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Chaos and complexity by design

Daniel A. Roberts, Beni Yoshida

TL;DR

This paper builds a rigorous link between quantum chaos and pseudorandomness by showing that higher-point OTO correlators probe unitary $k$-designs through the frame potential. It proves a precise correspondence between averaged $2k$-point OTOs and the $k$-fold channel, and derives a frame-potential-based bound connecting chaoticity to circuit complexity. The authors compute Haar averages for multi-point correlators to characterize when ensembles mimic Haar randomness and discuss time-averaged dynamics versus true randomness. They also extend these ideas to continuous and thermal ensembles, outlining implications for holography and complexity growth in chaotic systems. Overall, the work provides a quantitative framework linking chaos, randomness, and computational complexity in quantum dynamics.

Abstract

We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the "frame potential," which is minimized by unitary $k$-designs and measures the $2$-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order $2k$-point correlators is proportional to the $k$th frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these $2k$-point correlators for Pauli operators completely determine the $k$-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.

Chaos and complexity by design

TL;DR

This paper builds a rigorous link between quantum chaos and pseudorandomness by showing that higher-point OTO correlators probe unitary -designs through the frame potential. It proves a precise correspondence between averaged -point OTOs and the -fold channel, and derives a frame-potential-based bound connecting chaoticity to circuit complexity. The authors compute Haar averages for multi-point correlators to characterize when ensembles mimic Haar randomness and discuss time-averaged dynamics versus true randomness. They also extend these ideas to continuous and thermal ensembles, outlining implications for holography and complexity growth in chaotic systems. Overall, the work provides a quantitative framework linking chaos, randomness, and computational complexity in quantum dynamics.

Abstract

We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the "frame potential," which is minimized by unitary -designs and measures the -norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order -point correlators is proportional to the th frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these -point correlators for Pauli operators completely determine the -fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.

Paper Structure

This paper contains 28 sections, 8 theorems, 257 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Measures of Haar
  3. Measures of chaos and design
  4. Chaos and $k$-designs
  5. Frame potentials
  6. Ensemble vs. time averages
  7. Measures of complexity
  8. Discrete ensembles
  9. Continuous ensembles
  10. Measures of correlators
  11. Haar random averages
  12. Dissipation vs. scrambling
  13. Discussion
  14. More proofs
  15. Proof of Theorem \ref{['theorem:OTO_chaos']}
  16. ...and 13 more sections

Key Result

Theorem 1

[Schur-Weyl duality] Let $L(\mathcal{H}^{\otimes k})$ be the algebra of all the operators acting on $\mathcal{H}^{\otimes k}$. Let $U(\mathcal{H})$ be the unitary group on $\mathcal{H}$. An operator $A \in L(\mathcal{H}^{\otimes k})$ commutes with all operators $V^{\otimes k}$ with $V\in U(\mathcal{

Figures (7)

  • Figure 1: Schematic form of the $2k$-point OTO correlation functions Eq. \ref{['eq-form-of-correlator']}, interpreted as a correlation function on the enlarged $k$-copied system. The dotted line diagram surrounds the $k$-fold channel $\Phi_{\mathcal{E}}(B_{1} \otimes \cdots \otimes B_{k})$, which is probed by $A_{1}\otimes \ldots \otimes A_{k}$. (Periodic boundary conditions are implied to take the trace.)
  • Figure 2: Thermalization and scrambling in the decay of a four-point OTO correlator. These correlators typically decay to $\langle AC \rangle \langle B D \rangle$ in a thermal dissipation time $t_d$, and then they decay to a floor value $\sim d^{-2}$ at around the scrambling time $t_*$. These regimes are very well captured by replacing the dynamics with $1$-designs or $2$-designs, respectively.
  • Figure 3: A cartoon of the unitary group, with operators arranged by design. We pick the identity operator to be the reference operator of zero complexity and place it at the center. Typical operators have exponential complexity and live near the edge. Operators closer to the center have lower complexity, which makes them both atypical and more physically realizable in a particular computational model.
  • Figure 4: A $6$-qubit tensor network model for the geometry of the interior of a black hole. Via holography, the growth of the interior is expected to correspond to chaotic time evolution of a strongly coupled quantum theory. Here, each node corresponds to a perfect tensor and the numbers label the qubit.
  • Figure 5: A unitary operator $U$ representing the dynamics in the Hayden-Preskill thought experiment.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 4 more