Introduction to Quantum Error Correction Nielsen Chuang Quantum Information and Quantum Computation CUP 2000 Ch 10 Gottesman quant ph 0004072 Steane quant ph 0304016 Gottesman quant ph 9903099 Errors in QIP unitary 0 1 0 e 1 M p 0 non unitary 0 1 general pure mixed states i U f tr f 2 1 f Ek Ek Ek Ek k k from f trenv U envU k ek U e0 e0 U ek k Ek Ek Ek ek U e0 U sys env k trace preserving Ek Ek 1 k k take and randomly replace by Ek E k k with probability pk tr Ek Ek Quantum noise channel representation Ek Ek k Bit flip channel E0 1 0 0 1 p E1 1 p 0 1 1 0 Phase flip channel E0 1 0 1 0 p E 1 p 1 0 1 0 1 Bit phase flip channel E0 1 0 0 i p E1 1 pY 1 p 0 1 i 0 Amplitude damping channel 1 E0 0 0 E1 2 1 0 0 Depolarizing channel p 1 p X X Y Y Z Z 3 0 Geometrical interpretation Bloch sphere in r space NC p 376 I r x y z r rx ry rz 2 Discrete errors as Pauli matrices 1 0 0 1 I X 0 1 1 0 0 i 1 0 Y Z i 0 0 1 I a a X a a 1 Y a i 1 a a 1 Z a 1 a a N qubit Pauli matrices and 1 i Pauli group Pn 4n 1 elements 4n tensor products overall phase 1 i notation X Y I XYI eigenvalues 1 1 all pairs either commute or anticommute X Y Z anticommute X Z 0 X I etc commute X I 0 P2 spans 2x2 matrices Pn spans 2nx2n matrices e g general phase error 1 0 i 2 e i 2 R 2 e i 0 e 0 cos 2 I i sin 2 Z 0 ei 2 Repetition codes classical 0 000 1 111 error e g 010 corrected to majority value 000 note learned value of bits in doing so prob for bit error p 1 multi bit error prob 3p2 1 p p3 3p2 2p3 p when p 0 5 0 00000 n bits majority n 2 1 1 11111 error prob pn 2 1 error prob as n p 0 5 quantum No cloning theorem suppose and then but by linearity cannot copy unknown quantum states Encode Error Recovery quantum information is encoded into C an error occurs C Ek C E k k recovery procedure undertaken R C Rl Ek C E R k l k regain the encoded state C R C C l Encoding and Recovery R encode error diagnose error fix encoding ancillas Unitary qubits measurement operations error and recovery are superoperators S Ak Ak k Recovery operator R restores state to the code after error from environment encode into a subspace no meaurement of state only of error achieve by adding ancilla qubits measure ancillas syndrome of error perform unitaries conditional on syndrome to correct erroneous qubits Encoding e g 3 qubit bit flip code 0L 000 1L 111 0 1 C 0 L 1L 0 0 0 1 0 00 11 00 11 0 000 111 C 0 L 1L Measurement Pauli ops 0 ancilla H u H U H 0 1 0 1 2 eu eigenvalue of U 1 1 c U 0 1 u 0 u 1 eu u 2 2 1 0 eu 1 u 2 1 1 1 H 0 1 0 1 eu u 2 2 2 1 0 1 0 eu 1 eu u 2 Measure qubit 1 ancilla 1 2 result 0 with prob 2 1 eu syndromes result 1 with 1 prob 2 1 eu 2 eu 1 result 1 with prob 1 unit result 0 with prob 0 eigenvalues of U Pn eu 1 result 1 with prob 0 result 0 with prob 1 Continuous Errors 1 0 i 2 e i 2 R 2 e i 0 e 0 cos 2 I i sin 2 Z 0 i 2 e add ancilla s transfer error info to ancilla c U Z 0 L 1L 0anc Z 0 L 1L Z anc I 0 L 1L 0anc I 0 L 1L noerroranc ancilla superposition cos I 0 L 1L no erroranc 2 i sin Z 0 L 1L Z anc 2 measure ancilla prob sin 2 Z 0 L 1L Z anc 2 2 prob cos I 0 L 1L no erroranc 2 invert either one restore initial state 3 qubit Bit Flip Code 0L 000 1L 111 Error X with prob p R 0 1 0 0 M 0 anc encode error I II M diagnose III X fix IV I 0 1 0 0 000 111 II 8 possibilities from errors XII IXI IIX XXI XIX IXX XXX III state after error 000 111 100 011 010 101 001 110 110 001 101 010 011 100 111 000 Prob of getting state 1 p 3 p 1 p 2 p 1 p 2 1 or no p 1 p 2 error p2 1 p p2 1 p p2 1 p p3 a perform CNOT between qubits 1 2 with ancilla 1 b perform CNOT between qubits 1 3 with ancilla 2 000 111 00 1 p 3 2 100 011 11 p 1 p 010 101 10 p 1 p 2 001 110 01 p 1 p 2 110 001 01 p2 1 p 101 010 10 p2 1 p 011 100 11 p2 1 p 111 000 00 p3 syndrome syndrome redundant for 1 and 2 0 and 3 errors but unequal probabilities III III c M measure ancillas assume only 1 or 0 error syndrome uniquely identifies error failure rate of code rate of 2 errors 3p2 1 p p3 3p2 2p3 p for p 0 5 IV fix by applying unitary conditional on M syndrome 00 do nothing 01 apply x to 3rd qubit 10 apply x to 2nd qubit 11 apply x to 1st qubit 000 111 00 100 011 11 010 101 10 001 110 01 recover encoded state 000 111 Decoding e g from syndrome 10 after IV have 000 111 with p 1 p 2 extract original qubit 0 1 with circuit i ii i 000 111 0 00 1 10 ii 0 00 1 10 0 00 1 00 0 1 00 get correct qubit state with prob 1 p prob of failure 3p2 2p3 p for p 0 5 success 100 if no 2 or 3 errors error prob reduced from p to O p2 3 bit Phase Code z 0 1 0 1 not classical change basis 1 2 0 1 1 2 0 1 1 1 1 1 1 1 1 1 H 1 2 then z z like bit flip H zH x or H 0 1 R 0 0 H H H H H H M Z I II M X effectively encoded into 0L 1L I II phase errors ZII IZI ZII act …