Quantum Codes G. Eric Moorhouse, UW Math Corrected copies of - PDF document

Quantum Codes G. Eric Moorhouse, UW Math Corrected copies of transparencies for this sem- inar series should soon be available at http://math.uwyo.edu/~moorhous/quantum/

References P. Shor, ‘Quantum computing’, proceedings of the International Congress of Mathemati- cians, 1998. http://www.research.att.com/~shor/ papers/ICM.pdf A.R. Calderbank and P.W. Shor, ‘Good quan- tum error-correcting codes exist’, Phys. Rev. A 54 (1996), 1098–1105. A.R. Calderbank, E.M. Rains, P.W. Shor and N.J.A. Sloane, ‘Quantum error correction via codes over GF (4)’, preprint, 1998. J. Preskill, ‘Fault-tolerant quantum computa- tion’, in Introduction to Quantum Computa- tion and Information, H.-K. Lo, S. Popescu, and T. Spiller, 1998, pp.213–269. A.M. Steane, ‘Quantum error correction’, in Introduction to Quantum Computation and Information, H.-K. Lo, S. Popescu, and T. Spiller, 1998, pp.184–212.

The No-Cloning Theorem Unlike conventional bits, qubits cannot be cloned. Therefore ‘repetition codes’ have no analogue in quantum computation. Nevertheless quan- tum information can be spread throughout many qubits so that a small number of indi- vidual qubit errors do not obliterate the data.

Discretized Errors For simplicity we assume that any qubit error consists in the application of one of the Pauli operators � � 0 1 σ x = (‘bit flip error’) 1 0 � � 1 0 σ z = (‘phase error’) 0 − 1 � � 0 − i σ y = = iσ x σ z (combination of bit i 0 flip and phase errors) and that the three errors σ x , σ y , σ z are all equally likely. Moreover errors in different qubits are statistically independent. Codes which correct these three errors can also correct more general errors [Bennett, Di- Vincenzo, Smolin and Wooters (1996); Ekert and Macchiavello (1996); Knill and LaFlamme (1997)].

We’ll see how a single qubit can be encoded (A) as 3 qubits, to allow correction of at most one bit flip error σ x ; or (B) as 7 qubits, to allow correction of at most one arbitrary error σ x , σ y , σ z .

Controlled-Not (CNOT) Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . � . . . � . . . . . . . Control qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input . . . Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Target qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 00 � �→ | 00 �  1 0 0 0  | 01 � �→ | 01 � 0 1 0 0       | 10 � �→ | 11 � 0 0 0 1     | 11 � �→ | 10 � 0 0 1 0