Quantum Error CorrectionYamini Singh.

ContentsWhat is error correction•Intention to study this•Why do we need this (Quantum) Error

Correcting (QEC)•Classical error correction•Barriers to QEC•Types of Errors•Quantum Error Correcting Codes

What is error correction•To correction the error in the system is

known as error correction •In what frame of reference :- to process the quantum information without errors even in the noisy environment.

Intention to study…•How to process the information in the

presence of noise correctly.•We begin by developing the basic theory

of qec codes, which protect the quantum information against noise.

•These codes work by encoding quantum states in a special way that make them resilient against the effect of noise and then decoding when it is wished to recover the orginal state.

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.

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.

• 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.

0 + 1 00 + 11

(0 + 1)(0 + 1)

Barriers to Quantum Error Correction• Must correct multiple types of

errors (not just bit flips).• Must correct continuous errors

and decoherence.

Classical Error•Say, probability failure per gate = p•Probability getting right answer with n

gates:np)1(

Classical Error Correction

0|1|)1(1|1|0|)1(0|

pppp

If you have a possible bit-flip error like:

Can try an error correction code like:

01|11|)1(11|10|00|)1(00|

pppp

Take a example•Error detection or syndrome diagnosis•Let us examine more closely the error

syndrome for the classical repetition code.

•We performed a measurement which tells us what error,if any,occurred on quantum state.

•Mesurement result is called quantum syndrome

•For bit flip channel there are four error syndrome.

Syndrome mesurement•Does not cause any change to the state:it is

a100 +b011 both before and after syndrome measurement.

•Contain only information about what error has occurred

•Does not allow us to infer anything about the value of aor b.i.e it contain no information about the state being protected.

•recovery: value of error syndrome to tell us what produre is used to recover initial state

Know thy enemy - Errors

Phase Flip Z:

Complete dephasing: (depolarisation)

Rotation R: R0 = 0, R1 =ei1

At R->Z

1|0|1|0|

2/)1( pIp

Know thy enemy - Errors• A general operator :

•A density matrix ρ describes the statistical state of a system.

*

||

AA

A

Correcting Phase (Z) ErrorsHadamard transform H exchanges bit flip and phase errors:

H (0 + 1) = + + -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!

2/)1|0(||

2/)1|0(||

The shor’s code:-•This is the simple quantum code which

can protect against the effects of arbitrary error on a single qubit!

•This code is known as shor code, after its inventor.

•Combination of three qubit phase flip and bit flip code.

Shor code•First we encoded each of these qubit

using the phase flip code:|0> |+++>,|1>|--->

•Next we encoded each of these qubits using the three qubit bit :|+> is encoded as (|000>+|111>)sqrt2 and |-> is encoded as (|000>-|111>)sqrt2 .

•The result is a nine qubit code,with codewords given by

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?