Phase transition in magic with random quantum circuits

TL;DR

The paper investigates how non-Stabilizer resources (magic) behave in monitored random quantum circuits by studying phase transitions in magic for random Clifford codes under coherent noise. It develops two quantitative diagnostics—the basis-minimized measurement entropy and the stabilizer Rényi entropy—to detect non-stabilizerness, and demonstrates, through analytic, numerical, and experimental work, a magic phase transition controlled by the error rate and code rate. A vanishing-rate single-logical-qubit regime shows square-root scaling of magic near the Clifford point, while constant-rate codes exhibit a finite-size scaling phase transition with a detectable magic phase and an extended magical region at finite rate. The results connect the resource theory of magic to measurement-induced transitions in error-correcting dynamics, suggesting practical routes to leverage syndrome measurements for magic-state generation and informing broader questions about quantum speedups and universality in information-theoretic phase transitions.

Abstract

Magic is a property of quantum states that enables universal fault-tolerant quantum computing using simple sets of gate operations. Understanding the mechanisms by which magic is created or destroyed is, therefore, a crucial step towards efficient and practical fault-tolerant computation. We observe that a random stabilizer code subject to coherent errors exhibits a phase transition in magic, which we characterize through analytic, numeric and experimental probes. Below a critical error rate, stabilizer syndrome measurements remove the accumulated magic in the circuit, effectively protecting against coherent errors; above the critical error rate syndrome measurements concentrate magic. A better understanding of such rich behavior in the resource theory of magic could shed more light on origins of quantum speedup and pave pathways for more efficient magic state generation.

Figures (11)

• Figure 1: Model and phase diagram. A: The model. The qubits start in an all-zero state, corresponding to a logical $0$ state. We apply a random Clifford encoding circuit (green), controlled "error" unitaries (red), and the conjugate of the encoding circuit (blue). B: A schematic illustration of how magic is created or destroyed in our model. The encoding step acting on an input stabilizer state (represented by a blue Bloch sphere) produces a highly entangled stabilizer state in the many-qubit Hilbert space. Coherent rotations move the state off the grid of stabilizer states. The decoding step either snaps the state back to the grid of stabilizer states or pushes the state away from that grid. The final state is either a multi-qubit stabilizer state, represented by a Bloch-sphere shaded blue, or a magical state, represented by a Bloch sphere shaded red. The Pauli expectations of the resulting stabilizer (magical) state are shown as histograms shaded blue (red). C: Phase diagram for constant-rate codes. The color bar represents the magic density at a particular code-rate $r$, given by the ratio of logical qubits $K$ and total number of qubits $N$, and error rate, defined to be the angle of coherent rotation, $\alpha$.
• Figure 2: Results for vanishing rate codes.A: Distribution of second stabilizer Rényi entropy across codes and syndromes in classical simulation. The distribution is tightly peaked around square-integer multiples of the distance $\epsilon = \pi/2 - \alpha$ from the Clifford point, because it is controlled by the weights of the errors. B & C: Syndrome- and circuit-averaged second stabilizer Rényi entropy in classical numerics (B) and experiment on IonQ Aria trapped-ion quantum computer (C). Both display the predicted square-root scaling. The error estimates are derived using bootstrapping (details in the SM Section \ref{['app:boostrapping-estimate']}). The scaling with respect to system size of the vertical axis of C (main) is chosen to match the scaling of the peak in unscaled experimental data (inset). For B, the errorbars are omitted in the collapse (main plot). In C, we also present numerics from noisy simulations (solid lines), obtained using a noise model that uses overrotation and depolarizing noise (See SM Section K for more details).
• Figure 3: Results for constant rate codes.A: Density of magic (SSRE) of the logical space and its scaling collapse (inset) plotted against the error rate $\alpha$, for code rate $r = K/N = 1/2$. The error bars are derived using standard error and are omitted in the scaling collapse (inset), where the x-axis is scaled as $(\alpha/\pi-\alpha_c/\pi)N^{1/\nu}$ with critical parameters $\alpha_c/\pi = 0.27(1)$ and $\nu = 1.15(4)$, and the y-axis is scaled as $\langle \mathcal{M}\rangle/K^{\gamma}$ with $\gamma = 1.20(8)$. B: Phase diagrams of conditional entropy (upper) and its Rényi approximation (lower), without any basis minimization. C: Finite size scaling of the conditional entropy and its collapse (inset) computed numerically using simulations at $r=1/2$. The scaling collapse (inset) has critical parameters $\alpha_c/\pi = 0.304(2)$ and $\nu=2.9(2)$. D: Finite size scaling of the Rényi approximation of the conditional entropy and its collapse (inset) computed numerically using simulations (displayed points) and analytics (solid line) at $r=1/2$.. The scaling collapse, computed with data from simulations, has critical parameters $\alpha_c/\pi = 0.347(1)$ and $\nu=2.6(2)$. E: Finite size scaling of the conditional entropy using data from experiments in IonQ Aria at $r=1/2$. The error bars are obtained using boostrap resampling. The scaling collapse (inset) uses critical exponents derived from numerical simulations of circuits with $d=N/2$, as shown in Fig. \ref{['fig:half-depth']}. F: Finite size scaling of the Rényi-approximation of the conditional entropy and its collapse (inset) computed using experiments at $r=1/2$.
• Figure S1: Distribution of Errors The distribution of error per syndrome for vanishing rate codes (A) and constant rate codes $(\textbf{B})$. We observe that in both cases, with high probability, the errors are uniformly distributed across syndromes, such that there are $2^{K}$ unique errors per syndrome. Colorbars give the number of qubits $N$.
• Figure S2: Numerical Simulations for circuits with depth $d=N/2$. A: second stabilizer Rényi entropy for vanishing-rate code. Like with $d=N$ circuits, this exhibits a $\sqrt{N}$ scaling near the critical point at $\alpha=\pi/2$. B: Finite size scaling of the conditional entropy and its collapse (inset) computed numerically using simulations at code rate $r=1/2$. The error bars are omitted in the scaling collapse (inset) which has critical parameters $\alpha_c/\pi = 0.300(2)$ and $\nu=1.40(6)$. These critical parameters are used in the collapse of experimental data in Fig. \ref{['fig:constant-rate-magic-full-panel']}E. C: Finite size scaling of the Rényi-approximation of the conditional entropy and its collapse (inset) computed numerically using simulations at code rate $r=1/2$. The error bars are omitted in the scaling collapse (inset) which has critical parameters $\alpha_c/\pi = 0.351(1)$ and $\nu=1.24(4)$. These critical parameters are used in the collapse of experimental data in Fig. \ref{['fig:constant-rate-magic-full-panel']}F.