Quantum memory advantage for quantum process tomography

2026-07-15

AI Summary

This work investigates whether quantum memory confers a query complexity advantage in quantum process tomography. Addressing the problem of learning an unknown quantum channel, the authors establish—for the first time—a rigorous polynomial separation between protocols with and without quantum memory. By combining information-theoretic lower bounds, explicit constructions of non-adaptive protocols without auxiliary systems, and diamond-norm accuracy analysis, they prove that even when memoryless protocols are allowed to be fully adaptive and employ arbitrary ancillary systems, quantum-memory-assisted protocols remain strictly more efficient. Specifically, the optimal query complexity for memoryless protocols scales as Θ(d_in³d_out³/ε²), whereas protocols equipped with quantum memory achieve Θ(d_in²d_out²/ε²), thereby demonstrating a clear learning separation rooted in the presence of quantum storage.

Abstract

Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be $Θ(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)$, where $d_{\mathrm{in}}$ and $d_{\mathrm{out}}$ are the channel input and output dimensions, respectively, and $\varepsilon$ is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires $Ω(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)$ queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound $O(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)$. Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case $d_{\mathrm{in}}=1$. By contrast, coherent protocols with quantum memory achieve query complexity $Θ(d_{\mathrm{in}}^2 d_{\mathrm{out}}^2/\varepsilon^2)$. Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.

Problem

• quantum process tomography
• quantum memory
• query complexity
• learning separation
• incoherent protocols

Innovation

• quantum memory
• quantum process tomography
• query complexity
• learning separation
• incoherent protocols