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Extremal jumps of circuit complexity of unitary evolutions generated by random Hamiltonians

Marcin Kotowski, Michał Oszmaniec, Michał Horodecki

TL;DR

This work rigorously demonstrates an extremal jump phenomenon for circuit complexity in random Hamiltonian evolutions. By decomposing randomness into eigenbasis and spectrum and leveraging concentration of measure on the unitary group, the authors show that unitaries generated by $H\sim\mathrm{GUE}(d)$ or random-basis Gaussian ensembles remain within the identity neighborhood only up to a short escape time $t_{escape}$ and then rapidly attain near-maximal complexity at a nearby time $t_{jump}$, with $t_{jump}/t_{escape}=\Theta(1)$. The results extend to quantum states evolving under these ensembles and to diagonal Gaussian Hamiltonians, revealing a robust, ensemble-dependent symmetry in the growth of both unitary and state complexity. The analysis advances beyond high-mmoments/frame-potentials techniques by exploiting unitary invariance and torus-geometry arguments, providing sharper, dimension-dependent control and highlighting a deeper link between spectral data and complexity growth. These findings support the Brown-Susskind-type intuition about complexity growth in chaotic quantum systems and offer a tractable framework for studying late-time complexity in non-local random Hamiltonians.

Abstract

We investigate circuit complexity of unitaries generated by time evolution of randomly chosen strongly interacting Hamiltonians in finite dimensional Hilbert spaces. Specifically, we focus on two ensembles of random generators -- the so called Gaussian Unitary Ensemble (GUE) and the ensemble of diagonal Gaussian matrices conjugated by Haar random unitary transformations. In both scenarios we prove that the complexity of $\exp(-it H)$ exhibits a surprising behaviour -- with high probability it reaches the maximal allowed value on the same time scale as needed to escape the neighborhood of the identity consisting of unitaries with trivial (zero) complexity. We furthermore observe similar behaviour for quantum states originating from time evolutions generated by above ensembles and for diagonal unitaries generated from the ensemble of diagonal Gaussian Hamiltonians. To establish these results we rely heavily on structural properties of the above ensembles (such as unitary invariance) and concentration of measure techniques. This gives us a much finer control over the time evolution of complexity compared to techniques previously employed in this context: high-degree moments and frame potentials.

Extremal jumps of circuit complexity of unitary evolutions generated by random Hamiltonians

TL;DR

This work rigorously demonstrates an extremal jump phenomenon for circuit complexity in random Hamiltonian evolutions. By decomposing randomness into eigenbasis and spectrum and leveraging concentration of measure on the unitary group, the authors show that unitaries generated by or random-basis Gaussian ensembles remain within the identity neighborhood only up to a short escape time and then rapidly attain near-maximal complexity at a nearby time , with . The results extend to quantum states evolving under these ensembles and to diagonal Gaussian Hamiltonians, revealing a robust, ensemble-dependent symmetry in the growth of both unitary and state complexity. The analysis advances beyond high-mmoments/frame-potentials techniques by exploiting unitary invariance and torus-geometry arguments, providing sharper, dimension-dependent control and highlighting a deeper link between spectral data and complexity growth. These findings support the Brown-Susskind-type intuition about complexity growth in chaotic quantum systems and offer a tractable framework for studying late-time complexity in non-local random Hamiltonians.

Abstract

We investigate circuit complexity of unitaries generated by time evolution of randomly chosen strongly interacting Hamiltonians in finite dimensional Hilbert spaces. Specifically, we focus on two ensembles of random generators -- the so called Gaussian Unitary Ensemble (GUE) and the ensemble of diagonal Gaussian matrices conjugated by Haar random unitary transformations. In both scenarios we prove that the complexity of exhibits a surprising behaviour -- with high probability it reaches the maximal allowed value on the same time scale as needed to escape the neighborhood of the identity consisting of unitaries with trivial (zero) complexity. We furthermore observe similar behaviour for quantum states originating from time evolutions generated by above ensembles and for diagonal unitaries generated from the ensemble of diagonal Gaussian Hamiltonians. To establish these results we rely heavily on structural properties of the above ensembles (such as unitary invariance) and concentration of measure techniques. This gives us a much finer control over the time evolution of complexity compared to techniques previously employed in this context: high-degree moments and frame potentials.
Paper Structure (17 sections, 25 theorems, 173 equations, 2 figures)

This paper contains 17 sections, 25 theorems, 173 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of results
  4. Unitary complexity
  5. Avoiding low complexity unitaries by random tori
  6. Leaving the neighborhood of identity
  7. Gaussian Hamiltonians in computational basis
  8. Lower bound on unitary complexity
  9. Upper bound on complexity via explicit compilation for qubits
  10. State complexity
  11. Random tori in state spaces
  12. Leaving the neighborhood of the initial state
  13. Technical details for GUE
  14. Concentration of measure bounds for GUE
  15. Computation of Gaussian average for states
  16. ...and 2 more sections

Key Result

Lemma 4.1

For any unitary $X$ we have:

Figures (2)

  • Figure 1: (a) Diagram presenting the complexity jump, with threshold $t\sim \varepsilon$ appearing for evolutions under GUE Hamiltonian. Initially the complexity is zero and after a threshold it becomes nearly maximal. (b) Idea of our proof for unitary invariant ensembles proofs: we show that after the threshold time the evolution leaves the ball around identity, and that it avoids balls around low complexity unitaries $\boldsymbol{G}_i\in \mathcal{G}^{\leq k}$. Technically we show that this avoidance holds with high probability for the whole torus $\mathbb{T}_V$ on which the evolution is localized ($V$ is Haar random unitary).
  • Figure 2: Compilation of $Z^\alpha$ from $Z_1$ and c-not gates.

Theorems & Definitions (60)

  • Definition 1: Unitary complexity
  • Definition 2: State complexity
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 50 more