Low depth amplitude estimation without really trying

TL;DR

This work tackles the depth limitation of quantum amplitude estimation on near-term devices by proposing a universal protocol to build low-depth estimators from unbiased quantum estimators. The key idea is to run unbiased QAEs (or QAEs for phase estimation) at shallow depth and aggregate their outputs, ensuring that bias decays slowly enough with depth so that averaging yields high precision while preserving quantum advantage in total queries. Two prototype unbiased estimators, Type I and Type II, underpin the construction, with rigorous results showing how to set bias and variance (or bias and failure probability) and the number of parallel runs to achieve a target precision $\epsilon_0$ under hardware depth constraints controlled by $\beta$. The method extends to low-depth phase estimation and offers concrete parameter regimes illustrating depth-precision-variance tradeoffs, though it relies on QFT-free unbiased estimators, which motivates future work to develop such primitives for practical near-term deployment.

Abstract

Standard quantum amplitude estimation algorithms provide quadratic speedup to Monte-Carlo simulations but require a circuit depth that scales as inverse of the estimation error. In view of the shallow depth in near-term devices, the precision achieved by these algorithms would be low. In this paper we bypass this limitation by performing the classical Monte-Carlo method on the quantum algorithm itself, achieving a higher than classical precision using low-depth circuits. We require the quantum algorithm to be weakly biased in order to avoid error accumulation during this process. Our method is parallel and can be as weakly biased as the constituent algorithm in some cases.

Figures (7)

• Figure 1: A hybrid construction of low-depth algorithms. It avoids the worst of both worlds: a lower depth compared to standard quantum amplitude estimation algorithm and a smaller query complexity compared to the classical Monte-Carlo method.
• Figure 2: A standard black box QAE
• Figure 3: Hardware depth characterised by a single number $\beta$.
• Figure 4: Low-depth QAE comes with an extra controlling knob that can tune the maximum depth of the algorithm without compromising $\epsilon$ and $\delta$, allowing it to fit into arbitrary hardware.
• Figure 5: The convergence of the sample mean.

Theorems & Definitions (31)

• Definition 1
• Definition 2: Unbiased QAE type I
• Definition 3: Unbiased QAE type II
• Lemma 1: Bias and variance
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Definition 4: Unbiased QPE type II