PHYSICAL REVIEW X 11, 021019 (2021)
Quantum Advantage in Simulating Stochastic Processes
‡ ‡ † Kamil Korzekwa 1,2, ,* and Matteo Lostaglio 3,4, , 1Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland 2International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland 3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain 4QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
(Received 20 May 2020; revised 5 November 2020; accepted 3 March 2021; published 22 April 2021)
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 P to the quantum domain, we prove an advantage of the quantum cost of simulating P 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.
DOI: 10.1103/PhysRevX.11.021019 Subject Areas: Quantum Physics, Quantum Information
I. INTRODUCTION used to simulate a stochastic process that classically requires three bits [8]) and theoretically for a certain class A. Memory advantages of Poisson processes [9]. What tasks can we perform more efficiently by employ- Here, we take a complementary approach starting ing quantum properties of nature? And what are the from the following simple observation: Although all funda- quantum resources powering them? These are the central mental interactions are memoryless, the basic information- questions that need to be answered not only to develop processing primitives (such as the bit-swap operation) cannot novel quantum technologies but also to deepen our under- be performed classically in a time-continuous fashion with- standing of the foundations of physics. Over the last few out employing implicit microscopic states that act as a decades, these questions were successfully examined in the memory [10]. We show that this picture changes dramati- context of cryptography [1], computing [2], simulations cally if instead we consider memoryless quantum dynamics. [3], and sensing [4], proving that the quantum features of This difference is due to quantum coherence, arising from nature can indeed be harnessed to our benefit. the superposition principle, which can effectively act as an More recently, an area of active theoretical and exper- internal memory of the system during the evolution. imental interest focused on the memory advantages offered by quantum mechanics for the simulation of stochastic B. Classical vs quantum processes in the setting of classical causal models [5–8].An experimentally accessible and relevant measure of such an But what do we really mean when we say that a bit-swap advantage is the dimensionality of the memory required for (or other information-processing tasks) cannot be per- the simulation [8,9]. These dimensional advantages have formed classically in a memoryless way? First, when we been identified experimentally (a qubit system has been speak of a bit, we mean a fundamentally two-level system, i.e., a system with only two microscopic degrees of freedom (e.g., a spin-1=2 particle) and not a macroscopic *[email protected] object with a coarse-grained description having two states † [email protected] (e.g., a piece of iron magnetized along or against the z axis). ‡ These authors contributed equally to this work. Otherwise, if the system merely implements a bit in a higher-dimensional state space of dimension d>2, the Published by the American Physical Society under the terms of internal degrees of freedom can be used as a memory and a the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to bit swap can be performed, as illustrated in Fig. 1(a) for the author(s) and the published article’s title, journal citation, d ¼ 3. Thus, when we speak of classical systems, we take and DOI. them to be fundamentally d dimensional, and when we
2160-3308=21=11(2)=021019(20) 021019-1 Published by the American Physical Society KAMIL KORZEKWA and MATTEO LOSTAGLIO PHYS. REV. X 11, 021019 (2021)
information about the past. For classical systems with a continuous phase space, it is then difficult to properly assess the number of memory states used during a given evolution. Hence, in this work, we only focus on discrete systems. Ultimately, we are then interested in a stochastic process in which discrete outputs i are observed for given discrete inputs j. This process is characterized by a matrix of transition probabilities Pijj. In the classical setting, we want to know whether there exists a classical memoryless dynamics (described by a Markovian master equation) involving these states that outputs Pijj after some time. Quantum mechanically, we are similarly asking whether a quantum memoryless dynamics (described by a Markovian quantum master equation) can output Pijj after some time. In this work, we highlight that these two questions admit very different answers, both in terms of which Pijj can arise from memoryless processes and in terms of the memory FIG. 1. Space-time cost for classical and quantum bit swap. required to achieve a given P . (a) Space-time optimal realization of a bit swap, i.e., a trans- ijj position between two states (solid-line boxes), using one memory For concreteness, assume Pijj results from a thermal- state (dashed-line boxes) and three time steps. Each time step is ization process, which typically satisfies the so-called composed of a continuous memoryless dynamics that does not detailed balance condition. Physically, we are then asking affect one of the states and maps the remaining two to one of whether the observed Pijj is compatible or not with a process them. (b) In the quantum regime, a bit swap can be performed involving no memory effects, such as information backflows without any memory, simply by a time-continuous unitary from the environment [11]. Classically, P originates from σ ijj process expði xtÞ that continuously connects the identity oper- incoherent jumps induced by interacting with the environ- ation at time t ¼ 0 with the bit swap, represented by Pauli x ment (absorbing or emitting energy). Alternatively, we can operator σx, at time t ¼ π=2. During the process, the information about the initial state of the system is preserved in quantum see classical dynamics as the evolution of a quantum coherence. system that undergoes very strong decoherence at all times, so that any nonclassical effects are killed right away. In fact, standard quantum thermalization models speak of memoryless dynamics, we mean probabilistic (weak coupling with a very large thermal bath) can also be jumps between these discrete states occurring at rates understood in this way since they are unable to generate ’ independent of the system s past. Of course, this is just quantum superpositions of energy states. As soon as we the standard setting for classical Markov processes. Then, move away from this semiclassical limit, however, we see for every finite dimension d, there exist processes that that more exotic thermalization processes can generate cannot be performed in a memoryless fashion. P that, classically, would necessarily signal memory This setting should also be contrasted with classical ijj effects but that, quantum mechanically, can emerge from systems with a continuous phase space. The simplest memoryless processes due to quantum coherence. Note example is probably given by an isolated classical pendu- that the term “memoryless” is used throughout the paper lum, i.e., a harmonic oscillator. The state of the system is as a synonym of Markovian, i.e., that the evolution only then given by a point in (more generally, distribution over) depends on the current state of the system and not on its a two-dimensional phase space x; p describing its position ½ history. Such evolution may still require an auxiliary clock x relative to the equilibrium position, and its momentum p. system (used, e.g., to know how much longer the system If we now identify the pendulum’s state −x0; 0 with a bit ½ should be coupled to an external control field) and a state 0 and x0; 0 with a bit state 1, the time evolution of the ½ counter system (used to record the current channel in the system clearly performs a bit-swap operation within half a sequence of channels necessary to implement the given period. However, a pendulum is not a simple two-level dynamics). These constitute extra resources that one may system but rather a system with a continuously infinite want to separately account for, e.g., using the framework number of states. Thus, the bit swap is performed by of quantum clocks (see, e.g., Ref. [12]). employing infinitely many ancillary memory states: ½−x0; 0 evolves through states with x>−x0 and positive C. Summary of results momenta p to ½x0; 0 , while ½x0; 0 evolves through states with x