Householder methods for quantum circuit design

• Published in: Canadian journal of physics Vol. 94; no. 2; pp. 150 - 157
• Main Authors: Urias, Jesus, Quinones, Diego A
• Format: Journal Article
• Language: English
• Published: Ottawa NRC Research Press 01.02.2016; Canadian Science Publishing NRC Research Press
• Subjects: 03.65.Fd; 03.67.Ac; algorithmic synthesis of quantum circuits; Algorithms; Assembling; Circuit design; Combinatorial analysis; factorisation de Householder dans les espaces de produits tensoriels; Factorization; Gates; Gates (circuits); Householder factorizations in tensor-product spaces; Methods; Physics; Quantum computing; Quantum physics; quantum simulators; Qubits (quantum computing); simulateurs quantiques; Simulators; State vectors; Subsystems; synyhèse algorithmique des circuits quantiques; Tensors; Canada
• ISSN: 0008-4204; 1208-6045; 1208-6045
• DOI: 10.1139/cjp-2015-0490

Algorithms to resolve multiple-qubit unitary transformations into a sequence of simple operations on one-qubit subsystems are central to the methods of quantum-circuit simulators. We adapt Householder’s theorem to the tensor-product character of multi-qubit state vectors and translate it to a combinatorial procedure to assemble cascades of quantum gates that recreate any unitary operation U acting on n-qubit systems. U may be recreated by any cascade from a set of combinatorial options that, in number, are not lesser than super-factorial of 2 n , ∏ j = 1 2 n ( j ! ) . Cascades are assembled with one-qubit controlled-gates of a single type. We complement the assembly procedure with a new algorithm to generate Gray codes that reduce the combinatorial options to cascades with the least number of CNOT gates. The combined procedure —factorization, gate assembling, and Gray ordering — is illustrated on an array of three qubits.