Quantum sequential parameter testing

Abstract

Sequential strategies in hypothesis testing use a variable number of measurement rounds, allowing a decision to be made as soon as the observed data provide a prescribed level of error tolerance. Although sequential testing is well established for a discrete set of hypotheses, extending this framework to a continuous parameter space poses additional challenges and has remained largely unexplored. In this article, we introduce sequential \textit{parameter testing}, a framework for determining an unknown parameter up to a prescribed tolerance by ruling out sufficiently distant competing values with a target error tolerance. We clarify its operational distinction from conventional parameter estimation and develop sequential tests for continuous families of hypotheses, introducing the \textit{twin-peaks test} as a natural and computationally efficient analog of sequential likelihood-ratio testing. We apply the framework to two paradigmatic quantum tasks: testing the phase and the purity of a qubit. For phase testing, we show numerically that adaptive projective measurements achieve the same average sample cost as collective covariant measurements with a fixed number of copies. For purity testing, local measurements are optimal, and sequential parameter testing yields significant average sample savings over fixed sample-size protocols. Our results establish parameter testing as an operationally meaningful framework for resource-efficient certification tasks involving continuous parameters.

The paper studies sequential inference of a continuous parameter encoded in quantum states: measurements continue until an estimate is sufficiently separated from alternatives outside a prescribed tolerance $\delta$. It formulates three stopping rules: a complement test based on posterior mass, a concentration test comparing peak and average density, and a “twin-peaks” test comparing the MAP density with the largest density outside the tolerance region. For regular models, the twin-peaks mean stopping time is asymptotically $\mathbb E[N\mid\vartheta]\sim\log A/\inf_{|\vartheta-\vartheta_0|\geq\delta}D(\vartheta\|\vartheta_0)$, reducing locally to $2\log A/[I(\vartheta)\delta^2]$. Laplace-principle arguments relate the three tests at exponential order. For equatorial qubit phase testing, the covariant single-copy measurement has KL divergence $2\sin^2(\delta/2)$. Simulations find that adaptive projective measurements match the average copy cost of a fixed-size collective covariant measurement. For qubit purity, represented by Bloch length $r$, measurements along a known Bloch direction are optimal, with $\mathbb E[N\mid r]\sim2(1-r^2)\log A/\delta^2$. Relative to the parameter-uniform fixed-size benchmark, prior-averaged costs are factors $2/3$, $2/5$, and $1/4$ for flat, Hilbert–Schmidt, and Bures priors. With unknown direction, weak Schur measurements on growing blocks asymptotically recover the known-direction rate. The twin-peaks threshold is an evidence criterion, not an exact finite-sample error guarantee. Stopping-time claims require regularity and integrability assumptions; finite-threshold corrections remain visible. Fixed-size comparisons use an averaged ambiguity surrogate and concentration approximations. Some purity simulations are capped at 15,000 copies, and block-size optimization is left open.