Integrable Digital Quantum Simulation: Generalized Gibbs Ensembles and Trotter Transitions

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• Computational Physics

Background:

• Digital quantum simulation (DQS) approximates continuous dynamics using discrete Trotter steps.
• Previous studies indicated a sharp Trotter transition where errors become uncontrolled due to quantum chaos.
• Integrable systems, possessing exact conservation laws, offer a different perspective on DQS fidelity.

Purpose of the Study:

• To investigate the behavior of Trotterized evolution in an integrable system, specifically the XXZ Heisenberg spin chain.
• To contrast the typical Trotter transition with phenomena observed in integrable DQS.
• To identify novel signatures of Trotter transitions in digital quantum simulations of integrable models.

Main Methods:

• Focusing on a quench from a spin-wave state in the XXZ Heisenberg spin chain.
• Analyzing the integrable Trotterized evolution as a function of Trotter step duration (τ).
• Employing exact calculations to determine the properties of the late-time dynamics, described by a discrete generalized Gibbs ensemble (dGGE).

Main Results:

• For small τ, the dGGE depends analytically on the Trotter step, keeping discretization errors bounded.
• A novel Trotter transition occurs at a threshold τ_{th}, where the dGGE changes abruptly.
• This transition is detectable locally via the emergence of nonzero staggered magnetization, dependent on τ.

Conclusions:

• Integrable DQS does not necessarily lead to uncontrolled errors, challenging previous assumptions about Trotter transitions.
• A new type of Trotter transition exists in integrable systems, characterized by abrupt changes in the dGGE.
• Discrete GGEs represent unique nonequilibrium states specific to digital quantum platforms.