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arXiv:1503.05028 · Physical Review A

Universal simulation of Markovian open quantum systems

Ryan Sweke, Ilya Sinayskiy, Denis Bernard, Francesco Petruccione

Published Jun 8, 2015• 67 citations
arXivHTMLPDF

Abstract

We consider the problem of constructing a ``universal set'' of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension $d$, we explicitly construct a universal set of semigroup generators, parametrized by ${d}^{2}\ensuremath{-}3$ continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a $d$-dimensional Markovian quantum system is the ability to implement (a) quantum channels from the semigroups generated by elements of the universal set of generators, and (b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems.

DOI
10.1103/physreva.91.062308

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We consider the problem of constructing a ``universal set'' of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension ddd, we explicitly construct a universal set of semigroup generators, parametrized by d2\ensuremath−3{d}^{2}\ensuremath{-}3d2\ensuremath−3 continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a ddd-dimensional Markovian quantum system is the ability to implement (a) quantum channels from the semigroups generated by elements of the universal set of generators, and (b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems.
A quantum system that has interaction with external resources, such as probability distribution, dissipation, and noise, is referred to as an open quantum system. Not only do open quantum systems play a vital role in the field of quantum physics, but they are also fundamental objects in quantum information and quantum computing. In this thesis, we focus on computational problems related to open quantum systems. In particular, we study efficient constructions of open quantum systems and their algorithmic applications. \n \nA unitary 2-design is a quantum analogue of universal 2-hash functions. It is an example of open quantum systems in the sense that it is a probability distribution of unitaries. As unitary 2-designs inherit many properties of the Haar randomness on the unitary group, they have many applications in quantum information, such as benchmarking and decoupling. We study the structures of unitary 2-designs and present efficient methods for their constructions. \n \nThe continuous-time evolution of a closed quantum system can be described by the Schrödinger equation. A natural generalization of the Schrödinger equation to Markovian open quantum systems, in the sense of generating dynamical semigroups, is called the Lindblad equation. We show that it is impossible for a simple reductionist approach to simulate Lindblad evolution with gate complexity that has linear dependence in evolution time. Moreover, we use a novel variation of the "linear combination of unitaries" construction that pertains to quantum channels to achieve the desired linear dependence in evolution time and poly-logarithmic dependence in precision. \n \nOpen quantum systems can also be used as building blocks of quantum algorithms. We present a dissipative query model, which is based on the amplitude damping process. With this dissipative query model, we provide a quantum algorithm that performs a fixed-point quantum search while preserving the quadratic speedup against classical algorithms.
Abstract The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators.
A sequence of controlled collisions between a quantum system and its environment (composed of a set of quantum objects) naturally simulates (with arbitrary precision) any Markovian quantum dynamics of the system under consideration. In this paper we propose and study the problem of simulation of an arbitrary quantum channel via collision models. We show that a correlated environment is capable to simulate non-Markovian evolutions leading to any indivisible qubit channel. In particular, we derive the corresponding master equation generating a continuous time non-Markovian dynamics implementing the universal NOT gate being an example of the most non-Markovian quantum channels.
A simple collision model is employed to introduce elementary concepts of open system dynamics of quantum systems. In particular, within the framework of collision models we introduce the quantum analogue of thermalization process called quantum homogenization and simulate quantum decoherence processes. These dynamics are driven by partial swaps and controlled unitary collisions, respectively. We show that collision models can be used to prepare multipartite entangled states. Partial swap dynamics generates W-type of entanglement saturating the CKW inequalities, whereas the decoherence collision models creates GHZ-type of entangled states. The considered evolution of a system in a sequence of collisions is described by a discrete semigroup E_1,...,E_n. Interpolating this discrete points within the set of quantum channels we derive for both processes the corresponding Lindblad master equations. In particular, we argue that collision models can be used as simulators of arbitrary Markovian dynamics, however, the inverse is not true.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.