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lecture_2 - Lecture # 2, Quantum Computation 2: QEC...

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Lecture # 2, Quantum Computation 2: QEC Criteria Lecture notes of Isaac Chuang, transcribed by Jennifer Novosad Outline: 0. Review 1. Classical Coding 2. Q. Coding 3. Operator Measurement and Error Syndromes 4. Shor 9 Qubit Code 5. Quantum Error correction Codes Criteria (QEC criteria)
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2 0. Review χ χ β ( χ ) = E k χE k where E k E = I k k k
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3 1. CLASSICAL CODING 0 1 0 1 1-p p p 1-p FIG. 1: a binary symmetric channel P = prob of error DeFnition: A Classical [n,k,d] code is a set of 2 k n-bit strings which have a minimum Hamming distance d. DeFnition: A Hamming distance between two bit strings is d ( x, y ) = w ( x y ) where is the x-or operator, and w is an operation that counts the number of ones. Example: 0 L ( ogical ) = 000, 1 L = 111 is a [3,1,3] code could send could receive prob decode prob. of error 0 L = 000 000 (1 p ) 3 0 001 p (1 p ) 2 0 010 p (1 p ) 2 0 100 p (1 p ) 2 0 011 p 2 (1 p ) 1 p 2 (1 p )+ 101 p 2 (1 p ) 1 p 2 (1 p )+ 110 p 2 (1 p ) 1 p 2 (1 p )+ 111 p 3 1 p 3 3 So, the total probability of error is 3 p 2 2 p = O ( p 2 )
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( ± ) 4 2. QUANTUM CODING 1995: Thought error correction to be impossible! 1. States collapse on measurement 2. Classically error occurs or does not occur. In Q. M., errors are continuous: 0 + | → ± 0 + ... | 1 → ≡ ( + β ) | → 3. No cloning Thm prohibits copying, so cannot create | 0 + ± | 1 → ≡ ( | 0 + ± 1 )( 0 + | | → ± 1 )( 0 + ± 1 ) | | | The Solutions: 1. Measure only the effect of the environment, not the state (i.e. did an error
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lecture_2 - Lecture # 2, Quantum Computation 2: QEC...

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