Introduction toQuantum Error

Correction

Nielsen & ChuangQuantum Information and

Quantum Computation, CUP2000, Ch. 10

Gottesman quant-ph/0004072Steane quant-ph/0304016

Gottesman quant-ph/9903099

Errors in QIP

• unitary• non-unitary• general: pure → mixed states

( )0 1 0 1U ie

! "# $ # $ ++ %%& +

0 1 0M

p!! "+ ##$

21f ftr! " "# <

† †

f k k k k

k k

E E E E! " " ! ! " "= # = =$ $

k!

†

†

0 0

†

from f env env

k k

k

k k

k

tr U U

e U e e U e

E E

! ! !

!

!

" #= $% &

= $

=

'

' 0 sys+env, ( )k kE e U e U=

†

k

trace preserving: 1k kE E =!

≡ take ρ and randomly replace by

with probability

†

k k k kE E! " "=

( )†k k kp tr E E!=

Quantum noise:†

k k

k

E E! !"#

0 1

1 0 0 1, 1 ,

0 1 1 0E p E p

! " ! "= = #$ % $ %

& ' & '

0 1

1 0 1 0, 1

0 1 0 1E p E p

! " ! "= = #$ % $ %#& ' & '

channel representation

0 1

1 0 0, 1 1

0 1 0

iE p E pY p

i

!" # " #= = ! = !$ % $ %

& ' & '

( ) ( ) ( )13

pp X X Y Y Z Z! " " " " "= # + + +

0 12

1 0 0,

0 00 1E E

!!

" # " #= =$ % $ %$ %& ' (' (

Bit flip channel

Phase flip channel

Bit-phase flip channel

Amplitude damping channel

Depolarizing channel

, , , , , ,2

x y z x y z

I rr r r r

!" ! ! ! !

+ #= = =

Geometrical interpretation: Bloch sphere in r-space (NC p. 376)

Discrete errors as Pauli matrices1 0 0 1

0 1 1 0

0 1 0

0 0 1

I X

iY Z

i

! " ! "= =# $ # $% & % &

'! " ! "= =# $ # $'% & % &

• N-qubit Pauli matricesand ±1, ±i ≡ Pauli group, Pn: 4n+1 elements(4n tensor products, overall phase ±1, ±i)• notation: X⊗Y⊗I≡XYI• eigenvalues +1, -1• all pairs either commute or anticommute

X, Y, Z, anticommute X,Z=0X, I etc commute [X,I]=0

I|a>=|a>X|a>=|a⊕1>Y|a>=i(-1)a|a ⊕1>Z|a>=(-1)a|a>

P2 spans 2x2 matricesPn spans 2nx2n matricese.g. general phase error

( ) ( )

/ 2

/ 2

/ 2 / 2

1 0 0

0 0

cos / 2 sin / 2

i

i

i i

eR e

e e

I i Z

!!

! ! !

! !

"# $# $= = % &% &' ( ' (

= "

Repetition codesclassical 0→000

1→111

error, e.g. 010, corrected to majority value → 000note: learned value of bits in doing so

prob. for bit error p < 1: multi-bit error prob. = 3p2(1-p)+p3=3p2-2p3

< p when p < 0.50→00000…1→11111…

n bits, majority n/2+1⇒ error prob. ≅ pn/2+1 +…⇒ error prob. ↓ as n ↑ (p < 0.5)

No cloning theorem!

( ) ( )( )

suppose and

then

but bylinearity

! ! ! " " "

! " ! " ! "

!! "" "! !"

! " !! ""

# #

+ # + +

= + + +

+ # +

quantum? ?! ! ! !""#

cannot copy unknown quantum states

• quantum information is encoded into ρC

• an error occurs

• recovery procedure undertaken

• regain the encoded state ρC

Encode/Error/Recovery

[ ] † †( )C l k C k l

l k

R E E R! " "=# #R

†( )C k C k

k

E E! " "=#

[ ]( )C C

! " "=R

error and recovery are superoperators

( ) †A Ak k

k

! " !#= =S

Encoding and Recovery

error diagnoseerror fixencode

encoding qubits

ancillasmeasurement

Unitary operations

Recovery operator R restores state to the codeafter error from environment

• encode into a subspace• no meaurement of state, only of error• achieve by adding ancilla qubits• measure ancillas → syndrome of error• perform unitaries conditional on syndrome to

correct erroneous qubits

R

Encoding

e.g., 3-qubit bit flip code|0L>=|000>|1L>=|111>

0 1 0 1C L L

! " # ! " #= + $ = +

!

0

0

( )

( )

0 1 0 00 11

00 11 0 000 111 0 1L L

! " ! "

! " ! " ! "

+ # $ +

+ # $ + % +

C!

1

0 0 12

H!!" +

1 10 1 0 1

2 2

10 1

2

c

u

u

Uu

e

u e u

u

!""#

+

" +

=

+

( ) ( )

1 1 10 1 0 1

2 2 2

10 1 0 1

2

H

u

u u

e u

e ue

! "+ + #$ %

& '

+ + #

(()

=

H H

U

|0>

|u>

Measure qubit 1 (ancilla):result |0> with prob.

result |1> with prob.

( )2

11

2ue

! "+# $% &

( )2

11

2ue

! "#$ %& '

unit eigenvaluesof U (∈ Pn)

eu=-1 result 1 with prob. 1 result 0 with prob. 0

eu=+1 result 1 with prob. 0 result 0 with prob. 1

Measurement (Pauli ops.)

eu=eigenvalue of U

ancilla

syndromes

Continuous Errors

( ) ( )

/ 2

/ 2

/ 2 / 2

1 0 0

0 0

cos / 2 sin / 2

i

i

i i

eR e

e e

I i Z

!!

! ! !

! !

"# $# $= = % &% &' ( ' (

= "

add ancilla(s), transfer error info to ancilla (c-U)( ) ( )0 1 0 0 1

L L anc L L ancZ Z Z! " ! "+ # $ + #

( ) ( )0 1 0 0 1L L anc L L anc

I I noerror! " ! "+ # $ + #

ancilla → superposition

( )

( )

cos 0 12

sin 0 12

L L anc

L L anc

I noerror

i Z Z

!" #

!" #

$ %+ &' (

) *$ %+ + &' () *

measure ancilla

( )2prob. sin 0 12

L L ancZ Z

!" #$ %

+ &' () *

( )2prob. cos 0 12

L L ancI noerror

!" #$ %

+ &' () *

invert either one → restore initial state

3-qubit Bit Flip Code

|ψ>|0>

|0>

R

• • ••

•

•M

MX

|0L>=|000>|1L>=|111>

Error X with prob. p

encode diagnose fix

α|0>+β|1>

I II III IVerror

|0>anc

I: (α|0>+β|1>)⊗|0> ⊗|0>→ α|000>+β|111>

II: 8 possibilities from errors XII, IXI, IIX, XXI,XIX, IXX, XXX, III

α|000>+β|111> (1-p)3

α|100>+β|011> p(1-p)2

α|010>+β|101> p(1-p)2

α|001>+β|110> p(1-p)2

α|110>+β|001> p2(1-p)α|101>+β|010> p2(1-p)α|011>+β|100> p2(1-p)α|111>+β|000> p3

Prob. of getting statestate after error

III: a) perform CNOT between qubits 1 & 2with ancilla 1b) perform CNOT between qubits 1 &3 with ancilla 2

α|000>+β|111>|00> (1-p)3

α|100>+β|011>|11> p(1-p)2

α|010>+β|101>|10> p(1-p)2

α|001>+β|110>|01> p(1-p)2

α|110>+β|001>|01> p2(1-p)α|101>+β|010>|10> p2(1-p)α|011>+β|100>|11> p2(1-p)α|111>+β|000>|00> p3

→

1 or noerror

syndromesyndrome redundant for 1 and 2 (0 and 3) errors, but unequal probabilities

III. c) M = measure ancillas:assume only 1 (or 0) error ⇒ syndromeuniquely identifies error

failure rate of code = rate of ≥ 2 errors = 3p2(1-p)+p3

= 3p2-2p3 < p for p < 0.5

IV. fix by applying unitary conditional on Msyndrome: 00 do nothing

01 apply σx to 3rd qubit 10 apply σx to 2nd qubit 11 apply σx to 1st qubit

α|000>+β|111>|00>α|100>+β|011>|11>α|010>+β|101>|10>α|001>+β|110>|01>

recover encoded stateα|000>+β|111>

Decodinge.g. from syndrome 10

after IV. have α|000>+β|111> with p(1-p)2

extract original qubit α|0>+β|1> with circuit:

i) ii)

i) α|000>+β|111> → α|0>|00>+β|1>|10>ii) α|0>|00>+β|1>|10> → α|0>|00>+β|1>|00>

= (α|0>+β|1>)|00>

⇒ get correct qubit state with prob. > 1-p prob. of failure = 3p2-2p3 < p for p < 0.5 success = 100% if no 2 or 3 errors

error prob. reduced from p to O(p2)

3-bit Phase Codeσz(α|0>+β|1>) = α|0>-β|1> not classical!

change basis: |+> =1/√2(|0>+|1>) |->=1/√2(|0>-|1>)

1 1 1 11

1 1 1 12H

+! " ! "! " ! "= =# $ # $# $ # $% %& ' & '& ' & '

then σz|+>=|-> σz|->=|+>

like bit flip!H σzH=σx or H=|+><0|+|-><1|

|ψ>|0>

|0>

R

• • ••

•

•M

MX

H

H

H

H

H

H

I II

Z

I, II → α|+++> + β|--->

effectively encoded into |0L>=|+++>, |1L>=|--->

phase errors ZII, IZI, ZII act as Z on |000>, |111>

e.g., ZII|000> = |000> ZII|111> = -1|111>

but as X on |+++>, |--->

Both bit flip and phase errors:

concatenate these two codes:

|0L>=(|000>+|111>)(|000>+|111>)(|000>+|111>)|1L>=(|000>-|111>)(|000>-|111>)(|000>-|111>)

inner layer corrects bit flips 000, 111outer layer corrects phase flips +++, ----

Shor PRA 52, R2493 (1995)

define Bell basis:|000>±|111>|001> ±|110>|010> ±|101>|100> ±|011>

consider decoherence of qubit 1:e|0> → a0|0>+a1|1>e|1> → a2|0>+a3|1>

e, a0,…a3 =states of env

first triple:|000>+|111>→(a0|0>+a1|1>)|00>+

(a2|0>+a3|1>)|11>= a0|000>+a1|100|+a2|011>+a3|111>

put in Bell basis →=1/2 (a0+a3) (|000> + |111>) +1/2 (a0-a3) (|000> - |111>) +1/2 (a1+a2) (|100> + |011>) +1/2 (a1-a2) (|100> - |111>)

similarly |000> - |111> goes to=1/2 (a0+a3) (|000> - |111>) +1/2 (a0-a3) (|000> + |111>) +1/2 (a1+a2) (|100> - |011>) +1/2 (a1-a2) (|100> + |111>)

output 2 (syndrome 2)

assume 1 error only:compare all 3 triples, see which differsmajority sign indicates |0L> or |1L>find which qubit decohered(measure 9 ancillas → which syndrome)restore qubit state with a unitary operation

1/2 (a0+a3) (|000> - |111>) ⇒ no erroroutput 2 1/2 (a0-a3) (|000> + |111>) ⇒ Z error

1/2 (a1+a2) (|100> + |011>) ⇒ X error 1/2 (a1-a2) (|100> - |011>) ⇒ ZX=Y error

e.g. from |000> - |111>)

have diagnosed error on 1st qubit→ correct with appropriate unitary

Encoder:

|ψ>

|0>

|0>

H

H

H

|0>|0>

|0>|0>

|0>

|0>

[9,1,3] code: 9 physical qubits1 logical qubit(3-1)/2=1 arbitrary error corrected

not most efficient code: [7,1,3] and [5,1,3]cannot compute easily (logical X, Z OK

logical H, CNOT, T hard)

Fault Tolerant QC

• compute directly on encoded states (blocks)• periodic error correction on encoded states• need gates on encoded qubits• design encoded gates to avoid propagation of errors• no new errors introduced by error correction protocol• 1 physical error → at most 1 qubit error per block

successful (with high probability) computation even when all operations (channels, gates, measurements) are imperfect

x

x xeg. propagation of error via CNOT gate

( )

( )1 0

0 1 0 0 0 1 1

if Xerror on qubit1

0

Xerrorson both qubits1and 2

1 1 0 0

CNOT

CNOT

α β α β

α β α β

+ ⊗ → +

+ ⊗ → +

→

( )1 1 2

†1 1 2

CNOT X X X CNOT

UX U U X X U

=

= proliferationby conjugation

Circuit for Fault tolerant CNOT

transverse operations between qubitsin different blocks, i.e., implement inbitwise fashion⇒prevents spread of errors between qubitswithin block (but still single qubit propagation…)

|0L>

|1L>

≡

|0L>

|1L>

|0L>

|0>anc measure

error correct for FT:multiple measurementsto distinguish erroron data or ancilla

qu-ph/9903099

H|0>

|0>

Each component fails with prob. p⇒ Circuit error prob. O(p)

FT prepare |0L>

FT prepare |1L>

FT EC

FT EC

FTH

FT EC

FT EC

FT EC

FT EC

FTCNOT

FTmeasure

FTmeasure

FT = fault tolerantEC = error correct

FT Circuit fails with prob. O(p2)

assume errors independent, X, Y, Z, I (XX etc. on CNOT)

Fault Tolerant CNOT gateSteane [7,1,3] code:

Measuresyndrome

Measuresyndrome

recover

recover

|0L>

|1L>

1 2 3 4each component has error O(p)sources of 2 or more errors:• 2 pre-existing errors, 1 in each block: (c0p)2

(c0p if fault tolerant previously, c0 = # poss. error locations)• 1 pre-existing error + 1 failure in CNOT: c1p2

• 2 failures in CNOT: c2p2

• 1 failure in CNOT, 1 failure in measure syndrome: c3p2

• 2 failures in measure syndrome: c4p2

• 1 failure in measure syndrome, 1 failure in recovery: c5p2

• 2 failure in recovery: c6p2

count # places where failure can occur→c0, c1, c2, …, c6for Steane [7,1,3] code, c=c0

2+c1+c2+c3+c4+c5+c6=104

⇒ successful gate, prob. 1-cp2

if p < 10-4, encoded does better than 1-p

Concatenation & Threshold Theorem

reduce effective error for arbitrary accuracy ε

idea – concatenate circuitqubitsGatesmeasurements

k=0 k=1 k=2k

level of concatenation

3qubits 9

qubits

e.g.,Shor[9,1,3]codek=2

k=0 k=1 k=2level of concatenation, k

apply encoded(FT) CNOT

encoded(FT) CNOT

failure rate:

pk=0 k=1 k=2

cp2 c(cp2)2

k=3c(c3p4)2

…k…(cp)2k/c

k=3

encoded(FT) CNOT

…k

note: circuit size grows as ~dk

(d = # operations in encoded gates + EC)

→ for accuracy ε in circuit with p(n) gates:each gate accurate to ε/p(n)⇒ concatenate k times (cp)2k/c ≤ ε/p(n)

solution exists for p < pth ≡1/c

size: ( ) ( )log

log ( ) /(log ( ) / )

log(1/ )

dk p n cd O poly p n

pcε

ε

= =

i.e., polylog larger than original circuit

Steane [7,1,3] code:c ~ 104 ⇒ pth ~ 10-4

k=1: if p ~ 10-6 → failure rate cp2 ~ 10-8

can improve on this by concatenation, but …increasing overhead associated with recursion:

Fault tolerance issues:

Oskin, Chong & Chuang, IEEE Jan. 2002

• need transversal gates• can perform elementary operations in parallel• can couple any 2 qubits regardless of distance• source of fresh ancillas• larger problem size requires larger k

parallelism vs communicationlarger encodings (Steane [127,29,15])much current research – Steane, Terhal, Knill, …

QECC – active correction

>ψ| |Φ> error

extra qubits for encoding

unitary“encoder”

measure, diagnose error

Quantum Computation

Quantum Computation

fix error

decode

|Ω>

|Φ>|Uψ>

cold ancillaqubits

entropy dump(cool ancillas)

error>ψ| |Φ>

Quantum Computation

Quantum Computation

encode into decoherence-free subspace

|Φ>

decode

DFS – passive correction

Decoherence-free subspaces/subsystems:

• collective decoherenceZanardi & Rasetti

Mod. Phys. Lett. B 11, 1085 (1997)Lidar, Chuang, & Whaley

PRL 81, 2594 (1998)• modulated (striped) collective decoherence

K. Brown, Ph. D. Thesis, UCB 2003• correlated errors

Lidar, Bacon, Kempe, Whaley PRA 63, 022306 (2001)

• generalization to subsytemsKnill et al. PRL 84, 2525 (2000)

∑ ⊗=α

αα BSHI

DFS condition for unitary evolution on a subspace:

αααα iciS ~~ =

with the system-bath interaction

system bath

SBSystem-bath Hamiltonian: S BH α αα

⊗=∑

zσ

xσ

yσ

SBH

zσ

xσ

yσ

SBH−

SBH

Apply rapid pulsesflipping sign of Sα

SB

"time reversal",averaged to zero.

XZX Z

H⇒

=−

SBH Z Bλ= ⊗

SBH SBH

X X

Decoupling (bang-bang) example:Time-Reversal

Decoupling Methods: engineer effective interaction with a symmetry, dynamically generate a symmetry

Decoherence-Free Subspace: encode quantum information into a protected subspace of system. Protected due to symmetry ofsystem-environment interaction.

Symmetry protects Quantum Information

QECC: use symmetry of noise model.