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Random unitaries from Hamiltonian dynamics

Laura Cui, Thomas Schuster, Liang Mao, Hsin-Yuan Huang, Fernando Brandao

TL;DR

The paper tackles whether time-independent Hamiltonian dynamics can emulate Haar-random unitaries, showing a no-go for constant-$q$ local Hamiltonians to form unitary $2$-designs or PRUs, yet revealing that quasi-local models with $q=omega( )$ locality can achieve both, even at constant evolution time. The approach combines a simple two-copy distinguishing test with a path-recording framework and a novel gluing lemma to connect local random blocks into a global Haar-like unitary, leveraging strong unitary $k$-designs and PRFs. The authors provide concrete constructions in one dimension and establish conditions under which these ensembles form $$-approximate unitary $k$-designs and PRUs, with efficient simulation under cryptographic assumptions. Altogether, the results delineate the limits of random-unitary descriptions for time-independent Hamiltonians and point to a nuanced understanding of universal chaotic behavior in local quantum systems, with implications for learning, benchmarking, and fundamental quantum complexity.

Abstract

The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a $2$-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary $k$-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.

Random unitaries from Hamiltonian dynamics

TL;DR

The paper tackles whether time-independent Hamiltonian dynamics can emulate Haar-random unitaries, showing a no-go for constant- local Hamiltonians to form unitary -designs or PRUs, yet revealing that quasi-local models with locality can achieve both, even at constant evolution time. The approach combines a simple two-copy distinguishing test with a path-recording framework and a novel gluing lemma to connect local random blocks into a global Haar-like unitary, leveraging strong unitary -designs and PRFs. The authors provide concrete constructions in one dimension and establish conditions under which these ensembles form -approximate unitary -designs and PRUs, with efficient simulation under cryptographic assumptions. Altogether, the results delineate the limits of random-unitary descriptions for time-independent Hamiltonians and point to a nuanced understanding of universal chaotic behavior in local quantum systems, with implications for learning, benchmarking, and fundamental quantum complexity.

Abstract

The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a -design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary -design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.

Paper Structure

This paper contains 28 sections, 26 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Impossibility of random unitaries in constant-local Hamiltonians
  4. Formation of random unitaries in nearly-local Hamiltonians
  5. Discussion
  6. Review of Haar-random unitaries, unitary designs, and pseudorandom unitaries
  7. Haar-random unitaries
  8. Unitary designs
  9. Additive error and parallel indistinguishability
  10. Measurable error and adaptive indistinguishability
  11. Pseudorandom unitaries
  12. Review of the path-recording framework
  13. Relation state registers
  14. Restrictions on relation states
  15. The path-recording oracle
  16. ...and 13 more sections

Key Result

Theorem 1

The ensemble $\mathcal{E}$ cannot form a unitary 2-design for any additive error $\varepsilon \leq \mathcal{O}(1/12^q n)$ in one-dimensional systems, nor for any $\varepsilon \leq \mathcal{O}(1/12^q n^q)$ in all-to-all connected systems.

Figures (3)

  • Figure 1: Our work investigates when time-independent Hamiltonian dynamics are asymptotically indistinguishable from random unitary transformations. We find that pseudorandomness cannot arise in any family of Hamiltonians with constant-local interactions, but can be achieved in quasi-local models.
  • Figure 2: Illustration of our no-go result for constant-local dynamics. We show that it is possible to efficiently distinguish any ensemble $\mathcal{E}$ generated by evolving under $q$-local Hamiltonians up to arbitrarily long times from a Haar-random unitary by measuring random Pauli operators of weight up to $q$ on two copies of the system, as (a) the output distribution of $\mathcal{E}$ retains local correlations, while (b) the output distribution of the Haar ensemble appears completely uniform.
  • Figure 3: Graphical depiction of the asymptotic decomposition of random matrix ensembles into independent components. (a) Classical random matrix ensembles arising from Wigner matrices with i.i.d. elements converge to a deterministic diagonal distribution, conjugated by Haar-random eigenbasis transformations. (b) Our results show that Haar-random unitaries are indistinguishable from certain ensembles generated by transformations that act on subsystems of size $\mathcal{O}(\mathop{\mathrm{poly}}\nolimits\log n)$.

Theorems & Definitions (61)

  • Theorem 1: Time-evolution under constant-local Hamiltonians cannot form unitary designs
  • Theorem 2: Time-evolution under constant-local Hamiltonians cannot form PRUs
  • Theorem 3: Unitary $k$-designs from time-evolution under nearly-local Hamiltonians
  • Theorem 4: PRUs from time-evolution under nearly-local Hamiltonians
  • Theorem 5: Additive error designs from generic spectra and evolution times
  • Theorem 6: Non-adaptive PRUs from generic spectra and evolution times
  • Proposition 1: Impossibility of unitary designs from efficient temporal ensembles
  • Definition 1: Moments of the unitary group
  • Definition 2: Distinct subspace metger2024simple
  • Definition 3: Local distinct subspace cui2025unitary
  • ...and 51 more