Phase transition in magic with random quantum circuits

The study addresses when and how quantum circuits generate non-stabilizer resources ("magic") necessary for universal, fault-tolerant quantum computation. While entanglement is essential, Clifford circuits acting on stabilizer states are efficiently classically simulable, so non-Clifford resources are required for genuine quantum advantage. The work situates magic within the resource theory of stabilizer computation and connects it to monitored (measurement-interspersed) quantum circuits, where measurement-induced phase transitions in entanglement are known. The central hypothesis is that, in error-correcting circuits with coherent errors, syndrome measurements compete with error-induced magic generation, producing a phase transition in the logical-state magic as a function of error rate and measurement rate (or code rate).

Prior work established that Clifford circuits with stabilizer inputs are efficiently simulable, whereas non-stabilizer states or non-Clifford gates are believed to be classically hard (Gottesman; Aaronson & Gottesman; Bravyi et al.; Bu & Koh). The resource theory of magic formalizes non-stabilizerness and links it to computational power and magic state distillation (Veitch et al.; Campbell et al.). Magic has informed quantum complexity bounds and holographic tensor-network models and is necessary to simulate quantum chaos. Monitored quantum circuits display measurement-induced entanglement phase transitions with connections to percolation, stabilizer codes, and statistical mechanics (Skinner et al.; Li, Chen & Fisher; Bao, Choi & Altman; Jian et al.; Li et al.; Barratt et al.). Recent studies have explored transitions beyond entanglement (Agrawal et al.) and theoretical predictions of magic transitions in hybrid circuits (Leone et al. 2023). This paper extends these ideas by demonstrating a measurement-induced phase transition in magic within random stabilizer codes under coherent errors and relates it to decoder breakdown in QEC.

Model: Start with an N-qubit product state |0>^N. Apply a random Clifford encoding circuit C to realize a random stabilizer code. Apply coherent single-qubit Z-rotations on every qubit, Rz(α)=exp(-i σ_z α/2), modeling coherent errors (error-rate parameter α). Apply the decoding C†. Measure N−K qubits in the computational basis as stabilizer syndromes, leaving K logical qubits. Code rates considered: (i) vanishing rate K=1 (r=1/N→0), and (ii) constant rate K=rN (0<r≤1/2 in simulations and experiments). Circuit ensemble: Interleaved layers—d layers of single-qubit Clifford gates (uniform over 24 elements) and d layers of two-qubit Mølmer-Sørensen entangling gates MS(π/2)=exp(iπ σ_x⊗σ_x/4) on random disjoint pairs. Numerics use d=N; experiments use d=N/2 (also compared to d=2N in SM) to mitigate hardware noise. Measures of magic: (1) Second stabilizer Rényi entropy (SSRE) M2(ρ) = -log Σ_P ρ_P^4 Tr(P†P)^4; zero for stabilizer states and extensive for Haar-like states. (2) Basis-minimized measurement entropy: Shannon entropy of measurement outcomes minimized over stabilizer measurement bases (i.e., over Clifford basis changes), averaged over syndromes; its non-minimized version (computational basis) is used as an efficiently measurable upper bound in the non-magical phase. The Rényi-2 analogue of the conditional entropy is also analyzed. Analytics: For vanishing rate near the Clifford point α=π/2, perturbative analysis shows that for a given circuit and syndrome s, M2 ≈ (n ε)^2 with ε=π/2−α and n determined by the weights of the two Pauli errors producing the same syndrome; averaging over syndromes yields ⟨M2⟩ ≈ (1/4) N ε^2 for N ε^2 ≪ 1, implying a scaling collapse versus ε√N. For constant rate, closed-form analytics are provided for the Rényi conditional entropy via Clifford averaging using Schur-Weyl duality; in the r→1 limit (no measurements), the SSRE density approaches −log[1−(1/4) sin^2(2α)]. Classical simulations: Exact simulations up to N≈24 (varies by observable) across many random circuits (e.g., up to 5000 for smaller N). Finite-size scaling analyses perform free-parameter collapses to extract critical error rates and exponents; also performed under constrained exponent ranges in SM. Experiments: Implemented on IonQ’s Aria trapped-ion device via QLab (UMD). Up to N=16 qubits, depth d=N/2. Vanishing-rate experiments include logical qubit tomography using basis-change Pauli gates; postselection replaced by grouping syndrome outcomes into equivalence classes by effective logical action (identified via classical simulation). Error mitigation: project the single-qubit logical density matrix onto its maximum-eigenvalue eigenstate to mitigate incoherent noise. For constant rate, SSRE is impractical; instead measure computational-basis conditional entropies (Shannon and Rényi) using experimental distributions projected onto the support of ideal distributions and leveraging classical predictions p(x) (akin to linear cross-entropy benchmarking postprocessing). Statistical treatment: Error bars via bootstrap resampling; noisy simulation comparisons include overrotation and depolarizing noise on two-qubit gates (Gaussian overrotation ε~N(0,0.06) and depolarizing p=0.016).

- Existence of a magic phase transition: In random stabilizer codes with coherent single-qubit rotations and syndrome measurements, magic in the logical space undergoes a transition controlled by error rate α and measurement rate (via code rate r=K/N). Below threshold (small α or high measurement rate), syndrome measurements remove magic; above threshold, they concentrate magic into logical qubits. - Vanishing-rate (K=1) scaling near α=π/2: Analytically and numerically, SSRE exhibits a scaling collapse M2 = f((π/2−α)√N) with ⟨M2⟩ ≈ (1/4) N (π/2−α)^2 close to the Clifford point. The peak occurs at a distance O(1/√N) from α=π/2, and the Clifford point becomes a singularity at large N. Experimental data for N=8,12,16 qualitatively agrees, showing the predicted square-root finite-size scaling behavior after mitigation. - Constant-rate (e.g., r=1/2) magical phase: Classical simulations show an extended magical phase with extensive SSRE above a critical error rate. For SSRE at r=1/2, a free-parameter scaling collapse yields critical parameters (expressed in normalized units) α_c/π = 0.27(1), ν = 1.15(4), and scaling exponent γ = 1.20(8) for M/K ∝ (α/π − α_c/π)^{1/γ}. Constrained analyses placing γ within [0,1] are provided in SM. - Conditional entropy diagnostics: As an experimentally accessible proxy, the computational-basis conditional Shannon entropy shows a phase diagram consistent with a transition. Finite-size scaling at r=1/2 from simulations gives α_c/π = 0.304(2), ν = 2.9(2). The Rényi-2 conditional entropy approximation yields α_c/π = 0.347(1), ν = 2.6(2). Experimental measurements (N=8,12,16; d=N/2) exhibit scaling collapses consistent with numerically obtained critical parameters for d=N/2 (SM): Shannon α_c/π ≈ 0.300(2), ν ≈ 1.40(6); Rényi α_c/π ≈ 0.351(1), ν ≈ 1.24(4). - Analytical limits: In the r→1 limit (no syndrome measurements), the SSRE density is volume-law with magic density −log[1 − (1/4) sin^2(2α)] at large N. For vanishing rate near α=π/2, perturbation theory explains square-root scaling and predicts M2 distributions controlled by differences in Pauli error weights drawn from binomial statistics. - Decoder interpretation: The (basis-minimized) conditional measurement entropy directly captures decoder breakdown thresholds for classical information storage; large conditional entropies indicate an inability to recover stored information even with optimal Clifford decoding.

The results demonstrate a measurement-induced phase transition in non-stabilizerness (magic), paralleling known entanglement transitions in monitored circuits. The transition reflects competition between three noncommuting channels: correlation generation (random Clifford encoding), magic generation (coherent Rz rotations), and magic destruction (syndrome measurements). This connects resource-theoretic magic to quantum error-correction performance via decoder breakdown, as quantified by conditional entropies. The observed transitions suggest a broader landscape of information-theoretic critical phenomena governed by competing channels, with open questions about universality classes and whether critical points from different diagnostics (SSRE vs conditional entropies) coincide. Practically, operating in the magical phase could enable syndrome-informed concentration of magic into logical qubits, potentially improving downstream magic state distillation, contingent on efficiently computing syndrome-dependent Clifford decoders beyond special cases like zero-rate surface codes.

The study establishes and experimentally probes a phase transition in magic within random stabilizer codes under coherent errors and syndrome measurements. It introduces basis-minimized measurement entropy as a practical magic diagnostic, confirms vanishing-rate square-root scaling near the Clifford point, maps a finite-rate magical phase with critical behavior via SSRE and conditional entropies, and corroborates findings with trapped-ion experiments. Analytical treatments capture both the vanishing-rate scaling and the r→1 volume-law limit. The work bridges resource theory of magic and monitored-circuit phase transitions, motivates efficient experimental diagnostics (conditional entropies) for larger systems, and points to potential applications in generating useful logical magic states for more efficient distillation. Future research should clarify universality, reconcile critical parameters across diagnostics, scale to larger systems and deeper codes, and develop practical syndrome-dependent decoding strategies for magic concentration.

- Finite-size and depth constraints: Simulations and experiments are limited to N≲24 (sim) and N≤16 (exp) with shallow depths (d=N/2 in hardware), potentially biasing critical exponents and collapse quality. Critical parameters depend on scaling hypotheses and depth; SSRE exponents differed from those of conditional entropies. - Diagnostic limitations: SSRE is experimentally intractable at finite rate; conditional entropies (without basis minimization) serve as upper bounds and may not perfectly reflect basis-minimized magic. Typicality assumptions in analytics can break down for small N and α near π/2. - Experimental noise and mitigation: Hardware imperfections necessitated post-processing (projection onto ideal subspace, tomography projection), and results rely on noise mitigation and classical simulations. Postselection on exact syndromes was infeasible; grouping into equivalence classes approximates logical actions. - Unsettled universality and coincidence of transitions: It remains unclear whether different observables share the same critical point and exponents; universality class is not established.