Restrictions on Transversal Encoded Quantum Gate Sets

Abstract

Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.

4 pages, v2: minor changes

References in corpus (7)

• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• Methodology for quantum logic gate constructions
• Fault-Tolerant Postselected Quantum Computation: Schemes
• Introduction to Quantum Computation
• Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit
• Transversality versus Universality for Additive Quantum Codes

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