Quantum Complexity of Time Evolution with Chaotic Hamiltonians

TL;DR

The paper develops a geometric framework to study quantum circuit complexity for time evolution under chaotic Hamiltonians, using SU(2^{N/2}) as the unitary manifold and the Euler-Arnold formalism. It identifies the Eigenstate Complexity Hypothesis (ECH) as a key criterion ensuring off-diagonal energy-eigenstate projectors have exponentially small overlaps with k-local gates, which supports linear complexity growth up to times exponential in N. Through analytic results for N=2, numerical experiments, and general N arguments, the authors show a linear geodesic from the identity to e^{-iHt} remains locally minimizing for exponential times in chaotic systems like SYK, while integrable cases fail this behavior. These findings connect complexity growth to spectral properties, energy eigenstate structure, and have potential implications for holographic duality, saturation dynamics, and complexity-theoretic separations related to fast-forwarding. They also illuminate how locality constraints and geodesic topology shape the late-time behavior of quantum complexity.

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 $\nsubseteq$ BQP/poly and PSPACE $\nsubseteq$ BQSUBEXP/subexp, and the "fast-forwarding" of quantum Hamiltonians.

Figures (13)

• Figure 1: Schematic of the unitary manifold (gray disk). A geodesic path (black) is depicted from the identity to some target unitary $U$. The red straight lines represent construction of a circuit using some elementary gates $g_i$, and the final unitary is $U = g_3g_2g_1$. The geodesic approximates the circuit smoothly by varying a control velocity $V(s)$, analogous to an infinitesimal elementary gate, where $s$ parametrizes the curve.
• Figure 2: The complexity in chaotic systems is conjectured Brown2017 to grow linearly in time until a time of order $e^N$, after which it saturates to (and fluctuates around) its maximum value of $\mathcal{C}_{\text{max}}$. At doubly exponential time, the complexity is expected to exhibit recurrences. .
• Figure 3: Complexity over time in appropriate dimensionless units with sample parameters $J_1 = 1$, $J_2 = 2$, $J_3 = 0$, $\frac{1}{1+\mu} = 0.09$. The complexity demonstrates an initial linear growth with the slope $J = \sqrt{J_1^2 + J_2^2+J_3^2}$, attaining a maximum value of $\pi$, followed by linear oscillations (with slopes $\pm J$).
• Figure 4: The geodesic $\gamma_1$ (red) lies on a great circle of $S^2$, connecting $U(0)$ and $U(1)$. At the antipodal point $p$, the geodesic $\gamma_2$ oriented oppositely along the same great circle exchanges dominance with $\gamma_1$. This effect leads to the linear decrease in complexity in $S^3$, i.e. in $SU(2)$.
• Figure 5: Disorder-averaged complexity (blue) as a function of time. At early times the complexity grows linearly with slope $\sqrt{\frac{\pi}{2}} \sigma$ (green), while at late times it approaches the asymptotic value $\pi/2$ (orange).