Method for constructing a fault tolerant encode using a quantum computational model

Abstract

A method for constructing an n-qubit fault tolerant encode for any k-qubit quantum gate M, in any given quantum code [n, k, C], comprising: choosing a number n−k of independent spinors Sfrom the first stabilizer C and a first ordered set Sconsists of the independent spinors S; choosing a number n−k of independent spinors Ŝfrom a second stabilizer Ĉ in the intrinsic coordinate and a second ordered set Ŝconsists of the independent spinors Ŝconsist; implementing an encoding Q, wherein the encoding Qconverts the first ordered set Sto the second ordered set S, wherein the encoding Qis a sequential product provided by sequential operations of a number n−k of unitary operators Q; wherein each of the unitary operator Qis composed of a single s-rotation or a product of two s-rotations; and wherein the encoding Qconverts and maps the rindependent spinor Sin the first ordered set Sto the rindependent spinor Ŝin the second ordered set Scorrespondingly; a fault tolerant action Û in the quantum code [n, k, C] generated by the second stabilizer Ĉ in the intrinsic coordinate, wherein the fault tolerant action Û is a direct sum of a basis state operator Λ and a correction operator Ω; and acquiring a fault tolerant encode in the quantum code [n, k, C] generated by the first stabilizer C, wherein the fault tolerant encode is a sequential product of the encoding Q, the fault tolerant action Û and a complex conjugate Qof the encoding Q.