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

A computer-implemented method for constructing a fault tolerant encode using a quantum computational model at a hardware level, where the fault tolerant encode is an n-qubit fault tolerant encode for any k-qubit quantum gate, M, in any given quantum code [n, k, C], wherein

• the C is defined as a first stabilizer which is a k-th maximal bi-subalgebra;
• the quantum code [n, k, C] is consisting of a number 2n−k of cosets of the first stabilizer C and the quantum code is a Quotient Algebra Partition (QAP);
• the cosets of the first stabilizer C is able to group into a number 2n−k of blocks Γα, wherein each of the blocks Γα consists of 22k cosets of the first stabilizer C with same syndrome α;
• and
• the quantum code [n, k, C] is used for detecting and correcting chosen errors from different blocks Γα, and the errors are collected as an error set ε, and wherein the method comprises:
• (1) choosing a number n−k of independent spinors Sr from the first stabilizer C and a first ordered set SC consists of the independent spinors Sr;
• (2) choosing a number n−k of independent spinors Ŝr in a second stabilizer Ĉ in the intrinsic coordinate and a second ordered set SĈ consists of the independent spinors Ŝr in the intrinsic coordinate;
• (3) implementing an encoding Qen, wherein the encoding Qen converts the first ordered set SC to the second ordered set SĈ, and transforms the first stabilizer C to the second stabilizer Ĉ in the intrinsic coordinate wherein the code [n, k, C] is converted to [n, k, Ĉ] in the intrinsic coordinate;
• wherein the encoding Qen is a sequential product provided by sequential operations of a number n−k of unitary operators Qr;
• wherein each of the unitary operator Qr is composed of a single s-rotation or a product of two s-rotations;
• and wherein the encoding Qen converts and maps the rth independent spinor Sr in the first ordered set SC to the rth independent spinor Ŝrin the second ordered set SĈ correspondingly;
• (4) 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
• (5) 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 Qen, the fault tolerant action Û and a complex conjugate Qen† of the encoding Qen, and
• wherein the fault tolerant encode obeys two criteria, wherein one of the criteria is eigen-invariance that every codeword state is still a codeword after applying the encode, and the other one of the criteria is error-correctable that each corrupted state is a linear superposition of correctable states in [n, k, C] after applying the encode.