Structured Clifford+T Circuits for Efficient Generation of Quantum Chaos

TL;DR

The paper addresses the lack of universal chaos benchmarks in quantum circuits by proposing deterministic, causally connected Clifford+T architectures that realize chaotic signatures with two nonstabilizer resources. It introduces a causal cover framework and compares three entanglement heating architectures, showing that causal connectivity suffices to induce Wigner-Dyson entanglement spectra and OTOC decay at polylogarithmic circuit depths. The main contributions are evidence that deterministic, structured designs can emulate chaos reliably across hardware, and that a five-block causal-cover configuration ensures consistent scrambling, reducing reliance on depth or randomness. This work provides hardware-agnostic methods to generate and diagnose quantum chaos efficiently on near-term devices, with implications for benchmarking and understanding information scrambling in quantum circuits.

Abstract

We investigate the emergence of quantum chaos and unitary T-design behavior in derandomized Clifford+T circuits using causal cover architectures. Motivated by the need for deterministic constructions that can exhibit chaotic behavior across diverse quantum hardware platforms, we explore deterministic Clifford circuit architectures (random Clifford circuits with causal cover, bitonic sorting networks, and permutation-based routing circuits) to drive quantum circuits toward Wigner-Dyson (WD) entanglement spectrum statistics and OTOC decay.Our experiments demonstrate that causal connectivity, not circuit depth or randomness, is a critical feature that drives circuits to chaos. We show that initializing with n T-states and adding a second T-layer after a causally covered Clifford evolution yields consistent OTOC decay and WD statistics. This also enables deeper understanding of the circuit structures that generate complex entanglement behavior. Notably, our work suggests polylogarithmic-depth deterministic circuits suffice to approximate chaotic behavior, highlighting that causal connectivity is sufficient for operator spreading to induce Wigner-Dyson entanglement statistics and OTOC decay.

Figures (4)

• Figure 1: Causal cover: (a) Graph $G = (V, E)$; (b), (c), and (d) represent subgraphs $G_1$, $G_2$, and $G_3$ of $G$, where each $E_i$ (the edge set of $G_i$) is a matching in $G$; (e) shows the resulting circuit constructed from subgraphs $G_1$, $G_2$, and $G_3$. In the circuit, a connection $(u, v)$ is applied at time step $t = i$ if $(u, v) \in E_i$.$(1,4)$ is causally covered here but the circuit as a whole is not causally covered since $(4,2)$ is not causally covered.
• Figure 2: $\tilde{r}$ measured after the second $T$--gate layer. The circuit is initialized with $n$$T$--states, followed by a Clifford block, a second layer of $n$$T$--gates, and a final Clifford block. The entanglement heating section (Clifford block) consists of a random Clifford unitary that satisfies the causal cover condition. Subfigures (a), (b), and (c) correspond to increasing depths of this section: (a) $1\times$ causal cover, (b) $2\times$ causal cover, and (c) $3\times$ causal cover. $\tilde{r}$ is averaged over 15 different circuits for each depth.
• Figure 3: Comparison of $\tilde{r}$ values after the second $T$ gate layer for different entanglement heating architectures. Each circuit begins with $n$$T$--states in the initialization layer and $n$$T$ gates in the second $T$ layer. The three subfigures correspond to: (a) a causally covered random Clifford circuit, (b) a bitonic sorting network (restricted to $n=8$ and $n=16$ due to its $2^k$ structure), and (c) a Clifford circuit constructed from a random cyclic permutation. For each architecture, circuits of $n=8$, $12$, and $16$ qubits are tested (where applicable), and the results are averaged over 20 random circuit instances.For architectures (b) and (c), one unit of entanglement heating depth is defined as a pair of consecutive layers: one of CNOT gates and one of randomly chosen single--qubit Clifford gates ($H$ or $S$).
• Figure 4: OTOC decay comparison for different circuit architectures(n=20 qubits). The plot shows the OTOC value $real(F)$ as a function of circuit depth. (a) corresponds to a 4-block circuit with a random Clifford unitary as the entanglement heating section. (b) represents a 5-block architecture where the entanglement heating section is a random Clifford unitary satisfying the causal cover property. (c) also uses a 5-block structure, with the entanglement heating section constructed using a causally covered Clifford circuit based on a random permutation routing algorithm. Red dotted line denotes the depth of second layer of T-gates. The 5-block circuits show clear decay in OTOC after the second T-layer, indicating chaotic behavior, while the 4-block circuits do not consistently reach the chaotic regime.