Sequential Capacity of Quantum Processes with Finite Memory

2026-10-01

AI Summary

This study investigates the response complexity and distinguishability of fixed-memory quantum devices over extended runtimes. Through adaptive test sequence construction, time-varying phase rotation control, Pauli noise modeling, and information-theoretic analysis, we demonstrate that single-qubit phase rotations without auxiliary memory achieve logarithmic enhancements, while noisy capacity bounds are governed by the residual phase-flip probability. Furthermore, this work establishes that the sequential response capacity under fixed resolution scales as O(K log K), significantly surpassing the linear upper bound inherent to classical processes. By identifying the coherence timescale as the fundamental limiting factor, our findings provide a novel theoretical foundation for demonstrating quantum advantage in constrained-memory settings.

Abstract

How complex can the responses of a quantum device become as it runs longer with a fixed internal memory? We quantify this complexity through sequential response capacity: how many adaptive testing stages, each using a fresh run, can continue to separate possible processes by a prescribed gap in response probabilities. For fixed system and memory sizes, we establish a tight law relating this capacity to run length and probability resolution. At fixed resolution, the capacity grows on the order of $K\log K$, where $K$ is the number of time steps in each run. Our construction attains this growth using time-dependent phase rotations on a single visible qubit with no additional internal memory; its tests give response probabilities exactly zero or one. Under the same tests, classical stochastic processes that measure in a fixed basis at every step have only linear capacity at fixed sizes and resolution. For phase sequences selected by a stored classical label, we then quantify how known independent Pauli noise changes this logarithmic enhancement. With ideal controls and weak residual phase noise after correction, we prove matching capacity bounds at a fixed small probability gap. These bounds identify the inverse residual phase-flip probability as the coherence timescale that limits the extra logarithmic growth.

Problem

• Quantum processes
• Sequential capacity
• Finite memory
• Response complexity
• Coherence timescale

Innovation

• Sequential response capacity
• Quantum processes
• Finite memory
• Phase rotations
• Pauli noise