Strategy optimization for Bayesian quantum parameter estimation with finite copies: Adaptive greedy, parallel, sequential, and general strategies

TL;DR

This work presents a universal Bayesian framework for quantum parameter estimation with a finite number of channel uses, unifying parallel, sequential, ICO, and adaptive greedy strategies through testers and higher-order operations. It formulates the optimization as a semidefinite program, enabling efficient numerical design of probe states, control maps, measurements, and estimators across diverse priors and multi-parameter tasks. The authors demonstrate that performance hierarchies among protocol classes are highly problem-dependent, with strict separations in noisy or dissipative scenarios and occasional parity with parallel strategies in ideal unitary encoding; adaptive greedy strategies can closely approximate memory-assisted optimum in some finite-copy regimes. The results provide a practical toolkit for finite-data quantum metrology, clarifying when quantum memory is advantageous and offering scalable numerical methods to tailor protocols to specific tasks and priors.

Abstract

In this work, we study Bayesian quantum parameter estimation given a finite number of uses of the process encoding one or more unknown physical quantities. For multiple uses, it is conventional to classify quantum metrological protocols as parallel, sequential, or indefinite causal order. Within each class, the central question is to determine the optimal strategy -- namely, the choice of optimal input state, control operations, measurement, and estimator(s) -- to perform the estimation task. Using the formalism of higher-order operations, we develop an algorithm that looks for the optimal solution, and we provide an efficient numerical implementation based on semidefinite programming. Our benchmark examples, specifically those against existing analytical solutions, demonstrate how powerful and precise our method is. We further explore the potential of greedy adaptive strategies, which are based on classical feedforward to design the optimal protocol for the next round. Using this framework, we compare the optimal achievable Bayesian score across classes. We demonstrate the strength of our algorithm in several examples, from single to multiparameter estimation and with various prior distributions. Particularly, we find examples in which there is a strict hierarchy between different classes. Nonetheless, the performance of the different quantum memory-assisted classes are not significantly different, while they may significantly outperform the adaptive greedy strategy.

Figures (5)

• Figure 1: Schematic representation of the different strategies considered for Bayesian quantum parameter estimation with $k$ uses of a parameter-encoding channel $\Lambda_{\boldsymbol{\theta}}$. (a) Parallel strategy$W^{\mathrm{par}}$: a probe state $\rho$ (possibly entangled with an auxiliary system $\mathrm{aux}$) is sent through $k$ parallel copies of $\Lambda_{\boldsymbol{\theta}}$, followed by a joint measurement $M$. (b) Sequential strategy$W^{\mathrm{seq}}$: the channel uses are interleaved with intermediate operations $\Phi_1,\ldots,\Phi_{k-1}$ acting on the system and ancillary spaces, with a final measurement $M$. (c) General / ICO strategy$W^{\mathrm{gen}}$: a general higher-order operation connects the $k$ calls to $\Lambda_{\boldsymbol{\theta}}$ without a fixed causal order, before a final measurement $M$ is applied. (d) Adaptive greedy strategy: protocols are implemented in rounds with classical feedforward (no quantum memory between rounds), updating the prior and redesigning the next-round strategy based on previous measurement outcomes.
• Figure 2: Comparison between different strategies in the SU(2) multiparameter estimation. The optimal protocol (blue dashed) outperforms the greedy strategy (solid green) which in turn outperforms the non-adaptive and non-memory-assisted strategy (gray solid). The results suggest that the adaptive greedy strategy with $k=5$ calls is as good as the best quantum memory-assisted strategy with $k=4$ calls. The adaptive greedy and the non-adaptive scores are obtained using $N_m = 10^4$ random Monte Carlo simulations, with $N_\mathrm{H} = 8000$ and $N_\mathrm{O} = 27$.
• Figure 3: Minimum approximate score $\widetilde{\mathcal{S}}$ in a thermometry estimation task for a greedy algorithm (green solid line) compared to a sequential strategy (red solid line) for $2$ copies of the channel. For the optimization we have taken $N_\mathrm{H} = 2500$ and $N_\mathrm{O} = 20 ~(N_O=4)$ outputs for the quantum memory-assisted (adaptive greedy/non-adaptive) strategies, respectively. For the adaptive greedy/non-adaptive strategy, the simulation has been run for $N_m = 10^6$ Monte Carlo iterations. The prior is given by a uniform distribution in the range $\theta/\epsilon \in [1, 20]$.
• Figure 4: Maximum approximate score $\widetilde{\mathcal{S}}$ in a SU(2) estimation task followed by an AD channel as a function of $p$. We zoom in around $p=0.5$ to showcase the gaps between strategies. The simulation has been run for $N_m = 10^4$ Monte Carlo iterations, $N_\mathrm{H} = 8000$ and $N_\mathrm{O} = 1000 ~(N_O=27)$ outputs for the quantum memory-assisted (adaptive greedy/non-adaptive) strategies.
• Figure 5: Maximum approximate score $\widetilde{\mathcal{S}}_\mathrm{cos}$ in a phase estimation + AD task for $2$ copies of the channel, both for a (a) flat prior ($\alpha = -100$) and (b) sharp prior ($\alpha = 100$).