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Leibniz international proceedings in informatics

Quantum Probability Oracles & Multidimensional Amplitude Estimation

Joran van Apeldoorn

Published Jan 1, 2021• 5 citations
Publisher

Abstract

We give a multidimensional version of amplitude estimation. Let p be an n-dimensional probability distribution which can be sampled from using a quantum circuit U_p. We show that all coordinates of p can be estimated up to error ε per coordinate using Õ(1/(ε)) applications of U_p and its inverse. This generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2. Our results also imply a Õ(n/ε) query algorithm for 𝓁₁-norm (the total variation distance) estimation and a Õ(√n/ε) query algorithm for 𝓁₂-norm. We also show that these results are optimal up to logarithmic factors.

DOI
10.4230/lipics.tqc.2021.9

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We give a multidimensional version of amplitude estimation. Let p be an n-dimensional probability distribution which can be sampled from using a quantum circuit U_p. We show that all coordinates of p can be estimated up to error ε per coordinate using Õ(1/(ε)) applications of U_p and its inverse. This generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2. Our results also imply a Õ(n/ε) query algorithm for 𝓁₁-norm (the total variation distance) estimation and a Õ(√n/ε) query algorithm for 𝓁₂-norm. We also show that these results are optimal up to logarithmic factors.
Estimating statistical properties is fundamental in statistics and computer science. In this paper, we propose a unified quantum algorithm framework for estimating properties of discrete probability distributions, with estimating Rényi entropies as specific examples. In particular, given a quantum oracle that prepares ann-dimensional quantum state Σni=1√pi|i⟩, for α > 1 and 0Hα(p) to within additive error ϵ with probability at least 2/3 using Õ(n1-1/2α/ϵ + √n/ϵ1+ 1/2α) and Õ(n1/2α/ϵ1+ 1/2α) queries, respectively. This improves the best known dependence in ϵ as well as the joint dependence betweennand 1/ϵ. Technically, our quantum algorithms combine quantum singular value transformation, quantum annealing, and variable-time amplitude estimation. We believe that our algorithm framework is of general interest and has wide applications.
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 achieve the same level of precision εεε compared to state-of-the-art algorithms, but the classical part, i.e., the non-quantum part, has substantially lower computational complexity. We rigorously prove that the number of oracle queries achieves O(1/ε)O(1/ε)O(1/ε), i.e., a quadratic speedup over classical Monte Carlo sampling, and the computational complexity of the classical part achieves O(log⁡(1/ε))O(\log(1/ε))O(log(1/ε)), both up to a double-logarithmic factor.
We design and analyze two new low depth algorithms for amplitude estimation (AE) achieving an optimal tradeoff between the quantum speedup and circuit depth. For β∈(0,1] , our algorithms require N=O~(1ϵ1+β) oracle calls and require the oracle to be called sequentially D=O(1ϵ1−β) times to perform amplitude estimation within additive error ϵ . These algorithms interpolate between the classical algorithm (β=1) and the standard quantum algorithm ( β=0 ) and achieve a tradeoff ND=O(1/ϵ2) . These algorithms bring quantum speedups for Monte Carlo methods closer to realization, as they can provide speedups with shallower circuits.The first algorithm (Power law AE) uses power law schedules in the framework introduced by Suzuki et al \cite{S20}. The algorithm works for β∈(0,1] and has provable correctness guarantees when the log-likelihood function satisfies regularity conditions required for the Bernstein Von-Mises theorem. The second algorithm (QoPrime AE) uses the Chinese remainder theorem for combining lower depth estimates to achieve higher accuracy. The algorithm works for discrete β=q/k where k≥2 is the number of distinct coprime moduli used by the algorithm and 1≤q≤k−1 , and has a fully rigorous correctness proof. We analyze both algorithms in the presence of depolarizing noise and provide numerical comparisons with the state of the art amplitude estimation algorithms.
We present a near-optimal quantum algorithm, up to logarithmic factors, for estimating the Shannon entropy in the quantum probability oracle model. Our approach combines the singular value separation algorithm with quantum amplitude amplification, followed by the application of quantum singular value transformation. On the lower bound side, we construct probability distributions encoded via Hamming weights in the oracle, establishing a tight query lower bound up to logarithmic factors. Consequently, our results show that the tight query complexity for estimating the Shannon entropy within εεε-additive error is given by Θ~(nε)\tildeΘ\left(\tfrac{\sqrt{n}}ε\right)Θ~(εn​​).
We consider the problem of estimating the expected outcomes of Monte Carlo processes whose outputs are described by multidimensional random variables. We tightly characterize the quantum query complexity of this problem for various choices of oracle access to the Monte Carlo process and various normalization conditions on its output variables and the estimates to be made. More specifically, we design quantum algorithms that provide a quadratic speed-up in query complexity with respect to the precision of the resulting estimates and show that these complexities are asymptotically optimal in each of our scenarios. Interestingly, we find that these speed-ups with respect to the precision come at the cost of a (up to exponential) slowdown with respect to the dimension of the Monte Carlo random variables, for most of the parameter settings we consider. We also describe several applications of our algorithms, notably in the field of machine learning, and discuss the implications of our results.
In this work, we provide the first QFT-free algorithm for Quantum Amplitude Estimation (QAE) that is asymptotically optimal while maintaining the leading numerical performance. QAE algorithms appear as a subroutine in many applications for quantum computers. The optimal query complexity achievable by a quantum algorithm for QAE is log queries, providing a speedup of a factor of 1/ε over any other classical algorithm for the same problem. The original algorithm for QAE utilizes the quantum Fourier transform (QFT) which is expected to be a challenge for near-term quantum hardware. To solve this problem, there has been interest in designing a QAE algorithm that avoids using QFT. Recently, the iterative QAE algorithm (IQAE) [1] was introduced with a near-optimal query complexity and small constant factors. In this work, we combine ideas from the preceding line of work to introduce a QFT-free QAE algorithm that maintains the asymptotically optimal query complexity while retaining small constant factors. We supplement our analysis with numerical experiments comparing our performance with IQAE where we find that our modifications retain the high performance, and in some cases even improve the numerical results.