Five-qubit error correcting code

The five-qubit error correcting code, 5,1,3 code, or Laflamme–Miquel–Paz–Zurek code is the smallest quantum error correcting code that can protect a logical qubit from any arbitrary single qubit error. In this code, 5 physical qubits are used to encode the logical qubit. With X and Z being Pauli matrices and I the Identity matrix, this code's generators are \langle XZZXI, IXZZX, XIXZZ,ZXIXZ \rangle . Its logical operators are \bar{X} = XXXXX and \bar{Z} = ZZZZZ . Once the logical qubit is encoded, errors on the physical qubits can be detected via stabilizer measurements. A lookup table that maps the results of the stabilizer measurements to the types and locations of the errors gives the control system of the quantum computer enough information to correct errors.

The five-qubit error correcting code, 5,1,3 code, or Laflamme–Miquel–Paz–Zurek code is the smallest quantum error correcting code that can protect a logical qubit from any arbitrary single qubit error. In this code, 5 physical qubits are used to encode the logical qubit. With X and Z being Pauli matrices and I the Identity matrix, this code's generators are \langle XZZXI, IXZZX, XIXZZ,ZXIXZ \rangle . Its logical operators are \bar{X} = XXXXX and \bar{Z} = ZZZZZ . Once the logical qubit is encoded, errors on the physical qubits can be detected via stabilizer measurements. A lookup table that maps the results of the stabilizer measurements to the types and locations of the errors gives the control system of the quantum computer enough information to correct errors. It is possible to compute all the single qubit errors that can occur and how to correct them. This is done by calculating what errors commute with the stabilizers. For example, if there is an X error on the first qubit and no errors on the others ( X_{1} = XIIII ), it commutes with the first stabilizer [XIIII, XZZXI] = 0 . This means that if an X error occurs on the first qubit, the first ancilla qubit will be 0. The second ancilla qubit: [XIIII, IXZZX] = 0 , the third: [XIIII, XIXZZ] = 0 and the fourth [XIIII, ZXIXZ] \neq 0 . So if an X error occurs on the first qubit, the syndrome will be 0001 ; which is shown in the table below, to the right of X_{1} . Similar calculations are realized for all other possible errors to fill out the table. To correct an error, the same operation is performed on the physical qubit based on its syndrome. If the syndrome is 0001 , an X gate is applied to the first qubit to reverse the error. |0_{\rm L} \rangle = \frac{1}{4}[ &|00000\rangle + |10010\rangle + |01001\rangle + |10100\rangle + |01010\rangle - |11011\rangle - |00110\rangle - |11000\rangle \\ - &|11101\rangle - |00011\rangle - |11110\rangle - |01111\rangle - |10001\rangle - |01100\rangle - |10111\rangle + |00101\rangle], \end{align} \begin{align} |1_{\rm L} \rangle=\frac{1}{4}[ &|11111\rangle + |01101\rangle + |10110\rangle + |01011\rangle + |10101\rangle - |00100\rangle - |11001\rangle - |00111\rangle \\- &|00010\rangle - |11100\rangle - |00001\rangle - |10000\rangle - |01110\rangle - |10011\rangle - |01000\rangle + |11010\rangle ]. \end{align} Stabilizer measurements followed by a \bar{Z} measurement can be used to encode a logical qubit into 5 physical qubits. To prepare |0_{\rm L}\rangle , perform stabilizer measurements and apply error correction. After error correction, the logical state is guaranteed to be a logical codeword. If the result of measuring \bar{Z} is 0 , the logical state is |0_{\rm L}\rangle . If the result is 1 , the logical state is |1_{\rm L}\rangle and applying \bar{X} will transform it into |0_{\rm L}\rangle .

Text derived from https://en.wikipedia.org/w/index.php?oldid=1344265420 under CC BY-SA 4.0; modifications have been made.