Theory of Magic Phase Transitions in Encoding-Decoding Circuits

Abstract

Quantum magic resources, or nonstabilizerness, are a central ingredient for universal quantum computation. In noisy many-body systems, the interplay between these resources and errors leads to sharp magic phase transitions. However, the microscopic mechanism behind these critical phenomena is still an open question, especially since early empirical evidence showed conflicting results regarding their universality classes. In this work, we provide a comprehensive picture of magic phase transitions for the class of encoding-decoding quantum circuits to resolve these ambiguities. We analytically show that the behavior of magic resources is fundamentally dictated by the chosen measurement protocol. When we fix, or post-select, the class of measurement syndromes, the magic transition inherits the universal features of the error-resilience phase transition in the circuits. Interestingly, this clean transition survives even for fully random Haar encoders showing that it is a consequence of initial's state retrieval, and not an artifact of the Clifford encoders. On the other hand, if we consider realistic Born-rule sampling, the intrinsic statistical fluctuations of a given syndrome measurement act as a relevant perturbation. This brings in strong finite-size drifts and an apparent multifractality, which end up altering the critical behavior of the system. We reveal that magic phase transitions are actually direct manifestations of error-resilience thresholds, rather than independent critical phenomena, reconciling conflicting observations from the earlier literature. Ultimately, our framework clarifies how the quantum computational power can survive, or be irreversibly destroyed, due to the competition between scrambling, measurements, and errors.

Figures (9)

• Figure 1: Encoding--decoding circuit. The unitary $U$ encodes a $k$-qubit logical state into the $N$-qubit code space. The circuit comprises the encoder $U$, a layer of local errors $E_i$, and the decoder $U^\dagger$. The decoded logical state $|\psi_U(\mathbf{s})\rangle$ is obtained by projecting the ancilla qubits onto $|{\mathbf{s}}\rangle$.
• Figure 2: Error-resilience phase transition in encoding-decoding circuits: fidelity and self-averaging. (a) Fidelity $F$ as function of error strength $\alpha$ for system size $N$. (b) Finite-size scaling collapses of fidelity. Panels (c) and (d) show numerator and denominator fluctuations for Haar encoders at fixed error strenght $\alpha$, as a function of system size $N$. (e) and (f) show numerator and denominator fluctuations for the Clifford encoders. The insets in (c-f) show the corresponding data but plotted as function of $\alpha$ for various system sizes $N$. In all panels the solid lines show analytical expressions obtained with the replica method, while markers denote the numerical results averaged over more than 1000 circuit realizations.
• Figure 3: Phase transition in magic resources for Haar-random encoders with the postselected syndrome $\mathbf{s}=\mathbf{0}$. (a) The SRE $M_2$ as a function of the error strength $\alpha$ for various system sizes $N$ and a fixed code rate $r=k/N=1/2$. Solid lines represent the analytic formula \ref{['eq:SRE2good']}, while symbols denote the quenched averages $\overline{M}_2$ computed via state-vector simulations averaged over 1000 circuit realizations. The inset shows the collapse of $M_2$ onto the universal curve \ref{['eq:M2_haar_scaling']} when plotted against the rescaled error strength $(\alpha-\alpha_c)N^{1/\nu}$, with $\nu=1$. (b) Analogous results for the SRE $M_3$. Panels (c) and (d) show the SRE densities $M_q/N$ for $q=2$ and $q=3$, respectively, with the red dashed lines indicating the behavior in the thermodynamic limit ($N\to\infty$).
• Figure 4: Phase transition in magic resources for Clifford encoders with the postselected syndrome $\mathbf{s}=\mathbf{0}$. (a) The SRE $M_2$ as a function of the error strength $\alpha$ for various system sizes $N$ and a fixed code rate $r=k/N=1/2$. Solid lines represent the analytic formula \ref{['eq:SRE2cliffgood']}, while symbols denote the quenched averages $\overline{M}_2$ computed with Pauli propagation method [Sec.\ref{['sec:PauliPropaCliff']}], averaged over more than 4000 circuit realizations. The inset demonstrates the collapse of $M_2$ onto the universal curve \ref{['eq:M2_haar_scaling']} (which coincides with the Haar case) when plotted against the rescaled error strength $(\alpha-\alpha_c)N^{1/\nu}$, with $\nu=1$. (b) The SRE density $M_2/N$, with red dashed lines indicating the $N\to\infty$ limit. (c) Numerically calculated quenched averages of the SRE $M_3$; dashed lines are shown to guide the eye.
• Figure 5: Phase transition in magic resources for Clifford encoders at a fixed code rate $r=k/N=1/2$ across varying syndrome classes. The fidelity $F$ as a function of the error strength $\alpha$ is shown for the first and second syndrome classes in panels (a) and (c), respectively. The SRE $M_2$ as a function of $\alpha$ undergoes a transition between area-law and volume-law phases at $\alpha=\alpha_c$, shown for the first and second syndrome classes in panels (b) and (d), respectively. The insets in (a) and (c) demonstrate the collapse of the fidelity $F$ upon rescaling $\alpha \mapsto (\alpha-\alpha_c)N^{1/\nu}$ (with $\nu=1$). Meanwhile, the insets in (b) and (d) highlight the crossing point in the $M_2$ versus $\alpha$ curves, which converges to $\alpha=\alpha_c$ with increasing system size $N$. Results are computed with Pauli propagation [Sec.\ref{['sec:PauliPropaCliff']}] and averaged over more than 2000 circuit realizations.