Optimizing Quantum Models of Classical Channels: The Reverse Holevo Problem
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Optimizing Quantum Models of Classical Channels: The Reverse Holevo Problem Samuel P. Loomis1 · John R. Mahoney1 · Cina Aghamohammadi1 · James P. Crutchfield1 Received: 29 October 2019 / Accepted: 28 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Given a classical channel—a stochastic map from inputs to outputs—the input can often be transformed into an intermediate variable that is informationally smaller than the input. The new channel accurately simulates the original but at a smaller transmission rate. Here, we examine this procedure when the intermediate variable is a quantum state. We determine when and how well quantum simulations of classical channels may improve upon the minimal rates of classical simulation. This inverts Holevo’s original question of quantifying the capacity of quantum channels with classical resources: We determine the lowest-capacity quantum channel required to simulate a classical channel. We also show that this problem is equivalent to another, involving the local generation of a distribution from common entanglement. Keywords Quantum information · Classical channel · Complexity
1 Introduction One speaks of a quantum advantage when a computational task is performed more efficiently (in memory, time, or both) using quantum mechanical hardware than classical hardware. Quantum advantages appear in the simulation of a variety of classical systems [1]: thermal states [2], fluid flows [3,4], electromagnetic fields [5], diffusion processes [6,7], Burger’s equation [8], and molecular dynamics [9]. Quantum advantage also has been found in more
Communicated by Christian Maes.
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James P. Crutchfield [email protected] Samuel P. Loomis [email protected] John R. Mahoney [email protected] Cina Aghamohammadi [email protected]
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Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA
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S. P. Loomis et al.
mathematical contexts. The most well-known problems include the factorization of prime numbers (Shor’s integer factoring algorithm [10]), database search (Grover’s algorithm [11]), and the efficient solution of linear systems [12]. A recent but rich area of study is the quantum advantage for simulating classical stochastic processes. By stochastic process, we mean a source which probabilistically generates sequences of symbols, x0 x1 . . . xt . When the probability of each new symbol xt depends on the previous symbols, the process is said to have memory. Computational mechanics has previously studied the memory resources required for simulating and predicting stochastic processes using classical hardware [13–15]. Quantum computational mechanics now proposes to use sequential measurements of a quantum system as a more resource-efficient means of simulating stochastic processes [16–20]. Recent work has shown that many important results about optimal simulation and prediction do not carry over from the classical to the quantum domain. For instance, it is a comm
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