Quantum algorithm for simulating the dynamics of an open quantum system

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.

Figure 1

(a) Quantum circuit for simulating the Markovian dynamics of an open quantum system coupled to a time-independent environment. The first register represents the open system and the second register represents the environment. (b) Quantum circuit for simulating the Markovian dynamics of an open system coupled to a time-dependent environment.

Figure 2

Quantum circuit for a quantum estimator. $H$ represents the Hadamard gate and swap represents the swap gate.

Figure 3

The evolution of the matrix element ${\rho }_{\mathit{ee}}$ of a two-level system in a spin bath, where ${\rho }_{\mathit{ee}}$ denotes the diagonal element of the density matrix that describes the population of the excited state. The frequency of the environment mode in the region $\omega /{\Delta }_0\in [0.8\text{,}1.15]$ is divided into eight elements with equal width and each element is represented by a qubit. The Hamiltonian for the global system is given by Eq. (41). The unitary evolution time ${\Delta }_0\tau =30$. The red solid line represents the analytical result for the evolution of ${\rho }_{\mathit{ee}}$ with the initial state $|e\rangle$. The black square dots represent the simulated results for the evolution of ${\rho }_{\mathit{ee}}$.

Figure 4

The relaxation rate of the two-level system as a function of its frequency. The black squares represent the results obtained from the simulation of the algorithm; the blue dotted line represents the relaxation rate obtained by using the spectral density that is spanned by eight $\delta$ peaks in Eq. (44); and the red solid line represents the relaxation rate obtained from the analytical spectral density in Eq. (42). Obviously only the first two ones agree well with each other.

Figure 5

The relaxation rate of the two-level system as a function of the frequency of the two-level system. The black square dots represent the results obtained from the simulation of the algorithm using the improved coupling coefficients as discussed in the text; the blue dotted line represents the relaxation rate obtained by using the spectral density that is produced by eight $\delta$ peaks in Eq. (44) using the improved coupling coefficients; and the red solid line represents the relaxation rate obtained from the analytical spectral density in Eq. (42).