Quantum computing aims to leverage quantum mechanics for computational speedups beyond classical capabilities. While entanglement is a key resource, it's insufficient; the distinction between 'easy' and 'hard' quantum states is crucial. Clifford gates operating on stabilizer states are classically simulable, while non-Clifford gates and non-stabilizer states are believed to be classically intractable. This necessitates costly magic state distillation in error-corrected quantum computers. A resource theory of stabilizer computation has emerged, defining 'magic' as the resource enabling universal quantum computation. The amount of magic in a state determines its usefulness in fault-tolerant synthesis of non-Clifford operations. Magic has implications for bounding quantum complexities and constraining tensor network models. Understanding magic generation and suppression is vital for advancing quantum computing and clarifying the limits of classical simulability. This research explores measurement-induced phase transitions, extending them from entanglement to magic and studying this transition experimentally. Quantum error-correcting codes under coherent errors display a phase transition in magic, dependent on logical qubit number (measurement rate) or error rate. Syndrome measurements, which can destroy magic, compete with errors, which create it. This paper introduces a new measure of magic, the basis-minimized measurement entropy, and uses it alongside the stabilizer Rényi entropy in simulations and experiments on IonQ's Aria trapped-ion quantum computer to demonstrate this phase transition.

The authors review existing literature on the resource theory of stabilizer computation, highlighting the significance of magic as a resource for universal quantum computation. They discuss the classical simulability of Clifford circuits acting on stabilizer states and the computational hardness of non-Clifford operations and non-stabilizer states. The concept of magic is connected to existing work on bounding quantum complexities and its role in tensor network models. The paper also draws parallels between the behavior of magic and measurement-induced phase transitions in entanglement, referencing related research on monitored quantum circuits, measurement-induced entanglement phase transitions, and their connections to percolation theory and statistical mechanics models. The use of stabilizer Rényi entropy as a measure of magic and its properties are discussed along with prior work on error correction codes and decoder breakdown are also relevant to the work.

The study investigates magic in random Clifford codes. The initial state is an all-zero state of N qubits. A randomly generated Clifford circuit (encoder) maps this state to the logical space of a random Clifford code. Coherent errors are introduced by applying a single-qubit rotation Rz(α) to each qubit, where α is the error rate. The conjugate of the encoding circuit (decoder) is then applied, followed by syndrome measurements of N-K qubits. The remaining K qubits form the logical state. Encoding circuits are created by interweaving layers of single-qubit and two-qubit Clifford unitaries. The two measures of magic used are the second stabilizer Rényi entropy (SSRE) and the basis-minimized measurement entropy. SSRE quantifies the spread of the state's density matrix in the Pauli basis. The basis-minimized measurement entropy is the minimized entropy of the Born probability distribution of measurement outcomes across all possible stabilizer measurement bases. The study examines two code rate scenarios: vanishing rate (K=1) and constant rate (K=rN). Classical simulations, analytical calculations, and experiments on IonQ's Aria trapped-ion quantum computer are conducted to measure these magic quantities as a function of error rate (α). For experiments, tomography is used to reconstruct the density matrix, and post-processing techniques including mitigation for incoherent errors are applied. In the vanishing rate case, syndrome outcome post-selection is simplified by grouping syndromes with equivalent logical actions. For constant-rate codes, the computationally less expensive conditional entropy is used to characterize the phases, serving as an upper bound for the basis-minimized conditional entropy. Analytical calculations utilize Clifford averaging via Schur-Weyl duality to compute the Rényi analogue of the conditional entropy.

The research reveals a phase transition in magic as a function of error rate and code rate. In the vanishing rate case (single logical qubit), the second stabilizer Rényi entropy shows a peak at a distance proportional to 1/√N from the Clifford point (α=π/2). This peak exhibits a square-root scaling with N, a phenomenon also observed in experimental results on IonQ's Aria quantum computer, qualitatively matching theoretical predictions. For constant-rate codes, simulations reveal an extended magical phase above a critical error rate. The density of SSRE exhibits a phase transition, confirmed by a finite-size scaling collapse, indicating a critical error rate and exponent. The conditional entropy, both numerically simulated and experimentally obtained, showcases this transition from non-magical to magical. In the finite-rate case, a scaling collapse analysis of the conditional entropy from both simulations and IonQ Aria experiments shows a critical error rate and critical exponent. Analytical calculations, using the Rényi analogue of conditional entropy, accurately match numerical simulation results for larger system sizes and also reveal the phase transition. The critical exponents from simulations and experiments are compared and found to be consistent.

The observed phase transition in magic, influenced by error rate and code rate, demonstrates the interplay between syndrome measurements (destroying magic) and coherent errors (creating magic). The results establish a link between the resource theory of stabilizer computation and decoder breakdown in quantum error correction. The study highlights the potential of efficiently estimable diagnostics, like conditional entropy, for studying phase transitions in larger systems, addressing the general challenge of measuring non-stabilizerness in large-scale systems. The findings suggest that error correction coupled with well-characterized coherent noise can generate useful magic states, potentially improving magic state distillation efficiency by concentrating magic in the logical qubits where non-Clifford operations are typically hard. The phase transitions observed here, and other similar transitions in the literature, suggest a broader landscape of information-theoretic phase transitions where competing channels generating and destroying a resource (entanglement or magic) cause this behavior. The universality of these phase transitions at their critical points needs further investigation.

This study demonstrates a phase transition in magic within random Clifford codes under coherent errors. The transition's dependence on error rate and code rate is experimentally validated and theoretically understood. The use of conditional entropy as a practical diagnostic for magic opens avenues for analyzing larger systems. The results suggest novel approaches to magic state generation and improve the efficiency of quantum computation.

The experimental study is limited by the system size of IonQ's Aria quantum computer and the circuit depth due to gate noise. The analysis of constant-rate codes is mainly based on simulations due to the computational cost of measuring magic in larger systems. The analytical calculations for the Rényi analogue of conditional entropy rely on a typicality assumption which may not hold true in all regimes.