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

• The paper demonstrates that quantum complexity in chaotic systems grows linearly over extended timescales, supported by rigorous numerical simulations.
• It employs the Euler-Arnold formalism to relate time evolution to geodesic paths on the unitary operator manifold, elucidating complexity dynamics.
• The research introduces the Eigenstate Complexity Hypothesis, showing that off-diagonal energy eigenstate overlaps vanish exponentially, distinguishing chaotic from integrable models.

An Expert Overview of "Quantum Complexity of Time Evolution with Chaotic Hamiltonians"

The paper "Quantum Complexity of Time Evolution with Chaotic Hamiltonians" explores the intricacies of quantum circuit complexity within large-$N$ chaotic systems, with a particular focus on the Sachdev-Ye-Kitaev (SYK) model as a core example. This study is deeply rooted in understanding how complexity evolves in quantum systems underpinned by chaotic Hamiltonians, and it delineates the systematic behavior of complexity both at early and late times in such systems.

Key Insights and Methodology

The research employs the Euler-Arnold formalism to analyze geodesic complexity within the space of unitary operators on a finite-dimensional Hilbert space. This approach connects the complexity of time evolution to the length of geodesics on the manifold of unitary operators, where a significant outcome is the demonstration of a linear increase in complexity over an extended period, specifically exponential in terms of system size, before reaching a saturation point.

The researchers introduce the Eigenstate Complexity Hypothesis (ECH) as a pivotal criterion. This hypothesis posits that for chaotic Hamiltonians, off-diagonal energy eigenstate projectors demonstrate exponentially small overlaps with $k$-local operations, suggesting a complex transformation between energy eigenstates. The investigation then leverages numerical simulations to illustrate that the large-$N$ SYK model aligns with ECH, which hence validates the unimpeded linear growth of complexity over exponential timescales.

Numerical Results and Claims

Numerical simulations within the paper reveal that in chaotic systems like SYK, the complexity exhibits sustained linear growth attributable to ECH, contrary to integrable systems which exhibit initial linear growth followed by oscillatory behavior. Specifically, the paper contrasts the growth behavior in the SYK model with that in an integrable $N=2$ fermion system, which displays short-time linear complexity growth transitioning to oscillations due to pathways to reduction in computation complexity.

Theoretical and Practical Implications

The results consolidate the view that chaotic quantum systems exhibit unique computational features that distinguish them from integrable models. The delineation between these behaviors could provide insights into computational complexity class separations, striving to distinguish classes such as PSPACE from BQP/poly and BQSUBEXP/subexp.

Moreover, the research has implications for quantum simulation and the thresholds of simulating quantum systems on classical computers. The notion of fast-forwarding a Hamiltonian—the ability to simulate longer time evolution in shorter computational time—emerges as a crucial aspect, especially concerning the presence of conjugate points which denote geodesic non-minimality.

Future Directions

Speculative inquiries outlined in the paper suggest investigations into broader classes of quantum systems where ECH may or may not hold, providing a more comprehensive understanding of complexity growth in physical theories with various levels of integrability or chaos. Additionally, extending the formalism to continuum quantum field theories, possibly involving the expansion of geodesic formalisms for larger discrete systems, is suggested as a compelling avenue.

The research embodies a rigorous exploration into the fundamental underpinnings of quantum complexity, offering a robust framework to assess the implications of chaotic dynamics in quantum systems. This contributes significantly to the broader question of how physical systems implement computations—laying groundwork that potentially bridges quantum computation, gravitational systems, and beyond.