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Apparent Universal Behavior in Second Moments of Random Quantum Circuits

Daniel Belkin, James Allen, Bryan K. Clark

TL;DR

This work analyzes how quickly random quantum circuits converge to $\epsilon$-approximate $2$-designs by focusing on the $t=2$ second moment. It derives a tractable, exact formula for the multiplicative error $\mathcal{M}(\Phi_\varepsilon,\Phi_\text{Haar})$, enabling precise numerical evaluation up to $n=50$ via tensor-network contractions and permutation-basis methods. The study reveals that many architectures scramble in depth $d=\Theta(\log n)$, but certain graph geometries impose slower, sometimes polynomial-depth scaling (e.g., $\Omega(n^2)$ gates); it also uncovers fast architectures (e.g., PB-based) that reach $0.01$-approximate $2$-designs with shallow depths largely independent of $n$ in practical sizes. Additionally, the paper examines the relationship between anticoncentration and universal $2$-design convergence, showing both alignments and divergences depending on architecture and boundary conditions, and proposes conjectures about universal bounds tied to graph connectivity and design depth.

Abstract

Just how fast does the brickwork circuit form an approximate 2-design? Is there any difference between anticoncentration and being a 2-design? Does geometry matter? How deep a circuit will I need in practice? We tell you everything you always wanted to know about second moments of random quantum circuits, but were too afraid to compute. Our answers generally take the form of numerical results for up to 50 qubits. Our first contribution is a strategy to determine explicitly the optimal experiment which distinguishes any given ensemble from the Haar measure. With this formula and some computational tricks, we are able to compute $t = 2$ multiplicative errors exactly out to modest system sizes. As expected, we see that most families of circuits form $ε$-approximate $2$-designs in depth proportional to $\log n$. For the 1D brickwork, we work out the leading-order constants explicitly. For graphs, we find some exceptions which are much slower, proving that they require at least $Ω(n^2)$ gates. This answers a question asked by ref. 1 in the negative. We explain these exceptional architectures in terms of connectedness. Based on this intuition we conjecture universal upper and lower bounds for graph-sampled circuit ensembles. For many architectures, the optimal experiment which determines the multiplicative error corresponds exactly to the collision probability (i.e. anticoncentration). However, we find that the star graph anticoncentrates much faster than it forms an $ε$-approximate $2$-design. Finally, we show that one needs only ten to twenty layers to construct an approximate $2$-design for realistic parameter ranges. This is a large constant-factor improvement over previous constructions. The parallel complete-graph architecture is not quite the fastest scrambler, partially resolving a question raised by ref. 2.

Apparent Universal Behavior in Second Moments of Random Quantum Circuits

TL;DR

This work analyzes how quickly random quantum circuits converge to -approximate -designs by focusing on the second moment. It derives a tractable, exact formula for the multiplicative error , enabling precise numerical evaluation up to via tensor-network contractions and permutation-basis methods. The study reveals that many architectures scramble in depth , but certain graph geometries impose slower, sometimes polynomial-depth scaling (e.g., gates); it also uncovers fast architectures (e.g., PB-based) that reach -approximate -designs with shallow depths largely independent of in practical sizes. Additionally, the paper examines the relationship between anticoncentration and universal -design convergence, showing both alignments and divergences depending on architecture and boundary conditions, and proposes conjectures about universal bounds tied to graph connectivity and design depth.

Abstract

Just how fast does the brickwork circuit form an approximate 2-design? Is there any difference between anticoncentration and being a 2-design? Does geometry matter? How deep a circuit will I need in practice? We tell you everything you always wanted to know about second moments of random quantum circuits, but were too afraid to compute. Our answers generally take the form of numerical results for up to 50 qubits. Our first contribution is a strategy to determine explicitly the optimal experiment which distinguishes any given ensemble from the Haar measure. With this formula and some computational tricks, we are able to compute multiplicative errors exactly out to modest system sizes. As expected, we see that most families of circuits form -approximate -designs in depth proportional to . For the 1D brickwork, we work out the leading-order constants explicitly. For graphs, we find some exceptions which are much slower, proving that they require at least gates. This answers a question asked by ref. 1 in the negative. We explain these exceptional architectures in terms of connectedness. Based on this intuition we conjecture universal upper and lower bounds for graph-sampled circuit ensembles. For many architectures, the optimal experiment which determines the multiplicative error corresponds exactly to the collision probability (i.e. anticoncentration). However, we find that the star graph anticoncentrates much faster than it forms an -approximate -design. Finally, we show that one needs only ten to twenty layers to construct an approximate -design for realistic parameter ranges. This is a large constant-factor improvement over previous constructions. The parallel complete-graph architecture is not quite the fastest scrambler, partially resolving a question raised by ref. 2.

Paper Structure

This paper contains 55 sections, 3 theorems, 90 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Basics
  4. Constraining the Optimal Experiment
  5. Constraining the Optimal Experiment
  6. Explicit Form for Numerics
  7. Graphs
  8. Prior Work
  9. Error vs. Circuit Size
  10. Critical Depth vs. Qubit Count
  11. Connectedness
  12. Why is the lollipop special?
  13. Results by connection count
  14. Conjectures
  15. Brickwork
  16. ...and 40 more sections

Key Result

Theorem 1

Let $\Phi_\varepsilon$ be the $2$nd moment operator of a locally invariant distribution over $\mathcal{U}\left(q^n\right)$ with a PSD vectorization. Define The multiplicative error between $\Phi_\varepsilon$ and the 2nd moment operator $\Phi_{\text{Haar}}$ of the Haar distribution over $\mathcal{U}\left(q^n\right)$, as given by the maximization in Equation eq:multerr_expdef, is saturated by the c

Figures (17)

  • Figure 1: Multiplicative error $\mathcal{M}$ at $t = 2$ vs. gate count $s$ for four graphs on 12 qubits.
  • Figure 2: Circuit size needed to reach an $0.01$-approximate $2$-design for linear, circle, complete, and lollipop graphs.
  • Figure 3: Circuit size needed to reach an $0.01$-approximate $2$-design for some other families of graphs. Results for complete and linear graph are repeated for reference. (a) Tree and star graphs. (b) Several random $d$-regular Ramanujan graphs.
  • Figure 4: (a) $0.01$-approximate $2$-design depths for each of three families of graphs. Although the hourglass and bridge look very similar, their scrambling rates are very different. (b) Illustrations of the hourglass, bridge, and lollipop graph families. In each case we assign $\lceil \frac{n}{2} \rceil$ nodes to the upper clique, such that the two regions are of roughly equal size.
  • Figure 5: Mean gates per connection per qubit for each of three architectures, as estimated by the greedy algorithm described in Appendix \ref{['app:connections_of_architectures']}. We see that the hourglass is regularly-connected even at large $n$, while the bridge and lollipop become poorly connected as $n$ grows.
  • ...and 12 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Conjecture 3
  • Conjecture 4
  • Lemma 5
  • proof
  • proof
  • Definition 6