Harnessing Bayesian Statistics to Accelerate Iterative Quantum Amplitude Estimation

TL;DR

The paper presents BIQAE, a Bayesian enhancement of Iterative Quantum Amplitude Estimation (IQAE), demonstrating that injecting Bayesian inference into QAE reduces quantum sample complexity and yields reliable interval estimates across iterations. By formulating a unified statistical framework for Classical QAE, AAE, and IQAE, it quantifies a double-digit percentage speedup attributable to Bayesian updating and provides two concrete implementations—Normal-BIQAE and Beta-BIQAE—along with a noninformative-prior linkage to existing IQAE variants. Numerical simulations show Beta-BIQAE outperforms state-of-the-art approaches across amplitude estimation and molecular ground-state energy tasks, achieving substantial reductions in measurement cost (often orders of magnitude) and tighter, more reliable interval estimates. The results highlight the potential of Bayesian strategies to accelerate quantum utility, motivate further Bayesian optimization of scheduling and hyperparameters, and suggest broader applicability to quantum algorithms beyond QAE in noisy and fault-tolerant settings.

Abstract

We establish a unified statistical framework that underscores the crucial role statistical inference plays in Quantum Amplitude Estimation (QAE), a task essential to fields ranging from chemistry to finance and machine learning. We use this framework to harness Bayesian statistics for improved measurement efficiency with rigorous interval estimates at all iterations of Iterative Quantum Amplitude Estimation. We demonstrate the resulting method, Bayesian Iterative Quantum Amplitude Estimation (BIQAE), accurately and efficiently estimates both quantum amplitudes and molecular ground-state energies to high accuracy, and show in analytic and numerical sample complexity analyses that BIQAE outperforms all other QAE approaches considered. Both rigorous mathematical proofs and numerical simulations conclusively indicate Bayesian statistics is the source of this advantage, a finding that invites further inquiry into the power of statistics to expedite the search for quantum utility.

Figures (20)

• Figure 1: Pipeline of enhanced-sampling amplified estimation with identifiability challenge highlighted in green.
• Figure 2: Schematic illustrations of IQAE and BIQAE algorithms. (a) In IQAE, measurements of amplified-amplitude quantum circuits (top) are fed to a classical computer (bottom), which both iteratively improves the estimate of the confidence interval of $\theta$ (ComputeCI) and increments the number of Grover operators $k$ of the quantum circuit (FindNextK) until the quantum amplitude is estimated with the desired accuracy. (b) BIQAE introduces three new modules to improve IQAE's efficiency via Bayesian inference (highlighted in red): BayesianUpdate and PreparePrior, which maintain and refine the prior and posterior distribution of the amplified target probability; and ComputeCRI, the Bayesian substitute to ComputeCI, which calculates a credible interval for $\theta$ based on the posterior distribution in place of confidence interval.
• Figure 3: Quantum sample complexity comparison of BIQAE to benchmark QAE approaches for quantum amplitude $a=0.5$ over 200 repetitions. Legend is ordered from highest to lowest performance, with Bayesian and IQAE methods emphasized in color (BIQAE, purple triangles; IQAE-CP, green circles; BAE, blue circles; and IQAE-CH, pink circles).
• Figure 4: Quantum sample complexity comparison between Beta-BIQAE (dashed purple line with triangles) and IQAE-CP (solid green line with circles) across six orders of magnitude of target accuracy $\varepsilon$ (percentage improvements of Beta-BIQAE over IQAE-CP accentuated in text and as yellow bars).
• Figure 5: Radius ratio progression across stages in Beta-BIQAE (average value, solid blue line; values within one standard deviation, shaded gray area; $5^\text{th}$ percentile, dashed blue line; $95^\text{th}$ percentile, dotted blue line). Results are aggregated from 1000 repetitions with a target accuracy of $\varepsilon=10^{-8}$. A reference line (dashed gray line) is included to mark the theoretically expected ratio from Eq. \ref{['complexity_bound']} corresponding to the observed 14.3% improvement in the quantum sample complexity analysis.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Example 1
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 1