probability distribution. An independent selection may be a selection which does not depend on the result of any previous or subsequent selections. An independent selection may comprise a selection made according to a random seed. Such selections may be referred to as a random selection.

[0029] The performing a plurality of selections of a quantum circuit from the circuit library according to the probability distribution may comprise random sampling according to the probability distribution. For example, the performing a plurality of selections of a quantum circuit from the circuit library according to the probability distribution may comprise weighted random sampling, where the probability distribution forms the weights used in the weighted random sampling.

[0030] Whilst in some examples each selection from the circuit library may be an independent and / or random selection, in other examples deterministic selections may be made according to the probability distribution. For example, a deterministic method may be used to determine a number of times which each quantum circuit in the library is selected. In at least some examples, each quantum circuit may be selected a number of times which is approximately proportional to its associated probability in the probability distribution.

[0031] The determined control sequence may comprise a sequence of the quantum circuits as selected by the plurality of selections. For example, the control sequence may comprise a list of the outcome of all of the plurality of selections. In some examples, the control sequence may be determined to include additional circuits other than the plurality of selections and / or to include one or more repetitions of one or more of the circuit selections. For example, it may be more efficient and / or cheaper to execute the same quantum circuit multiple times than to execute the same number of different quantum circuits once each. In such examples, it may be more efficient and / or cheaper to include repetitions of the same quantum circuit in the control sequence.

[0032] As was explained above, the target quantum operation can be synthesised by a weighted summation over the plurality of circuit libraries according to the solution weight vector. Such a summation cannot be performed directly in hardware but can be implemented on average by executing the control sequence comprising the plurality of selections of a quantum circuit from the circuit library. By determining a probability distribution comprising probabilities proportional to the magnitude of a respective weight coefficient (in the solution weight vector), and performing the plurality of selections according to the probability distribution, results generated by executing the resulting control sequence on average serve to synthesise the target quantum operation. It has been shown that on average measurement results obtained after executing the control sequence correspond to directly implementing the target quantum operation. By using a circuit library to synthesise the target quantum operation on average, a circuit library can be constructed of quantum circuits which are relatively easy and / or inexpensive to implement in hardware, whereas the target quantum operation itself may be impossible, difficult and / or expensive to implement directly in hardware. The methods described herein are therefore motivated by considerations of the underlying quantum hardware on which a target quantum operation is to be executed and provide a control sequence which is more efficient to execute on the hardware. Furthermore, the methods described herein may be utilised for any form of target quantum operation and thus the methods and their advantages are not specific to any particular form of operation or data.

[0033] To obtain a synthesis of the target quantum operation on average, the control sequence comprises a plurality of selections of quantum circuits such that a plurality of quantum circuits are implemented to obtain the synthesis. The plurality of selections may, for example, comprise many selections such as hundreds, thousands or even tens of thousands of selections. That is, the control sequence may comprise hundreds, thousands or even tens of thousands of quantum circuits to execute on the quantum computer. In any case, near-term quantum computers produce noisy results which already requires many repetitions of a computation to determine an average result. When compared to implementing the target quantum operation itself, implementing a control sequence to synthesise the target quantum operation on average according to methods described herein, may introduce a measurement overhead in order to obtain a result to corresponding accuracy. That is, to obtain a result to a given accuracy, a control sequence determined according to methods disclosed herein may comprise execution of a greater number of quantum circuits and performing a greater number of measurements (which may referred to as a number of shots) to execute than a number of times (a number of shots) that the target quantum operation would need to be repeatedly executed. A difference in number of shots used in a control sequence determined according to methods disclosed herein and a number of shots of repeatedly executing the target quantum operation to obtain results to corresponding accuracy may be referred to as a measurement overhead.

[0034] According to methods disclosed herein, a solution weight vector may be determined which has a relatively small (and possibly minimum) vector norm (e.g. LI norm). It has been shown that this serves to introduce only a small variance into results generated by executing a control sequence determined according to methods disclosed herein. Accordingly, the measurement overhead described above is relatively small. Despite introducing a relatively small measurement overhead, the methods described herein are often advantageous (when compared, for example, to implementing the target quantum operation itself) because the circuit library can be constructed from circuits which are relatively easy and / or inexpensive to implement in hardware.

[0035] The method may further comprise: causing the quantum computer to execute the control sequence of quantum circuits; and performing a measurement of an outcome which results from execution of the quantum circuit for each execution of a quantum circuit in the control sequence so as to produce a plurality of measurement outcomes.

[0036] Causing the quantum computer to execute the control sequence of quantum circuits may comprise transmitting the control sequence to the quantum computer. The quantum computer may comprise a quantum information processor comprising a plurality of controllable qubits and an interaction module arranged to interact with the quantum information processor to control and measure quantum states of the qubits. The control sequence of quantum circuits may be executed directly on the quantum computer. Alternatively, the control sequence of quantum circuits may be further translated or decomposed into operations which can be directly executed on the quantum computer. For example, the control sequence may be translated or decomposed into a sequence of control pulses which can be directly produced in the quantum computer.

[0037] Performing a measurement of an outcome which results from execution of the quantum circuit may comprise measuring the quantum state of one or more qubits after execution of the quantum circuit. The measurement of the outcome may produce a numerical result representing the outcome of the circuit executed on the quantum computer. A measurement is performed for each execution of a quantum circuit in the control sequence to produce the plurality of measurement outcomes.

[0038] The method may further comprise determining an expectation value of an observable as an average over the plurality of measurement outcomes.

[0039] The expectation value may be determined as an average of the plurality of measurement outcomes each multiplied by a measurement factor. In the context of measurements of qubits, a typical measurement outcome corresponds to measurement of an expected value of a Pauli matrix in which measurement of each qubit yields one of two possible values (e.g. -1 or +1). The measurement factor for each measurement outcome may comprise the sign of the weight coefficient in the solution weight vector which corresponds to the quantum circuit which was executed to produce the measurement outcome. That is, for quantum circuits having a negative weight coefficient in the solution weight vector, the measurement factor may be negative. For quantum circuits having a positive weight coefficient in the solution weight vector, the measurement factor may be positive. The measurement factor for each measurement outcome may comprise a vector norm (e.g., an LI norm) of the solution weight vector.

[0040] In some examples, the measurement factor for each measurement outcome may comprise a vector norm (e.g., an LI norm) of the solution weight vector multiplied by the sign of the weight coefficient in the solution weight vector which corresponds to the quantum circuit which was executed to produce the measurement outcome. That is, each measurement outcome may be multiplied by a vector norm (e.g., an LI norm) of the solution weight vector and multiplied by the sign of the respective weight coefficient in the solution weight vector (which corresponds to the quantum circuit which was executed to produce the measurement outcome) to determine a modified measurement outcome. The expectation value may then be determined as an average over the modified measurement outcomes.

[0041] The solution weight vector may exactly synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library.

[0042] The solution weight vector may exactly synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library to an arbitrary numerical precision. The determining a solution weight vector from the plurality of candidate weight vectors may comprise determining a solution weight vector which satisfies the minimum condition to an arbitrary numerical precision.

[0043] Each of the plurality of candidate weight vectors may exactly synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library.

[0044] Each of the plurality of candidate weight vectors may exactly synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library to an arbitrary numerical precision. In such examples, the error term is zero. The method may therefore comprise determining a solution weight vector from the plurality of candidate weight vectors, wherein the solution weight vector has a vector norm (e.g., LI norm) which is a minimum of the vector norms (e.g., LI norms) of the candidate weight vectors.

[0045] The plurality of candidate weight vectors may include at least one weight coefficient which is less than zero.

[0046] Weight coefficients may be allowed to take on any value. Weight coefficients may be allowed to take on negative values, positive values or be zero. Consequently, at least one candidate weight vector may include at least one weight coefficient which is less than zero. Such an approach may allow for exact synthesis of the target quantum operation and / or may allow there to be a larger solution space from which to determine the solution weight vector (which may allow for a solution with a smaller vector norm and thus a smaller measurement overhead).

[0047] The determining a circuit library comprising a plurality of quantum circuits may comprise determining a plurality of quantum circuits each comprising a sequence of Clifford gates and at least one T gate. Each quantum circuit of the circuit library may comprise a determined sequence of Clifford gates and at least one T gate.

[0048] The quantum computer may be a fault tolerant quantum computer. In such instances the Clifford gates and T gates may form a universal gateset. Any quantum operation may be decomposed into a sequence of Clifford and T gates. However, an exact decomposition of the target quantum operation may result in a sequence of Clifford and T gates which is too long and / or which includes too many T gates for practical implementation. The plurality of quantum circuits each comprising a sequence of Clifford gates and at least one T gate may comprise sequences which include less than a threshold number of T gates (so as to impose an upper bound on a cost and / or complexity of executing the sequence). Imposing limitations on the circuits (e.g., by imposing a maximum number of T gates) to be included in the circuit library may mean that the target quantum operation cannot be exactly decomposed as any one of the circuits in the circuit library. Each circuit in the circuit library instead comprises an approximation of the target quantum operation.

[0049] The determining a circuit library comprising a plurality of quantum circuits may comprise determining a plurality of rotation gates. Each rotation gate may comprise rotation by a different angle. Each quantum circuit of the circuit library may comprise a determined rotation gate.

[0050] In the context of continuously parameterised rotation gates, the quantum circuits included in the circuit library may comprise different rotation gates corresponding to a discrete number of rotation angles. The rotation gates included in the circuit library may therefore not include all possible rotation angles. Consequently, the target quantum operation may not directly correspond to any one of the circuits in the circuit library. Each circuit in the circuit library instead comprises an approximation of the target quantum operation.

[0051] The determining a circuit library comprising a plurality of quantum circuits may comprise determining a plurality of different control pulses. Each quantum circuit of the circuit library may comprise a determined control pulse.

[0052] In some examples, the target quantum operation may comprise a control pulse (e.g., to which a qubit is subjected) of a particular shape. It may not be possible, or it may be too difficult, costly and / or expensive to implement the control pulse directly in the hardware of the quantum computer. The circuit library may therefore include control pulses having pulse shapes which approximate the control pulse of the target quantum operation, but which are not the same as the control pulse of the target quantum operation. The control pulses included in the circuit library may be directly implementable and / or may be implemented at lower cost and / or complexity than the control pulse of the target quantum operation.

[0053] The number of quantum circuits included in the circuit library may be greater than a total number of elements included in a process matrix representing the target quantum operation.

[0054] A dimension of the process matrix representing the target quantum operation may depend on the number of qubits which the target quantum operation operates on. The dimensions of the process matrix may be 2211 x 22n, where n is the number of qubits which the target quantum operation operates on. The process matrix may be a square matrix such that the column dimension and the row dimension are the same. Reference to the dimension of the process matrix may therefore comprise a reference to both the number of columns and the number of rows, which are the same as each other. A process matrix for a one qubit operation may be a 4 x 4 matrix such that the dimension of the process matrix is 4 and the total number of elements in the process matrix is 16. In which case the number of quantum circuits included in the circuit library may be greater than 16. A process matrix for a two qubit operation may be a 16 x 16 matrix such that the dimension of the process matrix is 16 and the total number of entries of the process matrix is 256. In which case the number of quantum circuits included in the circuit library may be greater than 256.

[0055] Including a greater number of quantum circuits in the circuit library than the dimension of the process matrix may mean that finding a weight vector which synthesises the target quantum operation comprises finding a solution to an underdetermined system of equations. The underdetermined system of equations may have a plurality of different solutions (corresponding to the plurality of candidate weight vectors). In some examples, the underdetermined system of equations may have an infinite number of different solutions (and thus an infinite number of candidate weight vectors).

[0056] The determining a circuit library comprising a plurality of quantum circuits may comprise determining the plurality of quantum circuits subject to hardware criteria of the quantum computer on which the control sequence is to be executed.

[0057] The hardware criteria may, for example, comprise a maximum cost and / or complexity of circuit to be included in the circuit library. For example, the hardware criteria may comprise a maximum number of T gates to be included in a quantum circuit in the circuit library. The hardware criteria, may for example, comprise a list of gates, gate types and / or pulse shapes which can be included in quantum circuits in the circuit library. The hardware criteria may, for example, comprise a number of discrete rotation angles corresponding to rotation gates that can be included in quantum circuits in the circuit library. The hardware criteria may, for example, comprise a discrete set of rotation angles corresponding to rotation gates that can be included in quantum circuits in the circuit library.

[0058] In general, the hardware criteria may place limitations on operations, gates and / or circuits which can be included in quantum circuits in the circuit library. The hardware criteria may place limitations on a complexity or cost associated with operations. Gates and / or circuits which can be included in quantum circuits in the circuit library. The hardware criteria may be dependent on the hardware of the quantum computer on which the control sequence is to be performed.

[0059] The method may further comprise receiving hardware criteria for the quantum computer on which the control sequence is to be executed and determining the plurality of quantum circuits subject to the received hardware criteria.

[0060] According to a second aspect of the present disclosure there is provided a computer implemented method of determining a combined control sequence for execution on a quantum computer. The method comprises: determining a plurality of control sequences for a plurality of target quantum operations wherein each of the plurality of control sequences is determined for each of the target quantum operations using a method according to the first aspect; and combining each of the plurality of determined control sequences to determine a combined control sequence.

[0061] The plurality of target quantum operations may each comprise a quantum gate or quantum circuit (e.g., representing a combination of a few quantum gates) to be combined into the same quantum circuit. For example, the plurality of target quantum operations may comprise different gates or circuits from the same quantum circuit to be executed.

[0062] A control sequence is determined for each target quantum operation as was described above. For example, a separate circuit library may be determined for each target quantum operation as was described above. Furthermore, a solution weight vector may be determined for each target quantum operation and circuit library. Similarly, a probability distribution may be determined for each target quantum operation and in dependence on the respective solution weight vector for that target quantum operation. For each of the target quantum operations, a plurality of selections of a quantum circuit from the respective circuit library and according to the respective probability distribution are made to form a control sequence for each target quantum operation. The control sequences determined for each of the plurality of target quantum operations are combined to form the combined control sequence.

[0063] The control sequences for each of the target quantum operations may be combined such that they are interleaved with one another. For example, as was explained above, the target quantum operations may comprise different parts of single quantum circuit to be executed before performing a measurement. In such examples, a selection of a quantum circuit from a respective circuit library may be performed for each target quantum operation to form a quantum circuit to be executed on the quantum computer. For example, a quantum circuit may comprise a first target quantum operation, a second target quantum operation and a third target quantum operation. To synthesise the quantum circuits first, second and third circuit libraries, solution weight vectors and probability distributions may be determined for each of the target quantum operations.

[0064] The combined control sequence may be determined by performing a selection of a first quantum circuit from the first circuit library and according to the first probability distribution (each corresponding to the first target operation), a selection of a second quantum circuit from the second circuit library and according to the second probability distribution (each corresponding to the second target operation) and a selection of a third quantum circuit from the third circuit library and according to the third probability distribution (each corresponding to the third target operation). The first quantum circuit, the second quantum circuit and the third quantum circuit may be executed in sequence as a single quantum circuit before performing a measurement of a quantum state which results from the circuit. This process may be repeated a plurality of times to obtain a plurality of measurement outcomes, where each measurement outcome corresponds to an execution of a quantum circuit comprising a selection from the first circuit library, a selection from the second circuit library and a selection from the third circuit library. The plurality of measurement outcomes may be averaged over to determine an expectation value resulting from the quantum circuit. In this way, a quantum circuit comprising a plurality of gates and / or smaller circuits can be synthesised in a corresponding way to the synthesis which was described above.

[0065] According to a third aspect of the present disclosure there is provided a computer implemented method of determining a combined control sequence for execution on a quantum computer. The method comprises: receiving a plurality of target quantum operations for execution on the quantum computer; for each of the target quantum operations, determining a circuit library comprising a plurality of quantum circuits, each of which approximate the target quantum operation, wherein the target quantum operations can each be synthesised by a weighted summation over the plurality of circuits in their respective circuit libraries, the weighted summation being performed according to a respective weight vector comprising a weight coefficient corresponding to each circuit in the respective circuit library, wherein the plurality of different circuits in each of the plurality of circuit libraries are determined such that, for each target quantum operation, there are a plurality of candidate weight vectors which synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the respective circuit library; for each of the target quantum operations, determining a respective solution weight vector from the respective plurality of candidate weight vectors, wherein, for each target quantum operation, the respective determined solution weight vector satisfies a minimum condition of all of the respective plurality of candidate weight vectors, wherein the minimum condition comprises having a minimum of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector, wherein the error term represents a difference between the target quantum operation and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the respective circuit library using the candidate weight vector; for each of the plurality of solution weight vectors determined for a respective target quantum operation, determining a probability associated with each quantum circuit in the respective circuit library, wherein each probability is proportional to the magnitude of the respective weight coefficient in the respective solution weight vector, wherein the probabilities associated with each quantum circuit in the circuit library form a probability distribution for the respective solution weight and the respective target quantum operation; for each of the target quantum operations, performing a plurality of selections of a quantum circuit from the respective circuit library according to the respective probability distribution; and determining a control sequence for execution on the quantum computer wherein the control sequence comprises the plurality of selections of a quantum circuit from the respective circuit libraries for each of the target quantum operations.

[0066] According to a fourth aspect of the present disclosure there is provided a computing device comprising one or more processors configured to perform a method according to any of the first, second or third aspects.

[0067] According to a fifth aspect of the present disclosure there is provided a computing system comprising, a computing device according to the fourth aspect and a quantum information processor configured to execute the control sequence.

[0068] According to a sixth aspect of the present disclosure there is provided a computer implemented method of determining a sequence of pulses of electromagnetic radiation to apply to a sample, the method comprising: receiving a target operation to be applied to the sample; determining a pulse library comprising a plurality of pulses of electromagnetic radiation, each of which approximate the target operation, wherein the target operation can be synthesised by a weighted summation over the plurality of pulses in the pulse library, the weighted summation being performed according to a weight vector comprising a weight coefficient corresponding to each pulse in the pulse library, wherein the plurality of different pulses in the pulse library are determined such that there are a plurality of candidate weight vectors which synthesise the target operation when applied in a weighted summation over the plurality of pulses in the pulse library; determining a solution weight vector from the plurality of candidate weight vectors, wherein the determined solution weight vector satisfies a minimum condition of all of the plurality of candidate weight vectors, wherein the minimum condition comprises having a minimum of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector, wherein the error term represents a difference between the target operation and an operation synthesised by a weighted summation over the plurality of pulses in the pulse library using the candidate weight vector; determining a probability associated with each pulse in the pulse library, wherein each probability is proportional to the magnitude of the respective weight coefficient in the solution weight vector, wherein the probabilities associated with each pulse in the pulse library form a probability distribution; performing a plurality of selections of a pulse from the pulse library according to the probability distribution; and determining a sequence of pulses to apply to the sample, wherein the sequence of pulses comprises the plurality of selections of a pulse from the pulse library.

[0069] Any of the features described herein with reference to quantum computing applications and / or any of the first, second, third, fourth or fifth aspects described above may also apply to a method according to the sixth aspect and / or other methods described herein related to determining a sequence of pulses of electromagnetic radiation. Methods described herein of determining a sequence of pulses of electromagnetic radiation need not necessarily be applied as part of a quantum computing process and may additionally or alternatively be applicable in processes such as Nuclear Magnetic Resonance (NMR) and / or Magnetic Resonance Imaging (MRI). The pulse library may correspond to the circuit library as described elsewhere herein. The plurality of pulses in the pulse library may correspond to the plurality of circuits in a circuit library as described elsewhere herein. The determined sequence of pulses may correspond to the determined control sequence as described elsewhere herein.

[0070] The plurality of pulses included in the pulse library may each comprise a pulse which can be applied in practice by a radiation source (such as a radio frequency pulse generator, a microwave source and / or a laser). The plurality of pulses included in the pulse library may be determined in dependence on one or more properties of a radiation source which is to emit the pulses.

[0071] The target operation may correspond with an operation for which it is difficult or impossible to generate and emit a pulse of radiation which exactly performs the target operation. The target operation can instead be synthesised by emitting at least some of the pulses in the pulse library.

[0072] Each of the pulses in the pulse library may not exactly match the target operation. That is, each pulse in the pulse library may represent an approximation of the target operation and may differ from the target operation. The pulses in the pulse library may be determined as pulses which closely approximate the target quantum operation but which can be emitted from a radiation source (and / or are can be emitted relatively easily and / or at relatively low cost). The pulses in the pulse library may be generated using optimum control techniques. Such techniques may, for example, use numerical methods to generate pulses which approximate a target operation.

[0073] Whilst each pulse in the pulse library differs from the target operation, the target operation can be synthesised as a weighted summation of the pulses included in the pulse library. The weighted summation comprises a weighted summation over the plurality of pulses in the pulse library according to a weight vector. The weight vector comprises a weight coefficient corresponding to each pulse in the pulse library. For example, for a pulse library comprising N pulses, a corresponding weight vector will comprise N weight coefficients, wherein each of the N weight coefficients corresponds to a single pulse in the pulse library. The weight coefficients may comprise real numbers. Each weight coefficient may comprise a positive number, zero, or a negative number. A weighted summation of the pulses included in the pulse library comprises a summation of each pulse multiplied by its respective weight coefficient in a weight vector.

[0074] The pulses in the pulse library are determined such that there are a plurality of candidate weight vectors which synthesise the target operation when applied in a weighted summation of the pulses in the pulse library. That is, there are a plurality of solutions (each represented by a candidate weight vector) which lead to synthesis of the target operation. This may be achieved by including a sufficiently large number of pulses in the pulse library such that a plurality of solutions (candidate weight vectors) are available to synthesise the target operation. The weighted summation of the pulses (to synthesise the target operation) is not executed directly in practice. The weighted summation of the pulses (to synthesise the target operation) may instead be implemented on average through multiple emission of different pulses in the pulse library.

[0075] The determining a solution weight vector from the plurality of candidate weight vectors may comprise determining each of the plurality of candidate weight vectors and selecting one of the candidate weight vectors as the solution weight vector. Alternatively, determining a solution weight vector from the plurality of candidate weight vectors may not comprise determining each and every candidate weight vector. For example, the solution weight vector may be determined using a search or optimisation process which does not require determination of all possible candidate weight vectors.

[0076] The minimum condition comprises determining a solution weight vector having a minimum (of the candidate weight vectors) of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector. Each candidate weight vector has an associated vector norm and an associated error term and thus a weighted sum of the vector norm and error term may be determined for each candidate weight vector. However, in practice the weighted sum of the vector norm and the error term may not be determined for each candidate weight vector since a minimum may be found using an optimisation process which does not require determination of all possible candidate weight vectors (or weighted sum). The solution weight vector is determined as the candidate weight vector having a minimum weighted sum of its vector norm and error term, from all of the candidate weight vectors. The weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector may comprise multiplying one or both of the vector norm and the error term by a respective weight factor and summing the terms. For example, the weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector may comprise a sum of the vector norm multiplied by a weight factor and the error term. Alternatively, the weighted sum may comprise a sum of the vector norm and the error term multiplied by a weight factor.

[0077] The vector norm may, for example, comprise the LI norm of a candidate weight vector. The vector norm may represent a measurement overhead associated with synthesising the target operation. In particular, a variance which is introduced into measurements taken after execution of the sequence of pulses may depend (e.g., may be proportional to and / or may be limited by) the vector norm of the solution weight vector which is used. A candidate weight vector having a smaller vector norm may therefore produce results with smaller variance. As a result, emission of fewer pulses and fewer measurements may be required to synthesise the target operation to a given accuracy.

[0078] The error term for a given candidate weight vector represents a difference between the target operation and an operation synthesised by a weighted summation over the plurality of pulses in the pulse library using the given candidate weight vector. The error term may therefore be considered to be a measure of the accuracy with which a given candidate weight vector synthesises the target operation.

[0079] For example, some (or all) candidate weight vectors may in at least some examples, produce an exact synthesis of the target operation. That is, a candidate weight vector when applied in a weighted summation of the pulses in the pulse library may produce an exact synthesis of the target operation. For such a candidate weight vector, the error term may be substantially zero. Additionally or alternatively, some (or all) candidate weight vectors may not produce an exact synthesis of the target operation such that their error term is non-zero.

[0080] In some examples, all of the plurality of candidate weight vectors may represent an exact synthesis of the target operation. The error term for each candidate weight vector may therefore be zero. In such examples, the weighted sum of a vector norm of a candidate weight vector and an error term for the candidate weight vector will therefore be dependent only on the vector norm of the candidate weight vector. In such examples, determining a solution weight vector from the plurality of candidate weight vectors may comprise determining a solution weight vector which has a vector norm which is a minimum of the vector norms of all of the plurality of candidate weight vectors.

[0081] In some examples, a weight factor applied in the weighted sum of the vector norm and error term may be chosen to prioritise a small vector norm of the solution weight vector, whilst allowing for inexact synthesis of the target operation (i.e. a non-zero error term). For example, the weighted sum may comprise a sum of the vector norm multiplied by a weight factor and the error term, and the weight factor may be set to be relatively large. In such examples, the vector norm of candidate weight vectors will contribute more to the weighted sum (which is minimised to find the solution weight vector) such that a small vector norm is prioritised in determining the solution weight vector.

[0082] In some examples, a weight factor applied in the weighted sum of the vector norm and error term may be chosen to prioritise the accuracy of the synthesis of the target operation. For example, the weighted sum may comprise a sum of the vector norm multiplied by a weight factor and the error term and the weight factor may be set to be relatively smaller (than when a small vector norm is prioritised). As the weight factor is reduced, the contribution of the error term to the weighted sum (which is minimised to find the solution weight vector) increases. Accordingly, a smaller error term and a more accurate synthesis is prioritised in determining the solution weight vector.

[0083] As was explained above, the weight coefficients in the candidate weight vectors may take on negative or positive values. Allowing solutions with positive and negative weight coefficients may provide greater flexibility in determining a solution weight vector which has the desired combination of a small vector norm and accurate synthesis of the target operation.

[0084] The determining a probability associated with each pulse in the pulse library may comprise determining a probability in dependence on the magnitude of the respective weight coefficient in the solution weight vector. As was explained above, a weight vector includes a weight coefficient for each pulse in the pulse library. The probability associated with each pulse may be determined based on the weight coefficient (in the solution weight vector) which corresponds to that pulse. The probability associated with each pulse may be determined as a normalised magnitude of the weight coefficient (in the solution weight vector) which corresponds to that pulse. The normalised magnitude of a weight coefficient may comprise the magnitude of the weight coefficient divided by a normalisation factor. The normalisation factor may comprise a vector norm (e.g. the LI norm) of the solution weight vector.

[0085] The performing a plurality of selections of a pulse from the pulse library according to the probability distribution may comprise selecting pulses from the pulse library such that each pulse has a probability of being selected which corresponds to its associated probability in the probability distribution. That is, pulses with a relatively higher associated probability in the probability distribution are more likely to be selected in each selection than pulses with a relatively lower associated probability. Accordingly, when many selections are performed a pulse with a higher associated probability will be selected on average more times than a pulse with a lower associated probability.

[0086] In some examples, the performing a plurality of selections of a pulse from the pulse library according to the probability distribution may comprise performing a plurality of independent selections of a pulse from the pulse library according to the probability distribution. An independent selection may be a selection which does not depend on the result of any previous or subsequent selections. An independent selection may comprise a selection made according to a random seed. Such selections may be referred to as a random selection.

[0087] The performing a plurality of selections of a pulse from the pulse library according to the probability distribution may comprise random sampling according to the probability distribution. For example, the performing a plurality of selections of a pulse from the pulse library according to the probability distribution may comprise weighted random sampling, where the probability distribution forms the weights used in the weighted random sampling.

[0088] Whilst in some examples each selection from the pulse library may be an independent and / or random selection, in other examples deterministic selections may be made according to the probability distribution. For example, a deterministic method may be used to determine a number of times which each pulse in the library is selected. In at least some examples, each pulse may be selected a number of times which is approximately proportional to its associated probability in the probability distribution.

[0089] The determined sequence of pulses may comprise a sequence of the pulses as selected by the plurality of selections. For example, the sequence may comprise a list of the outcome of all of the plurality of selections. In some examples, the sequence of pulses may be determined to include additional pulses other than the plurality of selections and / or to include one or more repetitions of one or more of the pulse selections.

[0090] According to methods disclosed herein, a solution weight vector may be determined which has a relatively small (and possibly minimum) vector norm (e.g. LI norm). It has been shown that this serves to introduce only a small variance into results generated by executing a pulse sequence determined according to methods disclosed herein. Accordingly, a measurement overhead which may be introduced is relatively small. Furthermore, it has been found that the accuracy with which a given target operation can be performed on average can be improved by the methods described herein even without introducing any measurement overhead.

[0091] The method may form part of a Nuclear Magnetic Resonance (NMR) analysis.

[0092] The method may form part of a Magnetic Resonance Imaging (MRI) analysis.

[0093] The method may form part of an Electron Spin Resonance (ESR) analysis.

[0094] The method may further comprise: causing a radiation source to emit the determined sequence of pulses such that they are applied to the sample; and performing a measurement of an outcome which results from applying the pulse to the sample for each emission of a pulse in the pulse sequence so as to produce a plurality of measurement outcomes.

[0095] Causing the radiation source to emit the determined sequence of pulses may comprise transmitting the sequence to the radiation source and / or driving the radiation source with an electrical signal comprising the sequence of pulses. The radiation source may, for example, comprise a radio frequency (RF) source, a microwave source and / or a laser.

[0096] The method may further comprise determining an expectation value of an observable as an average over the plurality of measurement outcomes.

[0097] The expectation value may be determined as an average of the plurality of measurement outcomes each multiplied by a measurement factor. In some examples, (e.g., in which the sample comprises one or more qubits in a quantum computer), each measurement outcome corresponds to measurement of an expected value of a Pauli matrix in which measurement of each qubit yields one of two possible values (e.g. -1 or +1). In other examples, (e.g., in which the sample comprises a plurality of atomic nuclei or electron spins), each measurement outcome yields a time-dependent signal resulting from emission of radiation from the sample. The measurement factor for each measurement outcome may comprise the sign of the weight coefficient in the solution weight vector which corresponds to the pulse which was emitted to produce the measurement outcome. That is, for pulses having a negative weight coefficient in the solution weight vector, the measurement factor may be negative. For pulses having a positive weight coefficient in the solution weight vector, the measurement factor may be positive. The measurement factor for each measurement outcome may comprise a vector norm (e.g., an LI norm) of the solution weight vector.

[0098] In some examples, the measurement factor for each measurement outcome may comprise a vector norm (e.g., an LI norm) of the solution weight vector multiplied by the sign of the weight coefficient in the solution weight vector which corresponds to the pulse which was emitted to produce the measurement outcome. That is, each measurement outcome may be multiplied by a vector norm (e.g., an LI norm) of the solution weight vector and multiplied by the sign of the respective weight coefficient in the solution weight vector (which corresponds to the pulse which was executed to produce the measurement outcome) to determine a modified measurement outcome. The expectation value may then be determined as an average over the modified measurement outcomes.

[0099] The sample may comprise a plurality of atomic nuclei. The sample may comprise a plurality of electron spins. The performing a measurement of an outcome which results from applying the pulse to the sample may comprise measuring radiation emitted from the sample following applying the pulse to the sample.

[0100] The sample may comprise one or more qubits in a quantum computer. The performing a measurement of an outcome which results from applying the pulse to the sample may comprise measuring a quantum state of the one or more qubits following applying the pulse to the sample.

[0101] The solution weight vector may exactly synthesise the target operation when applied in a weighted summation over the plurality of pulses in the pulse library.

[0102] The solution weight vector may exactly synthesise the target operation when applied in a weighted summation over the plurality of pulses in the pulse library to an arbitrary numerical precision. The determining a solution weight vector from the plurality of candidate weight vectors may comprise determining a solution weight vector which satisfies the minimum condition to an arbitrary numerical precision.

[0103] Each of the plurality of candidate weight vectors may exactly synthesise the target operation when applied in a weighted summation over the plurality of pulses in the pulse library. Each of the plurality of candidate weight vectors may exactly synthesise the target operation when applied in a weighted summation over the plurality of pulses in the pulse library to an arbitrary numerical precision. In such examples, the error term is zero. The method may therefore comprise determining a solution weight vector from the plurality of candidate weight vectors, wherein the solution weight vector has a vector norm (e.g., LI norm) which is a minimum of the vector norms (e.g., LI norms) of the candidate weight vectors.

[0104] The plurality of candidate weight vectors may include at least one weight coefficient which is less than zero.

[0105] Weight coefficients may be allowed to take on any value. Weight coefficients may be allowed to take on negative values, positive values or be zero. Consequently, at least one candidate weight vector may include at least one weight coefficient which is less than zero. Such an approach may allow for exact synthesis of the target operation and / or may allow there to be a larger solution space from which to determine the solution weight vector (which may allow for a solution with a smaller vector norm and thus a smaller measurement overhead).

[0106] The number of pulses included in the pulse library may be greater than a total number of elements included in a process matrix representing the target operation.

[0107] According to a seventh aspect of the present disclosure there is provided a computing device comprising one or more processors configured to perform a method according to the sixth aspect.

[0108] According to an eighth aspect of the present disclosure there is provided a system comprising, a computing device according to the seventh aspect and a radiation source configured to emit the determined sequence of pulses such that they are applied to the sample.

[0109] It will be appreciated from the foregoing disclosure and the following detailed description of the examples that certain features and implementations described as being optional in relation to any given aspect of the disclosure set out above should be understood by the reader as being disclosed also in combination with the other aspects of the present disclosure, where applicable. Similarly, it will be appreciated that any attendant advantages described in relation to any given aspect of the disclosure set out above should be understood by the reader as being disclosed as advantages of the other aspects of the present disclosure, where applicable. That is, the description of optional features and advantages in relation to a specific aspect of the disclosure above is not limiting, and it should be understood that the disclosures of these optional features and advantages are intended to relate to all aspects of the disclosure in combination, where such combination is applicable. BRIEF DESCRIPTION OF THE DRAWINGS

[0110] Certain examples of the present disclosure will now be described, with reference to the accompanying drawings, in which: - FIG. 1 is a schematic illustration of a computing system; - FIG. 2 is a flowchart of a method 200 according to examples disclosed herein; - FIG. 3 is a schematic illustration of a state preparation circuit; - FIG. 4 is a schematic illustration of part of a control sequence which may be determined for the state preparation circuit illustrated in FIG. 3; - FIG. 5 is a schematic illustration of a further part of a control sequence which may be determined for the state preparation circuit illustrated in FIG. 3; - FIG. 6 is a graph illustrating members of a circuit library which were determined by generating sequences of Clifford and T gates; - FIG. 7 is a graph illustrating an approximation error of each of a plurality of circuits forming a circuit library; and - FIG. 8 is a graph in which a plurality of different solution weight vectors are represented. DETAILED DESCRIPTION

[0111] Hereinafter, examples of the disclosure are described with reference to the accompanying drawings. However, it should be appreciated that the disclosure is not limited to the described examples, and all changes and / or equivalents or replacements thereto also belong to the scope of the disclosure. The same or similar reference denotations may be used to refer to the same or similar elements throughout the specification and the drawings.

[0112] As used herein, the terms “have,” “may have,” “include,” or “may include” a feature (e.g., a number, function, operation, or a component such as a part) indicate the existence of the feature and do not exclude the existence of other features. Throughout the description and claims of this specification, the words “comprise” and “contain” and variations of them mean “including but not limited to”, and they are not intended to (and do not) exclude other components, integers or steps. Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

[0113] As used herein, the terms “A or B,” “at least one of A and / or B,” or “one or more of A and / or B” may include all possible combinations of A and B. For example, “A or B,” “at least one of A and B,” “at least one of A or B” may indicate all of (1) including at least one A, (2) including at least one B, or (3) including at least one A and at least one B.

[0114] As used herein, the terms “first” and “second” may modify various components regardless of importance and do not limit the components. These terms are only used to distinguish one component from another. For example, reference to a first component and a second component may indicate different components from each other regardless of the order or importance of the components.

[0115] It will be understood that when an element (e.g., a first element) is referred to as being (physically, operatively or communicatively) “coupled with / to,” or “connected with / to” another element (e.g., a second element), it can be coupled or connected with / to the other 26 element directly or via a third element. In contrast, it will be understood that when an element (e.g., a first element) is referred to as being “directly coupled with / to” or “directly connected with / to” another element (e.g., a second element), no other element (e.g., a third element) intervenes between the element and the other element.

[0116] The terms as used herein are provided merely to describe some embodiments thereof, but not to limit the scope of other embodiments of the disclosure. It is to be understood that the singular forms “a,” “'an,” and “the” include plural references unless the context clearly dictates otherwise. All terms including technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the embodiments of the disclosure belong. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.

[0117] FIG. 1 is a schematic illustration of a computing system 102. The computing system 102 comprises a computing device 112, an interaction module 108 and a quantum information processor 104.

[0118] The quantum information processor 104 is configured to store and manipulate qubits 106. The qubits 106 may be realised as physical qubits comprising controllable two-state quantum mechanical systems. For example, atoms, ions, photons, electrons, and / or superconducting electronic circuits may be used to realise physical qubits. Examples of methods for generating physical qubits include but are not limited to: ion traps, superconductors, quantum dots, electrons on liquid helium, solid-state spin spectroscopy, and cavity quantum electrodynamics (QED). The quantum information processor 104 may be controlled to maintain and manipulate a plurality of physical qubits 106. Logical qubits may be encoded onto the physical qubits to execute quantum algorithms and realise quantum computations. The quantum information processor 104 may process logical qubits encoded on multiple physical qubits 106 using a quantum error correcting code. The quantum information processor 104 may generate, manipulate and measure qubits 106 according to a control sequence. The term quantum information processor 104 and quantum computer may be used interchangeably herein.

[0119] A total of nine qubits 106 are shown in the quantum information processor 104 of FIG. 1. However, it will be appreciated that a quantum information processor 104 may comprise any number of qubits 106 and may, for example, comprise more than nine qubits 106.

[0120] The interaction module 108 is suitable for interacting with the quantum information processor 104. The interaction module 108 may be used to generate an initial quantum state on the quantum information processor 104. The interaction module 108 may be used to execute one or more quantum circuits to manipulate the quantum state of the qubits 106. Such one or more circuits may be referred to as state preparation circuits. The interaction module 108 may be used to read out a state of the quantum information processor 150. For example, the interaction module 108 may be used to perform one or more measurements of the quantum state of one or more of the qubits 106. Measurements of the quantum state of one or more qubits 106 may be used to determine an expectation of a value of an observable, which may comprise a result of quantum computation. Each measurement of a quantum state of one or more qubits 106 yields one of two values (e.g. +1 or -1).

[0121] The form and characteristics of the interaction module 108 may depend on the type of quantum information processor 104 being used to process quantum information. For example, the form and characteristics of the interaction module 108 may depend on the type of qubits 106 included in the quantum information processor 104.

[0122] In some examples, the quantum information processor 104 may comprise qubits 106 realised as trapped ions. In such examples, the interaction module 108 may comprise one or more lasers for preparing an initial quantum state via e.g. optical pumping. The interaction module 108 may comprise apparatus for manipulating magnetic dipole transitions or stimulated Raman transitions and / or apparatus for manipulating electric quadrupole transitions, in order to manipulate the quantum state of the ions. The interaction module 108 may comprise apparatus for measuring one or more photons emitted from ions so as to enable measurement of the state of an ion qubit 106.

[0123] In some examples, the quantum computer may comprise a superconducting quantum computer in which qubits 106 are realised as superconducting circuits. In such examples, the interaction means 140 may comprise circuitry to apply a voltage across a particular point in the superconducting circuit and / or apparatus for coordinating microwave pulses applied to the superconducting quantum computer.

[0124] In general, the quantum information processor 104 may comprise any type of quantum information processor, the qubits 106 may comprise any type of qubit 106 and the interaction module 108 may comprise any apparatus suitable for interaction with the qubits 106 of the quantum information processor 104.

[0125] The computing device 112 may comprise a classical computing device 112. For example, the computing device 112 may comprise a server, a personal computer and / or a mobile or portable computer. The computing device 112 may be distributed across multiple devices which are communicably coupled so as to enable communication between them. The computing device 112 has a communication link 114 with the interaction module 108. The communication link 114 may be a local communication link 114 such as a direct wired or wireless connection or a wired or wireless communication link 114 facilitated by a local area network. In some examples, the communication link 114 may be established over a wide area network such as the internet. As will be explained in further detail below, the computing device 112 may determine a control sequence for execution on the quantum information processor 104. The communication link 114 may be used to send a determined control sequence to the interaction module 108. The communication link 114 may be used to send measurements of quantum states in the quantum information processor 104 to the computing device 112.

[0126] In the example depicted in FIG. 1, the computing device 112 comprises a processor 116, a memory 120 and a communication module 118. The processor 116, memory 120 and / or communication module 118 may be classical components. The memory 120 is suitable for storing data within computing device 112. The memory 120 may include a volatile memory unit or units. The memory 120 may include a non-volatile memory unit or units. The memory 120 may additionally be another form of computer-readable medium, such as a magnetic or optical disk. The memory 120 may provide mass storage for the computing device 112. Instructions for performing a method as described herein may be stored within the memory 120. For example, the memory 120 may contain instructions for determining a control sequence as described herein and may contain instructions for processing measurements made in the quantum information processor 104.

[0127] The communication module 118 is suitable for sending and receiving communications between processor 210 and remote systems. For example, the communication module 118 may be used to send and / or receive communications over communication link 114. Additionally or alternatively communication module 118 may be used to send and / or receive communications via a communication network, such as a local area network and / or a wide area network such as the Internet.

[0128] The processor 116 is configured to receive data, access the memory 120, and act upon instructions such as instructions stored in memory 120. The processor 116 may be configured to control the interaction module 108 according to a control sequence. The processor 116 may determine the control sequence locally according to any method of determining a control sequence described herein, or may receive the control sequence from an external computer apparatus 120, or via a computer-readable medium with the control sequence stored thereon.

[0129] Whilst not shown in FIG. 1. the computing device 112 may include one or more user interfaces such as a visualising means (e.g., an electronic display) and / or a virtual or dedicated user input device such as keyboard, mouse etc.

[0130] In some examples, a user of computing device 112 may seek to execute a quantum operation (such as a quantum circuit, e.g., a state preparation circuit) on the quantum information processor 104. A quantum operation which is to be executed (and may, for example, be specified by a user input) may be considered to be and may be referred to as a target quantum operation. The computing device 112 may determine a control sequence for performance on a quantum information processor 104. For example, the computing device 112 may determine a control sequence for execution on the quantum information processor 104 such that the quantum information processor 104 returns an expectation value of an observable. The computing device 112 may communicate via communication link 114 to the interaction module 108 so as to cause the interaction module 108 to execute the determined control sequence. Results, such as measurement outcomes resulting from execution of the control sequence may then be communicated back to the computing device 112 for provision to the user and / or an algorithm executed by the computing device 112 or another device.

[0131] In at least some examples, it may be difficult or impossible to directly execute all or part of a target quantum operation on the quantum information processor 104. As will be described in further detail below, as an alternative to directly executing the target quantum operation, all or part of the target operation may be synthesised as a weighted summation over a plurality of quantum circuits.

[0132] The target quantum operation may comprise a single quantum gate. Additionally or alternatively, the target quantum operation may comprise a plurality of quantum gates which together form a quantum circuit. The target quantum operation may comprise a unitary gate U and / or may be represented as a unitary process matrix U. A unitary process matrix U representing a quantum operation may be considered to act on a density matrix p representing the quantum state of one or more qubits 106 which the quantum operation acts on. In particular, a process matrix U acts on a vectorised density matrix vec[p] isomorphically to the action on a density matrix via conjugation as represented in equation 1 below. Uvec[p] = ve^UpU'] (1)

[0133] A process matrix U as contemplated herein is assumed to be the Pauli transfer matrix representation of a unitary matrix U with complex elements and of dimensions 2N x 2N, where N is the number of qubits 106 that the unitary acts on. The process matrix U then has only real elements and dimensions 22N x 22N.

[0134] A target quantum operation for which a control sequence is to be determined may be represented herein as Utarget. A target quantum operation Utarget may be synthesised as a weighted summation as shown in equation 2 below, where Ui are quantum circuits belonging to a circuit library comprising a plurality of quantum circuits Ui, yi are weight coefficients where each weight coefficient is associated with a respective quantum circuit Ui and / is an index used for different members Ui of the circuit library and their corresponding weight coefficients yi. Utarget = Y / V'Ui i (2)

[0135] It has been found that if the circuit library includes enough quantum circuits Ui and that the quantum circuits Ui approximate the target quantum operation Utarget then a solution can be found for the weight coefficients 7; such that the target quantum operation Utarget is synthesised by the weighted summation of equation 2. A solution for the weight coefficients 7 / may be found such that the target quantum operation Utarget is exactly synthesised such that equation 2 is solved exactly.

[0136] The weighted summation of equation 2 cannot be directly implemented. However, the weighted summation of equation 2 can be implemented on average according to methods described herein in which circuits Ui are selected from the circuit library according to a determined probability distribution and the selected circuits are implemented on the quantum information processor 104.

[0137] FIG. 2 is a flowchart of a method 200 according to examples disclosed herein. All or part of the method 200 of FIG. 2 may be performed by a computing device (which may be a classical computing device) such as a computing device 112 as described above with reference to FIG. 1.

[0138] At step 202, a target quantum operation for execution on a quantum computer (e.g., the quantum information processor 104 as described above with reference to FIG. 1) is received. Receiving the target quantum operation may, for example, comprise receiving a target operation which is specified by a user (e.g., input using a user interface). Additionally or alternatively, receiving the target quantum operation may comprise retrieving the target quantum operation from memory. Additionally or alternatively, receiving the target quantum operation may comprise receiving a quantum operation which is generated and / or retrieved by an algorithm and / or software routine being executed by the computing device 112 or other device. Additionally or alternatively, receiving the target quantum operation may comprise receiving a quantum operation from another computing device (e.g., over a network connection).

[0139] The target quantum operation may be represented as a unitary process matrix Utarget as was described above. The target quantum operation may form part of a larger quantum circuit or operation. For example, the target quantum operation may comprise a single quantum gate. The target quantum operation may comprise a quantum circuit comprising a plurality of quantum gates. For example, the target quantum operation may comprise a small quantum circuit comprising a few quantum gates (e.g, less than ten gates). The target quantum operation may act on a single qubit 106 or may act on a plurality of qubit 106. For example, the target quantum operation may be a two or three qubit operation. In general, the target quantum operation may be a part or all of any quantum operation which can be performed on a quantum computer.

[0140] At step 204 a circuit library is determined comprising a plurality of quantum circuits which approximate the target quantum operation. The plurality of quantum circuits may comprise the circuits { / / described above in connection with equation 2. One or more of the quantum circuits in the circuit library may comprise a single quantum gate. One or more of the quantum circuits in the circuit library may comprise a quantum circuit comprising a plurality (e.g., a sequence) of quantum gates.

[0141] The plurality of quantum circuits Ui in the circuit library may be determined such that the target quantum operation t / torgetcan be synthesised by a weighted summation over the circuits Ui in the circuit library as described above in connection with equation 2. As was explained above, and as can be seen in equation 2 the weighted summation over the circuits { / / in the circuit library utilises weight coefficients yz corresponding to each circuit Ui. A given set of weight coefficients yz is referred to herein as a weight vector X where a weight vector X comprises all of the weight coefficients as shown below in equation 3 in which Z is the number of circuits in the circuit library and correspondingly the number of weight coefficients y / . A [71, 7a, 73,.-- - 7?:] (3)

[0142] As will be explained further below, for a given target quantum operation Utarget and a given circuit library of quantum circuits Ui, a weight vector X may be determined which synthesises the target quantum operation Utarget when applied in a weighted summation over the circuits Ui in the circuit library (e.g., according to equation 2). In some examples, there may exist at least one weight vector X which exactly solves equation 2 such that the target quantum operation Utarget is exactly synthesised by a weighted summation over the circuit library. In some examples, at least one weight vector X may provide an approximate solution to equation 2 such the target quantum operation Utarget is approximately synthesised by a weighted summation over the circuit library.

[0143] The circuits Ui in the circuit library may be determined (at step 204) such that there exist a plurality of different candidate weight vectors X which synthesise (exactly or approximately) the target quantum operation Utarget when applied in a weighted summation over the circuit library. That is, there may be a plurality of different candidate weight vectors X which provide a solution to equation 2.

[0144] A plurality of candidate weight vectors X may be provided by including a sufficient number of quantum circuits Ui in the circuit library. Increasing the number quantum circuits which are included in the circuit library, may increase a number of candidate weight vectors X which provide a solution (exact or approximate) to equation 2. In at least some examples, the number of quantum circuits Ui in the circuit library may be greater than a number of elements in the process matrix Utarget- As will be further explained below, this may provide an undetermined system of equations with multiple solutions (i.e., a plurality of candidate weight vectors). The process matrix Utarget has dimensions 22N x 22N, where N is the number of qubits 106 on which the target quantum operation operates. Accordingly, the number of elements in the process matrix Utarget is 24N. In at least some examples, the circuit library may include greater than 24N quantum circuits.

[0145] The quantum circuits in the circuit library may be determined to approximate the target quantum circuit. That is, each quantum circuit in the circuit library may represent an approximation of the target quantum operation and may differ from the target quantum operation. As was explained above, the target quantum operation may be difficult, expensive or in some cases impossible to implement directly on the quantum information processor 104. In contrast, the quantum circuits included in the circuit library may be implemented on the quantum information processor at relatively low cost and / or difficulty.

[0146] References herein to a cost, difficulty and / or complexity associated with an operation, circuit or gate may, for example, comprise a measure of time (e.g. time needed to execute) and / or hardware complexity associated with execution of a given quantum operation, circuit and / or gate on a given quantum computer. Additionally or alternatively, a cost, difficulty and / or complexity may, for example comprise a measure of an impact on an accuracy of a computation carried out on the quantum computer. For example, a cost, difficulty and / or complexity may comprise a measure of noise and / or error which may be introduced through execution of the given quantum operation, circuit and / or gate on the given quantum computer.

[0147] In general, a given quantum information processor 104 may have some operations, gates and / or circuits which may be executed relatively easily and / or at relatively low cost and some operations, gates and / or circuits which are relatively difficult to execute and / or are executed at relatively high cost. For example, in the context of a fault-tolerant quantum computer, Clifford gates may be executed at relatively low cost (and / or may be executed relatively easily) whereas a T gate may be executed at relatively high cost (and / or may be relatively difficult to execute). In the context of realising a continuous set of parameterised gates (e.g. parameterised Pauli gates) parameterised by a rotation angle, a relatively small number of discrete rotation angles may be executed at relatively low cost (and / or may be executed relatively easily) whereas a relatively higher number of discrete rotation angles may be executed at relatively high cost (and / or may be relatively difficult to execute). In the context of control pulses (e.g. pulses of microwave radiation and / or laser pulses) used to directly control qubits, there may be some pulse shapes which may be executed at relatively low cost (and / or may be executed relatively easily) whereas there may other pulse shapes which may only be executed at relatively high cost (and / or may be relatively difficult to execute). The cost and / or complexity associated with different operations, gates and / or circuits may depend on the hardware of the quantum computer.

[0148] As was explained above, determining the circuit library comprising a plurality of quantum circuits ( / / may comprise determining quantum circuits Ui which can be implemented relatively easily and / or with relatively low cost, complexity and / or difficulty. That is, the quantum circuits Uiin the circuit library may be determined as quantum circuits which closely approximate the target quantum operation but which are implementable (and / or are implementable relatively easily and / or at relatively low cost) on the quantum computer.

[0149] At step 206 of the method 200 of FIG. 2, a solution weight vector is determined which satisfies a minimum condition. The solution weight vector may be one of a plurality of candidate weight vectors which each synthesise (exactly or approximately) the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library. The solution weight vector may be determined by finding a solution to a set of equations.

[0150] As was explained above, the target quantum operation can be represented by a unitary process matrix Vtarget. Similarly, each quantum circuit in the circuit library may be represented by a unitary process matrix Vf. The process matrices Vtarget and each Ui have dimensions of 22Nx 22N where Ais the number of qubits 106 that the target quantum operation and the quantum circuits in the circuit library operate on. The process matrices may be vectorised by stacking their column vectors on top of each other to form vectors as shown below in equations 4 and 5. U target ~ VSC j (4) U i (5)

[0151] The vectorised process matrices of the quantum circuits in the circuit library (i.e., the vectors formed according to equation 5 above) may be combined to form columns of a library matrix R as defined by equation 6 below, in which (as above) L is the number of quantum circuits in the circuit library (and correspondingly the number of weight coefficients in a weight vector). R (jJi, U2, (6)

[0152] Using the above, equation 2 can be rewritten as equation 7 below, in which (as above) X is weight vector comprising L weight coefficients (as shown in equation 3). U target ~ R^ (7)

[0153] Equation 7 defines a linear system of equations which may be solved to find a solution weight vector X. The library matrix R as defined by equation 6 has dimensions 24N x L where L (the number of quantum circuits in the circuit library) is the row dimension and 24N (where N is the number of qubits on which the target quantum operation operates) is the column dimension. In at least some examples contemplated herein, a sufficient number of quantum circuits may be included in the circuit library such that the row dimension of R is greater than the column dimension of R. That is, the condition L >24N may be satisfied. In such examples, the system of equations defined by equation 7 may be an underdetermined system of equations with multiple solutions. That is, there may be a plurality of candidate weight vectors X which provide a solution (an exact solution or an approximate solution) to equation 7. In some examples, L may be significantly larger than 24N and the system of equations defined by equation 7 may be significantly underdetermined. In some examples, there may be an infinite number of solutions to equation 7 (i.e., an infinite number of candidate weight vectors).

[0154] In step 206, a solution weight vector is determined subject to a minimum condition. The minimum condition may comprise a property of the solution weight vector being a minimum of corresponding properties of the candidate weight vectors. For example, each candidate weight vector may have a given property associated with it (e.g., a vector norm) and the minimum condition may comprise the solution weight vector having a minimum of the given property of all of the candidate weight vectors. The minimum condition may comprise having a minimum of a weighted sum of a vector of the candidate weight vector and an error term for the candidate weight vector. The error term may represent a difference between the target quantum operation and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the circuit library and using the candidate weight vector. Such a minimum condition may be represented by equation 8 below, in which a is a weight factor. 1 "T 11RX U target):^ (8)

[0155] Determining a solution weight vector subject to the minimum condition may comprise finding a candidate weight vector X (a weight vector which solves equation 7 either exactly or approximately) which minimises (amongst all candidate weight vectors) the terms inside the brackets in equation 8. The terms inside the brackets in equation 8 represent a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector.

[0156] The first term inside the brackets in equation 8 is a weighted vector norm of a candidate weight vector X. The vector norm may comprise the LI norm of the candidate weight vector X. The weighted vector norm (LI norm) of the candidate weight vector X comprises the vector norm of the candidate weight vector X multiplied by the weight factor a.

[0157] The second term inside the brackets in equation 8 is the error term. The error term represents a difference or distance between the target quantum operation (in particular the vectorised target quantum operation {jtarqel) and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the circuit library using the candidate weight vector (represented by the term RX). The error term comprises a measure of the total magnitude of this difference (e.g., the vector norm of the vector which represents the difference).

[0158] The error term (the second term in the brackets of equation 8) is the Hilbert-Schmidt distance between the process matrices RX and ft . The Hilbert-Schmidt distance between v target two density matrices is an operationally invariant distance between two quantum fixed states. Similarly, the Hilbert-Schmidt distance between two process matrices expresses an average distance between two quantum processes.

[0159] In some examples, only candidate weight vectors X representing exact solutions to equation 7 are considered. In such examples, the error term is zero and the minimum condition of equation 8 becomes a condition of the vector norm of the candidate weight vector X being a minimum as shown below in equation 9. minOAX Hi) x 1 (9)

[0160] In some examples, candidate weight vectors X representing approximate solutions to equation 7 are also considered. In such examples, the size of the weight factor a determines the extent to which the accuracy of the solution is prioritised and the extent to which minimising the vector norm of the candidate weight vector X is prioritised in determining the solution weight vector. For example, if the weight factor a is chosen to be large then minimising the vector norm of the candidate weight vector X is prioritised over minimising the error term. A solution weight vector may therefore be determined which has a small vector norm of the weight vector but which may represent an approximate solution of equation 7 (that is, the error term may be non-zero). Solutions having small vector norms of the candidate weight vector X may be referred to as sparse solutions. Reducing the size of the weight factor a will increase the extent to which the accuracy of the solution is prioritised since the error term will contribute relatively more to the terms inside the brackets of equation 8.

[0161] A weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector is represented in equation 8 by multiplying the vector norm by a weight factor a. Additionally or alternatively, the error term could be multiplied by a weight factor which could bring about corresponding results through a suitable choice of weight factor(s).

[0162] As was explained above, determining a solution weight vector may comprise determining a solution weight vector which exactly solves equation 7 or may comprise determining a solution weight vector which provides an approximate solution to equation 7. As will be explained in further detail below it may be desirable to find a relatively sparse solution such that a vector norm of the solution weight vector is relatively small (and may be minimised). As was further explained above, the sparsity of the solution and / or the accuracy of the solution may be prioritised through appropriate choice of one or more weight factor(s).

[0163] Determining the solution weight vector may comprise using numerical solution methods. For example, methods which may be referred to as basis pursuit denoising (BPDN) in the field of signal processing may be used to find a solution weight vector which satisfies a minimum condition of the type shown in equation 8. Corresponding methods may be referred to as LASSO methods in the field of machine learning and may equivalently be used to find a solution weight vector which satisfies a minimum condition of the type shown in equation 8.

[0164] In at least some examples a least-angle regression (LARS) algorithm may be used to find a solution which satisfies a minimum condition of the type shown in equation 8. A LARS algorithm may initially find a solution subject to the minimum condition of equation 8 in the limit a —► co using a null vector X=(0,0,..0)T. Subsequently, the value of the weight factor a may be reduced such that a single vector entry is selected that has the highest potential of reducing the error term to find a new solution weight vector X which has only a single nonzero entry. The algorithm may then proceed by adding single non-zero entries to the solution weight vector and continually reducing the value of a until an exact solution is found (the error term goes to zero) or a approaches zero. Such methods provide an efficient and accurate way in which to determine an appropriate solution weight vector.

[0165] Determining the solution weight vector may comprise finding the solution weight vector to arbitrary numerical precision. The solution weight vector may include negative weight coefficients. That is, in finding the solution weight vector, the weight coefficients are allowed to take on real numbers which are less than zero, zero, or greater than zero.

[0166] Determining a solution weight vector has been described herein in terms of determining a solution weight vector from the plurality of candidate weight vectors. Such references are intended to indicate that there may be a plurality of different candidate weight vectors which provide a solution (approximate or exact) to equation 7 and that the solution weight vector represents one such solution (which also satisfies a minimum condition). It will be appreciated that the determining of the solution weight vector from the plurality of candidate weight vectors may not include determining each and every candidate weight vector (for example, there may be an infinite number of candidate weight vectors). For example, the solution weight vector may be determined using a search or optimisation process (which may for example comprise an iterative process such as a LARS algorithm as described above) which does not require determination of all possible candidate weight vectors.

[0167] At step 208 of the method 200 of FIG. 2, a probability associated with each quantum circuit Vi in the circuit library is determined. The probability associated with each quantum circuit may be proportional to the magnitude of the respective weight coefficient yi in the solution weight vector determined at step 206. For example, the probability associated with each quantum circuit may comprise the magnitude of the respective weight coefficient yi in the solution weight vector divided by a vector norm (e.g., LI norm) of the solution weight vector as defined by equation 10 below, in whichp(l) is the probability associated with the Ith quantum circuit Ui in the circuit library and yi is the Ith element of the solution weight vector X (and thus the weight coefficient in the solution weight vector which corresponds to the Ith quantum circuit in the circuit library). (10)

[0168] The plurality of probabilities together form a probability distribution comprising each probability p(l) determined for each quantum circuit Ui in the circuit library. The probabilities may take other forms than those indicated in equation 10 but are generally proportional to the magnitude of the corresponding weight coefficient yi in the solution weight vector. As was explained above, the weight coefficients yi in the solution weight vector are real values which may be negative, positive or zero. Using the magnitude of the weight coefficients yi ensures that the probabilities p(l) are positive real values or zero.

[0169] At step 210 of the method 200, a plurality of selections of a quantum circuit Ui from the circuit library are performed. The plurality of selections may be made according to the probability distribution. That is, the plurality of selections may be made according to the probabilities p(l) associated with each of the quantum circuits Ui in the circuit library. Making the plurality of selections according to the probabilities p(l) means that, at least on average, circuits Ui with relatively larger associated probabilities p(l) will be selected more times than circuits Ui with relatively smaller associated probabilities p(l).

[0170] In at least some examples, the plurality of selections may comprise independent selections. That is, each selection may be independent of any previous and / or subsequent selections. In at least some examples, the plurality of selections may be performed according to a random seed, which may be referred to as random sampling. For example, random weighted sampling may be used to perform the plurality of selections. Weighted random sampling may be used to select circuits Vi from the circuit library with a probability proportional to their associated probability p(l). A portion of a continuous range (e g., the range 0 to 1) may be assigned to each circuit Ui where the size of the portion of the range is proportional to its associated probability. A random number may then be generated in the continuous range (e.g., a random number between 0 and 1). The random number will fall within one of the portions of the range assigned to the circuits Ui and the circuit Ui assigned to the portion in which the random number falls may be selected. Such a process may be performed multiple times to arrive at a plurality of independent selections.

[0171] Whilst in some examples, independent and / or random selections may be made, in other examples deterministic methods may be used to perform the plurality of selections. For example, for each circuit Vi in the circuit library, a number of times that the circuit should be selected may be determined in dependence on its associated probability p(l). The number of times that a circuit Ui should be selected may be proportional to its associated probability p(l). Performing the plurality of selections of a quantum circuit may comprise selecting each quantum circuit t / / in the circuit library the determined number of times for that circuit. To provide a simplified illustrative example, a circuit library may comprise four quantum circuits Ui, U2, U3 and U4. The four quantum circuits may have associated probabilities p(l)=QA, p(2)=QA,p(3)=Q2 andp(4)=Q.3 respectively. If a total of 1000 selections are to be made then it may be determined that the first quantum circuit Ui is selected 0.1 x 1000 = 100 times, the second quantum circuit this selected 0.4 x 1000 = 400 times, the third quantum circuit U3is selected 0.2 x 1000 = 200 times, and the fourth quantum circuit tAis selected 0.3 x 1000 = 300 times. This example is however merely illustrative and in some examples, the circuit library may include more than four circuits and in some examples more than 1000 selections may be made.

[0172] At step 212 of the method 200 of FIG. 2 a control sequence of quantum circuits is determined comprising the plurality of selections of a quantum circuit from the circuit library as determined at step 210. The control sequence of quantum circuits is for execution on the quantum computer (e.g., quantum information processor 104). The control sequence may simply be determined at step 212 as the circuits selected at step 210 (e.g., in the order that they were selected or in a different order). In some examples, the control sequence may comprise the circuits selected at step 210 and may include at least some repetitions of at least some of the circuits selected at step 210.

[0173] In at least some examples of a quantum computer (e.g., quantum information processor 104) it may be more efficient and / or cheaper to execute the same quantum circuit multiple times than to execute the same number of different quantum circuits once each. For example, it may be more efficient and / or cheaper to execute the same quantum circuit X number of times than to execute X different quantum circuits once each. In such examples, it may be more efficient and / or cheaper to include repetitions of the same quantum circuit in the control sequence. For example, the control sequence may be determined to include X repetitions of a first quantum circuit selected at step 210, X repetitions of a second quantum circuit selected at step 210, X repetitions of a third quantum circuit selected at step 210 and so on. In this way, each of the selections of a quantum circuit made at step 210 may be included in the control sequence a plurality of times.

[0174] As was explained above, a control sequence for execution on a quantum computer may be determined according to the method 200 of FIG. 2. Whilst not shown in the method of FIG. 2, the control sequence may be executed on a quantum computer. For example, a determined control sequence may be sent to a quantum computer for execution on the quantum computer. In the example architecture of a computing system 102 as shown in FIG. 1, a determined control sequence may be sent from a computing device 112 to an interaction module 108 for causing the interaction module 108 to execute the control sequence on the quantum information processor 104.

[0175] Methods contemplated herein may further include performing measurements of outcomes which result from execution of quantum circuits on a quantum computer as part of executing a determined control sequence on the quantum computer. For example, in examples in which a single target quantum operation is considered, a method 200 as described above with reference to FIG. 2 may be performed to determine a control sequence for execution on a quantum computer. The control sequence includes a plurality of selections of quantum circuits Ui from a circuit library, where each quantum circuit Ui approximates the target quantum operation. The control sequence may be executed on a quantum computer such that a plurality of selections of quantum circuits Ui from the circuit library are executed on the quantum computer. A method may further include performing a measurement for each instance of execution of a selection of a quantum circuit Ui from the circuit library. For example, following each execution of a quantum circuit Ui from the circuit library, a measurement of a quantum state of one or more qubits 106 in the quantum computer may be performed. Such a measurement may be referred to as a measurement of an outcome which results from execution of a quantum circuit Ui. Each measurement outcome typically corresponds with measurement of an expected value of a Pauli matrix in which measurement of each qubit yields one of two possible values (e.g. -1 or +1).

[0176] Methods contemplated herein may further include determining an expectation value of an observable based on a plurality of measurement outcomes. It will be appreciated that for noisy intermediate-scale quantum (NISQ) computers it is common to execute a given quantum circuit a plurality of times and perform a plurality of measurements to obtain an expectation value of an observable as an average over the plurality of measurements (which each may comprise an expected value of a Pauli matrix). For example, if a target quantum operation were to be implemented directly on a quantum computer then it would be typical to execute the target quantum operation a plurality of times and perform a corresponding plurality of measurements. An expectation value of an observable which results from execution of the target quantum operation may therefore be determined as a simple average of the plurality of measurements (each of which may comprise an expected value of a Pauli matrix).

[0177] According to examples contemplated herein and as was described above with reference to FIG. 2, a target quantum operation may be synthesised as a weighted average over a plurality of circuits Ui in a circuit library rather than executing the target quantum operation directly. This is achieved in practice by selecting circuits Ui from the circuit library according to a probability distribution. As was explained above, this results in circuits Ui having a relatively higher associated probability p(l) being selected (and thus executed) more times than circuits Ui having a relatively lower associated probability p(l). If an average is taken over all measurement outcomes then, since the number of times that each circuit Ui is selected and executed is dependent on its associated probability p(l), the average corresponds to a weighted average where circuits Ui with relatively higher associated probabilities p(l) (and which are therefore selected and performed more times on average) contribute more to the average than circuits Vi with relatively lower associated probabilities p(l). In this way the methods contemplated herein serve to implement in practice a synthesis of a target quantum operation as described above with reference to equation 2.

[0178] When determining a solution weight vector, weight coefficient values in the solution weight vector may be allowed to take on negative values. The probabilities p(l) associated with each circuit Ui are however all positive (by virtue of being based on the magnitude of the coefficient values) and a given circuit Ui can only be selected and executed zero or a positive number of times. This can be accounted for by multiplying a measurement outcome by at least the sign of the weight coefficient (in the solution weight vector) corresponding to the quantum circuit ( / / used to obtain the measurement outcome. In at least some examples, each measurement outcome may be multiplied by a measurement factor comprising a vector norm (e.g., an LI norm) of the solution weight vector X multiplied by the sign of the weight coefficient yi in the solution weight vector X which corresponds to the quantum circuit which was executed to produce the measurement outcome. That is, each measurement outcome may be multiplied by a measurement factor according to equation 11 below. || A (11)

[0179] An expectation value of an observable may then be obtained by averaging over (e.g., determining the mean of) all measurement outcomes (each of which may comprise an expected value of a Pauli matrix) each multiplied by their respective measurement factor.

[0180] Each instance of executing a quantum circuit Ui selected from the circuit library and performing a measurement may be referred to herein as a shot. It has been found that for solution weight vectors X which provide an exact solution to equation 7, then if enough shots are performed then an average over measurement outcomes (multiplied by their respective measurement factor) produces an accurate estimation value of the observable. In particular, such results are as accurate as implementing a target quantum operation directly.

[0181] Whilst the methods described herein have been found to produce accurate expectation values, in at least some examples, a greater number of shots may need to be performed to achieve a corresponding level of accuracy. In particular, it has been found that using the methods described herein may introduce additional variance into the measurement outcomes which are averaged over to determine an expectation value (when compared for example to repeatedly executing the target quantum operation itself directly). In more detail, if the target quantum operation were to be implemented directly then averaging over each measurement outcome may comprise averaging over a sequence of + 1 and -1 measurement outcomes, which have a variance less than or equal to 1. In the approach described above, each measurement outcome is multiplied by a measurement factor given in equation 11 before averaging. The variance of such a sequence of measurement outcomes (after being multiplied by their respective measurement factor) may increase the variance of the results which are averaged over to obtain an expectation value. This variance however has an upper bound of the square of the LI norm of the solution weight vector (IIXIIi)2.

[0182] An increase in the variance of results which are averaged over to obtain an expectation value may mean that a greater number of shots is needed in order to obtain an expectation value to the same accuracy. Such an increase in number of shots to obtain results of corresponding accuracy (e.g., when compared to executing the target quantum operation itself directly) may be referred to herein as a measurement overhead. However, as was described above, the variance has an upper bound of the square of the vector norm (e.g. LI norm) of the solution weight vector such that the variance is less than or equal to (I 111 11)2. Any measurement overhead may therefore be reduced by finding sparse solutions to equation 7 in which the vector norm of the solution weight vector I 111 11 is relatively low.

[0183] As was explained above, a solution weight vector may be determined which satisfies a minimum condition such as the minimum condition shown in equation 8 and / or equation 9 above. As can be seen in equations 8 and equation 9 and as described above, finding solutions subject to the minimum condition serves to find sparse solutions represented by solution weight vectors having relatively small vector norms (IIXIIi). In examples, in which only exact solutions to equation 7 are considered, the error term in equation 8 becomes zero such that the minimum condition corresponds to finding a solution weight vector with a minimum vector norm (as expressed in equation 9). In examples, in which approximate solutions to equation 7 are considered, then a solution weight vector with an even smaller vector norm IIXIIi may be found. Correspondingly any measurement overhead may be further reduced by allowing approximate solutions. Using an approximate solution may introduce an error into resulting expectation values but, depending on the circumstances, such error may be acceptable in light of the smaller measurement overhead which may be achieved by allowing for an approximate solution.

[0184] As was explained above, methods presented herein may introduce a measurement overhead when compared, for example, to directly implementing a target quantum operation. However, as was also explained above, circuits Uzin the circuit library may be chosen to be circuits which may be implemented on the quantum computer with relatively low cost, difficulty and / or complexity. By contrast, it might be impossible or very difficult, costly and / or expensive to implement the target quantum operation itself on the quantum computer. Synthesising the target quantum operation using methods described herein may therefore in many circumstances be preferable to implementing the target quantum operation itself in spite of any additional measurement overhead which is introduced. Furthermore, in at least some examples sufficient circuits Vi may be included in the circuit library and / or approximate solutions may be allowed such that the vector norm IIXIIi of the solution weight vector is small enough that little or no measurement overhead is introduced.

[0185] Examples have been described above in the context of synthesising a single target quantum operation. However, it will be appreciated that in many examples, a state preparation circuit to be executed on a quantum computer prior to performing a measurement may comprise a plurality of target quantum operations. FIG. 3 is a schematic illustration of a state preparation circuit 302 comprising a first quantum operation 304, a second quantum operation 306, a third quantum operation 308 and a fourth quantum operation 310. The state preparation circuit 302 may, for example, comprise performing the first quantum operation 304, second quantum operation 306, third quantum operation 308 and fourth quantum operation 310 sequentially.

[0186] The state preparation circuit 302 may be considered to be a target state preparation circuit 302. The first quantum operation 304, second quantum operation 306, third quantum operation 308 and fourth quantum operation 310 may each be considered to be examples of a target quantum operation as described above with reference to the method 200 of FIG. 2. The first quantum operation 304, second quantum operation 306, third quantum operation 308 and fourth quantum operation 310 may each respectively comprise a single quantum gate or a plurality of quantum gates which together form a quantum circuit. The first quantum operation 304, second quantum operation 306, third quantum operation 308 and fourth quantum operation 310 may each act on a single qubit 106 or a plurality of qubits 106. The first quantum operation 304, second quantum operation 306, third quantum operation 308 and fourth quantum operation 310 may act on the same one or more qubits 106 and / or may act on different one or more qubits 106.

[0187] For each of the first quantum operation 304, second quantum operation 306, third quantum operation 308 and fourth quantum operation 310, a method 200 as described above with reference to FIG. 2 may be performed to determine a control sequence for synthesising each of the first 304, second 306, third 308 and fourth 310 quantum operations. For example, for each quantum operation a circuit library comprising a plurality of quantum circuits which approximate the quantum circuit may be determined as described above with reference to step 204 of FIG. 2 so as to determine a first circuit library for the first quantum operation 304, a second circuit library for the second quantum operation 306, a third circuit library for the third quantum operation 308 and a fourth circuit library for the fourth quantum operation 310. For each quantum operation (and correspondingly each circuit library), a solution weight vector may be determined which satisfies a minimum condition as described above with reference to step 206 and a probability associated with each quantum circuit in the circuit library may be determined as described above with reference to step 208. Furthermore, for each quantum operation a plurality of independent selections may be performed from the respective circuit libraries as described above with reference to step 210. A control sequence may then be determined comprising the plurality of selections of quantum circuits made for each quantum operation. Equivalently, a control sequence may be determined for each quantum operation, as was described above with reference to step 212 to determine a plurality of control sequences (e.g., a control sequence corresponding to each quantum operation 304 -310) and the plurality of control sequences may be combined to determine a combined control sequence.

[0188] Any of the features described above with reference to any of the steps of the method 200 of FIG. 2 may apply to a corresponding method performed for a state preparation circuit comprising a plurality of quantum operations as described with reference to FIG. 3.

[0189] FIG. 4 is a schematic illustration of part of a control sequence which may be determined for the state preparation circuit 302 illustrated in FIG. 3. FIG. 5 is a schematic illustration of a further part of a control sequence which may be determined for the state preparation circuit 302 illustrated in FIG. 3. The part of the control sequence shown in FIG. 4 may represent a first circuit selection 402 for execution on a quantum computer and the part of the control sequence shown in FIG. 5 may represent a second circuit selection 502 for execution on the quantum computer. The first circuit selection 402 and the second circuit selection 502 may equivalently be thought of as shots for running on the quantum computer as part of a method of synthesising the state preparation circuit 302.

[0190] The first circuit selection 402 comprises a first selection 404 from the first circuit library determined for the first quantum operation 304, a first selection 406 from the second circuit library determined for the second quantum operation 306, a first selection 408 from the third circuit library determined for the third quantum operation 308 and a first selection 410 from the fourth circuit library determined for the fourth quantum operation 310. The first selections 404, 406, 408, 410 from the respective circuit libraries may be performed as described above with reference to step 210.

[0191] Similarly, the second circuit selection 502 comprises a second selection 504 from the first circuit library determined for the first quantum operation 304, a second selection 506 from the second circuit library determined for the second quantum operation 306, a second selection 508 from the third circuit library determined for the third quantum operation 308 and a second selection 510 from the fourth circuit library determined for the the fourth quantum operation 310. The second selections 504, 506, 508, 510 from the respective circuit libraries may be performed as described above with reference to step 210.

[0192] The first circuit selection 402 and the second circuit selection 502 may form quantum circuits which approximate the state preparation circuit 302. The first circuit selection 402 may be executed on a quantum computer and a corresponding first measurement may be made of the quantum state of qubits 106 in the quantum computer following execution of the first circuit selection 402. Similarly, the second circuit selection 502 may be executed on a quantum computer and a corresponding second measurement may be made of the quantum state of qubits 106 in the quantum computer following execution of the second circuit selection 502. The first and second measurements form different measurement outcomes which may be averaged over (e.g., after multiplication by an appropriate measurement factor as described above) to determine an expectation value of an observable which results from execution of the state preparation circuit 302.

[0193] In practice many more circuit selections corresponding to the circuit selections 402, 502 of FIG. 4 and FIG. 5 may be performed to derive a plurality of quantum circuits for execution on the quantum computer to synthesise the state preparation circuit 302. Whilst the state preparation circuit 302 shown in FIG. 3 comprises four quantum operations, it will be appreciated that other state preparation circuits may include different numbers of quantum operations and may, for example, comprise more than four quantum operations.

[0194] In some examples, a state preparation circuit may include first one or more quantum operations which can be executed on the quantum computer relatively easily and / or with relatively low cost and second one or more quantum operations which are difficult, impossible and / or can only be implemented on the quantum computer at high cost. In such examples, the first one or more quantum operations may be implemented directly and the second one or more quantum operations may be synthesised using the methods described herein. A control sequence may therefore be determined which includes direct implementation of the first one or more quantum and a plurality of selections of circuits from quantum libraries for the second one or more quantum operations.

[0195] The methods described herein for determining a control sequence for execution on a quantum computer may be applied to any form of quantum operation and is not therefore limited to any specific form of quantum operation. Several examples, will now be described of applications of the methods described herein.

[0196] At least some examples of quantum computers and / or quantum computing techniques may be described as being fault tolerant. In the context of fault tolerant quantum computing (or otherwise) a universal gateset may be provided by using Clifford gates and T gates. Such a universal gateset allows any quantum circuit to be decomposed into a sequence of singlequbit Clifford and T gates. This allows any quantum operation to be performed as a sequence of single-qubit Clifford and T gates. Typically Clifford gates may be implemented on a quantum computer relatively easily and / or at relatively low cost. However, T gates may be difficult to implement and / or may only be implemented at relatively high cost. It is therefore typically desirable to limit the number of T gates which are used to decompose a quantum circuit. However, limiting the number of T gates in a decomposition of a quantum circuit may also limit the accuracy with which the quantum circuit can be decomposed and implemented leading to systematic errors.

[0197] It has been found that the methods described herein can be used to decompose any quantum circuit into a plurality of sequences of Clifford and T gates which approximate the quantum circuit and which can be used to form a circuit library as described above. For example, for a given target quantum operation ( / target many different sequences of Clifford and T gates may be determined which approximate the target quantum operation ( / target. Such sequences may, for example, be determined subject to an upper limit on the number of T gates included in each sequence. For example, a maximum number Nt of T gates which are allowed in a sequence may be set. A plurality of sequences of Clifford and T gates may then be determined, each of which approximate the target quantum operation ( / target to within a given precision 8 and each of which include no more than Nt T gates. For example, methods corresponding to those described in [1] may be used to generate suitable sequences of Clifford and T gates.

[0198] The maximum number of T gates Minay be set such that each sequence can be implemented on the quantum computer relatively easily and / or with relatively low cost. By imposing an upper limit Nt on the number of T gates this may mean that the generated sequences each differ from the target quantum operation ( / target but approximate the target quantum operation ( / target to within the given precision 8. Such a plurality of sequences of Clifford and T gates may be used to form circuits ( / / in a circuit library as was described above with reference to the method 200 of FIG. 2.

[0199] FIG. 6 is a graph illustrating members of circuit library which were determined by generating sequences of Clifford and T gates as was described above. In particular, the methods described in [1] were used to generate twenty different Clifford and T gate sequences which each approximate a target quantum operation ( / target to within a precision of e <10’2 4 and with a maximum number of T gates of Nt=36. Each generated sequence of Clifford and T gates was represented as a circuit Ui and included in a circuit library. The generated sequences Vi are illustrated in FIG. 6 in terms of their difference with respect to the target quantum operation ( / target. In particular, the y-axis in FIG. 6 represents an over-rotation with respect to the target quantum operation ( / target in the spherical angle d and the x-axis in FIG. 6 represents an over-rotation with respect to the target quantum operation ( / target in the spherical angle 9. The origin in FIG. 6 represents the target quantum operation ( / target and the dots and circles plotted in FIG. 6 each represent a generated Clifford and T gate sequence. The position of the dots and circles on the x and y-axes represent the difference of each sequence with respect to the target quantum operation tAargetin terms of their over-rotation in 0 and ¢. The grayscale dots in FIG. 6 represent sequences with non-zero weight coefficients in the solution weight vector and the shade of grey used for each dot represents the magnitude of the respective weight coefficient, with higher magnitude coefficients being represented with darker shades. The circles (with white-fill) in FIG. 6 represent sequences with weight coefficients in the solution weight vector which are equal to zero. The sequences represented by the circles in FIG. 6 will not therefore be selected in the selection process.

[0200] Additional elements of the circuit library were then generated by adding all of the single-qubit purely Clifford sequences to one of the members Ui of the circuit library to arrive at circuit library including 35 circuits th In this instance the target quantum operation ( / target is a one qubit operation represented by a 4 x 4 process matrix ( / target having a total of 16 elements. Similarly, each member of the circuit library is represented by a 4 x 4 process matrix Vi each having a total of 16 elements. As was described above the process matrices Vi can be vectorised and combined to form a library matrix R which in this example has dimensions of 35 x 16. As was explained above, this leads to an overdetermined set of equations to which a sparse solution may be found. In particular, a solution weight vector was found which provides an exact solution to equation 7 and which has a measurement overhead of IIXIIi-l=10'6 7 which compares favourably to other techniques. For example, by comparison Clifford recovery operations could be used to correct for differences between the target quantum operation ( / target and generated sequences as proposed, for example, in [2], [3] and [4], In this example, such techniques were found to introduce a measurement overhead of llklli-l=10’2 3 which is more than four orders of magnitude larger than the measurement overhead introduced using the methods described herein.

[0201] Whilst an example has been described above in which the circuit library is formed by determining a plurality of sequences of Clifford gates and T gates, in other examples other forms of gates or circuits may be used. For example, the utility of the methods described herein have also been demonstrated in examples in which the plurality of quantum circuits in the circuit library comprise a plurality of rotation gates. As is described, for example, in [5] which is incorporated herein by reference in its entirety, if quantum gates are regarded as rotations then in order to provide a universal set of gates, a rotation by any angle should be possible. However, in practice a practical quantum computer may only be capable of realising a limited set of discrete angle rotations. In particular, a practical quantum computer may only be capable of realising B bits of angular resolution such that it only supports 2B variants of a physical gate. Consequently, it may not be possible to directly implement all quantum operations. Furthermore, increasing B may generally increase the complexity of the hardware needed to realise the B bits of resolution.

[0202] In at least some examples, a circuit library may be determined as comprising a quantum circuits Ui in the form of a plurality of rotation gates as expressed by equation 12 below, in which R(B) is a continuous-angle rotation gate. Ui = Rte,) (12)

[0203] As was explained above, in practice only a discrete set of rotation angles Bi are allowed in hardware as expressed in equation 13 below. (B)

[0204] It has been found that using a circuit library comprising the available rotation gates as expressed in equations 12 and 13 above, any arbitrary continuous rotation angle can be synthesised using methods described herein (such as the method 200 described above with reference to FIG. 2). In particular, it was found that an exact synthesis of an arbitrary continuous rotation angle 9 can be realised with a sparse solution weight vector X in which the only non-zero weight coefficients yi correspond with the two nearest allowed discrete rotation angles 9k and 9k 1 to the target rotation and a polar opposite rotation angle 9k + n. This result corresponds with the analytical result arrived at in [5], Additionally the methods disclosed herein allow for finding more sparse solutions with a lower measurement overhead by allowing for approximate solutions to equation 7.

[0205] A further example of the utility of the methods described herein is to quantum optimum control methods. Such methods contemplate quantum operations at the level of external fields applied to qubits in order to realise a quantum gate. In particular, gate operations are typically realised in hardware in a quantum computer through exposure of a qubit to one or more pulses of radiation such as pulses in an electrical and / or magnetic signal, pulses of radio frequency (RF) electromagnetic radiation, pulses of microwave frequency electromagnetic radiation and / or pulses of radiation output from a laser. Several quantum optimum control techniques (e.g., [6] and [7]) have been proposed to numerically generate control fields and / or pulses which produce an optimal approximation of a target quantum operation L / target or gate. However, such techniques typically do not exactly realise the target quantum operation ( / target and often output different results when run multiple times. It has been realised that this can be utilised to build a circuit library of different optimum control solutions Vi and use the methods described herein to synthesise a target quantum operation ( / target by selecting optimum control solutions Vi from the circuit library.

[0206] To provide an illustrative example the methods described herein were used to synthesise a single qubit rotation of the form given by equation 14 below. However it will be appreciated that corresponding methods may be used for optimum control of any target quantum operation. Utarget ~ Rx (^ / 2) (14)

[0207] To illustrate the approach, we consider the Hamiltonian given in equation 15 below in which piecewise-constant, complex control parameters h(tk) set the strength of the Pauli X and Y interactions for each timestep tk and may, for example, correspond to the phase and amplitude of an applied pulse (e.g., an RF pulse). Ho in equation 15 below is a drift Hamiltonian term that cannot be controlled by hardware and which is both typically unknown and variable over time. ^(tk, d) - dHQ + Re[h(tk)]X + Im[h(tk)]Y (15)

[0208] Implementation of a piecewise constant pulse '^(^ <0 in which each piece has a length in time of At yields a unitary gate which depends on the drift d as expressed in equation 16 below. U M "" IIe \ / (16)

[0209] A fidelity F(d) given by equation 17 below can then be numerically maximised with respect to the target quantum operation ( / target by iteratively updating the shape of the piecewise constant function h(tk).

[0210] The fidelity F(d) depends on the strength of the drift term. A particular class of pulse schemes commonly referred to as broadband pulses may be used to guarantee that the fidelity F(d) is maximal for values of d within a given range.

[0211] The optimum control techniques described above were used to generate a circuit library of 100 circuits Ui to approximate a target quantum operation UtarSet(d) as given in equation 14 with minimal error for drift values within the range -2 <d <2 (using dimensionless units). FIG. 7 is a graph illustrating an approximation error of each of the 100 circuits Ui as a function of the drift d (which may be referred to as an offset). The approximation error shown on the y-axis of FIG. 7 is calculated as the distance given in equation 18 below and is shown for each generated pulse (member of the circuit library) by the solid grey lines shown in FIG. 7. ||2 (18)

[0212] A perfect approximation of the target quantum operation Utargetas given in equation 14 can only be realised if an arbitrarily large amplitude h(tk) is allowed. However, in practice quantum hardware is limited and in order to limit energy dissipation in hardware an upper limit is placed on the maximum possible amplitude. The pulses represented by solid grey lines in FIG. 7 were generated by placing an upper limit on the amplitude as Ihl <6. As a result of this limitation no single generated pulse (member of the circuit library) exactly matches the target quantum operation across the range of drift values, as can be seen by the non-zero error of the solid lines in FIG. 7. Each optimisation which generated the pulses (members of the circuit library) illustrated in FIG. 7 was initialised by a different random seed so as to generate a plurality of different solutions Vi(d) which form the circuit library.

[0213] As can be seen in FIG. 7 each member Vi yields different results for different values of the drift d. This may be accounted for by evaluating each Ui at a different value of the drift d. In particular, the drift d may be discretised into a set of q discrete values (di, d. .dq). Columns of the library matrix R (corresponding to the library matrix described above with reference to equation 6) may then be formed by stacking vectorised process matrices on top of each other for each discrete value of d as shown in equation 19 below which represents a single column of the library matrix R for a given I. [VecUdd-^X Vec[Ui(d2)] XVecXUitd^J (19)

[0214] By evaluating each member Ui of the circuit library at a plurality of values of the drift d the column dimension of the library matrix R is increased by a factor of q. To account for this the vector representing the target quantum operation is formed by stacking q copies of the vectorised process matrix of the target operation ( / target on top of each other as shown in equation 20 below. / Vec[Utarget]\ Vec{Utarget] (7 target \Vec[Utarget] / (20)

[0215] Using matrices and vectors as represented in equations 19 and 20 in equation 7 sets up a set of equations for which a solution weight vector X can be found for all values of d. In the example depicted in FIG. 7, the drift range was discretised into ^=7 points producing a library matrix R with a column dimension of 7 x 16 = 112. The circuit library was further expanded to include pulses equivalent to appending Clifford operations to the best performing pulse sequences to produce library matrix R with a row dimension of 340. A solution weight vector was then found which solves equation 7 exactly subject to the minimum condition of equation 9. This solution is shown in FIG. 7, as evaluated at each discretised value of d, by solid circles labelled 704 . This exact solution was found to introduce a measurement overhead of IIXIIi- 1.68.

[0216] As was described above, in some examples approximate solutions to 7 may be allowed to find solutions with a reduced measurement overhead. Such solutions can be found by iteratively decreasing the sparsity of the solution weight vector which monotonically decreases the error term whilst monotonically increasing the HAIli term. This approach is illustrated in the graph of FIG. 8 in which a plurality of different solution weight vectors are represented by points labelled 802. The x-axis of FIG. 8 shows the mean error (the error term described above with reference to equation 8). The y-axis of FIG. 8 shows the measurement overhead IIXIIi-1 associated with the solution. The measurement overhead associated with an exact solution is represented by the dotted line labelled 804 in FIG. 8.

[0217] As can be seen in FIG. 8, the measurement overhead can be reduced by finding sparser solutions which are more approximate (and thus less accurate). This approach can be used to find an approximate solution which has no measurement overhead (IIXIIi = 1), for which the mean error is represented by the dashed line labelled 806 in FIG. 8. Such a solution is also shown in FIG. 7, evaluated at each discretised value of d, by solid squares labelled 702. Whilst this solution is approximate, it is more accurate (at all values of d) than each of the pulses included in the circuit library and which were generated using existing optimum control techniques and without introducing any measurement overhead.

[0218] The approach described above can also be extended to other examples of target quantum operations. For example, the same approach may be used to synthesise band selective pulses, which apply a target operation Utarget within a range -B <d <B but apply an identity operation outside of this range. In such an approach the vector representing target operation can be constructed according to equation 21 below. / \ Vec[K] Vec{U^ Vec[Utarget] Vec[Utargfft] VecpI] V Vofl / (21)

[0219] Methods have been described above in the context of synthesising a target quantum operation in the context of quantum computing. However, corresponding methods may also find utility in synthesising (on average) any desired operation which is realised by generating a pulse of electromagnetic radiation. For example, as will be explained further below, methods described herein may find utility in synthesising target operations in Nuclear Magnetic Resonance (NMR) applications, Magnetic Resonance Imaging (MRI) and / or Electron Spin Resonance (ESR) applications.

[0220] In NMR applications a sample is subjected to a constant magnetic field. The constant magnetic field serves to align magnetic nuclear spins of a plurality of atomic nuclei in the sample. The sample (and atomic nuclei) is then subjected to a pulse of RF energy inducing energy state transitions in the atomic nuclei. After an RF pulse, nuclei which underwent an energy state transition may then relax to a lower energy state and emit electromagnetic radiation. This emitted radiation produces a measurable NMR signal which may be used to deduce information (such as a molecular structure and composition) about the sample. In MRI similar phenomena are used to generate images of a body (e.g., a human body) under observation.

[0221] In NMR and MRI, electromagnetic pulses (e.g., RF pulses) are generated which approximate a target operation. For example, similarly to the optimum pulse control techniques described above it may be desirable to produce a pulse of radiation represented as a unitary process matrix Ui which approximates a target operation ( / target. An example target operation may again be represented as ( / target = Rx (k / 2) as in equation 14 above.

[0222] A large number of nuclear spins may be considered as an average mixed state p which is transformed by a given pulse Ui into a state pi' as represented in equation 22 below. UiVec[p] = Vec^f) (22)

[0223] In an illustrative example, it may be assumed that the initial state p is proportional to the Pauli Z matrix (paZ). A target operation of the form ( / target = Rx (u / 2) may then transform the state p to a target state p'target which is proportional to the the Pauli Y matrix (p'target a Y). In practice a pulse ( / i is applied and the state freely evolves under the natural Hamiltonian of the spin system resulting in a time-evolved state pi'(t). The expected value of the Pauli X and Y operators may be measured which form the real and imaginary parts of a classical timedependent signal Si(t) as expressed in equation 23 below. Re[Si{ty\ - Tr\pp(t]X} + iTr\pif(t)Y]

[0224] Typically such an experiment may be repeated many (Ns) times and the resulting signals averaged to suppress random noise by a factor Ns1'2. A Fourier transform of the averaged signal may then produce an NMR spectrum.

[0225] It can be seen that the principles of NMR are therefore similar to the quantum computing optimum pulse control examples described above in which it is desired to apply a target operation in the form of a pulse of radiation a plurality of times to obtain an average measurement result. Similarly to the quantum computing examples described above, the applied pulses Ui in NMR may not exactly replicate the target operation [ / target, for example, due to the drift term dHo. Consequently the resulting state p / may not exactly match the target state p target a Y.

[0226] Given the similarities between the principles of NMR and the quantum computing optimum pulse control examples described above, corresponding methods may be used to synthesise a target operation [ / target in NMR and / or MRI applications. For example, a circuit library (which may be referred to as a pulse library in this context) may be determined comprising a plurality of pulses Ui which each approximate the target pulse [ / target. For example, the plurality of pulses [ / / which form the circuit (pulse) library may be determined using numerical optimum pulse control techniques as was described above. Values of the drift d may then be discretised across a desired range and suitable library matrix R formed as described above with reference to equation 19. Similarly, a target vector may be formed as described above with reference to equation 20. This then establishes a system of equations according to equation 7 above and a solution weight vector X may be determined subject to a minimum condition (e.g., the condition of equation 8 and / or 9). A probability distribution may then be determined from the solution weight vector X as described above with reference to equation 10. A plurality of selections of pulses [ / / may then be made from the circuit (pulse) library according to the probability distribution. Each selected pulse [ / / may be applied to the sample and a resulting signal 5 / (t) measured. As was explained above with reference to equation 11, each measurement outcome may be multiplied by a measurement factor comprising the sign of the corresponding weight coefficient Xi and a vector norm (LI norm) of the solution weight vector IIXIIi. Similarly, to the examples above, if a solution weight vector X which represents an exact solution of equation 7 is used then an average over all measurement results (after multiplication by their respective measurement factor) will produce the same result as exactly applying the target pulse [ / target (which may not be possible in practice) for each value of d for which the solution weight vector X was determined.

[0227] As was explained above, the methods described herein may be used to synthesise a target operation brought about by application of a pulse of electromagnetic radiation (e.g., an RF pulse) for applications such as NMR and / or MRI. Similarly, to the quantum computing examples which were described above (for example with reference to FIG. 3, FIG. 4 and FIG. 5) in which a sequence of quantum operations are synthesised, some more complex NMR pulse sequences may apply different target operations in a sequence. In such examples, circuit libraries may be determined for each target operation in a sequence in the same way as was described above with reference to FIG. 3. Similarly, a plurality of circuit selections may be made comprising a selection from each of the circuit libraries determined for each operation in the sequence as was described above with reference to FIG. 4 and FIG. 5.

[0228] Whilst examples, have been described herein with reference to NMR and MRI applications, corresponding principles can also be applied to ESR applications in which electron spins are considered rather than nuclear spins. Any of the features described herein may additionally or alternatively be applied to ESR applications.

[0229] Any of the features or steps described above with reference to the synthesis of a target quantum operation in the context of quantum computing may also apply to synthesis of a target operation through pulses of electromagnetic radiation (e.g., RF pulses) in the context of applications such as NMR and / or MRI.

[0230] Features, integers, characteristics or groups described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith. All of the features disclosed in this specification (including any accompanying claims, abstract and drawings), and / or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and / or steps are mutually exclusive. The invention is not restricted to the details of any foregoing embodiments. The invention extends to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed. In particular, any dependent claims may be combined with any of the independent claims and any of the other dependent claims.

[0231] Each feature disclosed in this specification (including any accompanying claims, abstract and drawings), may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise. Thus, unless expressly stated otherwise, each feature disclosed is one example only of a generic series of equivalent or similar features. The invention is not restricted to the details of any foregoing embodiments. The invention extends to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed. The claims should not be construed to cover merely the foregoing embodiments, but also any embodiments which fall within the scope of the claims. References [1] N.J. Ross and P. Selinger “Optimal ancilla-free Clifford + T approximation of z-rotations”, aeXiv prepring arXiv: 1403.2975 (2014). [2] S. Endo, S. C. Benjamin and Y. Li, “Practical Quantum Error Mitigation for NearFuture Applications”, Phys. Rev. X, 8, 031027 (2018), doi 10.1103 / PhysRevX.8.03 1027. [3] C. Piveteau, D. Sutter and S. Woerner, “Quasiprobability Decompositions with Reduced Sampling Overhead”, npj Quantum Information, 8, 12 (2022), doi 10.1038 / s41534-022-00517-3. [4] Y. Suzuki, S. Endo, K. FUjii and Y Tokunaga, “Quantum Error Mitigation as Universal Error Reduction Technique: Applications from the NISQ to the fault-tolerant quantum computing eras”, PRX Quantum 3, 010345 (2022), doi 10.1103 / PRXQuantum.3.010345. [5] B Koczor, J. Morton, and S. Benjamin, “Probabilistic Interpolation of Quantum Rotation Angles”, arXiv preprint arXiv:2305.19881 (2023). [6] J. Wershnik and E. Gross, “Quantum Optimal Control Theory”, Journal of Physics B: Atomic, Molecular and Optical Physics 40, R175 (2007). [7] C P Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S.j. Glaser, R Kosloff, S. Montangero, T. Schulte- Herbr'uggen, D. Sugny, et al., “Quantum Optimal Control in Quantum Technologies. Strategic Report on Current Status, Visions and Goals for Research in Europe”, EPJ Quantum Technology 9, 19 (2022).

Claims

1. A computer implemented method of determining a control sequence for execution on a quantum computer, the method comprising:receiving a target quantum operation for execution on the quantum computer;determining a circuit library comprising a plurality of quantum circuits, each of which approximate the target quantum operation, wherein the target quantum operation can be synthesised by a weighted summation over the plurality of circuits in the circuit library, the weighted summation being performed according to a weight vector comprising a weight coefficient corresponding to each circuit in the circuit library, wherein the plurality of different circuits in the circuit library are determined such that there are a plurality of candidate weight vectors which synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library;determining a solution weight vector from the plurality of candidate weight vectors, wherein the determined solution weight vector satisfies a minimum condition of all of the plurality of candidate weight vectors, wherein the minimum condition comprises having a minimum of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector, wherein the error term represents a difference between the target quantum operation and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the circuit library using the candidate weight vector;determining a probability associated with each quantum circuit in the circuit library, wherein each probability is proportional to the magnitude of the respective weight coefficient in the solution weight vector, wherein the probabilities associated with each quantum circuit in the circuit library form a probability distribution;performing a plurality of selections of a quantum circuit from the circuit library according to the probability distribution; anddetermining a control sequence of quantum circuits for execution on the quantum computer, wherein the control sequence comprises the plurality of selections of a quantum circuit from the circuit library.

2. The computer implemented method of claim 1, further comprising:causing the quantum computer to execute the control sequence of quantum circuits; andperforming a measurement of an outcome which results from execution of the quantum circuit for each execution of a quantum circuit in the control sequence so as to produce a plurality of measurement outcomes.

3. The computer implemented method of claim 2, further comprising determining an expectation value of an observable as an average over the plurality of measurement outcomes.

4. The computer implemented method of any one of claims 1 to 3, wherein the solution weight vector exactly synthesises the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library.

5. The computer implemented method of claim 4, wherein each of the plurality of candidate weight vectors exactly synthesises the target quantum operation when applied in a weighted summation over the plurality of circuits in the circuit library.

6. The computer implemented method of any one of claims 1 to 5, wherein the plurality of candidate weight vectors include at least one weight coefficient which is less than zero.

7. The computer implemented method of any one of claims 1 to 6, wherein the determining a circuit library comprising a plurality of quantum circuits comprises determining a plurality of quantum circuits each comprising a sequence of Clifford gates and at least one T gate and wherein each quantum circuit of the circuit library comprises a determined sequence of Clifford gates and at least one T gate.

8. The computer implemented method of any one of claims 1 to 7, wherein the determining a circuit library comprising a plurality of quantum circuits comprises determining a plurality of rotation gates, wherein each rotation gate comprises rotation by a different angle and wherein each quantum circuit of the circuit library comprises a determined rotation gate.

9. The computer implemented method of any one of claims 1 to 8, wherein the determining a circuit library comprising a plurality of quantum circuits comprises determining a plurality of different control pulses and wherein each quantum circuit of the circuit library comprises a determined control pulses.

10. The computer implemented method of any one of claims 1 to 9, wherein the number of quantum circuits included in the circuit library is greater than a total number of elements included in a process matrix representing the target quantum operation.

11. The computer implemented method of any one of claims 1 to 10, wherein the determining a circuit library comprising a plurality of quantum circuits comprises determining the plurality of quantum circuits subject to hardware criteria of the quantum computer on which the control sequence is to be executed.

12. The computer implemented method of claim 11, wherein the method further comprises receiving hardware criteria for the quantum computer on which the control sequence is to be executed and determining the plurality of quantum circuits subject to the received hardware criteria.

13. A computer implemented method of determining a combined control sequence for execution on a quantum computer, the method comprising:determining a plurality of control sequences for a plurality of target quantum operations wherein each of the plurality of control sequences is determined for each of the target quantum operations using a method according to any one of claims 1 to 12; andcombining each of the plurality of determined control sequences to determine a combined control sequence.

14. A computer implemented method of determining a combined control sequence for execution on a quantum computer, the method comprising:receiving a plurality of target quantum operations for execution on the quantum computer;for each of the target quantum operations, determining a circuit library comprising a plurality of quantum circuits, each of which approximate the target quantum operation, wherein the target quantum operations can each be synthesised by a weighted summation over the plurality of circuits in their respective circuit libraries, the weighted summation being performed according to a respective weight vector comprising a weight coefficient corresponding to each circuit in the respective circuit library, wherein the plurality of different circuits in each of the plurality of circuit libraries are determined such that, for each target quantum operation, there are a plurality of candidate weight vectors which synthesise the target quantum operation when applied in a weighted summation over the plurality of circuits in the respective circuit library;for each of the target quantum operations, determining a respective solution weight vector from the respective plurality of candidate weight vectors, wherein, for each target quantum operation, the respective determined solution weight vector satisfies a minimum condition of all of the respective plurality of candidate weight vectors, wherein the minimum condition comprises having a minimum of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector, wherein the error term represents a difference between the target quantum operation and a quantum circuit synthesised by a weighted summation over the plurality of circuits in the respective circuit library using the candidate weight vector;for each of the plurality of solution weight vectors determined for a respective target quantum operation, determining a probability associated with each quantum circuit in the respective circuit library, wherein each probability is proportional to the magnitude of the respective weight coefficient in the respective solution weight vector, wherein the probabilities associated with each quantum circuit in the circuit library form a probability distribution for the respective solution weight and the respective target quantum operation;for each of the target quantum operations, performing a plurality of selections of a quantum circuit from the respective circuit library according to the respective probability distribution; anddetermining a control sequence for execution on the quantum computer wherein the control sequence comprises the plurality of selections of a quantum circuit from the respective circuit libraries for each of the target quantum operations.

15. A computing device comprising one or more processors configured to perform a method according to any one of claims 1 to 14.

16. A computing system comprising, a computing device according to claim 15 and a quantum information processor configured to execute the control sequence.

17. A computer implemented method of determining a sequence of pulses of electromagnetic radiation to apply to a sample, the method comprising:receiving a target operation to be applied to the sample;determining a pulse library comprising a plurality of pulses of electromagnetic radiation, each of which approximate the target operation, wherein the target operation can be synthesised by a weighted summation over the plurality of pulses in the pulse library, the weighted summation being performed according to a weight vector comprising a weight coefficient corresponding to each pulse in the pulse library, wherein the plurality of different pulses in the pulse library are determined such that there are a plurality of candidate weight vectors which synthesise the target operation when applied in a weighted summation over the plurality of pulses in the pulse library;determining a solution weight vector from the plurality of candidate weight vectors, wherein the determined solution weight vector satisfies a minimum condition of all of the plurality of candidate weight vectors, wherein the minimum condition comprises having a minimum of a weighted sum of a vector norm of the candidate weight vector and an error term for the candidate weight vector, wherein the error term represents a difference betweenthe target operation and an operation synthesised by a weighted summation over the plurality of pulses in the pulse library using the candidate weight vector;determining a probability associated with each pulse in the pulse library, wherein each probability is proportional to the magnitude of the respective weight coefficient in the solution weight vector, wherein the probabilities associated with each pulse in the pulse library form a probability distribution;performing a plurality of selections of a pulse from the pulse library according to the probability distribution; anddetermining a sequence of pulses to apply to the sample, wherein the sequence of pulses comprises the plurality of selections of a pulse from the pulse library.

18. The computer implemented method of claim 17, further comprising:causing a radiation source to emit the determined sequence of pulses such that they are applied to the sample; andperforming a measurement of an outcome which results from applying the pulse to the sample for each emission of a pulse in the pulse sequence so as to produce a plurality of measurement outcomes.

19. The computer implemented method of claim 18, further comprising determining an expectation value of an observable as an average over the plurality of measurement outcomes.

20. The computer implemented method of claim 18 or 19, wherein the sample comprises a plurality of atomic nuclei and performing a measurement of an outcome which results from applying the pulse to the sample comprises measuring radiation emitted from the sample following applying the pulse to the sample.

21. The computer implemented method of claim 18, wherein the sample comprises one or more qubits in a quantum computer and performing a measurement of an outcome which results from applying the pulse to the sample comprises measuring a quantum state of the one or more qubits following applying the pulse to the sample.

22. The computer implemented method of any one of claims 17 to 21, wherein the solution weight vector exactly synthesises the target operation when applied in a weighted summation over the plurality of pulses in the pulse library.

23. The computer implemented method of any one of claims 17 to 22, wherein the plurality of candidate weight vectors include at least one weight coefficient which is less than zero.

24. The computer implemented method of any one of claims 17 to 23, wherein the number of pulses included in the pulse library is greater than a total number of elements included in a process matrix representing the target operation.

25. A computing device comprising one or more processors configured to perform a method according to any one of claims 17 to 24.

26. A system comprising, a computing device according to claim 25 and a radiation source configured to emit the determined sequence of pulses such that they are applied to the sample.Application No: GB2400854.2Examiner: Dr Stephen RichardsonClaims searched: 1-26Date of search: 29 July 2024Patents Act 1977: Search Report under Section 17Documents considered to be relevant:Category Relevant to claims Identity of document and passage or figure of particular relevance A - US 11373114 B1 (NAVEHetal.) A - US 2023 / 0289638 Al (PETERSON et al.) A - US 2020 / 0272926 Al (CEIAPLIN et al.) A - CN 117332868 A (UNIV ZHEJIANG) A - US 2022 / 0114468 Al (BRAVYI et al.) A - US 2021 / 0012233 Al (GAMBETTA et al.)Categories:X Document indicating lack of novelty or inventive step A Document indicating technological background and / or state of the art. Y Document indicating lack of inventive step if P Document published on or after the declared priority date but combined with one or more other documents of same category. before the filing date of this invention. & Member of the same patent family E Patent document published on or after, but with priority date earlier than, the filing date of this application.Field of Search:International Classification:Subclass Subgroup Valid From G06N 0010 / 80 01 / 01 / 2022 G06N 0010 / 20 01 / 01 / 2022

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