Simulating quantum chaos without chaos

Abstract

Quantum chaos is a quantum many-body phenomenon that is associated with a number of intricate properties, such as level repulsion in energy spectra or distinct scalings of out-of-time ordered correlation functions. In this work, we introduce a novel class of "pseudochaotic" quantum Hamiltonians that fundamentally challenges the conventional understanding of quantum chaos and its relationship to computational complexity. Our ensemble is computationally indistinguishable from the Gaussian unitary ensemble (GUE) of strongly-interacting Hamiltonians, widely considered to be a quintessential model for quantum chaos. Surprisingly, despite this effective indistinguishability, our Hamiltonians lack all conventional signatures of chaos: it exhibits Poissonian level statistics, low operator complexity, and weak scrambling properties. This stark contrast between efficient computational indistinguishability and traditional chaos indicators calls into question fundamental assumptions about the nature of quantum chaos. We, furthermore, give an efficient quantum algorithm to simulate Hamiltonians from our ensemble, even though simulating Hamiltonians from the true GUE is known to require exponential time. Our work establishes fundamental limitations on Hamiltonian learning and testing protocols and derives stronger bounds on entanglement and magic state distillation. These results reveal a surprising separation between computational and information-theoretic perspectives on quantum chaos, opening new avenues for research at the intersection of quantum chaos, computational complexity, and quantum information. Above all, it challenges conventional notions of what it fundamentally means to actually observe complex quantum systems.

Figures (3)

• Figure 1: We consider quantum ensembles that are indistinguishable from a) Gaussian unitary ensembles featuring signatures of genuine quantum chaotic dynamics by any computationally restricted observer: Yet, b) this new ensemble that is computationally efficiently preparable on quantum computers does in many ways not exhibit quantum chaotic features.
• Figure 2: Level spacing statistics for GUE Hamiltonians (blue) and pseudo-GUE Hamiltonians (orange) with $d=2^6$. The dashed curves show a fitted Wigner-Dyson and exponential distribution.
• Figure 3: Calculating the Haar average over the eigenbasis for time evolution by a GUE Hamiltonian. The first step simply rewires the diagram into a format such that the "vectorized form" of the Haar average can be calculated mele2024introduction. In the last diagram, the sum runs over all permutations $\pi,\sigma \in \mathbb{S}_2$, and the wiring on the left half of the gray box corresponds to $\pi$, while the right side corresponds to $\sigma$. $\text{Wg}(\pi^{-1} \sigma,d)$ is the Weingarten function associated with the permutation $\pi^{-1} \sigma$. Note that the right side of the diagram simply results in coefficients which reweight the terms of the sum according to different spectral form factors; the one show in the figure corresponds to $\abs{\tr(e^{-i \Lambda t})}^2$.

Theorems & Definitions (93)

• Definition 1: Pseudochaotic Hamiltonians
• Theorem 1: Pseudo-GUE is indistinguishable from GUE
• Theorem 2: Dichotomy of spectral statistics
• Theorem 3: Separation in probes of quantum chaos
• Corollary 1: Scrambling property testing
• Corollary 2: Eigenvalues spectrum statistics
• Theorem 4: No general Hamiltonian learning. Informal version of \ref{['th:hardnessdiamond']}
• Corollary 3: Stronger distillation bounds
• Lemma 1: Marginals of $p(\lambda_1,\ldots,\lambda_d)$ gotze_rate_2005fyodorov2010introductionrandommatrixtheory
• Lemma 2: Exponential gate complexity of GUE kotowski2023extremal