{"id":168248,"date":"2013-06-01T00:00:00","date_gmt":"2013-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/synthesis-of-unitaries-with-cliffordt-circuits\/"},"modified":"2018-10-16T20:11:37","modified_gmt":"2018-10-17T03:11:37","slug":"synthesis-of-unitaries-with-cliffordt-circuits","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/synthesis-of-unitaries-with-cliffordt-circuits\/","title":{"rendered":"Synthesis of Unitaries with Clifford+T Circuits"},"content":{"rendered":"
\n

We describe a new method for approximating an arbitrary n<\/span><\/span><\/span><\/span> qubit unitary with precision \u03b5<\/span><\/span><\/span><\/span> using a Clifford and T circuit with O<\/span>(<\/span>4<\/span>n<\/span><\/span><\/span><\/span>n<\/span>(<\/span>log<\/span><\/span>(<\/span>1<\/span>\/<\/span><\/span><\/span>\u03b5<\/span>)<\/span>+<\/span>n<\/span>)<\/span>)<\/span><\/span><\/span><\/span> gates. The method is based on rounding off a unitary to a unitary over the ring Z<\/span><\/span><\/span>[<\/span>i<\/span>,<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span><\/span>\u2013\u221a<\/span>]<\/span><\/span><\/span><\/span> and employing exact synthesis. We also show that any n<\/span><\/span><\/span><\/span> qubit unitary over the ring Z<\/span><\/span><\/span>[<\/span>i<\/span>,<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span><\/span>\u2013\u221a<\/span>]<\/span><\/span><\/span><\/span> with entries of the form (<\/span>a<\/span>+<\/span>b<\/span>2<\/span><\/span>\u2013\u221a<\/span>+<\/span>i<\/span>c<\/span>+<\/span>i<\/span>d<\/span>2<\/span><\/span>\u2013\u221a<\/span>)<\/span>\/<\/span><\/span><\/span>2<\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span> can be exactly synthesized using O<\/span>(<\/span>4<\/span>n<\/span><\/span><\/span><\/span>n<\/span>k<\/span>)<\/span><\/span><\/span><\/span> Clifford and T gates using two ancillary qubits. This new exact synthesis algorithm is an improvement over the best known exact synthesis method by B. Giles and P. Selinger requiring O<\/span>(<\/span>3<\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>k<\/span>)<\/span><\/span><\/span><\/span> elementary gates.<\/p>\n<\/div>\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

We describe a new method for approximating an arbitrary n qubit unitary with precision \u03b5 using a Clifford and T circuit with O(4nn(log(1\/\u03b5)+n)) gates. The method is based on rounding off a unitary to a unitary over the ring Z[i,1\/2\u2013\u221a] and employing exact synthesis. We also show that any n qubit unitary over the ring 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