Bayesian Iterative Quantum Amplitude Estimation

• The paper's main contribution is the integration of Bayesian inference into iterative amplitude estimation to reduce measurement complexity and improve confidence intervals.
• BIQAE employs adaptive Bayesian credible intervals that recycle prior measurement information, thereby achieving significant query reductions, especially with beta priors in low-sample regimes.
• Practical implementations in quantum chemistry, finance, and machine learning demonstrate two to three orders-of-magnitude efficiency improvements over traditional QAE methods.

Bayesian Iterative Quantum Amplitude Estimation (BIQAE) is a statistically grounded quantum algorithmic framework designed to estimate quantum amplitudes with superior efficiency by integrating Bayesian inference into the iterative quantum amplitude estimation protocol. By adopting rigorous Bayesian updates for interval estimation at each iteration, BIQAE consistently outperforms classical and frequentist quantum amplitude estimation (QAE) methods in measurement efficiency, interval accuracy, and sample complexity, with applications across quantum chemistry, quantum finance, and quantum-enhanced machine learning (Li et al., 30 Jul 2025). The following sections provide a comprehensive technical treatment of BIQAE’s mathematical foundations, algorithmic procedures, analytical performance, and practical implications.

1. Statistical Framework and Algorithmic Structure

BIQAE is formulated within a unified statistical model where the quantum amplitude $a$ of interest is parameterized by an angle $\theta$ such that $a = \sin^2(\theta)$. A crucial element of QAE and its variants is the use of amplitude amplification: after $k$ Grover iterations, the probability of measuring the "good" subspace is $p_k = \sin^2[(2k+1)\theta]$. Standard quantum amplitude estimation (QAE) yields mean squared error scaling as $O(1/N_\text{shots})$, requiring $O(1/\epsilon^2)$ oracle queries to reach accuracy $\epsilon$.

Iterative quantum amplitude estimation (IQAE) improves upon this by structuring the algorithm as a sequence of $T$ stages, where each stage $t$ uses a set quantum amplification factor $k_t$ and adapts the measurement decision using stagewise confidence intervals (CIs) for $p_{k_t}$. BIQAE replaces the frequentist confidence intervals in IQAE with Bayesian credible intervals, relying on a fully updated posterior at each stage. The algorithm proceeds by positing a prior $\pi_t^{\text{prior}}$ for $p_{k_t}$, updating via the binomial likelihood for $N_t$ quantum circuit measurements, and determining the posterior $\pi_t^{\text{post}}(p_{k_t}\mid\overline{X}_t)$:

$$\pi_{t}^{\text{post}}(p_{k_t}|\overline{X}_t) \propto \pi_{t}^{\text{prior}}(p_{k_t}) \cdot \mathcal{L}_t(\overline{X}_t|p_{k_t})$$

where $\overline{X}_t$ is the observed mean outcome and $\mathcal{L}_t$ denotes the appropriate binomial likelihood at amplification $k_t$. The credible interval is computed from this posterior and deterministically mapped via the relation $a = \sin^2(\theta)$ to an interval for $\theta$ and $a$ (Li et al., 30 Jul 2025).

This structure is implemented in two distinct modes: with normal priors ("Normal-BIQAE") and with beta priors ("Beta-BIQAE"). The latter is especially effective for small sample regimes due to the conjugacy of the beta prior with the binomial likelihood.

2. Bayesian Update and Credible Interval Construction

In each iteration, the Bayesian update incorporates both the current measurement outcome and all prior information, “recycling” information from previous stages. For example, in the normal approximation:

$$\mu_{\text{post}, t} = \frac{\mu_0/\Delta^2 + \overline{X}_t/(\widehat{\sigma}_t^2/N_t)}{1/\Delta^2 + 1/(\widehat{\sigma}_t^2/N_t)}$$

$$\sigma_{\text{post}, t}^2 = \left(\frac{1}{\Delta^2} + \frac{1}{\widehat{\sigma}_t^2/N_t}\right)^{-1}$$

where $\mu_0$ is the prior mean, $\Delta^2$ is the prior variance, and $$\widehat{\sigma}_t^2 = \overline{X}_t(1-\overline{X}_t)$$ is the empirical variance estimate for binomial trials.

Credible intervals $[p_{k_t}^l, p_{k_t}^u]$ are computed as the $\alpha_t/2$ and $1-\alpha_t/2$ quantiles of the posterior, truncated to $[0,1]$ if necessary, and then mapped to angle intervals using an invertible function derived from the sinusoidal form of amplitude amplification:

$$[\theta_t^l, \theta_t^u] = f_t\left([p_{k_t}^l, p_{k_t}^u]\right)$$

This Bayesian interval incorporates all previous measurement information, in contrast to IQAE where only the current batch is considered. As a consequence, the BIQAE intervals have superior coverage and typically narrower width for the same quantum resource cost (Li et al., 30 Jul 2025).

3. Sample Complexity and Efficiency Analysis

Analytical and numerical results demonstrate that, for target amplitude accuracy $\epsilon$, BIQAE achieves quantum sample complexity $O(1/\epsilon)$ (plus potential log factors), reducing the leading constant compared to IQAE. The per-stage sample complexity bound is:

$$N_{\text{oracle}, t} \leq \frac{z^2_{\alpha_t/2}}{4K_t} \left[\frac{1}{\epsilon_t^2} - \frac{1}{\epsilon_{t-1}^2}\right]$$

where $K_t = 2k_t + 1$, $\epsilon_t$ is the credible interval radius for $\theta$ at step $t$, and $z_{\alpha_t/2}$ is the normal quantile associated with the desired coverage. The subtraction of the $1/\epsilon_{t-1}^2$ term—enabled by Bayesian information carryover—directly reduces the number of additional queries required at each stage.

Empirical simulations for $a=0.5$ show that Beta-BIQAE achieves a 10–16% reduction in total quantum sample complexity versus canonical IQAE-CP across six orders of magnitude in $\epsilon$. The scaling exponent is close to $-1$, confirming the expected quadratic quantum speedup relative to classical Monte Carlo methods ($O(1/\epsilon^2)$) (Li et al., 30 Jul 2025). The improved resource savings are particularly evident in applications requiring a high degree of amplitude precision.

4. Mathematical Properties, Proof Techniques, and Numerical Benchmarks

The foundational properties of BIQAE—such as unbiasedness, interval coverage, and mean squared error—are established using standard tools, including the invariance property of the maximum likelihood estimator (MLE) and the delta method. The delta method is used for propagating Bayesian posterior uncertainty for the transformed amplitude parameter $a = \sin^2 \theta$, given a posterior distribution for $\theta$.

The identifiability of $\theta$ from observations of $p_k$ is guaranteed by keeping careful track of the “quadrant index” $l(k)$, indicating the unique solution for the amplified angle that corresponds to the measured value $p_k$. The posterior is then appropriately mapped to this branch to maintain consistency across iterations.

Pseudocode for key subroutines is provided in the reference implementation, including Bayesian update, credible interval computation (ComputeCRI), and prior preparation. Extensive mock quantum circuit simulations—using parameterized $R_Y$ rotations—validate theoretical error and coverage bounds against empirical performance. Applications to quantum chemistry (molecular energy estimation for H$_2$, LiH, HF, BeH$_2$) demonstrate that BIQAE can yield two to three orders-of-magnitude shot reduction compared to classical estimation at fixed target accuracy (Li et al., 30 Jul 2025).

5. Practical Implications in Chemistry, Finance, and Quantum Machine Learning

In quantum chemistry, BIQAE has been applied to molecular ground-state energy estimation, significantly reducing the number of quantum circuit evaluations (shots) required to obtain chemical accuracy. For example, rigorous benchmarking in molecular simulations shows that, for fixed accuracy, Bayesian interval recycling enables drastic query cost reductions compared to both classical amplitude estimation and frequentist IQAE.

In quantitative finance, where quantum Monte Carlo risk metrics (Value-at-Risk, Conditional Value-at-Risk) and option pricing are central applications, the improved efficiency of BIQAE translates into more tractable quantum resource requirements for probabilistic simulation tasks. Machine learning approaches relying on quantum expectation value estimation—such as quantum reinforcement learning or deep quantum neural networks—also benefit directly, as the “Bayesian acceleration” of QAE protocols reduces the cost of frequent amplitude estimations required throughout the learning process (Li et al., 30 Jul 2025).

6. Implementation Considerations and Limitations

BIQAE inherits the circuit resource requirements of IQAE: efficient iteration requires rapid, accurate application of Grover-based amplitude amplification circuits and a classical Bayesian update step at each stage. The benefit of information recycling is realized even in the regime of low to moderate sample size, provided conjugate priors (e.g., normal for approximate Gaussian likelihoods, beta for exact binomial) are maintained.

In noisy or pre-threshold (NISQ) devices, sample efficiency remains superior if accurate error characterization enables Bayesian likelihood correction. The reduction in quantum queries is limited by the quality of prior and posterior models; significant model mismatch may adversely affect the actual coverage of the credible intervals. For highly nonclassical or ill-conditioned circuits, advanced prior modeling (possibly using nonparametric Bayesian methods) may further improve BIQAE's performance, but remains an open research direction (Li et al., 30 Jul 2025).

7. Extensions and Open Directions

Mathematical analysis and empirical evidence indicate that the efficiency improvement of BIQAE is not solely a function of Bayesian inference, but of the posterior’s ability to aggregate and compress all available measurement evidence stagewise. Consequently, BIQAE suggests further research into:

• The joint Bayesian optimization of the (K, N) schedule, possibly employing Bayesian experimental design or reinforcement learning techniques to minimize overall measurement cost.
• Integration of advanced conjugate or flexible prior models (beyond normal and beta) for small $a$, low $N$, or multi-modal posterior settings.
• Robust Bayesian interval estimation under circuit noise by incorporating real-time noise parameter inference, or by leveraging recent advances in noise-adaptive Bayesian amplitude estimation (Ramôa et al., 5 Dec 2024).
• Empirical and analytic paper of BIQAE in the context of strongly correlated quantum systems and high-dimensional amplitude vectors, where Bayesian updating may play an even more decisive role.

The currently demonstrated 10–16% sample complexity reduction over IQAE (Li et al., 30 Jul 2025) underlines that Bayesian statistics is the operational source of advantage in BIQAE, opening a direction for further leveraging statistical inference to expedite quantum resource use in amplitude estimation and related quantum algorithms.

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In summary, Bayesian Iterative Quantum Amplitude Estimation is a statistically robust quantum algorithmic paradigm that leverages rigorous Bayesian information recycling at each step of iterative quantum amplitude estimation, yielding reduced query complexity, improved interval accuracy, and broad applicability for estimation and simulation tasks in quantum technologies (Li et al., 30 Jul 2025).