Fragility of Magic State Distillation under Imperfect Measurements

TL;DR

This work develops a general framework to analyze magic state distillation under imperfect measurements by modeling the stabilizer-projection step with a noisy projector and mapping the MSD dynamics to a dynamical system. A sharp measurement-threshold phenomenon is established: below a critical measurement strength $\beta^*$ distillation fails, while above it distillation remains possible but with a fidelity cap and linear (rather than polylogarithmic) efficiency, drastically increasing overhead. The authors also offer a universal, low-cost robustness strategy—measuring stabilizer generators in standard form—which reduces the impact of measurement noise and improves thresholds without extra overhead. Collectively, these results provide quantitative benchmarks and a practical design principle for implementing robust MSD on near-term quantum hardware and for guiding resource estimates in realistic devices.

Abstract

Magic state distillation (MSD) is the leading approach to generate the non-Clifford resources required for universal fault-tolerant quantum computation. While most analyses assume ideal measurements in the distillation process, this assumption breaks down on near-term hardware where measurement fidelity remains limited and large quantum error-correcting codes are unavailable. Here we establish a general framework to analyze MSD under imperfect measurements, and reveal a sharp threshold phenomenon that differs from previous known threshold on input state error: Below a critical measurement strength, MSD loses its distillation power entirely, whereas above the threshold, the \textit{target states} are at most first-order biased and the \textit{distillation efficiency} is reduced to linear, leading to exponentially higher distillation overheads. To mitigate this fragility, we present a universal method to maximize MSD robustness against imperfect measurements by choosing stabilizer generators in standard form, which applies to all known protocols without incurring additional costs. Our work reveal fundamental constraints on MSD protocols with measurement noise and provide insights for designing practically robust distillation protocols in the near-term era of quantum hardware.

Figures (7)

• Figure 1: Stabilizer reduction scheme. The input is multiple copies of noisy states. A Clifford circuit associated with the protocol code $\mathcal{Q}$ is applied on the input state and single Pauli measurements are performed on all ancillary qubits. Only certain desired measurement outcomes herald a better output state, and all other outcomes lead to discarding the state and restarting the protocol.
• Figure 2: Critical distillation behavior of the $[[15, 1, 3]]$ protocol for different measurement strength. (a) Iterative distillation behavior for the $[[15, 1, 3]]$ protocol. When measurement strength (inverse of noise) is above the critical threshold, distillation is still achievable but with a fidelity cap (dashed line). Distillation fails entirely if measurement strength is below the critical threshold. The initial state is set to be depolarized $\ket{T}$ state with error rate 0.1. (b) Flow diagram of the $[[15, 1, 3]]$ MSD protocol on the $x$-$y$ cross section of the Bloch sphere. The orange stars denote the target states to distill into and the orange area denotes the convergence region for the target states. $\beta=2$($\beta=1$) is used for the above(below) threshold simulation. The threshold is $\beta^*\approx 1.74$.
• Figure 3: Deviation of target states. For both $[[15, 1, 3]]$ and $[[14, 2, 2]]$ protocols, the target states deviates from the ideal $\ket{T}$ states under imperfect measurements. The deviation is also dominated by the $X$ noise as the $Y$ noise is significantly in the smaller order. $\exp(-2\beta)$ is the factor for first-order noise in the simulated noise model.
• Figure 4: Convergence and distillation cost. (a-b) Convergence of distilled states toward the target states for $[[15, 1, 3]]$ and $[[14, 2, 2]]$ protocols. The convergence under ideal measurements is exponentially faster than the imperfect cases. (c-d) Distillation cost under imperfect measurements. As imperfect measurements cause linear efficiency, the distillation cost is also exponentially higher than the ideal case.
• Figure 5: Noise-resilient MSD protocols under imperfect measurements. (a,c) Displacement between the noisy target states and the ideal state. (b,d) Scaling of the linear convergence prefactor. The grey line in all figures denotes the performance with the canonical generator choice.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4