The Mathematical Description of Steady-State Solutions for an Open Quantum System

Enej Ilievski, Bojan Žunkovič

We present a systematic approach for constructing steady state density operators of Markovian dissipative evolution for open quantum chain models with integrable bulk interaction and boundary incoherent driving. The construction is based on fundamental solutions of the quantum Yang–Baxter equation pertaining to quantum algebra symmetries and their quantizations ( q -deformations). In particular, we facilitate a matrix-product state description, by resorting to generic spin- s infinite-dimensional solutions associated with non-compact spins, serving as ancillary degrees of freedom. After formally deriving already known steady states for the anisotropic spin-1/2 Heisenberg chain from first symmetry principles, we obtain a class of solutions belonging to interacting quantum gases with SU ( N )-symmetric Hamiltonians, using a restricted set of incoherent boundary jump processes, and point out how new non-trivial generalizations emerge from twists of quantum group structures. Finally, we discuss the possibilities for the analytical calculation of observables by employing the algebraic properties of associated auxiliary vertex operators.

Rodrigo A. Vargas–Hernández, Ricky T. Q. Chen, Kenneth A. Jung +1 more

Inverse design of a property that depends on the steady-state of an open quantum system is commonly done by grid-search type of methods. In this paper we present a new methodology that allows us to compute the gradient of the steady-state of an open quantum system with respect to any parameter of the Hamiltonian using the implicit differentiation theorem. As an example, we present a simulation of a spin-boson model where the steady-state solution is obtained using Redfield theory.

Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu +1 more

The solutions of the problems related to open quantum systems have attracted considerable interest. We propose a variational quantum algorithm to find the steady state of open quantum systems. In this algorithm, we employ parameterized quantum circuits to prepare the purification of the steady state and define the cost function based on the Lindblad master equation, which can be efficiently evaluated with quantum circuits. We then optimize the parameters of the quantum circuit to find the steady state. Numerical simulations are performed on the one-dimensional transverse field Ising model with dissipative channels. The result shows that the fidelity between the optimal mixed state and the true steady state is over 99%. This algorithm is derived from the natural idea of expressing mixed states with purification and it provides a reference for the study of open quantum systems.

裕樹 真島