CS-count-optimal quantum circuits for arbitrary multi-qubit unitaries

Abstract

In quantum computing there are quite a few universal gate sets, each having their own characteristics. In this paper we study the Clifford+CS universal fault-tolerant gate set. The CS gate is used is many applications and this gate set is an important alternative to Clifford+T. We introduce a generating set in order to represent any unitary implementable by this gate set and with this we derive a bound on the CS-count of arbitrary multi-qubit unitaries. Analysing the channel representation of the generating set elements, we infer JnCS⊂JnT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {J}}_n^{CS}\\subset {\\mathcal {J}}_n^T$$\\end{document}, where JnCS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {J}}_n^{CS}$$\\end{document} and JnT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {J}}_n^T$$\\end{document} are the set of unitaries exactly implementable by the Clifford+CS and Clifford+T gate sets, respectively. We develop CS-count optimal synthesis algorithms for both approximately and exactly implementable multi-qubit unitaries. With the help of these we derive a CS-count-optimal circuit for Toffoli, implying JnTof=JnCS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {J}}_n^{Tof}={\\mathcal {J}}_n^{CS}$$\\end{document}, where JnTof\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {J}}_n^{Tof}$$\\end{document} is the set of unitaries exactly implementable by the Clifford+Toffoli gate set. Such conclusions can have an important impact on resource estimates of quantum algorithms.