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arXiv:1405.6049 · Physical Review A

Simulation of single-qubit open quantum systems

Ryan Sweke, Ilya Sinayskiy, Francesco Petruccione

Published Aug 27, 2014• 34 citations
arXivHTMLPDF

Abstract

A quantum algorithm is presented for the simulation of arbitrary Markovian dynamics of a qubit, described by a semigroup of single-qubit quantum channels ${{T}_{t}}$ specified by a generator $\mathcal{L}$. This algorithm requires only single-qubit and controlled-not gates and approximates the channel ${T}_{t}={e}^{t\mathcal{L}}$ up to the chosen accuracy $\ensuremath{\epsilon}$, with a slightly superlinear cost ${O((\ensuremath{\parallel}\mathcal{L}\ensuremath{\parallel}}_{(1\ensuremath{\rightarrow}1)}t{)}^{1+1/2k}/{\ensuremath{\epsilon}}^{1/2k})$ for any integer $k$. Inspired by developments in Hamiltonian simulation, a decomposition and recombination technique is utilized which allows for the exploitation of recently developed methods for the approximation of arbitrary single-qubit channels. In particular, as a result of these methods the algorithm requires only a single ancilla qubit, the minimal possible dilation for a nonunitary single-qubit quantum channel.

DOI
10.1103/physreva.90.022331

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A quantum algorithm is presented for the simulation of arbitrary Markovian dynamics of a qubit, described by a semigroup of single-qubit quantum channels Tt{{T}_{t}}Tt​ specified by a generator L\mathcal{L}L. This algorithm requires only single-qubit and controlled-not gates and approximates the channel Tt=etL{T}_{t}={e}^{t\mathcal{L}}Tt​=etL up to the chosen accuracy \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, with a slightly superlinear cost O((\ensuremath∥L\ensuremath∥(1\ensuremath→1)t)1+1/2k/\ensuremathϵ1/2k){O((\ensuremath{\parallel}\mathcal{L}\ensuremath{\parallel}}_{(1\ensuremath{\rightarrow}1)}t{)}^{1+1/2k}/{\ensuremath{\epsilon}}^{1/2k})O((\ensuremath∥L\ensuremath∥(1\ensuremath→1)​t)1+1/2k/\ensuremathϵ1/2k) for any integer kkk. Inspired by developments in Hamiltonian simulation, a decomposition and recombination technique is utilized which allows for the exploitation of recently developed methods for the approximation of arbitrary single-qubit channels. In particular, as a result of these methods the algorithm requires only a single ancilla qubit, the minimal possible dilation for a nonunitary single-qubit quantum channel.
In the study of open quantum systems, one typically obtains the decoherence dynamics by solving a master equation. The master equation is derived using knowledge of some basic properties of the system, the environment, and their interaction: One basically needs to know the operators through which the system couples to the environment and the spectral density of the environment. For a large system, it could become prohibitively difficult to even write down the appropriate master equation, let alone solve it on a classical computer. In this paper, we present a quantum algorithm for simulating the dynamics of an open quantum system. On a quantum computer, the environment can be simulated using ancilla qubits with properly chosen single-qubit frequencies and with properly designed coupling to the system qubits. The parameters used in the simulation are easily derived from the parameters of the system +++ environment Hamiltonian. The algorithm is designed to simulate Markovian dynamics, but it can also be used to simulate non-Markovian dynamics provided that this dynamics can be obtained by embedding the system of interest into a larger system that obeys Markovian dynamics. We estimate the resource requirements for the algorithm. In particular, we show that for sufficiently slow decoherence a single ancilla qubit could be sufficient to represent the entire environment, in principle.
One of the first proposals for the use of quantum computers was the simulation of quantum systems. Over the past three decades, great strides have been made in the development of algorithms for simulating closed quantum systems and the more complex open quantum systems. In this tutorial, we introduce the methods used in the simulation of single qubit Markovian open quantum systems. It combines various existing notations into a common framework that can be extended to more complex open system simulation problems. The only currently available algorithm for the digital simulation of single qubit open quantum systems is discussed in detail. A modification to the implementation of the simpler channels is made that removes the need for classical random sampling, thus making the modified algorithm a strictly quantum algorithm. The modified algorithm makes use of quantum forking to implement the simpler channels that approximate the total channel. This circumvents the need for quantum circuits with a large number of CNOT gates. Quanta 2023; 12: 131–163.
Quantum simulation, which is generically the task to employ quantum computers to simulate quantum physical models, has been one of the most significant motivations of quantum computing. Quantum dynamics, unitary or nonunitary Markovian dynamics driven by local interactions, has been proved to be efficiently simulatable on quantum computers. Extending the underlying models from unitary to nonunitary evolution, and from continuous-time to discrete-time evolution is essential not only for quantum simulation of more general processes, e.g., dissipative processes with evident non-Markovian effects, but also for developing alternative quantum computing models and algorithms. In this thesis, we explore quantum simulation problems mainly from the following three themes. First, we extend quantum simulation framework of Hamiltonian-driven evolution to quantum simulation of quantum channels, combined with the scheme of algorithmic simulation that accepts a promised simulation accuracy, hence algorithmic quantum channel simulation. Our simulation scheme contains a classical preprocessing part, i.e. a classical algorithm for quantum-circuit design, and a quantum part that is the execution of a quantum circuit for the quantum channel simulator. Second, we employ simulation algorithms for arbitrary quantum channels beyond the dilation method. We explore channel decomposition in terms of convex combination of smaller channels, known as generalized extreme channels, which is known as a nontrivial open problem. To attack this problem, we develop an optimization algorithm for approximate decomposition into a convex sum of generalized extreme channels. We provide an ansatz that proves to be able to yield arbitrary generalized extreme channels and allows a precise quantum circuit description. Furthermore, our numerical simulation has demonstrated the validity of our optimization algorithm for low-dimensional quantum channels. Third, by considering quantum simulation problems beyond the rough distinction between digital and analog simulations, and beyond the quantum-state generation problem, we define uniform, strong, and weak quantum simulations from the point view of operator topology. Besides strong simulation of quantum channels, we define a general weak quantum simulation problem, which simulates observable effects instead of the effects on state generation. Also we study the channel simulation problem in the quantum query model, and provide the query complexity by employing uniform quantum simulation method.
Digital quantum simulation on quantum systems require algorithms that can be implemented using finite quantum resources. Recent studies have demonstrated digital quantum simulation of open quantum systems on Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we develop quantum circuits for optimal simulation of Markovian and Non-Markovian open quantum systems. The circuits use ancilla qubits to simulate the environment, and memory effects are induced by storing information about the system on extra qubits. We simulate the amplitude damping channel and dephasing channel as examples of the framework and infer (Non-)Markovianity from the (non-)monotonic behaviour of the dynamics. Further, we develop a method to optimize simulations by decomposing complex open quantum dynamics into smaller parts, that can be simulated using a small number of qubits. We show that this optimization reduces quantum space complexity from O(l)O(l)O(l) to O(1)O(1)O(1) for simulating the environment.
Inspired by the Solovay-Kitaev decomposition for approximating unitary operations as a sequence of operations selected from a universal quantum computing gate set, we introduce a method for approximating any single-qubit channel using single-qubit gates and the controlled-not (cnot). Our approach uses the decomposition of the single-qubit channel into a convex combination of "quasiextreme" channels. Previous techniques for simulating general single-qubit channels would require as many as 20 cnot gates, whereas ours only needs one, bringing it within the range of current experiments.
We present a quantum algorithm for simulating open quantum systems coupled to Gaussian environments valid for any configuration and coupling strength. The algorithm is applicable to problems with strongly coupled, or non-Markovian, environments; problems with multiple environments out of mutual equilibrium; and problems with time-dependent Hamiltonians. We show that the algorithm can reproduce the true dynamics of such problems at arbitrary accuracy and, for a broad range of problems, only adds a minor resource cost relative to Trotterized time evolution; the cost is low-degree polynomial in the inverse target accuracy. The algorithm is based on the insight that any Gaussian environment can be represented as a train of ancillary qubits that sequentially interact with the system through a time-local coupling, given by the convolution square root of the bath correlation function; this is a secondary result of our work. In numerical simulations of a spin-boson model, we find accurate simulations can be achieved with moderate ( 10 1 – 10 3 ) ancilla counts in the intermediate to strong-coupling regimes. Our results open up applications of quantum computers for efficient simulation of nonequilibrium and open quantum systems.