Quantum Relative Entropy Decay Composition Yields Shallow, Unstructured k-Designs

TL;DR

This work addresses how rapidly quantum circuits converge to approximate $k$-designs under shallow-depth, potentially unstructured architectures. By framing design convergence as relative entropy decay under entropic SDPI, the authors derive composition rules that combine per-layer entropy contraction into global guarantees. They establish that parallel 2-layer architectures and random-gate graphs achieve $\lambda$-CSDPI with bounds scaling polylogarithmically in $n$, implying $O(\mathrm{polylog}(n))$ depth suffices for additive $k$-designs and, under related conditions, for multiplicative $k$-designs in the diamond norm. The results show that extra structure is not strictly necessary for sublinear-depth convergence and provide a flexible, entropically grounded framework for analyzing a broad class of quantum circuits. The techniques may inform near-term implementations by indicating that modest randomness can robustly generate useful randomness properties in quantum circuits.

Abstract

A major line of questions in quantum information and computing asks how quickly locally random circuits converge to resemble global randomness. In particular, approximate k-designs are random unitary ensembles that resemble random circuits up to their first k moments. It was recently shown that on n qudits, random circuits with slightly structured architectures converge to k-designs in depth O(log n), even on one-dimensional connectivity. It has however remained open whether the same shallow depth applies more generally among random circuit architectures and connectivities, or if the structure is truly necessary. We recall the study of exponential relative entropy decay, another topic with a long history in quantum information theory. We show that a constant number of layers of a parallel random circuit on a family of architectures including one-dimensional `brickwork' has O(1/logn) per-layer multiplicative entropy decay. We further show that on general connectivity graphs of bounded degree, randomly placed gates achieve O(1/nlogn)-decay (consistent with logn depth). Both of these results imply that random circuit ensembles with O(polylog(n)) depth achieve approximate k-designs in diamond norm. Hence our results address the question of whether extra structure is truly necessary for sublinear-depth convergence. Furthermore, the relative entropy recombination techniques might be of independent interest.

Figures (4)

• Figure 1:
• Figure 2: Illustration of how the 1-D brickwork architecture converts to a bipartite graph.
• Figure 3: Illutration of Corollary \ref{['cor:brick1d']}. We start from the "chunkwork" configuration as in Figur \ref{['fig:brickvschunk']}. Using Theorem \ref{['thm:unital']}, we replace the $O(\log n)$ relative-error convergence of each $O(\log n)$-size chunk by $O(1 / \log n)$-CSDPI of each layer. Via Corollary \ref{['cor:simplified']}, we obtain that composing two layers respectively corresponding to the two different chunkings yields a channel with $O(1 / \log n)$-CSDPI toward a global $k$-design. Finally, we apply pre- and post-processing to fill gaps left from the chunk boundaries.
• Figure 4: Illustration of Corollary \ref{['cor:lattice']} in two spatial dimensions. Each small square corresponds to one qudit. In the first layer, small squares are grouped into and interacted in fours. In the second layer, small squares are again grouped into fours, but shifted in each direction by one. The scheme forms a 2-layer parallel architecture with a Hamiltonian path. On this path, one may for instance start at the top-left of (a), then at each step move from a 4-square cluster in (a) to the lower-right overlapping square of (b), or from a 4-square cluster in (b) to the upper-right overlapping square in (a).

Theorems & Definitions (39)

• Definition 1.1
• Lemma 2.1: Chain Rule
• Theorem 2.2: gao_complete_2025 Thoerem 2.5
• Lemma 2.3: winter_tight_2016 Lemma 7
• Lemma 2.4: laracuente_quasi-factorization_2022, Corollary II.15. gao_complete_2025, Lemma 2.3
• Lemma 2.5: Pinsker's Inequality
• Lemma 2.6: schuster_random_2025 Lemma 2
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3