In this paper, we introduce an efficient algorithm for the quantum amplitude estimation task which works in noisy intermediate-scale quantum(NISQ) devices. The quantum amplitude estimation is an important problem which has various applications in fields such as quantum chemistry, machine learning, and finance. Because the well-known algorithm for the quantum amplitude estimation using the phase estimation cannot be executed in NISQ devices, alternative approaches have been proposed in recent literature. Some of them provide a proof of the upper bound which almost achieves the Heisenberg scaling. However, the constant factor is large and thus the bound is loose. Our contribution in this paper is to provide the algorithm such that the upper bound of query complexity almost achieves the Heisenberg scaling and the constant factor is small.
We propose a general-purpose quantum algorithm for preparing ground states of quantum Hamiltonians from a given trial state. The algorithm is based on techniques recently developed in the context of solving the quantum linear systems problem [Childs,Kothari,Somma'15]. We show that, compared to algorithms based on phase estimation, the runtime of our algorithm is exponentially better as a function of the allowed error, and at least quadratically better as a function of the overlap with the trial state. We also show that our algorithm requires fewer ancilla qubits than existing algorithms, makin…
This paper focuses on the quantum amplitude estimation algorithm, which is a core subroutine in quantum computation for various applications. The conventional approach for amplitude estimation is to use the phase estimation algorithm, which consists of many controlled amplification operations followed by a quantum Fourier transform. However, the whole procedure is hard to implement with current and near-term quantum computers. In this paper, we propose a quantum amplitude estimation algorithm without the use of expensive controlled operations; the key idea is to utilize the maximum likelihood …
Amplitude estimation algorithms are based on Grover's algorithm: alternating reflections about the input state and the desired outcome. But what if we are given the ability to perform arbitrary rotations, instead of just reflections? In this situation, we find that quantum signal processing lets us estimate the amplitude in a more flexible way. We leverage this technique to give improved and simplified algorithms for many amplitude estimation tasks: we perform non-destructive estimation without any assumptions on the amplitude, develop an algorithm with improved performance in practice, presen…
We show how phase and amplitude estimation algorithms can be parallelized. This can reduce the gate depth of the quantum circuits to that of a single Grover operator with a small overhead. Further, we show that for quantum amplitude estimation, the parallelization can lead to vast improvements in resilience against quantum errors. The resilience is not caused by the lower gate depth, but by the structure of the algorithm. Even in cases with errors that make it impossible to read out the exact or approximate solutions from conventional amplitude estimation, our parallel approach provided the co…
We consider performing phase estimation under the following conditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured. Most quantum estimation algorithms make assumptions that make them unsuitable for this 'coherent' setting, leaving only the textbook approach. We present novel algorithms for phase, energy, and amplitude estimation that are both conceptually and computationally simpler than the textbook method, featuring both a smaller query complexity and ancilla footprint. They do not requir…
We propose to perform amplitude estimation with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the context of Monte Carlo (MC) integration, we numerically show that shallow circuits can accurately approximate many amplitude amplification steps. 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 typically has larger computational requirements than classical MC sampling. To reduce the v…
We introduce the Real Quantum Amplitude Estimation (RQAE) algorithm, an extension of Quantum Amplitude Estimation (QAE) which is sensitive to the sign of the amplitude. RQAE is an iterative algorithm which offers explicit control over the amplification policy through an adjustable parameter. We provide a rigorous analysis of the RQAE performance and prove that it achieves a quadratic speedup, modulo logarithmic corrections, with respect to unamplified sampling. Besides, we corroborate the theoretical analysis with a set of numerical experiments.
Many quantum algorithms contain an important subroutine, the quantum amplitude estimation. As the name implies, this is essentially the parameter estimation problem and thus can be handled via the established statistical estimation theory. However, this problem has an intrinsic difficulty that the system, i.e., the real quantum computing device, inevitably introduces unknown noise; the probability distribution model then has to incorporate many nuisance noise parameters, resulting that the construction of an optimal estimator becomes inefficient and difficult. For this problem, we apply the th…
We provide a method for estimating spectral gaps in low-dimensional systems. Unlike traditional phase estimation, our approach does not require ancillary qubits nor does it require well characterised gates. Instead, it only requires the ability to perform approximate Haar-random unitary operations, applying the unitary whose eigenspectrum is sought out and performing measurements in the computational basis. We discuss application of these ideas to in-place amplitude estimation and quantum device calibration.
Amplitude estimation is a fundamental quantum algorithmic primitive that enables quantum computers to achieve quadratic speedups for a large class of statistical estimation problems, including Monte Carlo methods. The main drawback from the perspective of near term hardware implementations is that the amplitude estimation algorithm requires very deep quantum circuits. Recent works have succeeded in somewhat reducing the necessary resources for such algorithms, by trading off some of the speedup for lower depth circuits, but high quality qubits are still needed for demonstrating such algorithms…
We introduce a new variant of Quantum Amplitude Estimation (QAE), called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover's Algorithm, which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantl…
Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit
Mizuho Research & Technologies, Ltd.（瑞穗研究与技术株式会社）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）；Graduate School of Science and Technology, Keio University（庆应义塾大学理工学研究科）；Department of Applied Physics and Physico-Informatics, Keio University（庆应义塾大学应用物理与物性信息学系）
This paper generalizes the quantum amplitude amplification and amplitude estimation algorithms to work with non-boolean oracles. The action of a non-boolean oracle $U_φ$ on an eigenstate $|x\rangle$ is to apply a state-dependent phase-shift $φ(x)$. Unlike boolean oracles, the eigenvalues $\exp(iφ(x))$ of a non-boolean oracle are not restricted to be $\pm 1$. Two new oracular algorithms based on such non-boolean oracles are introduced. The first is the non-boolean amplitude amplification algorithm, which preferentially amplifies the amplitudes of the eigenstates based on the value of $φ(x)$. St…
In this paper, we propose a quantum amplitude estimation method that uses a modified Grover operator and quadratically improves the estimation accuracy in the ideal case, as in the conventional one using the standard Grover operator. Under the depolarizing noise, the proposed method can outperform the conventional one in the sense that it can in principle achieve the ultimate estimation accuracy characterized by the quantum Fisher information in the limit of a large number of qubits, while the conventional one cannot achieve the same value of ultimate accuracy. In general this superiority requ…
Outcome probability estimation via classical methods is an important task for validating quantum computing devices. Outcome probabilities of any quantum circuit can be estimated using Monte Carlo sampling, where the amount of negativity present in the circuit frame representation quantifies the overhead on the number of samples required to achieve a certain precision. In this paper, we propose two classical sub-routines: circuit gate merging and frame optimisation, which optimise the circuit representation to reduce the sampling overhead. We show that the runtimes of both sub-routines scale po…
Recently we find several candidates of quantum algorithms that may be implementable in near-term devices for estimating the amplitude of a given quantum state, which is a core sub- routine in various computing tasks such as the Monte Carlo methods. One of those algorithms is based on the maximum likelihood estimate with parallelized quantum circuits. In this paper, we extend this method so that it incorporates the realistic noise effect, and then give an experimental demonstration on a superconducting IBM Quantum device. The maximum likelihood estimator is constructed based on the model assumi…
Since the quantum amplitude estimation (QAE) was invented by Brassard et al., 2002, several advanced algorithms have recently been published (Grinko et al., 2019, Aaronson et al, and Suzuki et al., 2020). The main difference between the variants and the original algorithm is that the variants do not need quantum phase estimation (QPE), a key component of the canonical QAE (Brassard et al., 2002), that is composed of many expensive operations on NISQ devices. In this paper, we compare and analyze two of these new QAE approaches (Grinko et al., 2019, and Suzuki et al., 2020) by implementation us…
This study presents a quantum circuit for estimating the pi value using arithmetic circuits and by quantum amplitude estimation. We review two types of quantum multipliers and propose quantum squaring circuits based on the multiplier as basic arithmetic circuits required for performing quantum computations. The squarer realized by a quantum adder with the gate size of $ O(n) $ requires $ O(n^2) $ gates and at least one ancillary qubits, while that realized by using quantum Fourier transform (QFT) requires $ O(n^3) $ gates without ancillary qubit. The proposed quantum circuit to estimate pi is …
Since the publication of the Quantum Amplitude Estimation (QAE) algorithm by Brassard et al., 2002, several variations have been proposed, such as Aaronson et al., 2019, Grinko et al., 2019, and Suzuki et al., 2020. The main difference between the original and the variants is the exclusion of Quantum Phase Estimation (QPE) by the latter. This difference is notable given that QPE is the key component of original QAE, but is composed of many operations considered expensive for the current NISQ era devices. We compare two recently proposed variants (Grinko et al., 2019 and Suzuki et al., 2020) by…
Quantum Amplitude Estimation (QAE) -- a technique by which the amplitude of a given quantum state can be estimated with quadratically fewer queries than by standard sampling -- is a key sub-routine in several important quantum algorithms, including Grover search and Quantum Monte-Carlo methods. An obstacle to implementing QAE in near-term noisy intermediate-scale quantum (NISQ) devices has been the need to perform Quantum Phase Estimation (QPE) -- a costly procedure -- as a sub-routine. This impediment was lifted with various QPE-free methods of QAE, wherein Grover queries of varying depths / …
Quantum Amplitude Estimation (QAE) can achieve a quadratic speed-up for applications classically solved by Monte Carlo simulation. A key requirement to realize this advantage is efficient state preparation. If state preparation is too expensive, it can diminish the quantum advantage. Preparing arbitrary quantum states has exponential complexity with respect to the number of qubits, thus, is not applicable. Currently known efficient techniques require problems based on log-concave probability distributions, involve learning an unknown distribution from empirical data, or fully rely on quantum a…
Since the initial discovery of quantum teleportation, it is devoted to transferring unknown quantum states from one party to another distant partner. However, in the scenarios of remote sensing, what people truly care about is the information carried by certain parameters. The problem of multiparameter estimation in the framework of qutrit teleportation under amplitude damping (AD) noise is studied. Particularly, two schemes are proposed to battle against AD noise and enhance the precision of multiparameter estimation by utilizing weak measurement (WM) and environment-assisted measurement (EAM…
Linear regression is a basic and widely-used methodology in data analysis. It is known that some quantum algorithms efficiently perform least squares linear regression of an exponentially large data set. However, if we obtain values of the regression coefficients as classical data, the complexity of the existing quantum algorithms can be larger than the classical method. This is because it depends strongly on the tolerance error $ε$: the best one among the existing proposals is $O(ε^{-2})$. In this paper, we propose the new quantum algorithm for linear regression, which has the complexity of $…
Anomaly detection plays a critical role in fraud detection, health care, intrusion detection, military surveillance, etc. Anomaly detection algorithm based on density estimation (called ADDE algorithm) is one of widely used algorithms. Liang et al. proposed a quantum version of the ADDE algorithm [Phys. Rev. A 99, 052310 (2019)] and it is believed that the algorithm has exponential speedups on both the number and the dimension of training data point over the classical algorithm. In this paper, we find that Liang et al.'s algorithm doesn't actually execute. Then we propose a new quantum ADDE al…
We derive families of optimal and near-optimal probe states for quantum estimation of the coupling constants of a general two-mode number-conserving bosonic Hamiltonian describing one-body and two-body dynamics. We find that the optimal states for estimating the dephasing of the modes, the self-interaction strength, and the contact interaction strength are related to the NOON states, whereas the optimal states for estimation of the intermode single particle tunneling amplitude are superpositions of antipodal SU(2) coherent states. For estimation of the amplitude of pair tunneling and the ampli…
We present a method to model a discretized time evolution of probabilistic networks on gate-based quantum computers. We consider networks of nodes, where each node can be in one of two states: good or failed. In each time step, probabilities are assigned for each node to fail (switch from good to failed) or to recover (switch from failed to good). Furthermore, probabilities are assigned for failing nodes to trigger the failure of other, good nodes. Our method can evaluate arbitrary network topologies for any number of time steps. We can therefore model events such as cascaded failure and avala…
Quantum amplitude estimation is a key sub-routine of a number of quantum algorithms with various applications. We propose an adaptive algorithm for interval estimation of amplitudes. The quantum part of the algorithm is based only on Grover's algorithm. The key ingredient is the introduction of an adjustment factor, which adjusts the amplitude of good states such that the amplitude after the adjustment, and the original amplitude, can be estimated without ambiguity in the subsequent step. We show with numerical studies that the proposed algorithm uses a similar number of quantum queries to ach…
A universal fault-tolerant quantum computer holds the promise to speed up computational problems that are otherwise intractable on classical computers; however, for the next decade or so, our access is restricted to noisy intermediate-scale quantum (NISQ) computers and, perhaps, early fault tolerant (EFT) quantum computers. This motivates the development of many near-term quantum algorithms including robust amplitude estimation (RAE), which is a quantum-enhanced algorithm for estimating expectation values. One obstacle to using RAE has been a paucity of ways of getting realistic error models i…
