Distilling Magic States in the Bicycle Architecture

TL;DR

This work presents practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block that have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint.

Abstract

Magic State Distillation is considered to be one of the promising methods for supplying the non-Clifford resources required to achieve universal fault tolerance. Conventional MSD protocols implemented in surface codes often require multiple code blocks and lattice surgery rounds, resulting in substantial qubit overhead, especially at low target error rates. In this work, we present practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block. We formulate distillation circuit design as a joint optimization of logical qubit mapping, gate scheduling, measurement nativization, and protocol compression via qubit recycling. Based on detailed resource analysis and simulations, our BB factories have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint, and are particularly compelling as second-round distillers following magic state cultivations.

Figures (10)

• Figure 1: Magic-state factory design space. Target logical error rate as a function of available physical qubits for surface-code cultivation, BB-code distillation on gross and two-gross codes, and two-level protocols combining cultivation with two-gross or surface-code-only distillation at physical error rate $p_{\mathrm{phys}}=10^{-3}$. Our BB-based factories achieve lower output error at similar qubit budgets, and the cultivation + two-gross pipeline extends to lower error regimes than cultivation can reach.
• Figure 2: One module in the bicycle architecture, consisting of a Bivariate Bicycle (BB) code and a Logic Processing Unit (LPU). BB codes can be presented bravyi2024high-threshold2 as a lattice of physical qubits on a torus, consisting of X and Z check qubits and L and R data qubits. Connections between qubits are within the lattice and also along certain long-range connections. The LPU, which consists of two modules, is connected to the $X$ and $Z$ operators of the pivot qubit $L_0$ and its dual $L_6$. The LPU is capable of measuring any product of operators $X_{L_0}, Z_{L_0}, X_{L_6},$ and $Z_{L_6}$. See Section \ref{['subsec:cliffordsynth']} for more details.
• Figure 3: Fault-tolerant implementation of a shift-automorphism generator and its impact on logical operators. Shift automorphisms permute data qubits via successive swap operations (green, then red) between data and check qubits along edges in the connectivity graph. Logical operators $X_{L_0}, X_{L_1}, X_{L_2}$ supported on shaded regions are permuted so that their overlap with the pivot's $Z_{L_0}$ support changes. After conjugation, multi-qubit Paulis that were not directly accessible through the pivot become measurable via an LPU $Z_{L_0}$ measurement.
• Figure 4: (a–c) Magic state injection schemes for implementing $\exp(i \frac{\pi}{8} P)$ in a BB architecture. The schemes differ in how the magic state is teleported to the target qubits and how the resulting conditional Clifford correction is handled, leading to different latency and error profiles. Inter-module measurements are shown in orange, conditional Clifford corrections in blue. A $P$ label denotes a Pauli operator, which is cheap to track in fault-tolerant architectures. (d) Measurement-to-rotation circuit implementing $\exp(i \frac{\pi}{4} P)$ using a designated pivot qubit and BB's toric symmetry.
• Figure 5: (a) Masking technique to nativize a Pauli measurement by allowing $Z$ to act on an idle logical qubit initialized to $\ket{0}$ within the BB code. (b) Native and non-native measurements in the 15-to-1 distillation circuit, which becomes fully nativized after masking. (c) Scheduling the 15-to-1 rotations in an order that minimizes automorphism rounds between successive measurements.