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Uploaded byPhelim Bradley
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Quantum error correction

This document discusses quantum error correction. It explains that while quantum states and operators are theoretically perfect, in reality approximations must be made which can cause errors. Quantum error correction deals with these imperfections. It describes different types of quantum errors and discusses barriers to quantum error correction, such as the no-cloning theorem. The document introduces classical error correction techniques and explains how similar techniques can be applied to encode quantum states to correct bit flip and phase flip errors by measuring the parity of qubits without collapsing their superpositions. Specific quantum error correcting codes are presented, including Shor's code which can correct both types of errors.

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Quantum Error Correction Phelim Bradley
Contents • Why we need (Quantum) Error Correcting (QEC) • Classical error correction • Barriers to QEC • Types of Errors • Quantum Error Correcting Codes
Why do we need (Quantum) Error Correction? • Theoretically operators and states are pristine and perfect. • In a lab: ▫ Approximations must be made. ▫ Operators don’t always do as they should. ▫ Fidelity of the prepared states and the theoretical states is not perfect . • Quantum Error Correction (QEC) deals with the imperfection of the real world.
Classical Error • Say, probability failure per gate = p • Probability getting right answer with n gates: (1 p)n
Classical Error Correction If you have a possible bit-flip error like: |0 (1 p) | 0 p |1 |1 (1 p) | 1 p|0 Can try an error correction code like: | 00 (1 p) | 00 p | 10 | 11 (1 p) | 11 p | 01
Classical Error Correction But this hasn’t done much, because it’s possible to have a bit-flip in either bit. | 00 (1 p)(1 q) | 00 p | 10 q | 01 pq | 11 | 11 (1 p)(1 q) | 11 p | 01 q | 10 pq | 11 So now we can’t distinguish between states.
Classical Error Correction However, with 3 bits we can make progress: Let p=q for simplicity. 3 | 000 (1 p) | 000 ... { | 001 2 (1 p) p | 010 ... | 100 { 2 | 011 (1 p) p | 101 ... | 110 3 p | 111
Classical Error Correction • Recover by looking at the majority of bits. • i.e. | 001 | 000 |0 L | 101 | 111 |1 L • Ideally p 0 in reality p 2 c1p c2 p3 o( p 2 ) 2 • Gain in accuracy available for cp p • Increase in resources to reduce error • e.g. p p17 needs 18 bits.
Barriers to Quantum Error Correction • Measurement of error destroys superpositions. (Classically we can observe all bits)
Barriers to Quantum Error Correction • No-cloning theorem prevents repetition. 0 + 1 0 0 + 1 1 ( 0 + 1 )( 0 + 1) • Not the same state. => contradiction proves a cloning operator cannot exist. • Means you can’t just keep “back up” of states. Must protect original. • Also prevents some error correction codes.
Barriers to Quantum Error Correction • Must correct multiple types of errors (not just bit flips). • Must correct continuous errors and decoherence.
Know thy enemy - Errors • A general operator : | A | * A A • A density matrix ρ describes the statistical state of a system.
Know thy enemy - Errors • A density matrix ρ describing the ensemble: * p A A • Normalise * p A A 1
Know thy enemy – Errors Bit Flip X |0 |1 |1 |0 •2 possibilities => 2 operators 10 01 A0 A1 01 10 * * (1 p) A0 A0 pA A 1 1 00 01 11 00 10 01 (1 p) p 10 11 01 10 00 11
Know thy enemy - Errors Phase Flip Z: |0 |1 |0 |1 Complete dephasing: (1 p) pI / 2 (depolarisation) Rotation R: R 0 = 0 , R 1 =ei 1 At R->Z
Quantum Error Correction •Let us apply the classical repetition code to a quantum state to try to correct a single bit flip error: 0 + 1 000 + 111 •This is just a tensor product of three qubits, and different from cloning a state. •Bit flip error (X) on 2nd qubit: 010 + 101 Can’t measure the data!
Measure the error, not the data. • Simultaneously measure: (1) ( 2) z z ( 2) ( 3) On 010 + 101 z z •This will check for the orthogonality of the qubits. •We will measure -1 for both, showing a bit- flip in the 2nd qubit. •Correct by applying a bit-flip X operator.
Classical Quantum Error Correction • Recover by looking at the majority parity of bits. • i.e. | 001 | 000 |0 L | 101 | 111 |1 L • Ideally p 0 in reality p c1p 2 c2 p3 o( p 2 ) • Gain in accuracy available for cp2 p • Increase in resources to reduce error • e.g. p p17needs 18 bits (per error type).
Correcting Phase (Z) Errors Hadamard transform H exchanges bit flip and phase errors: | (| 0 |1 ) / 2 H( 0 + 1 )= + + - | (| 0 |1 ) / 2 X + = + , X - = - - (acts like phase flip) Z + = - , Z - = + (acts like bit flip) Repetition code corrects a bit flip error The same code in a new basis corrects a phase error! + + - +++ + ---
Shor’s Code •This is simply a combination of the two codes above. •Had to stick to one basis, so it’s a little less intuitive. •Correct for both X (bit-flip) and Z(phase-flip) •Also, get Y errors for free Y=iXY!
Summary •Applied concepts of classical error correction to QEC. •Learnt about quantum errors. •Circumvented the problem caused by no cloning and superposition. •Learnt codes to correct for multiple types of errors that can occur in quantum computing.
Questions?