On temporal entropy and the complexity of computing the expectation value of local operators after a quench

TL;DR

This work links the computational cost of simulating time-dependent local observables after a quench to the growth of operator entanglement by bounding the rank of reduced transition matrices with the OE of Heisenberg-evolved operators. By casting the problem in terms of temporal matrix product states on a Keldysh contour and introducing efficient truncation schemes, the authors show that integrable systems—with OE growing only logarithmically—allow polynomial-time simulations, while non-integrable cases tend to require exponential resources. Numerical evidence from Ising models supports the theoretical bound, illustrating when tMPS offers a real advantage over standard MPS methods. The results clarify when temporal approaches are advantageous and propose practical algorithms for compressing the time-evolved local observable dynamics.

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.

Figures (8)

• Figure 1: (a) The time-dependent expectation value of a local operator acting on a one-dimensional system in the Keldysh representation (b) can be described by a double sheet 2D TN contraction. (c) Upon using a Trotter approximation and the locality of the operator, it simplifies to a triangular TN.
• Figure 2: (a): The triangular TN can be contracted from the sides, identifying a left $\bra{L_{\mathcal{O}}}$ and a right $\ket{R_\mathcal{Q}}$ tMPS. The upper-most tensors of the temporal MPS are dictated by the initial state, while the lower-most ones by the choice of the operator. (b) The reduced transition matrix. (c) In the absence of operators, the folded tensors of the time evolution resolve to identities. (d) The partial transpose of the time-evolved left-right density matrix of the system, where forwards and backwards legs are swapped.
• Figure 3: Left: Bond dimensions $D$ obtained by imposing a truncation error of $10^{-4}$ in the singular value decomposition of the RTMs, as function of time (see main text). The shaded areas are delimited by the minimum and maximum $D$ we found by varying the initial state. The thick solid lines represent the bond dimension curves for the operator entanglement (OE): they lie consistently above the corresponding ones for the tMPS. Blue colors denotes the integrable case (int), orange-red the non-integrable one (non-int).
• Figure 4: Bond dimensions $D^*$ obtained by imposing a $10^{-4}$ fidelity for the tMPS $\bra{L_\mathcal{O}}$, again the shaded area is delimited by the minimum and maximum bond dimension of the tMPS for a given initial state, and the thick solid lines denote the operator entanglement (OE) (blue: integrable case, red: non-integrable)
• Figure 5: Illustrations of the methods used for building the tMPS. (a) Power method (b) Light cone method (c) The matrix involved in our low-rank approximation of $\mathcal{T}$ and the role of the gauge transformation in our calculation.