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Physical Review A

Reversibility and quantum coherence in one-dimensional quantum cellular automata

Federico Centrone, Camillo Tassi, Marco Barbieri, Alessio Serafini

Published Jul 3, 2018• 1 citation
Publisher

Abstract

Quantum cellular automata are important tools in understanding quantum dynamics, thanks to their simple and effective list of rules. Here we consider a class of noisy, one-dimensional quantum cellular automata that allow one to shift from unitary dynamics to completely positive maps, investigating the evolution of coherence as well as the notion of reversibility in such a setting. To this aim, we associate an approximate reverse automaton to each noisy automaton, and assess its effect, and we define an irreversibility time based on the distance from the maximally mixed state, which is shown to be the only attractor of the automaton map in the presence of dephasing. Our analysis illustrates the interplay between unitary and noisy dynamics.

DOI
10.1103/physreva.98.012105

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Quantum cellular automata are important tools in understanding quantum dynamics, thanks to their simple and effective list of rules. Here we consider a class of noisy, one-dimensional quantum cellular automata that allow one to shift from unitary dynamics to completely positive maps, investigating the evolution of coherence as well as the notion of reversibility in such a setting. To this aim, we associate an approximate reverse automaton to each noisy automaton, and assess its effect, and we define an irreversibility time based on the distance from the maximally mixed state, which is shown to be the only attractor of the automaton map in the presence of dephasing. Our analysis illustrates the interplay between unitary and noisy dynamics.
We model the transport of an unknown quantum state on one dimensional qubit lattices by means of a quantum cellular automata evolution. We do this by first introducing a class of discrete noisy dynamics, in the first excitation sector, in which a wide group of classical stochastic dynamics is embedded within the more general formalism of quantum operations. We then extend the Hilbert space of the system to accommodate a global vacuum state, thus allowing for the transport of initial on-site coherences besides excitations, and determine the dynamical constraints that define the class of noisy quantum cellular automata in this subspace. We then study the transport performance through numerical simulations, showing that for some instances of the dynamics perfect quantum state transfer is attainable. Our approach provides one with a natural description of both unitary and open quantum evolutions, where the homogeneity and locality of interactions allow one to take into account several forms of quantum noise in a plausible scenario.
We model the transport of an unknown quantum state on one dimensional qubit lattices by means of a quantum cellular automata (QCA) evolution. We do this by first introducing a class of discrete noisy dynamics, in the first excitation sector, in which a wide group of classical stochastic dynamics is embedded within the more general formalism of quantum operations. We then extend the Hilbert space of the system to accommodate a global vacuum state, thus allowing for the transport of initial on-site coherences besides excitations, and determine the dynamical constraints that define the class of noisy QCA in this subspace. We then study the transport performance through numerical simulations, showing that for some instances of the dynamics perfect quantum state transfer is attainable. Our approach provides one with a natural description of both unitary and open quantum evolutions, where the homogeneity and locality of interactions allow one to take into account several forms of quantum noise in a plausible scenario.
Abstract Modeling noisy discrete systems utilizing conservative reversible elementary cellular automata (CRECA) is introduced. Reversibility in the ECA evolution rule and reversibility in the ECA circuit implementation are both achieved using a new Swap-based algorithm called Swap-based CRECA (SCRECA). The new method results in adding variable redundancy to counteract the effect of noise. The problem of obtaining a reversible map from an irreversible map is important because quantum circuits are inherently reversible and thus does not consume power, while irreversible circuits (due to either an irreversible mapping or to noise within a reversible mapping) cannot exist in the quantum domain and its circuits consume power. Since noise is an integral part of any real process and since the reduction of power consumption is a main requirement for the circuit design of future technologies such as in quantum computing (QC), the main features of several future technologies will include reversibility, and thus the...
We consider discrete and integer-valued cellular automata (CA). A particular class of which comprises "Hamiltonian CA" with equations of motion that bear similarities to Hamilton's equations, while they present discrete updating rules. The dynamics is linear , quite similar to unitary evolution described by the Schrödinger equation. This has been essential in our construction of an invertible map between such CA and continuous quantum mechanical models, which incorporate a fundamental discreteness scale. Based on Shannon's sampling theory, it leads, for example, to a one-to-one relation between quantum mechanical and CA conservation laws. The important issue of linearity of the theory is examined here by incorporating higher-order nonlinearities into the underlying action. These produce inconsistent nonlocal (in time) effects when trying to describe continuously such nonlinear CA. Therefore, in the present framework, only linear CA and local quantum mechanical dynamics are compatible.