Quantum gates generated by rotationally invariant operators in a decoherence-free subsystem

Abstract

In recent investigations, Ozawa showed that the angular momentum conservation laws led to a precision limit on quantum computing in single-spin qubit systems and suggested the importance of encoding of a logical qubit by multiple-spin qubits for circumventing this difficulty. Subsequently, Lidar pointed out that the universal encoding on decoherence-free (noiseless) subspaces or subsystems can be used effectively for this purpose. In this paper, we investigate the three-qubit decoherence-free subsystem proposed by Knill, Laflamme, and Viola to construct error-free quantum logic gates in the serial mode satisfying the angular-momentum conservation law. This paper shows that the class of quantum gates defined by the 19-gate sequence with the repetition and correlation symmetries precisely coincides with the class of controlled-$U$ operations up to local equivalence. As a result, any controlled rotation, frequently used in quantum Fourier transforms, can be performed by the 26-gate sequence, the same as the controlled-NOT operation.