Generators and Relations for Un(Z[1/2,i])

Abstract

Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i].

arXiv commentIn Proceedings QPL 2021, arXiv:2109.04886

References in corpus (7)

• Improved Simulation of Stabilizer Circuits
• Automated optimization of large quantum circuits with continuous parameters
• Exact synthesis of multiqubit Clifford+T circuits
• Reducing T-count with the ZX-calculus
• T-count optimization and Reed-Muller codes
• Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits
• Techniques to Reduce $\pi /4$-Parity-Phase Circuits, Motivated by the ZX Calculus

Cited by in corpus (1)

• Symbolic Synthesis of Clifford Circuits and Beyond