Quantum complexity of time evolution with chaotic Hamiltonians

Abstract

We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e −iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion -the Eigenstate Complexity Hypothesis (ECH) -which bounds the overlap between off-diagonal energy eigenstate projectors and the k-local operators of the theory, and use it to show that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE BQP/poly and PSPACE BQSUBEXP/subexp, and the "fast-forwarding" of quantum Hamiltonians.

Renormalized Circuit Complexity

“…where, γ 1 , γ 2 are two Majorana operators and J 1 , J 2 , J 3 are the random couplings. Following the analysis of [23] if we suppress the contribution of the γ 1 γ 2 by a large penalty factor, then we can show that in this case d(p) ∼ p 0 . This means that as k becomes large, the circuit will involve less of these gates.…”

Extended eigenstate thermalization and the role of FZZT branes in the Schwarzian theory

“…It would be very interesting to investigate what more can be said about the properties of the AQECC hosted by the eigenstates of SYK-like models in the light of the OTOC version of ETH we found. Recent investigations linking ETH to AQECC in chaotic theories, including holographic ones, have appeared in [94], while [95] link complexity of time evolution to ETH-type behavior.…”

Variational fast forwarding for quantum simulation beyond the coherence time

“…Previous results analyze fast-forwarding of Hamiltonians mostly in a computational complexity setting 27,28,33,34 , in which the asymptotic scaling of the runtime of quantum circuits implementing a large-scale simulation is important. However, near-term devices are constrained to simulating intermediate-scale systems using finite depth circuits.…”