Iteratively decoded magic state distillation

TL;DR

The paper tackles the bottleneck of producing high-fidelity magic states for fault-tolerant quantum computation by introducing an iteratively decoded scheme using transversal CNOTs across surface-code patches to perform 7-to-1 and 15-to-1 distillations. It combines stabiliser-proxy analysis, ZX-calculus benchmarking, Pauli webs, and Stim-based simulations to demonstrate that, under circuit-level noise, the distilled states achieve cubic error suppression $p_{\text{out}} = \mathcal{O}(p^3)$ with leading constants $7$ and $35$ for the two protocols, respectively. The authors show that, with re-configurable hardware, these distillations can run in $\mathcal{O}(1)$ code cycles, requiring roughly $47d^2$ qubit-cycles for 7-to-1 and $111d^2$ for 15-to-1 (excluding certain costs), and that only a few iterations of decoding are needed. This work provides a path toward time-optimal magic-state distillation on flexible architectures and outlines future steps including hardware realism and enhanced benchmarking via Pauli webs and ZX-calculus.

Abstract

We present numerical simulation results for the 7-to-1 and 15-to-1 state distillation circuits, constructed using transversal CNOTs acting on multiple surface code patches. The distillation circuits are decoded iteratively using the method outlined in [arXiv:2407.20976]. We show that, with a re-configurable qubit architecture, we can perform fast magic state distillation in $\sim\mathcal{O}(1)$ code cycles. We confirm that both circuits suppress an injected input error rate $p$ to $\mathcal{O}(p^3)$ in the presence of additional circuit-level noise. We also outline how ZX-calculus and Pauli webs can be used to benchmark stabiliser proxies for these distillation circuits.

Figures (9)

• Figure 1: Tanner graph for the Steane code, encoded using 7 qubits arranged in the usual geometric manner on a triangle, rectangular nodes: $\{A_0,A_1,A_2\}$ are the self-dual $X$ or $Z$ stabiliser checks, while circular nodes 1-7 are the qubit labels.
• Figure 2: The 7-to-1 distillation circuit from zhou2024algorithmicfaulttolerancefast with the first and fourth redundant CNOTs removed as the CNOTs were acting on either $\ket{{+},{+}}$ or $\ket{{0},{0}}$. A single round of syndrome extraction is inserted at every time step indicated by blue dotted lines in the surface code simulations.
• Figure 3: The $[[15,1,3]]$ Reed-Muller code's geometric representation with encoding qubits as circular nodes, rectangular stabiliser check nodes and its edges are excluded for representation purposes.
• Figure 4: One of the four weight-8 $X$ stabiliser check cells in the $[[15,1,3]]$ Reed-Muller code with the rectangular stabiliser check node explicitly shown.
• Figure 5: The 15-to-1 ($\ket{T}$ state) distillation circuit from costofuni with all the redundant CNOTs removed. A single round of syndrome extraction is inserted at every time step indicated by blue dotted lines in the surface code simulations.