Random-Walk Bayesian IQPE

• Random-walk Bayesian IQPE is an adaptive Gaussian inference method that estimates an unknown eigenphase by mapping Bayesian updates to a random-walk process.
• The algorithm adaptively selects experimental parameters based on the current Gaussian posterior to optimize information gain and minimize estimation uncertainty.
• It attains Heisenberg-limited scaling with minimal computational overhead, enabling real-time FPGA-driven adaptive experiments.

Random-walk Bayesian Iterative Quantum Phase Estimation (RW-IQPE) is an adaptive, Gaussian-based online Bayesian inference algorithm for @@@@1@@@@. This approach estimates the unknown eigenphase $\omega$ of a unitary family $U(t)$, defined by $U(t)|\psi\rangle = e^{i\omega t}|\psi\rangle$, by performing a series of controlled unitary experiments, updating beliefs about $\omega$ after each measurement, and adaptively optimizing future experimental settings. RW-IQPE achieves Heisenberg-limited scaling in estimation error, while requiring exponentially less classical processing time compared to existing Bayesian particle filter methods, making it suitable for real-time, FPGA-driven adaptive experiments (Granade et al., 2022).

1. Bayesian Formulation of Iterative Quantum Phase Estimation

RW-IQPE addresses the problem of learning an unknown eigenphase $\omega$ through a sequence of controlled-$U(t)$ experimental steps and measurements. In the Bayesian framework, a prior probability distribution over $\omega$ is maintained and updated to a posterior after each measurement outcome $d\in\{0,1\}$.

This approach offers several advantages:

• Automatic adaptation: Incorporates prior knowledge and adapts to experimental drift.
• Adaptive design: Posterior guides the choice of experimental parameters to minimize uncertainty.
• Optimality: Capable of achieving the Heisenberg limit, i.e., estimation error scaling as $1/T$ where $T$ is total evolution time, even in the presence of adaptive feedback.

Bayesian methods in this context contrast with non-adaptive (or non-Bayesian) approaches, which often lack flexibility or optimality in adapting to variable experimental conditions (Granade et al., 2022).

2. Gaussian Prior as a Random Walker

RW-IQPE exclusively models the prior and posterior distribution of $\omega$ as Gaussian,

$$P_n(\omega) = \mathcal{N}(\mu_n, \sigma_n^2) = \frac{1}{\sqrt{2\pi}\sigma_n} \exp\left(-\frac{(\omega-\mu_n)^2}{2\sigma_n^2}\right).$$

This low-parametric representation requires only $\mathcal{O}(1)$ memory.

The algorithm interprets the mean $\mu_n$ as the "position" of a random walker and $\sigma_n$ as the spread. On each measurement, the walker deterministically steps to the left or right by an amount proportional to the current standard deviation, dictated by the observed datum $d$; the spread contracts multiplicatively. Thus, the Bayesian inference process is mapped onto a one-dimensional Gaussian random walk with exponentially decaying step size (Granade et al., 2022).

3. Adaptive Experimental Protocol

Each adaptive step involves selecting experiment parameters ($t$, $\omega_\text{inv}$) based on the current Gaussian posterior. The experimental datum $d$ is sampled according to the likelihood:

$$L(d|\omega; t, \omega_\text{inv}) = \Pr(d|\omega) = \cos^2\left(\frac{t(\omega - \omega_\text{inv})}{2} + d \frac{\pi}{2} \right).$$

To minimize next-step posterior variance, optimal choices at step $n$ are:

$$t_n = \frac{1}{\sigma_n}, \qquad \omega_{\text{inv},n} = \mu_n.$$

Longer evolution times $t$ increase phase sensitivity but risk ambiguity unless the prior is sufficiently narrow. Centering the likelihood at $\mu_n$ maximizes information gain for the current posterior (Granade et al., 2022).

4. Bayesian Update and Random-Walk Recursion

After observing $d$ at step $n$, the (generally non-Gaussian) posterior is approximated by matching the first two moments to a new Gaussian. With rescaling to $\mu=0$, $\sigma=1$ and $s = (-1)^d$, the general update is:

• Mean:

$$\mu' = \frac{t\,\sin(t\,\omega_\text{inv})}{s\,e^{t^2/2} + \cos(t\,\omega_\text{inv})}$$

• Variance:

$${\sigma'}^2 = 1 - s t^2 \frac{e^{t^2/2}\cos(t\,\omega_\text{inv}) + s}{\left(e^{t^2/2} + s \cos(t\,\omega_\text{inv})\right)^2}$$

For the optimal adaptive choices $t = 1/\sigma$, $\omega_\text{inv} = \mu$, these reduce to the canonical random-walk update:

$$\begin{aligned} \mu_{n+1} &= \mu_n + (-1)^d \frac{\sigma_n}{\sqrt{e}} \ \sigma_{n+1} &= \sigma_n \sqrt{\frac{e-1}{e}} \end{aligned}$$

Thus, each measurement event deterministically shifts $\mu_n$ by $\pm \sigma_n/\sqrt{e}$, with $\sigma_n$ shrinking by a constant factor at every step (Granade et al., 2022).

5. Heisenberg-limited Scaling and Fisher Information

The Fisher information for each measurement is $$I = \mathbb{E}[(\partial_\omega \ln \Pr(d|\omega))^2] = t^2$$. As $\sigma_k = \sigma_0((e-1)/e)^{k/2}$ under the random-walk update, $t_k = 1/\sigma_k$ grows geometrically. The accumulated Fisher information over $n$ measurements is approximately

$$I_{\text{total}} = \sum_{k=0}^{n-1} t_k^2 \sim \frac{r^n - 1}{r-1},\ \text{where}\ r = e/(e-1).$$

Total experimental evolution time $T = \sum_k t_k$ grows similarly. Eliminating $n$, the estimation error satisfies $\sigma_n \sim 1/T$, manifesting Heisenberg-limited scaling—a fundamental lower bound for quantum parameter estimation (Granade et al., 2022).

6. Classical Computational Complexity and Online Realization

RW-IQPE achieves constant-time online updates. Each step requires only a few floating point operations: one reciprocal ($t=1/\sigma$), one multiplication for $\omega_\text{inv} = \mu$, two additions for the mean update, and a single multiplication for the variance contraction.

This yields the following performance characteristics:

Algorithm	Time per update (CPU)	Memory	Scaling with $\varepsilon$
RW-IQPE (Gaussian)	$\lesssim 1~\mu$s	$\mathcal{O}(1)$ ($\approx 250$ bits)	Constant
Particle filter (SMC)	$\sim 10$ ms	$N_\text{part} \in \mathcal{O}(1/\varepsilon^2)$	Linear in $N_\text{part}$

The low computational overhead and $\mathcal{O}(1)$ memory requirement directly enable real-time and embedded operation in FPGA-accelerated experiments, contrasting with the scaling overhead of classical SMC filters (Granade et al., 2022).

7. Practical Implementation, Limitations, and Safeguards

RW-IQPE presumes that posteriors remain approximately Gaussian and unimodal throughout inference. Rare failures may occur in strongly multimodal or very high-uncertainty regimes (initial uncertainty $\gg 2\pi$). To mitigate these, periodic "consistency checks" and "unwinding" procedures (as described in Alg. 2 of the source) can reverse a few steps and re-evaluate consistency.

Further requirements and constraints:

• Implementation must allow arbitrary $U(t)$ for real $t$, and ancilla rotation $\phi = -t\omega_\text{inv}$.
• The analysis holds under idealized qubit conditions (negligible decoherence). Moderate noise can be accommodated by adapting the likelihood, at the cost of reduced effective Fisher information.
• The method excludes cases where the Gaussian approximation fails persistently; performance remains near-optimal when this approximation holds.

In summary, RW-IQPE reduces the computational expense of online Bayesian phase estimation to minimal, constant per-datum operations, by mapping quantum inference onto a random-walk process over a Gaussian parameterization, and achieves Heisenberg-limited performance when applied adaptively (Granade et al., 2022).