[Paper Review] On temporal entropy and the complexity of computing the expectation value of local operators after a quench

Stefano Carignano, Carlos Ramos Marimón+1 more|arXiv (Cornell University)|Jul 21, 2023

Quantum many-body systems46 references4 citations

TL;DR

This paper establishes a theoretical link between temporal matrix product state (tMPS) complexity and operator entanglement in one-dimensional quantum systems after a quench. It shows that the bond dimension of tMPS—critical for simulating time-dependent expectation values of local operators—is upper-bounded by the operator entanglement of the Heisenberg-picture evolution, implying polynomial scaling for integrable systems and exponential scaling for non-integrable ones.

ABSTRACT

We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states. We argue that such cost is intimately related to that of encoding temporal transition matrices and their partial traces. In particular, we show that we can upper-bound the rank of these reduced transition matrices by the one of the Heisenberg evolution of local operators, thus making connection between two apparently different quantities, the temporal entanglement and the local operator entanglement. As a result, whenever the local operator entanglement grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal matrix product states is likely advantageous with respect to computing the same quantities using standard matrix product states techniques.

Motivation & Objective

• To understand the computational cost of simulating time-dependent expectation values of local operators after a quantum quench using temporal matrix product states (tMPS).
• To identify conditions under which tMPS provides a computationally efficient alternative to standard matrix product state (MPS) techniques for out-of-equilibrium dynamics.
• To establish a theoretical connection between temporal entanglement in tMPS and operator entanglement in the Heisenberg picture.
• To determine whether the growth of operator entanglement can predict the scaling of tMPS bond dimension and thus the feasibility of efficient simulations.

Proposed method

• Formalizes a modified tMPS algorithm based on earlier work by Bañuls et al., using a Keldysh contour to encode the influence matrix of the system.
• Introduces reduced transition matrices (RTMs) that capture the evolution of local observables and relate their rank to the operator entanglement of the Heisenberg-evolved operator.
• Uses singular value decomposition (SVD) of RTMs to truncate the tMPS, with fidelity-based error control to determine required bond dimensions.
• Compares the bond dimension scaling of tMPS with that of operator entanglement, showing that the former is upper-bounded by the latter.
• Employs numerical simulations on integrable and non-integrable spin chains to validate the theoretical bound, using fidelity thresholds to extract effective bond dimensions.
• Analyzes the scaling of bond dimension $ D^* $ required to maintain a fixed truncation error $ \epsilon $, comparing it to the operator entanglement scaling.

Experimental results

Research questions

• RQ1 Can the computational cost of simulating time-dependent local observables via tMPS be bounded by the entanglement properties of the Heisenberg-evolved operator?
• RQ2 Under what conditions does tMPS offer a polynomial scaling advantage over standard MPS for out-of-equilibrium dynamics?
• RQ3 How does the growth of operator entanglement relate to the required bond dimension of the tMPS in the time evolution of local observables?
• RQ4 Is there a systematic connection between temporal entanglement (in tMPS) and operator entanglement (in Heisenberg picture) that explains numerical observations of small tMPS bond dimensions?
• RQ5 Can the theoretical bound on tMPS rank be saturated in realistic models, and does it match numerical results for integrable and non-integrable systems?

Key findings

• The bond dimension of the tMPS is upper-bounded by the rank of the Heisenberg-evolved operator, establishing a direct theoretical link between tMPS complexity and operator entanglement.
• For integrable systems, where operator entanglement grows logarithmically, the tMPS bond dimension grows polynomially in time, enabling efficient simulations.
• For non-integrable systems, where operator entanglement grows linearly, the tMPS bond dimension grows exponentially, making simulations infeasible with standard tMPS methods.
• Numerical results confirm that the required tMPS bond dimension $ D^* $ scales identically with the operator entanglement bond dimension when maintaining a fixed fidelity threshold.
• The scaling of the bond dimension needed to achieve a fixed error in tMPS simulations matches the scaling required to simulate the Heisenberg evolution of the operator with constant error.
• The bound is saturated in specific cases, indicating that the theoretical limit is tight and not merely a loose estimate.