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.