Characterization and Verification of Trotterized Digital Quantum Simulation via Hamiltonian and Liouvillian Learning

Abstract

The goal of digital quantum simulation is to approximate the dynamics of a given target Hamiltonian via a sequence of quantum gates, a procedure known as Trotterization. The quality of this approximation can be controlled by the so called Trotter step, that governs the number of required quantum gates per unit simulation time, and is intimately related to the existence of a time-independent, quasilocal Hamiltonian that governs the stroboscopic dynamics, refered to as the Floquet Hamiltonian of the Trotterization. In this work, we propose a Hamiltonian learning scheme to reconstruct the implemented Floquet Hamiltonian order-by-order in the Trotter step: this procedure is efficient, i.e., it requires a number of measurements that scales polynomially in the system size, and can be readily implemented in state-of-the-art experiments. With numerical examples, we propose several applications of our method in the context of verification of quantum devices, from the characterization of the distinct sources of errors in digital quantum simulators to the design of new types of quantum gates. Furthermore, we show how our approach can be extended to the case of non-unitary dynamics and used to learn Floquet Liouvillians, thereby offering a way of characterizing the dissipative processes present in NISQ quantum devices.

Digital quantum simulation, learning of the Floquet Hamiltonian, and quantum chaos of the kicked top

“…We emphasize that the method presented here applies also to the case of DQS of genuinely many-body systems [24], of which the kicked top can be seen as a limit where the dynamics, for suitable initial conditions, admits a description in terms of collective spin operators [see Sec. 2.1].…”

“…A detailed analysis of the influence of measurement noise and imperfections is beyond the scope of this work, and we refer the reader to Ref. [24] for a detailed study of realistic applications of this protocol in many-body DQS.…”

“…The discussion is based on the HL framework for the characterization of Trotterized DQS of many-body systems developed by us in Ref. [24] ‡, which we adapt and extend here to study the Trotter threshold and the transition to quantum chaos in the kicked top [2]. HL is presently being developed as a new tool in quantum information theory in the context of quantum many-body systems and quantum simulation [25,26,27,28,29,30,31,32,33,34,35,36,37].…”

Non-Abelian Floquet Spin Liquids in a Digital Rydberg Simulator

“…Finally, we note that Hamiltonian learning methods [40,41] could be used to confirm the preparation of the non-abelian phase without the need for sophisticated control of the excitations. Similarly, variational state preparation combined with effective Hamiltonian evolution suffice to probe the chiral nature of the edge modes.…”

Recovering models of open quantum systems from data via polynomial optimization: Towards globally convergent quantum system identification

“…Several methods have been developed for identifying quantum systems in various contexts [12][13][14][15][16][17][18][19][20][21][22][23] including approaches based on active learning [24]. These approaches may involve starting the system from a set of initial states and then making measurements at a given final time, with the purpose of determining the linear map from the initial to the final time.…”