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Absence of censoring inequalities in random quantum circuits

Daniel Belkin, James Allen, Bryan K. Clark

TL;DR

This work addresses whether censoring inequalities hold for random quantum circuits by constructing a counterexample where deleting gates speeds up achieving an $\epsilon$-approximate $t$-design, specifically for $t=2$. The authors analyze the second-moment operator (moment operator) spectra and relate the subleading eigenvalues $\lambda$ and $\lambda'$ to design accuracy, deriving a depth bound $d \ge 1 + \left\lfloor\dfrac{2 N t \log q}{\log(\lambda/\lambda')}\right\rfloor$ that demonstrates the violation. The results extend to spectral and singular-value gaps and to non-deterministic graph-edge architectures, while showing that deletions of boundary gates can still uphold censoring inequalities. Overall, the paper reveals that local scrambling does not universally dictate global scrambling and that the relationship between local dynamics and global mixing is nuanced, with implications for 1D brickwork designs and scrambling-rate bounds across architectures.

Abstract

Ref. 1 asked whether deleting gates from a random quantum circuit architecture can ever make the architecture a better approximate $t$-design. We show that it can. In particular, we construct a family of architectures such that the approximate $2$-design depth decreases when certain gates are deleted. We also give some intuition for this construction and discuss the relevance of this result to the approximate $t$-design depth of the 1D brickwork. Deleting gates always decreases scrambledness in the short run, but can sometimes cause it to increase in the long run. Finally, we give analogous results for spectral gaps and when deleting edges of interaction graphs.

Absence of censoring inequalities in random quantum circuits

TL;DR

This work addresses whether censoring inequalities hold for random quantum circuits by constructing a counterexample where deleting gates speeds up achieving an -approximate -design, specifically for . The authors analyze the second-moment operator (moment operator) spectra and relate the subleading eigenvalues and to design accuracy, deriving a depth bound that demonstrates the violation. The results extend to spectral and singular-value gaps and to non-deterministic graph-edge architectures, while showing that deletions of boundary gates can still uphold censoring inequalities. Overall, the paper reveals that local scrambling does not universally dictate global scrambling and that the relationship between local dynamics and global mixing is nuanced, with implications for 1D brickwork designs and scrambling-rate bounds across architectures.

Abstract

Ref. 1 asked whether deleting gates from a random quantum circuit architecture can ever make the architecture a better approximate -design. We show that it can. In particular, we construct a family of architectures such that the approximate -design depth decreases when certain gates are deleted. We also give some intuition for this construction and discuss the relevance of this result to the approximate -design depth of the 1D brickwork. Deleting gates always decreases scrambledness in the short run, but can sometimes cause it to increase in the long run. Finally, we give analogous results for spectral gaps and when deleting edges of interaction graphs.

Paper Structure

This paper contains 10 sections, 4 theorems, 12 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Intuition
  4. Proof of Theorem \ref{['thm:depth']}
  5. Conclusion
  6. Numerical results
  7. Exact multiplicative error
  8. Small examples
  9. Architectures sampled from graphs
  10. Proof of Table \ref{['table:status']}

Key Result

Theorem 1

For sufficiently large $N$, there exist values of $\epsilon$ and $d$ such that $C'^d$ is an $\epsilon$-approximate $2$-design and ${C}^{d}$ is not.

Figures (7)

  • Figure 1: (a) A single period of the architecture on $N = 5$ sites. When the red gates are deleted, the ensemble becomes more scrambled. (b) Spectral gap with and without red gates. (c) Multiplicative error for the second moment operator of a single period of the architecture, with and without red gates.
  • Figure 2: Classical analogy for the preservation of quantum information in the circuits $C$(left) and $C'$(right). The scrambling of quantum information through gates is analogous to the mixing of a pigment over the whole system. We start with all the red pigment on the 1st site of the system, and track how much of that pigment gets spread to the other sites by the action of each gate. Each gate is a perfect mixer on its support, i.e. it replaces the amount of pigment on each site in its support with the average across the support. We assume $N \gg 1$ and give all pigment levels to leading order in $N$. Despite the fact that $C$ has more gates, those extra gates allow it to temporarily store the pigment on the 2nd site. As a result, the level of mixing is not as strong as it is in $C'$, where the mixing is so strong that the first site only differs from the rest of the system by an $O(\frac{1}{N^2})$ amount of pigment.
  • Figure 3: Crossover of multiplicative error for the circuit of Figure \ref{['fig:architecture']} with $5$ qubits.
  • Figure 4: (a) A circuit which shows a dramatic crossover in the multiplicative error over time. (b) Evolution of multiplicative error. Because the error is initially very large, we omit the first five points from this graph.
  • Figure 5: A small circuit on $5$ qubits which violates the censoring inequality for spectral gaps. The largest non-unit eigenvalue of the second moment operator is $0.2512$ with the final gate (red) and $0.2423$ without.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof