quantum error correction - lu

Quantum Error Correction

• In principle: whole chapter 10

• What is error correction? (classically)• Introduction to quantum errors• Some formalism to help us• What are the boundries for correctability

• Fault-tolerant threshold• Example: Surface codes with superconducting qubits

Today’s lecture:

In the book:

• TL;DR version: sec. 10.1-10.2, 10.6

Classical Error Correction

Everyday example: Noisy phoneline

B D VRAVO

ELTA

ICTOR

Classical Error Correction

Example:

Error:With probability pany 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 021

232 pp

Repetition better if

21 p

When do we gain by error correction?

Parity check:

Classical Error CorrectionRepetition code has heavy cost, +200 % of original message length

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

Error: byte and parity bit missmatch = resend

Works well, if p is very lowExample: computers, where ε < 10-17

1010 001 1

1010 101 1Cost is only +~15%


Differences from classical:

• No cloning: Cannot use repetition coding directly, becausewe cannot duplicate arbitrary states

• Measurement collapse: Everytime we try to detect whatstate 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!


00000 L

11111 L

0110

XQuantum bit flip = Pauli X operator,

Define logical qubit:

11100010

No cloning… But what does this circuit do?

First task: correct for bit flip

Cannot copy… but can still add redundancy

One quantum bit flipError-detection circuit for original state 111000 orig

Applycorrection: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 CorrectionHow much improvement from the error correction?

Measured by fidelity: ,origF

Without error correction:

XpXp )1(

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

F XXpp )1( p 1

With error correction:

...)1(3)1( 23 pppF 32 231 pp

21 p


Quantum phase flip:

10 orig 10

Change basis:

2

10

2

10

L0

L1

1001

Z

Z,

Same procedure as before! )( HZHX

10, X


Both phase and bit flip at the same time?

First, encode01

, 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 11001001

,

Z

2121 11001100ZZ

1010010111110000

+1-1

+1-1

1111110000000 P

0110111001001 P

1011010100102 P

1101100010013 P

Projective measurement set:

Eigenvalues +1 and -1

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?


Generators (classical)

Definition: a linear code C, encoding k bits of information into an n bitcode space, is specified by an n by k generator matrix G

Example: 3-bit repetion code

111

G code Gxy , where x is the k bit message

0x Ly 0000

1x Ly 1

111

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


Parity check matrix, H (classical)

0Hy

Example: 3-bit repetion code

, where H is an n – k by k matrix

0HGx 0 HG , so the rows of H must be orthogonalvectors to the columns of G(modulo 2)

111

G

110

,011

11 vv

110011

HHy is only zero forthe code words(0,0,0) and (1,1,1)


Error detection with parity matrix

Gxy eyy HeHeHyyH

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

101010111001101111000

HIf ej is an erroron the j’th bit

Hej is the binaryrepresentation of j


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:

21100

EPR

EPR

XX 21

EPR

EPRZZ 21

EPR ?

?

Stabilizers: example


},,,{3

313221 ZZZZZZISn

21ZZ

111

110

001

000

32ZZ

111

011

100

000

111,000Common base (Vs):

221ZZI

322131 ZZZZZZ

3221 , ZZZZS

Z1Z2 Z2Z3tot

3-qubit flip code!

Generator (quantum):

The stabilizers that generate our logical qubits tellus how to measure the error syndrome!

Realization:

(not unique)

Check matrix (quantum parity)


101010111001101111000

H

101010111001101111000000000000000000000000

000000000000000000000101010111001101111000

H

Name Operator1g IIIXXXX2g IXXIIXX3g XIXIXIX4g IIIZZZZ5g IZZIIZZ

6g ZIZIZIZ

1101001011110010110100001111

1100110011001110101010000000810

L

0010110100001101001011110000

0011001100110001010101111111811

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 findthe generators:

Error Correction Measurement

Measure arbitrary operator M, as a controlled-M

Specific case: X as controlled-X

:temporary ancilla qubit

𝑍

𝑍

Z1Z2 Z2Z3

tot 3-qubit flip code!

Error Correction Measurement7 qubit Steane code (standard form):

Name Operator

𝑔 𝑍𝑍𝑍𝐼𝐼𝑍𝐼

𝑔 𝑋𝐼𝐼𝐼𝑋𝑋𝑋𝑔 𝐼𝑋𝐼𝑋𝐼𝑋𝑋

𝑔 𝐼𝑍𝑍𝐼𝑍𝐼𝑍𝑔 𝑍𝐼𝑍𝑍𝐼𝐼𝑍𝑔 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 qubitsn is the encoded number of qubitst 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 ~10 ways• Improvement if 𝑐𝑝 𝑝

Fault-tolerant threshold, important for experiments

→ 𝑝 ~10

Example: Surfaces codesSuperconducting qubits are designed two-dimensionally:

Gambetta et al, npj Quantum Information 3, 2 (2017)

Intel’s chip

• Fault tolerant error threshold: 1% but…

• Trade-off between error threshold and number of qubits

• Only nearest neighbor interactions

Example: Surfaces codes

Detailed walkthrough from Fowler et al, PRA 86, 032324 (2012)

How many qubits are required for a technologically relevant implementation of Shor’s algorithm?

Classical record for integer factorization:

• 768-bit RSA took 2 years running of several hundred machines• 1024-bit RSA would take a factor of 1000 longer

Quantum test:Find prime factors for a 2000-bit RSA number in ~1 day

Trade-off between space and time

Shor’s:


Number of logical qubits: 2N = 4000

Total number of logical gates: 40N3

Ancilla qubits per (Toffoli) gate: 7

Total number of ancilla qubits: 7 ⋅ 40𝑁 280 ⋅ 4000 10

Final error per operation must be1

280𝑁 10

Estimating the total number of gates:

Fowler et al, PRA 86, 032324 (2012)

N = 2000

(Logical)

(Code is limited by error correction)


Realistically:

For perfect gates: 13 physical qubits for 1 logical qubit

0.1% errors

2) 3600 physical qubits for 1 logical qubit

Operation times:Readout time: ~100 nsGate time: ~0.1 – 10 µs (Limited partly by classical post

processing)

Starting from 0.1% = 10-3 , two stages of error correction are needed:

Final error per logical operation must be1

280𝑁 10

Scheme can work with 1% physical qubit error

How much error correction is needed to get there?

1) 900 physical qubits for 1 logical qubit

Total footprint of 1 logical ancilla qubit: 200 000 physical qubits

(low error limit)

Example: Surfaces codesSumming up

→ ~1000 logical ancilla qubits needed in parallel:

1013 logical ancilla qubits are needed

Fast gate time ~100 ns → ~1010 logical ancilla qubits within a day

Total qubits needed:

For the error correction…qubits involved in calculation = 20 million

1000 logical ⋅ 200 000 physical/logical

= 200 million physical qubits

Fowler et al, PRA 86, 032324 (2012)

From before:

Quantum Error CorrectionSummary

• 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• Technologically relevant algorithms require a

lot (200 million) qubits!

(Steane code)

quantum error correction - lu

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