Brickwork Random Clifford Circuits

• Brickwork random Clifford circuits are quantum circuits with a staggered, bipartite arrangement of two-qubit Clifford gates that yield analytically tractable dynamics and classical simulability.
• They demonstrate measurement-induced phase transitions and classical percolation phenomena, where entanglement shifts from area law to volume law as measurement rates vary.
• The architecture supports studies on operator spreading, localization, and shadow estimation, highlighting applications in quantum chaos, error mitigation, and NISQ device implementations.

Brickwork random Clifford circuits are a canonical class of quantum circuits composed of Clifford gates arranged in a staggered, bipartite ("brickwork") pattern, with circuit dynamics driven by random choices of local two-qubit gates and/or measurements. Owing to the Clifford group’s stabilizer structure, such circuits are classically simulable yet can exhibit rich entanglement, localization, and critical phenomena—including measurement-induced phase transitions, operator fragmentation, and diffusive spreading of quantum "magic." Their precise brickwork geometry also enables analytic results for classical shadow estimation and deep connections to percolation theory elucidated through ZX-calculus simplifications.

1. Circuit Architecture and Dynamics

Brickwork random Clifford circuits are built from alternating layers of parallel two-qubit Clifford gates (such as CNOT, CZ, SWAP, FSWAP) acting on a one-dimensional (or higher-dimensional) lattice, typically with open or periodic boundary conditions. Each period consists of two half-steps:

• On even (or odd) bonds, all two-qubit gates are applied in parallel, e.g., to qubit pairs $(1,2), (3,4), \dots$.
• On the subsequent layer, gates shift by one site to act on the odd (or even) bonds, e.g., $(2,3), (4,5), \dots$.

In monitored circuits, identity or Bell-pair measurements may be randomly interspersed at each bond and time. Gate selection is governed by explicit stochastic rules, for example, with probabilities for CNOT/SWAP/identity/Bell-pair measurement determined by parameters $(p, r)$ such as

$$p_{\rm meas}=p,\quad p_{\rm CNOT}=r(1-p),\quad p_{\rm SWAP}=(1-r)(1-p),\quad p_{\rm id}=p/2,\quad p_{\rm BP}=p/2,$$

with normalization $$p_{\rm CNOT} + p_{\rm SWAP} + p_{\rm id} + p_{\rm BP} = 1$$ (Buznach et al., 18 Feb 2025, Arienzo et al., 2022, Farshi et al., 2022, Kovács et al., 2 Aug 2024).

Such ensembles are well-suited for simulating local quantum circuits, benchmarking information scrambling, and assessing the interplay of quantum chaos, localization, and measurement.

2. Measurement-Induced Phase Transitions and Percolation

Brickwork Clifford circuits exhibit measurement-induced phase transitions (MPTs) between "area law" (low-entanglement) and "volume law" (high-entanglement) regimes, controlled by measurement density $p$ or the ratio of entangling to non-entangling gates. The transition is diagnosed via tripartite mutual information, partitioning the final qubits into regions $A,B,C$ and computing the second-Rényi mutual information: $I_2(A:C) = S_A + S_C - S_B,$ where $S_X$ is the entanglement entropy of region $X$.

The transition displays scaling collapses with the critical exponent $\nu\approx 4/3$, matching two-dimensional classical percolation: $$I_2 \sim \begin{cases} e^{-aN} & \text{(area law)} \ \text{constant} & \text{(critical)} \ bN & \text{(volume law)} \ \end{cases}$$ The crossing point $p_c$ of size-dependent $I_2$ curves locates the transition (Buznach et al., 18 Feb 2025).

By representing the circuit as a ZX-diagram and applying fusion, copy, and Hopf rewrite rules, one reduces the dynamics to a random graph whose entanglement structure is dictated by classical percolation: the existence of a top-to-bottom path corresponds to an entangling ("volume law") phase. The percolation threshold $p_c^{\rm (perc)}$ extracted from the ZX graph matches the MPT threshold $p_c^{\rm (MI)}$ within uncertainty, establishing that the entanglement transition in brickwork Clifford circuits is controlled by a hidden classical percolation transition in the ZX-simplified network.

3. Operator Spreading, Localization, and Fragmentation

Operator dynamics in brickwork random Clifford circuits are governed by the interplay of local Clifford structure and circuit geometry:

• In 1D, operator spreading is limited by the formation of "blocking walls"—random configurations that prevent Pauli strings from crossing certain bonds, leading to exact localization. The probability of a wall per bond yields a mean localization length (e.g., $\sim8$ for $N=1$ per site). The circuit then decomposes into disconnected fragments, restricting information propagation and entanglement growth (Farshi et al., 2020, Farshi et al., 2022, Kovács et al., 2 Aug 2024).
• In 2D, the directed percolation picture in the brickwork geometry is supercritical, with Pauli operators growing ballistically to the light-cone edge with probability $\gtrsim0.44$. The butterfly velocity is $v_B^{(2d)}=1$, and no localization occurs even at maximal Clifford disorder.

Perturbations by single-qubit non-Clifford unitaries with probability $p$ control the crossover from fragmentation to global mixing: for $p<1$, operator space splits into fragments separated by walls with mean size scaling as $1/(1-p)$; at $p=1$, the circuit regains global ergodicity and CUE-like spectral statistics (Kovács et al., 2 Aug 2024).

4. Entanglement and Chaos Diagnostics

Entanglement dynamics reveal sharp distinctions between localized and ballistic regimes:

• In 1D localized phases, entanglement entropy across a cut asymptotes to a finite value independent of system size ("area law"), with explicit bottlenecks across walls. For pure Clifford circuits, the entropy across a wall is bounded by one qubit; mixed Clifford-non-Clifford perturbations yield stationary values $\leq2/3$.
• In 2D, entanglement grows linearly within the light-cone ("light-cone front") and saturates to volume law, $S(\infty)\sim|A|/2$, matching chaotic (Haar) dynamics (Farshi et al., 2022).

The spectral form factor (SFF), defined as

$K(t)=\langle|\operatorname{Tr}\,U^t|^2\rangle,$

serves as a probe of chaos and ergodicity. In 1D fragmented circuits, $K(t)$ grows quadratically at early times, plateaus quickly, and exhibits large group-dependent fluctuations. In 2D, the exponential ramp $K(t)\sim2^{t/2}$ up to $t\sim 2L$ signals ergodicity in the symplectic phase space, though not full Haar randomness (Farshi et al., 2022, Kovács et al., 2 Aug 2024).

5. Shadow Estimation and Analytical Tractability

Brickwork Clifford circuits of constant (e.g., depth 2) provide a tractable ensemble for classical shadow estimation of quantum observables. The induced shadow channel is diagonal in the Pauli basis, with explicit eigenvalues $\lambda(v)$ given in closed form in terms of the brickwork partition of Pauli support (Arienzo et al., 2022): $S(W(v)) = \lambda(v) W(v)$ This enables tight sample complexity bounds for Pauli observable estimation: $m = O(\lambda(v)^{-1}\epsilon^{-2}\ln(1/\delta))$ For global observables, brickwork shadows outperform local-Clifford shadows whenever the observable acts nontrivially on more than $\sim0.68n$ qubits; for sparse observables, local-Clifford shadows retain smaller sample complexity. Such circuits interpolate between global random Clifford (depth $O(n)$) and single-site (depth 1) schemes, facilitating shadow tomography on NISQ devices (Arienzo et al., 2022).

6. Magic and Stabilizer Rényi Entropy Dynamics

The nonstabilizerness ("magic") of quantum states in brickwork Clifford circuits is quantified by stabilizer Rényi entropy (SRE) of single-qubit marginals. Injecting initial magic (e.g., T state) into a product state and evolving under brickwork random Clifford dynamics yields:

• Strict confinement of single-qubit SRE within a ballistic light cone ($|i-m| \leq t$)
• Exponential decay of the total magic footprint $\mathcal{M}(t)\sim e^{-\Gamma t}$, with decay rate $\Gamma\approx0.44$
• When normalized to remove total decay, the SRE profile diffuses within the light cone under Haar-random Clifford gates ($a_i(t)$ solves a discrete diffusion equation), and superdiffuses under restricted Clifford circuits ($\sigma(t)\sim t^\beta,\,\beta\approx0.65$) (Maity et al., 11 Nov 2025).

This demonstrates the emergence of hydrodynamic laws for magic spreading in brickwork Clifford circuits despite the absence of explicit conservation laws.

7. Implementation and Outlook

The schematic brickwork arrangement of two-qubit Clifford layers is naturally compatible with current superconducting and ion-trap quantum hardware, readily extends to monitored dynamics via Bell-pair measurements, and enables precise control of entanglement structure and operator spreading through tunable gate selection and measurement rates (Arienzo et al., 2022, Kovács et al., 2 Aug 2024). Brickwork Clifford architectures therefore form a minimal and analytically tractable platform for studying quantum information dynamics, measurement-induced transitions, shadow estimation, and the emergence of classical criticality and localization phenomena in random circuits. Future directions include exploring ZX-calculus simplification in non-Clifford circuits, generalizing magic dynamics, and leveraging operator fragmentation for robust error mitigation and quantum simulation.