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TRANSCRIPT
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