Abstract
We investigate the problem of simulating classical stochastic processes through quantum dynamics and present three scenarios where memory or time quantum advantages arise. First, by introducing and analyzing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory. Second, by extending the notion of space-time cost of a stochastic process to the quantum domain, we prove an advantage of the quantum cost of simulating over the classical cost. Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.
- Received 20 May 2020
- Revised 5 November 2020
- Accepted 3 March 2021
DOI:https://doi.org/10.1103/PhysRevX.11.021019
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
For the past three decades, scientists have been harnessing the quantum features of nature to challenge the best-known classical methods of computing, metrology, and cryptography. In all these cases, quantum superposition allows one to perform certain tasks faster, more precisely, or more securely. Here, we report and investigate a novel quantum advantage that reduces the amount of memory and time needed to implement certain dynamical processes, such as thermodynamic cooling or information processing.
In our work, we introduce a unifying framework to theoretically analyze memory and time costs from a mathematical, computational, and thermodynamical point of view, and we apply it to study three distinct scenarios where quantum advantages arise. First, we show that there exist stochastic processes that cannot be simulated classically without memory but can be implemented “quantumly” in a memoryless fashion. Second, we prove that for processes that require memory even in the quantum regime, there is an improvement over classical computers in the number of times the memory must be accessed or in its size. Third, we demonstrate that memoryless quantum processes allow for much better control of the system’s state than their classical counterparts.
Our work provides new tools to quantify the extent to which quantum superposition can replace the role that classically is played by memory. This shows that quantum mechanics provides powerful models to simulate stochastic processes and paves the way to the investigation of practical advantages for quantum information processing.
