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Vanishing correlations in (bi)stochastic controlled circuits

Pavel Kos, Bruno Bertini, Tomaž Prosen

TL;DR

The paper shows that in one-dimensional brickwork circuits built from two-site controlled gates that are either stochastic or bistochastic, two-point correlations vanish off the equal-space line: $C(x,t)=0$ for $x>0$ under CS, and $C(x,t)=0$ for $x<0$ under BCS, with $C(x,t)=0$ for $x\neq 0$ when both hold. This strong simplification extends to multi-point correlations, which vanish unless the two rightmost operators act on the same site. Autocorrelations $C(0,t)$ are harder to compute and typically decay exponentially with time, though special initial states or finite-size scars can yield non-vanishing long-time limits. The authors illustrate the framework with two families: random quantum controlled gates that reduce to an effective classical stochastic process, and controlled bistochastic gates including deterministic cellular automata. They discuss extensions to multi-replica analyses (e.g., entanglement and OTOCs) and potential applications to quantum quenches and inactive-region protocols.

Abstract

We study the dynamics of circuits composed of stochastic and bistochastic controlled gates. This type of dynamics arises from quantum circuits with random controlled gates, as well as in stochastic circuits and deterministic classical cellular automata. We prove that stochastic and bistochastic controlled gates lead to two-point spatio-temporal correlation functions that vanish everywhere except when the two operators act on the same site. More generally, for multi-point correlations the two rightmost operators must act on the same site. We argue that autocorrelation, while hard to compute, typically decays exponentially towards a value that is exponentially small in the system size. Our results reveal a broad class of quantum systems that exhibit surprisingly simple correlation structures despite their complex microscopic dynamics.

Vanishing correlations in (bi)stochastic controlled circuits

TL;DR

The paper shows that in one-dimensional brickwork circuits built from two-site controlled gates that are either stochastic or bistochastic, two-point correlations vanish off the equal-space line: for under CS, and for under BCS, with for when both hold. This strong simplification extends to multi-point correlations, which vanish unless the two rightmost operators act on the same site. Autocorrelations are harder to compute and typically decay exponentially with time, though special initial states or finite-size scars can yield non-vanishing long-time limits. The authors illustrate the framework with two families: random quantum controlled gates that reduce to an effective classical stochastic process, and controlled bistochastic gates including deterministic cellular automata. They discuss extensions to multi-replica analyses (e.g., entanglement and OTOCs) and potential applications to quantum quenches and inactive-region protocols.

Abstract

We study the dynamics of circuits composed of stochastic and bistochastic controlled gates. This type of dynamics arises from quantum circuits with random controlled gates, as well as in stochastic circuits and deterministic classical cellular automata. We prove that stochastic and bistochastic controlled gates lead to two-point spatio-temporal correlation functions that vanish everywhere except when the two operators act on the same site. More generally, for multi-point correlations the two rightmost operators must act on the same site. We argue that autocorrelation, while hard to compute, typically decays exponentially towards a value that is exponentially small in the system size. Our results reveal a broad class of quantum systems that exhibit surprisingly simple correlation structures despite their complex microscopic dynamics.
Paper Structure (13 sections, 4 theorems, 50 equations, 1 figure)

This paper contains 13 sections, 4 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting
  3. Strategy and results
  4. Simplifications of correlation functions
  5. Examples
  6. Random quantum controlled gates
  7. Controlled bistochastic gates
  8. Autocorrelation functions in (bi)stochastic controlled circuits
  9. Discussion
  10. Acknowledgments
  11. Characterization of the family fulfilling Eqs. \ref{['eq:CS']} and \ref{['eq:BCS']}
  12. Generalized condition
  13. An example of random controlled gates leading to tilted East gate

Key Result

Theorem 1

Consider a brickwork circuit of two-site gates $U$ and let be the infinite-temperature two-point correlation functions of diagonal traceless observables at $(0,0)$ and $(x,t)$ (subscripts correspond to the operators' position). Then the following facts hold

Figures (1)

  • Figure 1: Exponential decay of the autocorrelation function for different system sizes and averaged controlled gate. The difference between $L=27$ and $L=29$ is almost indistinguishable. We deduced finite size contribution arising from a scar state $|00 \ldots 0 \mathbin{\scalerel*{\newmoon}{G}} \rangle$.

Theorems & Definitions (6)

  • Theorem 1: Vanishing of two-point correlation functions
  • Theorem 2: Vanishing-multi-point correlation functions
  • Theorem 3
  • proof
  • Theorem 4
  • proof