Quantum Error Correction

• In principle: whole chapter 10

• TL;DR version: sec. 10.1-10.2, 10.6

• What is error correction? (classically)

• Introduction to quantum errors

• Some formalism to help us

• What are the boundries for correctability

Classical Error Correction

Everyday example:

Noisy phoneline

B D VR

A

V

O

E

L

T

A

I

C

T

O

R

Classical Error Correction

Example:

Error:

With probability p

any bit is flipped

000

Repetition coding:

00001111

31 pProbability: 213 pp pp 13 2 3p

001

100010

110

011101000 111

Majority voting Scheme fails

?13 32 pppp 02

1

2

32 pp

Repetition better if

2

1 p

Parity check:

Classical Error Correction

Repetition code has heavy cost, 300 %

of original message length

1  001  1010

byte parity bit = 1 if odd number of 1’s

Error:

1  101  1010

byte and parity bit missmatch = resend

Works well, if p is very low

Example: computers, where ε < 10-17

Quantum Error Correction

Differences from classical:

• No cloning: Cannot use repetition coding directly, because

we cannot duplicate arbitrary states

• Measurement collapse: Everytime we try to detect what

state we have, it collapses to the basis

• Continuous errors: Infinite ways that errors can occur,

think rotations in the Bloch sphere

Despite all this, quantum error correction still works!

Quantum Error Correction

00000 L

11111 L

01

10XQuantum bit flip = Pauli X operator,

Define logical qubit:

11100010

No cloning… But what does this circuit do?

First task: correct for bit flip

Quantum Error Correction

Error-detection circuit for original state 111000 orig

Apply

correction:

bit flip

I X1 X2 X3

3210 eeeeeeorigtot cccc

1111110000000 P

0110111001001 P

1011010100102 P

1101100010013 P

Projective measurement set:

orig , α and β are untouched!

Quantum Error Correction

How much improvement from the error correction?

Measured by fidelity: ,F

Without error correction:

XpXp )1(

XXpppppp 3223 )1(3)1(3)1(

F XXpp )1( p 1

With error correction:

...)1(3)1( 23 pppF32 231 pp

2

1 p

Quantum Error Correction

Quantum phase flip:

10 orig 10

Change basis:

2

10

2

10

L0

L1

10

01Z

Z,

Same procedure as before! )( HZHX

10, X

Quantum Error Correction

Both phase and bit flip at the same time?

First, encode0

1, then, encode each of these according to the bit flip:

22

11100011100011100000

L

22

11100011100011100011

L

The Shor code:

9 qubit code!

XZeZeXeIeE 3210

Continuous errors?

Saved by the projective measurement!

Ex: How to measure the error syndrome

111000 orig

tot P0 P1 P2 P3Z1Z2 Z2Z3

32121 IZZZZ 110010

01,

Z

2121 11001100ZZ

1010010111110000

+1-1

+1-1

1111110000000 P

0110111001001 P

1011010100102 P

1101100010013 P

Projective measurement set:

Error Correction Formalism

• How can we find better codes?

• Can we know if we have found the best code?

• How can we build real circuits from the theory?

• How big error is allowed for a full scale fault-tolerant

quantum computer?

Why do we need to develop formalism?

Error Correction Formalism

Generators (classical)

Definition: a linear code C, encoding k bits of information into an n bit

code space, is specified by an n by k generator matrix G

Example: 3-bit repetion code

1

1

1

G code Gxy , where x is the k bit message

0x Ly 0

0

0

0

1x Ly 1

1

1

1

[n,k] code = [3,1] here

Error Correction Formalism

Parity check matrix, H (classical)

0Hy

Example: 3-bit repetion code

, where H is an n – k by k matrix

0HGx 0HG , so the rows of H must be orthogonal

vectors to the columns of G

(modulo 2)

1

1

1

G

1

1

0

,

0

1

1

11 vv

110

011H

Hy is only zero for

the code words

(0,0,0) and (1,1,1)

Error Correction Formalism

Error detection with parity matrix

Gxy eyy HeHeHyyH

Special case: (classical) Hamming code, a [7,4] code

1010101

1100110

1111000

HIf ej is an error

on the j’th bit

Hej is the binary

representation of j

Error Correction Formalism

Generators (quantum)Stabilizers

},,,,,,,{1 iZZiYYiXXiIIG Pauli group:

Suppose S is a subgroup of Gn and define VS to be the set

of n qubit states which are fixed by every element of S.

S is then said to be the stabilizer of the space VS.

Definition:

2

1100

EPR

EPRXX 21

EPR

EPRZZ 21

EPR ?

?

Stabilizers: example

Error Correction Formalism

},,,{

3

313221 ZZZZZZIS

n

21ZZ

111

110

001

000

32ZZ

111

011

100

000

111,000Common base (Vs):

221ZZI

322131 ZZZZZZ

3221 , ZZZZS

Z1Z2 Z2Z3

tot

3-qubit flip code!

Generator (quantum):

The stabilizers that generate our logical qubits tell

us how to measure the error syndrome!

Realization:

Check matrix (quantum parity)

Error Correction Formalism

1010101

1100110

1111000

H

1010101

1100110

1111000

0000000

0000000

0000000

0000000

0000000

0000000

1010101

1100110

1111000

H

Name Operator

1g IIIXXXX

2g IXXIIXX

3g XIXIXIX

4g IIIZZZZ

5g IZZIIZZ

6g ZIZIZIZ

1101001011110010110100001111

11001100110011101010100000008

10

L

0010110100001101001011110000

00110011001100010101011111118

11

L

7 qubit Steane code:

Quantum case is bigger because errors are more complex,

not only bit flips but also phase errors

Can be used to find

the generators:

Error Correction Measurement

Measure arbitrary operator M, as a controlled-M

Specific case: X as controlled-X

:temporary ancilla qubit

𝑍1

𝑍2

Z1Z2 Z2Z3

tot3-qubit flip code!

Error Correction Measurement7 qubit Steane code (standard form):

Name Operator

𝑔6 𝑍𝑍𝑍𝐼𝐼𝑍𝐼

𝑔1 𝑋𝐼𝐼𝐼𝑋𝑋𝑋

𝑔2 𝐼𝑋𝐼𝑋𝐼𝑋𝑋

𝑔5 𝐼𝑍𝑍𝐼𝑍𝐼𝑍

𝑔4 𝑍𝐼𝑍𝑍𝐼𝐼𝑍

𝑔3 𝐼𝐼𝑋𝑋𝑋𝑋𝐼

Error Correction Bounds

What is the smallest possible code that protects against any errors?

Quantum Singelton Bound (ch12):

tkn 4k is the original number of qubits

n is the encoded number of qubits

t is the max number of qubit errors

141 nExample: 5 n

5-qubit code:

Error Correction BoundsFault-tolerant quantum computing:

Single error: p After block:

Fails with cp2

• c depends on number of components

• In how many ways can two components fail? c ~104 ways

• Improvement if 𝑐𝑝2 < 𝑝

Large overhead for few qubits, but fortunately scales only logarithmically!

→ 𝑝 < ~10−4

Quantum Error Correction

Summary

• Difference with quantum error correction• No cloning

• Continuous errors

• Measurement collapse

• Simple case: bit flip and phase flips

• Systematic treatment using stabilizers• 7-qubit Steane code

• Fault-tolerant bound: 𝑒~10−4