Low depth algorithms for quantum amplitude estimation

Abstract

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.

Fragmented imaginary-time evolution for early-stage quantum signal processors

“…In its standard form, QAE has similar hardware requirements as QAA 35 . However, recently, interesting algorithms have appeared 56 , 57 that perform partial QAE with circuit depths that can interpolate between the probabilistic and coherent cases. In contrast, here, we beat full QAA using circuit depths for most runs much lower than in the bare probabilistic approach.…”

Quantum advantage of Monte Carlo option pricing

“…where N shot is the sample count for the quantum circuit in every Grover step, N q is the number of queries per shot, c s ˜and q s ˜are the standard deviations of the classical and quantum distributions being sampled. In the presence of depolarizing noise, there is an upper threshold for the number of Grover steps (m k ), where the first term in the previous formula (11) remains constant for increasing N q [18]. One can easily see that fewer quantum shots (N shot ) lead to a greater quantum advantage Q.…”

“…, where N denotes the number of samples [10]. Over the last years, many different realizations were proposed for QAE with decreasing quantum circuitry footprint [11][12][13][14][15].…”

Noise tailoring for robust amplitude estimation

“…This lesser-acknowledged problem of infeasible runtimes, which was dubbed the 'measurement problem' [31], poses a major roadblock for hybrid algorithms, as well as many non-variational algorithms, towards achieving quantum advantage on devices in the NISQ era and beyond. In an effort to deal with the measurement problem on NISQ devices, recent works apply techniques of quantum amplification and quantum estimation borrowed from the field of fault-tolerant quantum computing [32][33][34][35]. As a first step towards this effort, Wang et al [32] achieves a reduction in the desired number of samples for VQE by improving the sample scaling from O(…”