Quantum Codes with Arbitrary Z-Rotation logical Gates and Applications to Fault-Tolerant Code Switching
2026-08-11 • Information Theory
Information Theory
AI summaryⓘ
The authors developed a new way to create quantum error-correcting codes that can perform small, precise rotations on quantum bits, which is important for quantum computing. They use a method called doubling to build better codes with fewer qubits and improved error correction compared to previous designs. Their approach works not only for color codes but also for rotated surface codes, a popular type of quantum code. They also extended techniques for switching between codes to perform fault-tolerant logical operations and demonstrated this with a practical example using only 45 physical qubits.
quantum error correctionquantum codescolor codesrotated surface codestransversal gateslogical qubitClifford hierarchyfault-tolerancecode switchingmagic state preparation
Authors
Reza Dastbasteh, Ruben M. Otxoa, Pedro M. Crespo, Josu Etxezarreta Martinez
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.