Quantum computer and method of operation thereof

Abstract

Quantum computer (10) includes: an array of qubits (30) provided with initial values depending on input data supplied in use to the quantum computer; a quantum computing arrangement (60,120) executing

Description

Conventional classical binary computers rely on Silicon-based semiconductor devices. Moreover, the conventional binary computers have become progressively 15 more powerful in recent years as integrated circuit minimum lateral feature sizes have approached 10 nm and smaller. As integrated circuit minimum lateral feature sizes are reduced to 5 nm and smaller, Silicon integrated circuits experience significant quantum effects, such that further miniaturization is not feasible. It has been appreciated that, to continue to increase computing power yet further, a 20 paradigm shift in computer technology is required. This paradigm shift has resulted in the innovation of quantum computers and optimal supercomputers, amongst other technologies. Referring to FIG. 5, a quantum computer is indicated generally by 10. The 25 quantum computer 10 includes a control unit 20 that is coupled to an array of qubits 30, for example to a given qubit 40. The qubits 30 are implemented, for example, using ion traps, Josephson junctions and so forth. The control unit 20 is coupled to a configuration unit 50 that is configured to set initial states of the array of qubits 30. Moreover, the control unit 20 is coupled to a process control 30 unit 60 that is configured to apply a sequence of quantum operations to the array of qubits 30. Furthermore, the control unit 20 is coupled to a measuring unit 70 that is configured to sense states of the array of qubits 30 after the sequence of quantum operations have been applied to the array of qubits 30. 30 04 25 In the quantum computer 10, information is represented by the array of qubits 30. A characteristic of the given qubit 40 is that it is destroyed as a measurement of its state is made using the measuring unit 70. The array of qubits 30 are thus 5 represented by a smallest unit of information storage that is feasible in the quantum computer 10, for example each qubit is represented by a single photon or a single ion. When data is represented by qubits of such small magnitude in the quantum computer 10, the quantum computer 10 becomes vulnerable to external sources of interference, as well as sources of noise that are internal to the quantum computer 10 10. In order to reduce such sources of noise from adversely affecting operation of the quantum computer 10, it is contemporary practice to cool the quantum computer 10 to temperatures approaching an absolute zero temperature, namely 0 Kelvin, namely -273.15 °C. Such low temperatures reduce, for example, a magnitude of thermal noise sources within the quantum computer 10. Noise artefacts in the array 15 of qubits 30 can also arise when the quantum computer 10 is in operation on account of residual signalling (namely, crosstalk) between qubits; such residual signalling can be a function of the sequence of quantum operations that is applied to the array of qubits 30. The quantum computer 10, when implemented in a contemporary manner as a 20 noisy intermediate scale quantum computer (NISQ computer), has in an order of 50 qubits, although larger quantum computers including 1000 qubits or more are envisaged to become available within the next 5 to 10 years. As aforementioned, the array of qubits 30 are then optionally implemented as ion traps, superconducting Josephson junction devices or similar. Each qubit is configured in 25 operation to have a quantum state between |0) and |1). Next, a method for using the quantum computer 10 to execute a quantum computation using will be described in overview with reference to FIG. 6. In a first step of executing the quantum computation on the array of qubits 10, initial values 30 are assigned to the qubits of the array of qubits 30; such initial assignment of values is achieved by using, for example, an Ansatz that is generated using a Hamiltonian via operation of the configuration unit 50. In a second step, a sequence of operations is executed on the array of qubits 30, wherein the operations can include interactions such as C-NOT gates, Hadamard transforms, rotations, entanglements and so forth, 35 wherein executions of the operations are shown in respect of a temporal abscissa 30 04 25 axis 100 in FIG. 6, namely from left to right, and wherein the qubits are represented by horizontal lines from left to right in FIG. 6, for example a line 110. The operations are, for example, executed on the qubits temporally in parallel. In a third step, when the sequence of operations has been completed, values of the array of qubits 5 20 are then read during a measurement phase using the measuring unit 70. The number of operations in the sequence is referred to a "depth" of a quantum circuit represented by FIG. 6, indicated generally by 120. As the depth of the of the quantum circuit 120 increases, there is risk of decoherence of the array of qubits 30 as well as errors occurring that are manifest as quantum noise. The quantum 10 noise therefore determines a limit to a complexity of quantum algorithm that can be executed in the quantum computer 10 on its array of qubits 30. It will be appreciated that data representative of a given real physical system, for example sensor signals provided from a plurality of sensors coupled to the given real physical system to be monitored or characterized, can be processed to form to the 15 aforesaid Ansatz to set initial states of the array of qubits 30, and the quantum circuit 120 can then be used to process the Ansatz to generate processed results representative of the given real physical system; the results are measured using the measuring unit 70. The quantum circuit 120, for example, can be configured to perform special types of signal processing, for example to compute minimum energy 20 states of the real physical system that provides insights, for example, into the stability of the real physical system, for example for safety purposes or optimal control purposes. The present disclosure is concerned with measurements implemented at the measuring unit 70. Whereas quantum errors accumulate in the array of qubits 30 as 25 quantum operation are performed upon them, further noise can arise within the measuring unit 70. The technical problem addressed by the present invention is to reduce a magnitude of the accumulated quantum errors and the further noise arising in the measuring unit 70. It will be appreciated that earlier inventions involving techniques to reduce 30 stochastic noise in sensed data representative of real physical systems have had patent rights granted in respect thereof; for example, reference is made to a granted United Kingdom patent GB2559437B, "Prenatal screening and diagnostic system and method", (Applicant: Congenica Ltd.), wherein PCR genetic readout data generated using known PCR genetic readout apparatus is subject to 30 04 25 mathematical operations to reduce stochastic noise in the readout data. In the case of this granted patent, stochastic noise reduction in sensed data was regarded by the UKIPO as being a technical effect and acceptable for patentability pursuant to section 1(2) Patents Act 1977. 5 Summary The present disclosure seeks to provide a technical solution to a technical problem of quantum noise arising within qubits when subject to a sequence of quantum operations to generate corresponding processed qubits, and noise arising in a 10 measuring unit that is used to read out values of the processed qubits of a quantum computer; the technical solution is concerned with configuring the quantum computer to perform additional operations on the processed qubits that is highly effective at reducing quantum noise arising in from fundamental physical effects occurring in the quantum computer when in use. 15 According to a first aspect, there is provided a quantum computer including: (i) an array of qubits that is configured to be provided with initial values depending on input data supplied in use to the quantum computer; (ii) a quantum computing arrangement that is configured to execute a 20 sequence of quantum operations on the array of qubits to generate corresponding processed qubits; and (iii) a measuring unit for determining states of the processed qubits to generate output data from the quantum computer, characterized in that the quantum computer is configured to use a variational 25 quantum amplitude estimation process, which includes performing a first step of applying a finite number of Grover iterations to generate a quantum circuit, the number at least partially determined according to a noise resilience of the quantum computer, and a subsequent step of performing a variational optimization to replace the generated circuit with a more compact circuit, in at least one of the 30 sequence of operations, to reduce an effect of quantum noise that arises when executing the sequence of quantum operations on the qubits and determining the states of the processed qubits. 30 04 25 The first aspect provides an invention that is of advantage in that use of the variational quantum amplitude estimation process enables quantum noise arising in the quantum computer to be suppressed. Optionally, the quantum computer is configured to receive the input data 5 representative of sensed data from a physical system, and the sequence of quantum operations is arranged to process the qubits so that the output data from the quantum computer is representative of state or characteristic of the physical system. Thus, the quantum computer is coupled to a sensor arrangement for measuring the physical system, and the quantum computer is combination with the 10 sensor arrangement form a measurement apparatus for sensing physical characteristics of the physical system. Optionally, in the quantum computer, the variational quantum amplitude estimation process is implemented as an adaptive variational amplitude estimation process that is adaptive to characteristics of the sequence of operations when applied to the 15 qubits and to characteristics of the measuring unit when determining the states of the processed qubits. More optionally, in the quantum computer, the adaptive variational amplitude estimation process includes a maximum likelihood process for learning characteristics of the sequence of operations when applied to the qubits and to characteristics of the measuring unit when determining the states of the processed 20 qubits. Yet more optionally, in the quantum computer, the maximum likelihood process is implemented as an iterative process applied to the qubits of the quantum computer. According to a second aspect, there is provided a method for operating a quantum computer, characterized in that the method includes: 25 (i) arranging for the quantum computer to include an array of qubits that is configured to be provided with initial values depending on input data supplied in use to the quantum computer; (ii) configuring a quantum computing arrangement to execute a sequence of quantum operations on the array of qubits to generate corresponding 30 processed qubits; and (iii) using a measuring unit to determine states of the processed qubits to generate output data from the quantum computer, 30 04 25 characterized in that the method further includes configuring the quantum computer to use a variational quantum amplitude estimation process, which includes performing a first step of applying a finite number of Grover iterations to generate a quantum circuit, the number at least partially determined 5 according to a noise resilience of the quantum computer, and a subsequent step of performing a variational optimization to replace the generated circuit with a more compact circuit, in at least one of the sequence of operations, to reduce an effect of quantum noise that arises when executing the sequence of quantum operations on the qubits and determining the states of the processed 10 qubits. Optionally, the method includes configuring the quantum computer to receive the input data representative of sensed data from a physical system, and arranging for the sequence of quantum operations to process the qubits so that the output data from the quantum computer is representative of state or characteristic of the physical 15 system. More optionally, the method includes implementing the variational quantum amplitude estimation process as an adaptive variational amplitude estimation process that is adaptive to characteristics of the sequence of operations when applied to the qubits and to characteristics of the measuring unit when determining the states of the processed qubits. Yet more optionally, the method includes arranging for the 20 adaptive variational amplitude estimation process to include a maximum likelihood process for learning characteristics of the sequence of operations when applied to the qubits and to characteristics of the measuring unit when determining the states of the processed qubits. Yet more optionally, in the method, the maximum likelihood process is implemented as an iterative process applied to the qubits of the quantum 25 computer. In a third aspect, embodiments of the present disclosure provide a computer program product comprising a non-transitory computer-readable storage medium having computer-readable instructions stored thereon, the computer-readable instructions being executable by a computerized device comprising processing hardware to 30 execute the method pursuant to the aforementioned second aspect. Additional aspects, advantages, features and objects of the present disclosure would be made apparent from the drawings and the detailed description of the illustrative embodiments construed in conjunction with the appended claims that 35 follow. 30 04 25 Description of the diagrams The summary above, as well as the following detailed description of illustrative embodiments, is better understood when read in conjunction with the appended 5 drawings. For the purpose of illustrating the present disclosure, exemplary constructions of the disclosure are shown in the drawings. However, the present disclosure is not limited to specific methods and apparatus disclosed herein. Moreover, those in the art will understand that the drawings are not to scale. Wherever possible, like elements have been indicated by identical numbers. 10 Embodiments of the present disclosure will now be described, by way of example only, with reference to the following patent diagrams wherein: FIGs. 1 to 4 are illustrations as described in the APPENDIX below; FIG. 5 is an illustration of a known type of quantum computer including an array of qubits; and 15 FIG. 6 is an illustration of the known quantum computer of FIG. 5 that is configured to execute a sequence of operation on the array of qubits to generate corresponding processed qubits, wherein the quantum computing includes a measuring unit that is configured to read out states of the processed qubits to provide output data from the quantum computer, and wherein 20 the qubits are susceptible to be being assigned initial states, prior to executed the sequence of operations, that are representative of sensed data from a real physical system. In the accompanying drawings, an underlined number is employed to represent an item over which the underlined number is positioned or an item to which the 25 underlined number is adjacent. When a number is non-underlined and accompanied by an associated arrow, the non-underlined number is used to identify a general item at which the arrow is pointing. Detailed description of embodiments 30 In overview, referring to FIGs. 5 and 6, there is shown the quantum circuit 120 of the quantum computer 10. The measuring unit 70 is configured to measure states of the array of qubits 30 after the sequence of operations has been executed 30 04 25 thereupon, for example by making single measurements, alternatively a plurality of measurements and then processing to try to reduce quantum noise in determinations of the states of the array of qubits 20; it is contemporary known practice to use Monte Carlo (MC) estimation of the states of the array of qubits 30. 5 In embodiments of the present disclosure, it is beneficial that the measuring unit 70 in combination with the array of qubits 30 is implemented in a particular advantageous configuration, as will next be elucidated. The measuring unit 70 is beneficially configured to implement quantum amplitude estimation (QAE); in other words, with learning and by using fixed depth quantum 10 circuit arrangements, it is feasible to implement quantum amplitude estimation (QAE) on a NISQ computer. In embodiments of the present disclosure, it is feasible to extend a speed-up of QAE to at least a further order of magnitude in precision of estimation values of the array of qubits 30 compared to other, non-variational approaches to determining the states of the array of qubits 30 after the aforesaid 15 of quantum operations have been executed. In embodiments of the present disclosure, exponential noise accumulation in the array of qubits 30 of the quantum computer 10 is suppressed; the exponential noise accumulation increases in magnitude as the depth of the quantum circuit 120 is increased. Embodiments of the present disclosure beneficially use fixed-depth 20 quantum circuits that have a fixed depth, wherein the fixed-depth quantum circuits can be fine-tuned based on characteristics of quantum hardware of the quantum computer 10. Thus, it is feasible to implement optimal hardware operations in the quantum computer 10 that are susceptible to achieve a full theoretical speed-up when executing operations on the array of qubits 30; such a benefit of achieved 25 by learning parameters of quantum gates that are applied to the array of qubits 30, despite sub-optimal quantum hardware control that may arise during operation of the control unit 20. In embodiments of the present disclosure, experimental noise arising in quantum 30 hardware of the quantum computer 10 are capable of being suppressed by using constant-depth quantum circuits when implementing the quantum circuit 120. Moreover, in the embodiments, there is used a learning strategy that mitigates sub-optimal qubit control, wherein quantum noise otherwise grows exponentially as function of quantum circuit depth, wherein the quantum noise can potentially cause 30 04 25 decoherence of the array of qubits 30, namely potentially a complete loss of quantum properties of the array of qubits 30. Furthermore, embodiments of the present disclosure improve sub-optimal control arising from operation of the control unit 20 that would otherwise reduce a speed-up that is feasible based on theoretical noise 5 limits. It will be appreciated that, in embodiments of the present disclosure, the aforesaid quantum computer 10 can be used to process data generated by sensors that is representative of one or more real physical systems; for example, the data can be: (i) satellite data of captured images of the Earth's surface; 10 (ii) aircraft movement data; (iii) projectile movement data when the quantum computer 10 is included as part of a military defence system, and so forth. However, in embodiments of the present disclosure, it will be appreciated that the 15 quantum computer 10 is also susceptible to being used in use applications of a more abstract nature (that may be susceptible to patent protection in the USA pursuant to 35 USC 101), such as: (i) machine learning, for example for training generative models for simulating real physical systems; 20 (ii) in finance, for performing risk estimation, for pricing portfolios, for performing value adjustments; and (iii) for performing quantum chemistry simulations, for example for estimating observables in quantum simulations. Embodiments of the present disclosure, when applied to the hardware of the quantum 25 computer 10, beneficially use adaptive variational quantum amplitude estimation (AVQAE) that is a special case of variational quantum amplitude estimation (VQAE). Optionally, the AVQAE makes use of a lognormal distribution, that potentially provides an order of magnitude reduction in quantum noise arising in the quantum computer 10. Moreover, embodiments of the present disclosure most effectively when the 30 quantum computer 10 is implemented as an ion-trap quantum computer, for example as manufactured by Honeywell, that benefits from a high connectively 30 04 25 between qubits of the array of qubits 30. Furthermore, such an ion-trap quantum computer allows deeper quantum circuits 120 to be used before running an optimisation step; such a manner of use potentially enables the quantum noise reduction to be improved a further order of magnitude in comparison to the 5 quantum computer 10 implemented in a known contemporary manner. However, embodiments of the present disclosure can be used with other type of quantum computers, for example superconducting Josephson junction quantum computers as manufactured by IBM Corp. USA. Next, a simple definition for VQAE and AVQAE will be provided. 10 Variational quantum amplitude estimation (VQAE) is a combination of two concepts: firstly, an original quantum amplitude estimation (QAE) algorithm and, secondly, a variational quantum algorithmic (VQA) method. The original QAE algorithm is susceptible to providing an advantage of a quadratic speedup over classical Monte Carlo (MC) sampling. However, this quantum algorithm has 15 quantum hardware requirements that exceed capabilities of current NISQ computers. The VQA approach has established itself as a powerful set of tools to make the most of current quantum devices. In the present disclosure, an aim of synergistically combining QAE and VQA is to obtain a quantum algorithm for current NISQ computing technologies that achieves a quantum advantage over MC 20 sampling. To make the most of current NISQ computers, it is crucial to reduce the noise arising when executing quantum calculations. To this end, VQAE starts from the aforesaid original QAE algorithm and, firstly, replaces the deep and noisy quantum circuits required for the quantum Fourier transform by a classical inference procedure based 25 on maximum likelihood estimation. Then, VQAE iterates the following two steps: (1) a first step of applying a finite number of Grover iterations, during which the depth of the quantum circuit 120 and its associated noise grows, to estimate an amplitude of the noise; and (2) a step of performing a variational optimization to reduce the circuit depth and 30 noise of the quantum circuit 120. 30 04 25 The number of Grover iterations in the step (1) is chosen to be as large as possible according to a noise resilience of the available quantum hardware of the quantum computer 10. Moreover, the step (2) replaces the deep and noisy circuit (denoted "120A") obtained after several Grover iterations by a more compact and less noisy 5 circuit (denoted "120B") that represents a same quantum state of the circuit 120A. Adaptive VQAE (AVQAE) is a special, improved case of VQAE that has reduced computational requirements when executed on the quantum computer 10. A naive VQAE implementation has a computational cost that exceeds a computation cost of a corresponding classical MC calculation. Such an increased computational cost arises 10 due to the VQA part in VQAE; in other words, it is a cost of implementing the variational optimization. To reduce this cost, AVQAE adapts the problem and rescales the amplitude to be estimated. Then, every variational optimization only needs to correct the current quantum state for a small deviation from the previous state. Such an approach can significantly reduce the computational cost, below the one of classical 15 MC sampling, so that AVQAE can obtain a quantum advantage more easily than a naive VQAE realization. After having completed the step (1) and (2) of AVQAE, there is thereby obtained an empirical estimation of the expectation value that goes beyond what hardware noise allows standard QAE and with the same polynomial quantum advantage. The 20 steps (1) and (2) can be repeated multiple times, namely in an iterative manner, for further refining the expected value estimation with quantum advantage. A more detailed description of operation of VQAE and AVQAE will be described next with reference to the APPENDIX included below. Whereas conventional quantum computers use Monte Carlo amplitude estimation when implementing the aforesaid 25 measuring unit 70 to determine states of the array of qubits 30 after executing of the quantum circuit 120 in the quantum computer 10, the quantum computer 10, when implemented pursuant to the present disclosure, implements a VQAE in the quantum circuit 120, wherein the VQAE is based on a maximum likelihood framework as proposed in reference

[15] with a linearly incremental sequence of iterations {m = 30 1, 2, 3, ..., M}, wherein m is an integer representative of iteration number and M is a maximum number of iterations. Such a framework has an advantage that the depth of the quantum circuit 120 implementing the state lxm)n+1 = 9“lxo>„+i increases linearly with an increase in the integer m. However, to prevent the circuit depth of the 30 04 25 quantum circuit 120 increasing indefinitely, there is included, as an innovation, a variational step during which the state lxm>n+i of depth O(m) is approximated by a quantum state of a constant depth 0(1). Beneficially, the variational approximation is performed every fc-th power of Q,with 0 <k <M. For all other iterations, a 5 correspond power of Q is applied to the variational state. In the APPENDIX, results are presented in respect of Algorithm 1. The maximum likelihood post-processing

[15] consists in maximizing the likelihood function L({h]m,x) = HmLm(hm,x) with 10 Lm(hm,x) = [sln2{(2m+T)x^]hm[cos2^(2m +l)x)]h hm, 12 So that the estimate of the phase 0 becomes 6 = arg maxx(lnL (_{hm},x)~). 13 15 Our implementations of the maximum likelihood estimation use h = 2 x 103 samples. The minimization of L0hm},x) is accomplished by means of a brute-force search algorithm that uses 5 x 103 grid points. We variationally approximate states Qfcl4>Jn+i by minimizing || l4>rar(A))n+1 ||2 which is equivalent to maximizing the objective function 20 TOC) = 14 With respect to the variational parameters X. The depth of the quantum circuits for TO] is 2k + 2. In general the variational quantum state |(K,ar(A))n+1 is a parameterized 25 quantum circuit (PQC) l^var(A))n+1 — Uvar^] — J '' J 15 j where Gj are Hermitian operators acting on the (n + 1)-qubit register and |0init)n+i is 30 some initial state. For our purposes, we are interested in hardware-efficient quantum circuits that produce real-valued quantum states. We use the PQC shown in Fig. 2 that is composed of d layers with 15 parameterized single-qubit rotation gates and 10 CNOT gates per layer. 30 04 25 One single variational update of a PQC consists of ns sweeps over all circuit parameters, during which all parameters are updated simultaneously. To perform the optimization, it is convenient to introduce a coordinate-wise version of Eq. (14) for the j-th parameter 5 = f 2} ■ ■" >— 1 >4" 19 1 ■ * ). 16 The optimization of the parameterized state in Eq. (15) can then be performed via a particle swarm approach [27, 28], the coordinate-wise update [29-33], or gradient 10 based methods with the parameter-shift rule [34-39] = + tt / 4) - 17 We obtained the best results using the gradient based approach with the Adam 15 optimizer

[40] , Therefore this technique is being used throughout this article for the computation of all results. Each gradient calculation requires two evaluations of the coordinate-wise objective function ^(^±^ / 4). On a quantum computer, / } can be determined via the Hadamard test

[41] , In our numerical simulations, we emulate the measurement of the Hadamard circuit by first evaluating the exact value of and 20 then sampling it using a binomial distribution with the probability (1 + / )) / 2 and nf independent Bernoulli trials

[26] . The variational approximation step significantly affects the total number of queries Nq used by VQAE. In MLAE with a linearly incremental sequence, the total number of queries is equal to 25 Nq = h(2m + 1) = hM(M + 2), 18 m=l where 2m+ 1 is the depth of the quantum circuit encoding \xm)n+i = Qm\Xo)n+i- In VQAE, the total number of queries is composed of two separate contributions. The 30 first one accounts for the sampling of the quantum circuits \xm)n+i and we denote it by Nsamp. The second contribution corresponds to the variational approximation cost, which we denote by Nvar. We assume that the number of queries required per variational approximation is independent of the iteration numberm and changes only 30 04 25 as a function of the desired variational error as well as the depth of the circuit for the objective function, which is equal to 2k + 2. For convenience, we denote the cost of a single variational update for k = 1 by Nvar / r, so that the total number of variational queries reads 5 Nvar = Nvar / 1(2k + 2)[M / ki~0(k[M / kl). 19 The number of sampling queries is equal to 10 Nsamp = hk(k + 2)[M / k\ + h(M%kXM%k + 2), 20 Where H denotes the floor function the % is the modulo operation. In the limit M » fc, and N..amp-0(kM). Note that both contributions scale like O(M) which is quadratically better than the scaling O(M2) of MLAE in Eq. (18). 15 The aforesaid VQAE is susceptible to being implemented in a naive manner in the quantum computer 10. In our naive implementation of the VQAE algorithm, the initial state of the PQC in Eq. 20 (15) and Fig. 2 'S I0imt)n+1 |0)n+l- Let us first explore the expressive power of the corresponding variational state ^varW)- To this end, we perform amplitude amplifications followed by variational approximations of the resulting state with k = 1 and M = 50. To evaluate the quality of the variational approximation, we calculate the infidelity 25 Im = 1 - ($i\n+iQj\Xm}n+1>m = i-k+j, 21 where for k = 1 we have j = 0 and i = m. Figure 3(a) shows the results of such calculations performed for different depths d of the PQC form = 10 and Fig. 3(b) shows 30 the infidelity as a function of m for d = 4. We observe that the accuracy of the variational ansatz increases with the depth and saturates at d « 4. The infidelity increases linearly with m. This behaviour is seen for all probability distributions considered. 30 04 25 Next, we present the amplitude estimation results of naive VQAE. Figure 4 shows the convergence of 89 as a function of Nq, under the assumption that Nvar / 1 = 0 and k = 1. The resulting error is compared with the one of classical MC sampling which scales like 89~o(Nq''2^ and the one of MLAE which scales like 89~O^N~3 / 4y Interestingly, we 5 find that the convergence of 89 changes as a function of M. For small values of M, it follows the ideal VQAE scaling 86-O^N,^2^ as if the variational approximation is performed without error. We emphasize that this scaling is cubically better than the one of classical MC sampling. The second convergence regime is observed for larger values of M. In this regime, the error follows the MC scaling with 89-0(^^2^. One 10 can understand this new scaling in terms of a one-dimensional classical random walk with an average step of length 29 and some finite variance induced by variational approximations. Provided that the variance is much smaller than 9, after M~Nq steps the total distance of the random walk converges to the value 2M9 with a relative error scaling as o(M-1 / 2). 15 Finally, we take into account the cost of the variational approximation, to obtain a more complete assessment of the algorithmic performance of naive VQAE. To estimate the cost of a single variational update, we use Nvar / 1 = 2nfnsnp, where np is the number of parameters of a PQC and the factor 2 comes from the two evaluations of the objective function in Eq. (17). For the PQC in Fig. 2 with d = 4, the number of 20 parameters is np = 60. Additionally, we choose nf~np~100 so that Nvar / l~1.2 x 106 and Nvar~4.8x 106M. This large variational cost is the dominant part in the calculation of the total number of queries Nq. Ultimately, it leads to a performance of naive VQAE that is worse than the one of classical MC sampling. Reducing any of nf, ns or np decreases the variational cost but also increases the variational error which then leads 25 to a worse final amplitude estimation error. By configuring the quantum circuit 120 to implement VQAE therein to assist with determining states of the array of qubits 30 when measured using the measuring unit 70, the state of the array of qubits 30 can be determined with less quantum noise. 30 Often, the noise reduction can be an order of magnitude in comparison to Monte Carlo amplitude estimation. In embodiments of the present disclosure, the quantum computer 10 configured to implement VQAE can be further improved to reduce quantum noise arising therein by 30 04 25 using adaptive variational quantum amplitude estimation (AVQAE); such AVQAE makes use of adaptive rescaling that is commonly used in Monte Carlo simulations [see reference 42], but has not hitherto been used for quantum noise reduction in the quantum computer 10. 5 To introduce the adaptive VQAE algorithm, we first note that the amplitude a = Ep[f] is linear in f, meaning that rescaling the function f with a proportionality constant r also rescales the amplitude a: 10 a' = Ep[f’] = Ep[rf] = ra. 22 The rescaled function f can then be used to encode a new quantum state |^)n+1, provided that f'(x) <1 for all x, which is required for the successful state preparation via Eq. (7). Next, a new Grover operator Q' can be defined and amplitude estimation 15 can be performed to estimate amplitude a'. To proceed further, we make the observation that after k iterations, the overlap between the states |%o)n+1 and \x'k}n+i = Q'k lzo)n+i is equal to cos(2k9'). Hence, one can always find a commensurate amplitude 0 <a' <1 and a phase 9' so that 20 a’ = sir2 (9'), 9' = nl / k, I G Z, 23 and the overlap between the corresponding states |%o>«+i and lx'k)n+i is equal to one. Combining these two observations, we conclude that for a proper choice of the renormalization factor r satisfying rf(x) <1 for all x, it is possible to rescale the 25 function f such that the / c-th power of the corresponding Grover operator becomes the identity operator Q'k =J. Finding the exact renormalization factor r requires exact knowledge of the initial amplitude a. However, as we show in the following, a loose estimate ah obtained from a moderate number of MC samples of the initial state U0>n+1 is sufficient. Assuming 30 that such a loose amplitude estimate is provided, the renormalization factor can then be approximately expressed as rL =a' / ax, with a! defined as in Eq. (23). As a result, the overlap between the states l / ok+i and lx(.)n+1 becomes (k59'Y Uo'ln+ilxDn+1 = cos{9' - k89') «1------, 24 35 30 04 25 where 60' = 0' - arcsin ^ / 77^. We see that for small deviations, the overlap decreases quadratically with k. Next, we use the VQAE algorithm to estimate the amplitude a' by means of the Grover operator Q' and the initial state \x'o)n+i- The variational approximation is performed every k-th step, when the overlap between | / o)n+1 and 5 I4)n+1 reaches maximum. Additionally, we assume that the PQC has the initial state \(pimt)n+i = lzo)n+i so that the variational quantum state of Eq. (15) reads 10 var On+1 = ‘^var (^)IZo)n+l- 25 10 Having the PQC initialized to identity at the beginning of each optimization step, the only role of the variational quantum circuit is to correct the deviation of Eq. (24) originating from the imprecise value of the renormalization constant Finally, at the end of the calculation, an estimation of the phase 0' corresponding to the renormalized amplitude a' = sin2 6' is obtained. To go back to the original formulation of the problem 15 and compare the results, we use the inverse transformation 6 = arcsin4d, a = a' / rb 26 where the renormalization constant rt has to be exactly the same as the one used for 20 the function rescaling. This last step concludes the adaptive VQAE algorithm. We analyze the performance of the adaptive VQAE algorithm with a simplified variational ansatz consisting of only six single-qubit rotation gates and four CNOT gates, as shown in Fig. 2 in dark blue color. This simplified ansatz has only six parameters in total, which significantly reduces the number of variational queries as well as the effects of 25 the noise due to finite sampling. We determine the loose estimate of the amplitude ar via 5 x 105 MC samples. As a result, much smaller values of infidelity are achieved for nf being an order of magnitude smaller than in our naive VQAE computations. We also note that for smaller values of a, the initial MC estimation of a' gets worse and, as a consequence, more sweeps are required to ensure the convergence of the variational 30 ansatz. Our results for adaptive VQAE are presented in Fig. 1, where we show the convergence of 86 as a function of Nq for k = 10. The simulations use Adam with the initial learning rate / ? = 10-3, nf = 100, ns = 100 and np = 6, resulting in Nvar / l = 2nfnsnp = 1.2 x 10s. As in Fig. 4, we compare with the classical MC scaling 66~O^Nq1 / 2^ and the MLAE scaling 35 66~O^Nq3 / 4y The major difference of the adaptive VQAE as compared to all previously 30 04 25 studied methods is a large starting cost which corresponds to the amount MC samples required for the evaluation of This starting cost, however, represents only an additive contribution to Nq and, hence, is insignificant in the regime of our interest when the number of queries gets large. Additionally, we find that, thanks to a 5 significant improvement of the number of query calls and overall precision of the variational state, the resulting final error 80 of the adaptive VQAE algorithm surpasses the classical MC error, therefore leading to a genuine quantum advantage. Interestingly, we observe that in the regime of small k, the performance of the adaptive VQAE algorithm decreases. This has several reasons. Firstly, the precision of 10 the maximum likelihood estimation decreases when the angle 6' = nl / k (where I = 1 in our case) becomes larger than n / 4, i.e. for k <4. Hence, to perform an estimation with such small values of k, a different statistical inference technique has to be considered. Secondly, for small values of k, the rescaling factor can become much larger than one and then leads to more efficient classical MC sampling. In calculations 15 with C = 0.1, this leads to a loss of quantum advantage for k = 5, corresponding to r &7.508. The quantum advantage of the adaptive VQAE algorithm holds true for other cases with k >5. By using the AVQAE, it is feasible to configure the quantum circuit 120 in 20 combination to using the measuring unit 70 so that the sequence of quantum operations applied to the array of qubits 30 generates qubit output results from the measuring unit 70 that are less noisy in respect of quantum noise in comparison to merely using known Monte Carlo amplitude estimation to implement the measuring unit 70. By making the VQAE adaptive, up to an order of magnitude 25 mitigation in quantum noise generated in the quantum computer 10 is feasible to comparison merely to using VQAE. VQAE is capable of providing up to an order of magnitude reduction in quantum noise relative to using known Monte Carlo amplitude estimation. An implementation of a VQAE quantum circuit implementation of the quantum circuit 30 120 is illustrated in FIG. 2, for example in relation to suitable establishment of the Ansatz for the circuit 120. The quantum circuit in FIG. 2 includes a configuration of rotation operators denoted by "R", and C-NOT gates denoted by circles with "+" symbols. 30 04 25 Modifications to embodiments of the present disclosure described in the foregoing are possible without departing from the scope of the present disclosure as defined by the accompanying claims. Expressions such as "including", "comprising", "incorporating", "consisting of", "have", "is" used to describe and claim the present invention are 5 intended to be construed in a non-exclusive manner, namely allowing for items, components or elements not explicitly described also to be present. Reference to the singular is also to be construed to relate to the plural; as an example, "at least one or indicates "one or in an example, and "a plurality or in another example; moreover, "two or, and similarly "one or more" are to be construed in a likewise 10 manner. Numerals included within parentheses in the accompanying claims are intended to assist understanding of the claims and should not be construed in any way to limit subject matter claimed by these claims. The phrases "in an embodiment", "according to an embodiment" and the like generally mean the particular feature, structure, or characteristic following the phrase is 15 included in at least one embodiment of the present disclosure, and may be included in more than one embodiment of the present disclosure. Importantly, such phrases do not necessarily refer to the same embodiment. 30 04 25 APPENDIX Scientific paper: "VARIATIONAL QUANTUM AMPLITUDE ESTIMATION" Authors: Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini and Michael 5 Lubasch The authors are associated with Cambridge Quantum Computing Ltd, London SW1E 6DR, United Kingdom To enhance performance of quantum computer hardware by reducing quantum noise arising in operation in the hardware, we propose to perform amplitude 10 estimation. The amplitude estimation is implemented with the help of constantdepth quantum circuits that are executed on the hardware, wherein the constantdepth quantum circuits variationally approximate states during amplitude amplification. In the context of Monte Carlo integration used in quantum computing hardware, we numerically show that shallow quantum circuits can accurately 15 approximate many amplitude amplification steps. In aforesaid embodiments of the present disclosure, we combine the variational approach with maximum likelihood amplitude estimation [Y. Suzuki et al., Quantum Inf. Process. 19, 75 (2020)] in variational quantum amplitude estimation (VQAE); VQAE is this an inventive concept of the present disclosure. Beneficially, VQAE can exhibit a cubic quantum speedup 20 over classic known Monte Carlo (MC) sampling if the variational cost is ignored. If this variational cost is taken into account, VQAE typically has larger computations requirements than classic Monte Carlo sampling. In embodiments of the present disclosure, we propose adaptive VQAE and numerically show that it can outperform classical Monte Carlo sampling. 30 04 25 Amplitude estimation [1] is a powerful algorithm that can achieve a quadratic quantum speedup over classical Monte Carlo (MC) methods [2]. It has a wide range of 5 applications, e.g. in quantum chemistry [3,4], machine learning [5-7], and finance [8-10] where it can help with tasks such as risk analysis [11, 12] and the pricing of financial derivatives [13, 14]. The original amplitude estimation procedure [1] has hardware requirements that are challenging for current quantum devices and, therefore, reducing these requirements 10 is currently an active area of research. Crucial breakthroughs were obtained in recent proposals which succeeded in replacing the hardware-intensive components of traditional amplitude estimation - controlled multi-qubit gates and quantum Fourier transform - by classical post-processing [15-18]. Alternatively, one can systematically reduce the circuit depth by interpolating between classical MC methods and amplitude 15 estimation

[19] , Additionally, classical pre-processing can replace costly quantum arithmetic

[20] . In this article, we address the question whether the quantum computation requirements for amplitude estimation can be further decreased by making use of variational quantum algorithms [21-23], We present variational quantum amplitude 20 estimation (VQAE) in which the depth of the entire quantum circuit is always kept below a desired maximum value by means of variational optimization. VQAE is based on maximum likelihood amplitude estimation (MLAE)

[15] . We present a naive and an adaptive VQAE algorithm. Adaptive VQAE rescales the amplitude to reduce the cost of the variational optimization. The advantage of VQAE over MLAE is that the maximum 25 circuit depth of VQAE is independent of the total number of MLAE steps, whereas in MLAE this depth grows linearly with the number of MLAE steps. The advantage of VQAE over classical MC sampling is that VQAE can have a lower computation cost. Figure 1 shows that, for the problems considered here, VQAE outperforms classical 30 MC sampling and additionally keeps the overall circuit depth below a fixed value. This article is organized as follows. Firstly, in Section II, we define the problems considered here and explain the original quantum algorithm for amplitude estimation as well as the classical MC approach. Then, in Section III we present our variational methods, study variational errors of constant-depth quantum circuits, and develop 35 naive and adaptive VQAE. We conclude this article and discuss potential next steps in Section IV. II. BACKGROUND In this section, we first define the problem that we are interested in. Next, we explain quantum amplitude estimation and classical MC sampling. A. Problem definition Throughout this article, we focus on the calculation of expectation values Epff]=^p(x)f(x) 1 X Where the sum runs over 2n equidistant values of x e [0,1), p(x) represents a probability distribution and f(x) a real-valued function. Here n is the qubit count of the wave function that encodes p(x) and f(x) in its amplitudes. We consider three probability distributions: a Gaussian , A 1 O - |02 pGW = —exp--’ Cauchy-Lorentz r a Q ^c_Ldx-py + ^ And log-normal distribution P;-n(x) 1 Qn(c0 + ctx) - n)2 + 2o2 The normalization constants Nc, Mc-L and W;_n are chosen so that £xp(x) = 1. We choose the following parameters for our analysis. We fix the total number of qubits encoding f(x) and p(x') to n = 5. In our calculations with the Gaussian and Cauchy-Lorentz distributions, we use p = 0.5 and a = 0.1. In our calculations with the lognormal distribution, we use c0 = 0, = 10, p = 1.5, and a = 0.2. For the function f(x), we use with some C >0. For this choice of parameters, the expectation value (1) is approximately Ep\f} « 0.5C for all chosen distributions. B. Quantum amplitude estimation 5 Let us present a way to encode the solution to (1) on a quantum computer. We assume / (x)and p(x) are functions that map [0,1) to [0,1]. We consider real numbers xe [0,1) that satisfy 6 10 And them we identify with n-bit strings {x£,i = 1,2, ....,nJ. Each bit string shall correspond to a quantum state |x) = \xr,x2,...,xn) in a computation basis of an n-qubit register. Additionally we have a quantum circuit JI that acts on a register of n + 1 qubits and produces a state lx0>n+i = <^1 0)n+i such that 15 IXo)n+l Vi 4^bad)n 10) + 4^good)n 11) 7 30 04 25 20 Here |Q / bad>n and |ipgOod)n are two normalized quantum states of a n-qubit register which is connected to one additional ancilla qubit. We define the good state |lPgood)n = So that a = Ep[f] of Eq. (1) coincides with the probability of measuring the ancilla qubit in the state 11>. To determine a, the amplitude estimation algorithm uses the Grover operator Q = —g00d [1, 24, 25] where 1 — 2\Xo)n+l(Xo\n¥l — 1 — 2c / ?|0)n+1(0|„+1c / l',' •Rgood — 1 — 2 |V[good)nl l){*Pgood ln(l| are reflections in a two-dimensional subspace Hx spanned by states \ipbad)n 1°) and 30 UPgoodtn |1>- We define a = sin2(6) and explicitly write out the action of Q: I \ =0^1 \ = C0S^2m + ^eWbad}n 1°) V lZo / n+i +sin[(2m+ 1)0] |^ood>n |1). 10 Therefore, the subspace Hx is stable under the action of Q is to rotate by an angle 20. The original amplitude estimation algorithm then uses quantum phase estimation to find the eigenvalues of Q equal to exp(±2i0) and provides an estimate of a with an error 30 04 25 with a probability of at least 8 / tt2 [1]. Traditional amplitude estimation has high requirements on quantum hardware because it uses quantum phase estimation. This algorithm needs the quantum Fourier 10 transform and multiple controlled Qm operations where {m = 1,2,4,...,2M}. The depth of the corresponding quantum circuit is mostly determined by the depth of the last controlled Qm operator for which m = 2M. In general, the total circuit depth scales like the total number of queries O(l / e)

[19] , One proposal is MLAE

[15] in which one combines measurements of the states |xm)n=1 with a maximum likelihood estimation 15 of a. For an exponential schedule {m = 1,2,3,...,M} increases the query cost to ^-^(l / ^’*1) for quantum circuits of reduced depth 0(1 / 61^). These algorithms are controlled by an external parameter which allows on to interpolate between the quantum regime at p = 0 and the classical MC regime at p = 1. 20 C. Classical MC sampling We perform classical MC sampling in the following way. We sample from the state lxo)n+i of Eq. (7) and measure the ancilla qubit. We compute a as the relative frequency of measuring the ancilla qubit in the state |1). This calculation of a has the 25 error e = 7^(1 - a) / Nq so that the total number of queries required for a certain error e is / Vc,-0(1 / 62)

[26] , Comparing this query cost with the previous ones, we find that traditional amplitude estimation as well as MLAE with exponential schedule achieve a quadratic quantum speedup over classical MC sampling. Both MLAE with linear schedule and the 30 algorithms in

[19] obtain a reduced quantum speedup. Note that, throughout this article, the query complexity is defined in terms of c / Z operators, with two applications of JI required per application of Q, see Eq. (9). Also, the depth of quantum circuits is measured in units of JI. 30 04 25 III. VARIATIONAL ALGORITHMS Here we present our variational algorithms. We first explain the general VQAE formalism, then our naive implementation, and finally the adaptive VQAE approach. 5 A. General formalism The VQAE algorithm is based on the maximum likelihood framework of Ref.

[15] with linearly incremental sequence {m = 1,2,3,...,M}. In this framework, the depth of the 10 quantum circuit implementing the state lxm)n+i = 2mlxo)n+i increases linearly with m. To prevent the circuit depth from increasing indefinitely, we add to this framework a variational step during which the quantum state lxm)n+i of depth O(m) is approximated by a quantum state of a constant depth 0(1). We perform the variational approximation every / c-th power of Q, with 0 <k <M. For all the other iterations, we 15 simply apply the corresponding power of Q to the variational state. This results in Algorithm 1 presented below. Algorithm 1 Variational quantum amplitude estimation Require: I4»i=o>„+1 = lxo)n+i< Q for 0 <i <[M / ki do for 0 <j <k do Sample the circuit (V'l^Jn+i and collect h samples. Save the number of times the ancilla qubit is |1) in A variable h.m, with m = i - k +j end for if i * [M / ki then Perform the variational approximation l^i+l^n+l w 2 end if end for Use {hm} to carry out the maximum likelihood estimation The maximum likelihood post-processing

[15] consists in maximizing the likelihood 20 function L({hm],x) = HmLm(hm,x) with Lm(hm,x) = [sin2((2m + l)x)]hm[cos2((2m + l)x)]h hm, 12 30 04 25 So that the estimate of the phase 0 becomes 0 = arg max^ / nL^J.x)). 13 5 Our implementations of the maximum likelihood estimation use h = 2 x 103 samples. The minimization of L({hm},x) is accomplished by means of a brute-force search algorithm that uses 5 x io3 grid points. We variationally approximate states Qfcl4>Jn+i by minimizing || |c|)var(A))n+1 -Qk\^i} ||2 which is equivalent to maximizing the objective function 10 W = H<4Wa)ln+12kk>n+1) 14 With respect to the variational parameters A. The depth of the quantum circuits for T(A) is 2k + 2. In general the variational quantum state |<Kar(K»n+i is a parameterized 15 quantum circuit (PQC) l^varW^n+l = = j |e 15 J where Gj are Hermitian operators acting on the (n + 1)-qubit register and |0init)n+1 is some initial state. For our purposes, we are interested in hardware-efficient quantum 20 circuits that produce real-valued quantum states. We use the PQC shown in Fig. 2 that is composed of d layers with 15 parameterized single-qubit rotation gates and 10 CNOT gates per layer. One single variational update of a PQC consists of ns sweeps over all circuit parameters, during which all parameters are updated simultaneously. To perform the 25 optimization, it is convenient to introduce a coordinate-wise version of Eq. (14) for the j-th parameter fj(x) = ^F(Ai,A2, ..., x,Aj+1, ...). 16 The optimization of the parameterized state in Eq. (15) can then be performed via a 30 particle swarm approach [27, 28], the coordinate-wise update [29-33], or gradient based methods with the parameter-shift rule [34-39] = + n / 4) n / 4). 17 30 04 25 We obtained the best results using the gradient based approach with the Adam optimizer

[40] , Therefore this technique is being used throughout this article for the computation of all results. Each gradient calculation requires two evaluations of the coordinate-wise objective function ^(^±^ / 4). On a quantum computer, fj can be 5 determined via the Hadamard test

[41] . In our numerical simulations, we emulate the measurement of the Hadamard circuit by first evaluating the exact value of f, and then sampling it using a binomial distribution with the probability (1 + / )) / 2 and nf independent Bernoulli trials

[26] . The variational approximation step significantly affects the total number of queries Nq 10 used by VQAE. In MLAE with a linearly incremental sequence, the total number of queries is equal to M Nq = h{2m + 1) = hM(M + 2), 18 m=l 15 where 2m+ 1 is the depth of the quantum circuit encoding = 2wlxo)n+i- In VQAE, the total number of queries is composed of two separate contributions. The first one accounts for the sampling of the quantum circuits \xm)n+i and we denote it by Nsamp. The second contribution corresponds to the variational approximation cost, which we denote by Nvar. We assume that the number of queries required per 20 variational approximation is independent of the iteration numberm and changes only as a function of the desired variational error as well as the depth of the circuit for the objective function, which is equal to 2k + 2. For convenience, we denote the cost of a single variational update for k = 1 by Nvar / r, so that the total number of variational queries reads 25 Nvar = Nvar / 1(2k + 2)[M / k\~O(k[M / k\), 19 The number of sampling queries is equal to 30 Nsamp = hk(k + 2)(M / k\ + h(M%k)(M%k + 2), 20 Where [ J denotes the floor function the % is the modulo operation. In the limit M » k, Nvar~0(M) and Nsamp~0(kM). Note that both contributions scale like O(M) which is quadratically better than the scaling O(M2) of MLAE in Eq. (18). 30 04 25 B. Naive VQAE In our naive implementation of the VQAE algorithm, the initial state of the PQC in Eq. (15) and Fig. 2 is ^lnit)n+1 = |0)n+1. 5 Let us first explore the expressive power of the corresponding variational state \4>varW)- To this end, we perform amplitude amplifications followed by variational approximations of the resulting state with k = 1 and M = 50. To evaluate the quality of the variational approximation, we calculate the infidelity 10 = 1 - \n+iQj\Xm)n+1^ = i -k+j, 21 where for k = 1 we have j = 0 and i = m. Figure 3(a) shows the results of such calculations performed for different depths d of the PQC form = 10 and Fig. 3(b) shows the infidelity as a function of m for d = 4. We observe that the accuracy of the 15 variational ansatz increases with the depth and saturates at d « 4. The infidelity increases linearly with m. This behaviour is seen for all probability distributions considered. Next, we present the amplitude estimation results of naive VQAE. Figure 4 shows the convergence of 80 as a function of Nq, under the assumption that Nvar / 1 = 0 and k = 20 1. The resulting error is compared with the one of classical MC sampling which scales like 80~O^Nq''2^ and the one of MLAE which scales like 8Q~O^Nq3 / 4y Interestingly, we find that the convergence of 80 changes as a function of M. For small values of M, it follows the ideal VQAE scaling 80~o(Nq3'2) as if the variational approximation is performed without error. We emphasize that this scaling is cubically better than the 25 one of classical MC sampling. The second convergence regime is observed for larger values of M. In this regime, the error follows the MC scaling with 80~o(N~1 / 2y One can understand this new scaling in terms of a one-dimensional classical random walk with an average step of length 20 and some finite variance induced by variational approximations. Provided that the variance is much smaller than 0, after M~Nq steps 30 the total distance of the random walk converges to the value 2M0 with a relative error scaling as o(M-1 / 2). Finally, we take into account the cost of the variational approximation, to obtain a more complete assessment of the algorithmic performance of naive VQAE. To estimate the cost of a single variational update, we use Nvar / 1 = 2nfnsnpf where np is the number 35 of parameters of a PQC and the factor 2 comes from the two evaluations of the 30 04 25 objective function in Eq. (17). For the PQC in Fig. 2 with d = 4, the number of parameters is np = 60. Additionally, we choose nf~np~100 so that Nvar / l~1.2 x 106 and Nvar~4.8x 106M. This large variational cost is the dominant part in the calculation of the total number of queries Nq. Ultimately, it leads to a performance of naive VQAE 5 that is worse than the one of classical MC sampling. Reducing any of nf, ns or np decreases the variational cost but also increases the variational error which then leads to a worse final amplitude estimation error. C. Adaptive VQAE 10 To reduce the variational cost of VQAE, in this section we present the adaptive VQAE algorithm. This algorithm makes use of adaptive rescaling of the function f, based on importance sampling by scaling, which is well known in statistics and commonly used in MC simulations

[42] . 15 To introduce the adaptive VQAE algorithm, we first note that the amplitude a = Ep[f] is linear in f, meaning that rescaling the function f with a proportionality constant r also rescales the amplitude a: a' = Ep[f'] = Ep[rf] = ra. 22 20 The rescaled function f can then be used to encode a new quantum state \x’o)n+i, provided that f'(x) <1 for all x, which is required for the successful state preparation via Eq. (7). Next, a new Grover operator Q' can be defined and amplitude estimation can be performed to estimate amplitude a'. To proceed further, we make the 25 observation that after k iterations, the overlap between the states | / o)n+1 and \x'k)n+i = Q'k lxo)n+i is equal to cos(_2kd'). Hence, one can always find a commensurate amplitude 0 <a' <1 and a phase 9' so that a' = sin2(9'),9' = nl / k,l G Z, 23 30 and the overlap between the corresponding states | / o)n+1 and \x’k)n+i is equal to one. Combining these two observations, we conclude that for a proper choice of the renormalization factor r satisfying rf(x) <1 for all x, it is possible to rescale the function f such that the / c-th power of the corresponding Grover operator becomes 35 the identity operator Q'k =1. 30 04 25 Finding the exact renormalization factor r requires exact knowledge of the initial amplitude a. However, as we show in the following, a loose estimate at, obtained from a moderate number of MC samples of the initial state |j0)n+1 is sufficient. Assuming that such a loose amplitude estimate is provided, the renormalization factor can then 5 be approximately expressed asr, = a' / a., with a’ defined as in Eq. (23). As a result, the overlap between the states | / o)n+1 and lxk)n+1 becomes / । . (kse')2 = cos^e -k86)^l-----, 24 10 where 60' = 0' - arcsinWe see that for small deviations, the overlap decreases quadratically with k. Next, we use the VQAE algorithm to estimate the amplitude a' by means of the Grover operator Q' and the initial state \x'o)n+i- The variational approximation is performed every k-th step, when the overlap between | / o)n+1 and I / / X+i reaches maximum. Additionally, we assume that the PQC has the initial state 15 I0mit)n+i = l / o)n+i so that the variational quantum state of Eq. (15) reads 10 var(.^-))n+l ^var(^)\Xo)n+l’ 25 Having the PQC initialized to identity at the beginning of each optimization step, the 20 only role of the variational quantum circuit is to correct the deviation of Eq. (24) originating from the imprecise value of the renormalization constant^. Finally, at the end of the calculation, an estimation of the phase S' corresponding to the renormalized amplitude a' = sin2 3' is obtained. To go back to the original formulation of the problem and compare the results, we use the inverse transformation 25 0 = arc sin a = a!26 where the renormalization constant r( has to be exactly the same as the one used for the function rescaling. This last step concludes the adaptive VQAE algorithm. We 30 analyze the performance of the adaptive VQAE algorithm with a simplified variational ansatz consisting of only six single-qubit rotation gates and four CNOT gates, as shown in Fig. 2 in dark blue color. This simplified ansatz has only six parameters in total, which significantly reduces the number of variational queries as well as the effects of the noise due to finite sampling. We determine the loose estimate of the amplitude ar 35 via 5 x 105 MC samples. As a result, much smaller values of infidelity are achieved for 30 04 25 nf being an order of magnitude smaller than in our naive VQAE computations. We also note that for smaller values of a, the initial MC estimation of a’ gets worse and, as a consequence, more sweeps are required to ensure the convergence of the variational ansatz. 5 Our results for adaptive VQAE are presented in Fig. 1, where we show the convergence of 89 as a function of Nq for k = 10. The simulations use Adam with the initial learning rate p = 10-3, nf = 100, ns = 100 and n_p = 6, resulting in Nvar / 1 = 2nfnsnp = 1.2 x 105. As in Fig. 4, we compare with the classical MC scaling and the MLAE scaling 89~o(N~3 / 4y The major difference of the adaptive VQAE as compared to all 10 previously studied methods is a large starting cost which corresponds to the amount MC samples required for the evaluation of This starting cost, however, represents only an additive contribution to Nq and, hence, is insignificant in the regime of our interest when the number of queries gets large. Additionally, we find that, thanks to a significant improvement of the number of query calls and overall precision of the 15 variational state, the resulting final error 80 of the adaptive VQAE algorithm surpasses the classical MC error, therefore leading to a genuine quantum advantage. Interestingly, we observe that in the regime of small k, the performance of the adaptive VQAE algorithm decreases. This has several reasons. Firstly, the precision of the maximum likelihood estimation decreases when the angle 9' = nl / k (where I = 1 20 in our case) becomes larger than n- / 4, i.e. for k <4. Hence, to perform an estimation with such small values of k, a different statistical inference technique has to be considered. Secondly, for small values of k, the rescaling factor can become much larger than one and then leads to more efficient classical MC sampling. In calculations with C = 0.1, this leads to a loss of quantum advantage for k = 5, corresponding to 25 r « 7.508. The quantum advantage of the adaptive VQAE algorithm holds true for other cases with k >5. 30 04 25 IV. DISCUSSION In this article, we provide numerical evidence that variational quantum algorithms and 5 constant-depth quantum circuits can lead to a quantum advantage over classical MC sampling in the context of amplitude estimation. Our results are proof of concept that it is feasible, in principle, for current quantum devices to achieve a quantum advantage by performing amplitude estimation. The quantum circuits used for our numerical demonstrations, however, are still challenging for this generation of gate-10 based quantum computers. Therefore, an exciting next step is to find other problems and applications for which VQAE has low quantum hardware requirements and can be realized on actual quantum devices. We can imagine future applications for VQAE in several areas, including combinatorial optimization, quantum machine learning, and quantum chemistry. In the context of 15 combinatorial optimization, VQAE enables us to use constant-depth quantum circuits to carry out Grover search, which can find the optimal solution with a quadratic quantum speedup over brute-force search. Here it is also enticing to study whether such a variational Grover search algorithm can benefit from filtering operators

[43] . In relation to quantum machine learning, VQAE has the potential to make it possible 20 for current quantum devices to accelerate inference in Bayesian networks

[44] , which can then be compared with state-of-the-art variational quantum algorithms for inference

[45] . With regards to quantum chemistry, the concept of VQAE can be combined with variational quantum phase estimation (VQPE) [46-48] to realize VQPE with shallow circuits on actual quantum hardware. 25 We anticipate that the efficiency of the VQAE algorithm can be further increased. Firstly, it would be interesting to analyse whether a local cost function - which can help mitigate the negative effect of barren plateaus

[49] - improves the variational optimization and reduces the required number of variational queries. Secondly, the performance of our variational algorithm crucially depends on the maximum likelihood 30 estimation procedure. It would be interesting to investigate whether alternative approaches perform better, e.g. iterative QAE

[18] or QoPrime AE

[19] . 30 04 25 REFERENCES [1] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Quantum amplitude amplification and estimation, Contemp. Math. 305, 53 (2002). [2] A. Montanaro, Quantum speedup of Monte Carlo methods, Proc. Math. Phys. Eng. 5 Sci. 471 (2015). [3] E. Knill, G. Ortiz, and R. D. Somma, Optimal quantum measurements of expectation values of observables, Phys. Rev. A 75, 012328 (2007). [4] I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, and A. Aspuru-Guzik, Polynomialtime quantum algorithm for the simulation of chemical dynamics, Proc. Natl. Acad. Sci. 10 USA 105, 18681 (2008). [5] N. Wiebe, A. Kapoor, and K. M. Svore, Quantum Algorithms for Nearest-Neighbor Methods for Supervised and Unsupervised Learning, Quantum Info. Comput. 15, 316 (2015). [6] N. Wiebe, A. Kapoor, and K. M. Svore, Quantum Deep Learning, Quantum Info. 15 Comput. 16, 541 (2016). [7] I. Kerenidis, J. Landman, A. Luongo, and A. Prakash, q-means: A quantum algorithm for unsupervised machine learning, Adv. Neural Inf. Process. Syst. 32 (2019). [8] R. Orus, S. Mugel, and E. Lizaso, Quantum computing for finance: Overview and 20 prospects, Rev. Phys. 4, 100028 (2019). [9] D. J. Egger, C. Gambella, J. Marecek, S. McFaddin, M. Mevissen, R. Raymond, A. Simonetto, S. Woerner, and E. Yndurain, Quantum Computing for Finance: State-of-the-Art and Future Prospects, IEEE Trans.Quantum Eng. 1, 1 (2020).

[10] A. Bouland, W. van Dam, H. Joorati, I. Kerenidis, and A. Prakash, Prospects and 25 challenges of quantum finance, arXiv:2011.06492 (2020).

[11] S. Woerner and D. J. Egger, Quantum risk analysis, npj Quantum Inf. 5, 15 (2019).

[12] M. C. Braun, T. Decker, N. Hegemann, S. F. Kerstan, and C. Sch’ afer, A Quantum Algorithm for the Sensitivity Analysis of Business Risks, arXiv:2103.05475 (2021). 30 04 25

[13] P. Rebentrost, B. Gupt, and T. R. Bromley, Quantum computational finance: Monte Carlo pricing of financial derivatives, Phys. Rev. A 98, 022321 (2018).

[14] N. Stamatopoulos, D. J. Egger, Y. Sun, C. Zoufal, R. Iten, N. Shen, and S. Woerner, Option Pricing using Quantum Computers, Quantum 4, 291 (2020). 5

[15] Y. Suzuki, S. Uno, R. Raymond, T. Tanaka, T. Onodera, and N. Yamamoto, Amplitude estimation without phase estimation, Quantum Inf. Process. 19, 75 (2020).

[16] T. Tanaka, Y. Suzuki, S. Uno, R. Raymond, T. Onodera, and N. Yamamoto, Amplitude estimation via maximum likelihood on noisy quantum computer, arXiv:2006.16223 (2020). 10

[17] S. Aaronson and P. Rall, Quantum approximate counting, simplified, in Symposium on Simplicity in Algorithms (SIAM, 2020) pp. 24-32.

[18] D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, Iterative quantum amplitude estimation, npj Quantum Inf. 7, 1 (2021).

[19] T. Giurgica-Tiron, I. Kerenidis, F. Labib, A. Prakash, and W. Zeng, Low depth 15 algorithms for quantum amplitude estimation, arXiv:2012.03348 (2020).

[20] S. Herbert, Quantum Monte-Carlo Integration: The Full Advantage in Minimal Circuit Depth, arXiv:2105.09100 (2021).

[21] M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019). 20

[22] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational Quantum Algorithms, arXiv:2012.09265 (2020).

[23] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C.Kwek, 25 and A. Aspuru-Guzik, Noisy intermediate-scale quantum (NISQ) algorithms, arXiv:2101.08448 (2021).

[24] G. Brassard and P. Hoyer, An exact quantum polynomial-time algorithm for Simon's problem, in Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems (IEEE, 1997) pp. 12-23. 30 04 25

[25] L. K. Grover, Quantum computers can search rapidly by using almost any transformation, Phys. Rev. Lett. 80, 4329 (1998).

[26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery, Numerical Recipes in C (Cambridge University Press, New York, USA, 1992). 5

[27] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proceedings of ICNN'95 - International Conference on Neural Networks, Vol. 4 (1995) pp. 1942-1948.

[28] M. R. Bonyadi and Z. Michalewicz, Particle Swarm Optimization for Single Objective Continuous Space Problems: A Review, Evol. Comput. 25, 1 (2017).

[29] J. Gil Vidal and D. O. Theis, Calculus on parameterized quantum circuits, 10 arXiv: 1812.06323 (2018).

[30] R. M. Parrish, J. T. losue, A. Ozaeta, and P. L. McMahon, A Jacobi Diagonalization and Anderson Acceleration Algorithm For Variational Quantum Algorithm Parameter Optimization, arXiv: 1904.03206 (2019).

[31] K. M. Nakanishi, K. Fujii, and S. Todo, Sequential minimal optimization for 15 quantum-classical hybrid algorithms, Phys. Rev. Research 2, 043158 (2020).

[32] M. Ostaszewski, E. Grant, and M. Benedetti, Structure optimization for parameterized quantum circuits, Quantum 5, 391 (2021).

[33] M. Benedetti, M. Fiorentini, and M. Lubasch, Hardware-efficient variational quantum algorithms for time evolution, Phys. Rev. Research 3, 033083 (2021). 20

[34] E. Hazan, K. Y. Levy, and S. Shalev-Shwartz, Beyond convexity: Stochastic quasi- convex optimization, arXiv: 1507.02030 (2015).

[35] Y. Suzuki, H. Yano, R. Raymond, and N. Yamamoto, Normalized Gradient Descent for Variational Quantum Algorithms, arXiv:2106.10981 (2021).

[36] J. Li, X. Yang, X. Peng, and C.-P. Sun, Hybrid Quantum-Classical Approach to 25 Quantum Optimal Control, Phys.Rev. Lett. 118, 150503 (2017).

[37] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018). 30 04 25

[38] M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).

[39] L. Banchi and G. E. Crooks, Measuring analytic gradients of general quantum evolution with the stochastic parameter shift rule, Quantum 5, 386 (2021). 5

[40] D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, arXiv: 1412.6980 (2014).

[41] D. Aharonov, V. Jones, and Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, Algorithmica 55, 395 (2009).

[42] B. Arouna, Adaptative Monte Carlo Method, A Variance Reduction Technique, 10 Monte Carlo Methods Appl. 10, 1 (2004).

[43] D. Amaro, C. Modica, M. Rosenkranz, M. Fiorentini, M. Benedetti, and M. Lubasch, Filtering variational quantum algorithms for combinatorial optimization, arXiv:2106.10055 (2021).

[44] G. H. Low, T. J. Yoder, and I. L. Chuang, Quantum inference on Bayesian 15 networks, Phys. Rev. A 89, 062315 (2014).

[45] M. Benedetti, B. Coyle, M. Fiorentini, M. Lubasch, and M. Rosenkranz, Variational inference with a quantum computer, arXiv:2103.06720 (2021).

[46] R. M. Parrish and P. L. McMahon, Quantum Filter Diagonalization: Quantum Eigendecomposition without Full Quantum Phase Estimation, arXiv: 1909.08925 20 (2019).

[47] N. H. Stair, R. Huang, and F. A. Evangelista, A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons, J. Chern. Theory Comput. 16, 2236 (2020).

[48] K. Klymko, C. Mejuto-Zaera, S. J. Cotton, F. Wudarski, M. Urbanek, D. Hait, M. 25 Head-Gordon, K. B. Whaley, J. Moussa, N. Wiebe, W. A. de Jong, and N. M. Tubman, Real time evolution for ultracompact Hamiltonian eigen-states on quantum hardware, arXiv:2103.08563 (2021).

[49] M. Cerezo, A. Sone, T. Volkoff, L Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parameterized quantum circuits, Nat. Commun. 12, 1791 (2021) 30 04 25 30 04 25

Claims

5 1. A quantum computer (10) including:(i) an array of qubits (30) that is configured to be provided with initial values depending on input data supplied in use to the quantum computer (10);(ii) a quantum computing arrangement (60, 120) that is configured to execute a sequence of quantum operations on the array of qubits (30) to10 generate corresponding processed qubits; and(iii) a measuring unit (70) for determining states of the processed qubits to generate output data from the quantum computer (10),characterized in that the quantum computer (10) is configured to use a variational quantum amplitude estimation process, which includes performing a first step of15 applying a finite number of Grover iterations to generate a quantum circuit, the number at least partially determined according to a noise resilience of the quantum computer, and a subsequent step of performing a variational optimization to replace the generated circuit with a more compact circuit, in at least one of the sequence of operations, to reduce an effect of quantum noise that arises when20 executing the sequence of quantum operations on the qubits (30) and determining the states of the processed qubits.

2. A quantum computer (10) of claim 1, wherein the quantum computer (10) is configured to receive the input data representative of sensed data from a25 physical system, and the sequence of quantum operations is arranged to process the qubits so that the output data from the quantum computer (10) is representative of state or characteristic of the physical system.

3. A quantum computer (10) of claim 1 or 2, wherein the variational quantum 30 amplitude estimation process is implemented as an adaptive variational amplitude estimation process that is adaptive to characteristics of the sequence of operations when applied to the qubits and to characteristics of the measuring unit (70) when determining the states of the processed qubits.35 4. A quantum computer (10) of claim 3, wherein the adaptive variationalamplitude estimation process includes a maximum likelihood process for learning30 04 25characteristics of the sequence of operations when applied to the qubits and to characteristics of the measuring unit (70) when determining the states of the processed qubits.5 5. A quantum computer (10) of claim 4, wherein the maximum likelihoodprocess is implemented as an iterative process applied to the qubits of the quantum computer (10).

6. A method for operating a quantum computer (10), characterized in that 10 the method includes:(i) arranging for the quantum computer (10) to include an array of qubits (30) that is configured to be provided with initial values depending on input data supplied in use to the quantum computer (10);(ii) configuring a quantum computing arrangement (60, 120) to execute a 15 sequence of quantum operations on the array of qubits (30) to generatecorresponding processed qubits; and(iii) using a measuring unit (70) to determine states of the processed qubits to generate output data from the quantum computer (10),characterized in that the method further includes configuring the quantum 20 computer (10) to use a variational quantum amplitude estimation process, which includes performing a first step of applying a finite number of Grover iterations to generate a quantum circuit, the number at least partially determined according to a noise resilience of the quantum computer, and a subsequent step of performing a variational optimization to replace the generated circuit with a more compact 25 circuit, in at least one of the sequence of operations, to reduce an effect of quantum noise that arises when executing the sequence of quantum operations on the qubits (30) and determining the states of the processed qubits.

7. A method of claim 6, wherein the method includes configuring the quantum 30 computer (10) to receive the input data representative of sensed data from a physical system, and arranging for the sequence of quantum operations to process the qubits so that the output data from the quantum computer (10) is representative of state or characterize of the physical system.35 8. A method of claim 6 or 7, wherein the method includes implementing thevariational quantum amplitude estimation process as an adaptive variational amplitude estimation process that is adaptive to characteristics of the sequenceof operations when applied to the qubits and to characteristics of the measuring unit (70) when determining the states of the processed qubits.

9. A method of claim 8, wherein the method includes arranging for the5 adaptive variational amplitude estimation process to include a maximum likelihood process for learning characteristics of the sequence of operations when applied to the qubits and to characteristics of the measuring unit (70) when determining the states of the processed qubits.10 10. A method of claim 9, wherein the maximum likelihood process isimplemented as an iterative process applied to the qubits of the quantum computer (10).15LDCM11 A computer program product comprising a non-transitory computer readable storage medium having computer-readable instructions stored thereon, the computer-readable instructions being executable by a computerized device comprising processing hardware to execute the method of any one of claim 6 to 10.

Images