Fault-tolerant conversion between adjacent Reed-Muller quantum codes based on gauge fixing

Abstract

We design forward and backward fault-tolerant conversion circuits, which convert between the Steane code and the 15-qubit Reed-Muller quantum code so as to provide a universal transversal gate set. In our method, only 7 out of total 14 code stabilizers need to be measured, and we further enhance the circuit by simplifying some stabilizers; thus, we need only to measure eight weight-4 stabilizers for one round of forward conversion and seven weight-4 stabilizers for one round of backward conversion. For conversion, we treat random single-qubit errors and their influence on syndromes of gauge operators, and our novel single-step process enables more efficient fault-tolerant conversion between these two codes. We make our method quite general by showing how to convert between any two adjacent Reed-Muller quantum codes $\overline{\text{RM}}(1,m)$ and $\overline{\text{RM}}\left(1,m+1\right)$, for which we need only measure stabilizers whose number scales linearly with m rather than exponentially with m obtained in previous work. We provide the explicit mathematical expression for the necessary stabilizers and the concomitant resources required.

References in corpus (21)

• Good Quantum Error-Correcting Codes Exist
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Multiple Particle Interference and Quantum Error Correction
• Quantum Computing with Very Noisy Devices
• Restrictions on Transversal Encoded Quantum Gate Sets
• Overhead and noise threshold of fault-tolerant quantum error correction
• Magic state distillation with low overhead
• Active stabilisation, quantum computation and quantum state synthesis
• Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
• Universal fault-tolerant quantum computation with only transversal gates and error correction
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Improved magic states distillation for quantum universality
• Universal transversal gates with color codes - a simplified approach
• Multilevel distillation of magic states for quantum computing
• Surface code implementation of block code state distillation
• Using concatenated quantum codes for universal fault-tolerant quantum gates
• Protected gates for topological quantum field theories
• Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit
• Stacked codes: universal fault-tolerant quantum computation in a two-dimensional layout
• Reducing the quantum computing overhead with complex gate distillation
• On triply even binary codes

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