Quantum Codes with Arbitrary Z-Rotation logical Gates and Applications to Fault-Tolerant Code Switching

2026-08-11

AI Summary

This work addresses a key limitation in fault-tolerant quantum computing: the inefficiency of code switching for implementing arbitrary-angle logical Z-rotations on rotated surface codes, which hinders universal fault-tolerant operations. To overcome this, the authors construct a new family of high-distance quantum color codes and r-orthogonal codes based on doubling techniques, enabling transversal implementation of arbitrarily small Z-rotations. They extend existing code-switching protocols to align with the geometric constraints of rotated surface codes. The proposed codes offer reduced qubit overhead and support single-shot Z-syndrome decoding, outperforming conventional triorthogonal codes. Notably, the scheme achieves the first code-switching-based fault-tolerant magic state preparation on a distance-3 rotated surface code using only 45 physical qubits, with numerical simulations confirming its superior performance.

Abstract

A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical $T$ gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer $Z$-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical $Z$-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of $Z$-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of $r$-orthogonal quantum codes, $r \ge 2$, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical $Z$-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of $Z$-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.

Problem

• fault-tolerant quantum computing
• code switching
• Z-rotation gates
• rotated surface codes
• quantum error correction

Innovation

• code switching
• Z-rotation gates
• doubling technique
• rotated surface codes
• fault-tolerant quantum computing