[Paper Review] On Quantum Complexity

Mohsen Alishahiha|arXiv (Cornell University)|Sep 29, 2022

Quantum many-body systems4 citations

TL;DR

This paper proposes a general condition on matrix elements of quantum operators in energy eigenstate basis that leads to late-time linear growth in expectation values—even after thermalization—by requiring a double pole structure in the operator's matrix elements. The key result is that such a condition reproduces the linear growth of quantum complexity, as seen in holographic and Krylov complexity, with saturation at times of order the Hilbert space dimension, D.

ABSTRACT

The ETH ansatz for matrix elements of a given operator in the energy eigenstate basis results in a notion of thermalization for a chaotic system. In this context for a certain quantity - to be found for a given model - one may impose a particular condition on its matrix elements in the energy eigenstate basis so that the corresponding quantity exhibit linear growth at late times. The condition is to do with a possible pole structure the corresponding matrix elements may have. Based on the general expectation of complexity one may want to think of this quantity as a possible candidate for the quantum complexity. We note, however, that for the explicit examples we have considered in this paper, there are infinitely many quantities exhibiting similar behavior.

Motivation & Objective

• To identify a general condition on operator matrix elements that leads to linear growth in quantum expectation values at late times, even after thermalization.
• To connect this condition to the notion of quantum complexity, particularly in chaotic systems with finite entropy.
• To show that the proposed condition reproduces known behaviors of holographic and Krylov complexity, including linear growth and saturation.
• To clarify the role of the A-function (matrix elements in energy basis) and its pole structure in determining the late-time dynamics of quantum observables.
• To establish a link between the ETH ansatz and the emergence of complexity growth via universal correlation structures in chaotic systems.

Proposed method

• Define a quantum object $\mathcal{A}_{\mathcal{O}}(\beta,t)$ as the thermal expectation value of a time-evolved operator $\mathcal{O}$, using a density matrix with inverse temperature $\beta$.
• Express $\mathcal{A}_{\mathcal{O}}(\beta,t)$ in terms of the energy eigenstate matrix elements $A(E_1,E_2) = \langle E_1|\mathcal{O}|E_2\rangle$ and the state density $\rho_\psi(E_1,E_2)$.
• Impose a double pole structure on $A(E_1,E_2)$, specifically $A \sim 1/(E_1 - E_2)^2$, to generate linear time dependence in the disconnected part of the correlation function.
• Use the sine-kernel form of the connected correlation function $\rho_c(\varepsilon,\omega) \approx -\sin^2(D\omega\rho(\varepsilon))/(D\omega)^2$ to model short-range correlations in chaotic systems.
• Show that the disconnected part leads to linear growth at late times ($t \ll D$), while the connected part causes saturation at $t \sim D$ due to the $\omega \ll 1$ regime.
• Demonstrate that the Krylov complexity framework arises as a special case when $\mathcal{O}$ is the label operator of the Krylov basis, with Lanczos coefficients saturating to a constant implying the double pole structure.

Experimental results

Research questions

• RQ1 What condition on the matrix elements of an operator in the energy eigenstate basis leads to linear growth in its time-evolved expectation value at late times?
• RQ2 How does the double pole structure in the matrix elements of an operator relate to the emergence of linear quantum complexity growth?
• RQ3 Can the saturation of complexity at $t \sim D$ (Hilbert space dimension) be derived from universal random matrix theory correlations in chaotic systems?
• RQ4 In what sense does the proposed condition reproduce known behaviors of Krylov complexity, particularly the linear growth phase?
• RQ5 How does the ETH ansatz relate to the late-time behavior of complexity when the system is already in thermal equilibrium?

Key findings

• A double pole structure in the matrix elements $A(E_1,E_2)$ of an operator $\mathcal{O}$—specifically $A \sim 1/(E_1 - E_2)^2$—leads to linear growth in the expectation value $\mathcal{A}_{\mathcal{O}}(\beta,t)$ at late times.
• The linear growth phase occurs when $t \ll D$, where $D = e^S$ is the Hilbert space dimension, and is dominated by the disconnected part of the density-density correlation.
• Saturation of complexity occurs at $t \sim D$, driven by the connected correlation term modeled by the universal sine-kernel $\rho_c(\varepsilon,\omega) \approx -\sin^2(D\omega\rho(\varepsilon))/(D\omega)^2$.
• The saturation mechanism is consistent with an ETH-like behavior of the $A$-function at very late times, where the disconnected part alone suffices to describe the constant late-time value.
• The proposed framework reproduces the Krylov complexity in the special case where $\mathcal{O}$ is the label operator of the Krylov basis, with the double pole structure arising from saturation of Lanczos coefficients.
• The full expression for $\mathcal{A}_{\mathcal{O}}(\beta,t)$ shows that at late times, the $1/\omega^2$ term dominates for $\omega \sim 1/t \ll 1$, yielding linear growth, while the sine-kernel term suppresses the integral at $t \sim D$, leading to saturation.