Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6...
-
Upload
nguyendang -
Category
Documents
-
view
221 -
download
0
Transcript of Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6...
![Page 1: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/1.jpg)
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
![Page 2: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/2.jpg)
Classical Error Correction
Everyday example:
Noisy phoneline
B D VR
A
V
O
E
L
T
A
I
C
T
O
R
![Page 3: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/3.jpg)
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
![Page 4: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/4.jpg)
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
![Page 5: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/5.jpg)
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!
![Page 6: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/6.jpg)
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
![Page 7: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/7.jpg)
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!
![Page 8: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/8.jpg)
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
![Page 9: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/9.jpg)
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
![Page 10: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/10.jpg)
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!
![Page 11: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/11.jpg)
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:
![Page 12: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/12.jpg)
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?
![Page 13: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/13.jpg)
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
![Page 14: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/14.jpg)
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)
![Page 15: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/15.jpg)
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
![Page 16: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/16.jpg)
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 ?
?
![Page 17: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/17.jpg)
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:
![Page 18: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/18.jpg)
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:
![Page 19: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/19.jpg)
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!
![Page 20: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/20.jpg)
Error Correction Measurement7 qubit Steane code (standard form):
Name Operator
𝑔6 𝑍𝑍𝑍𝐼𝐼𝑍𝐼
𝑔1 𝑋𝐼𝐼𝐼𝑋𝑋𝑋
𝑔2 𝐼𝑋𝐼𝑋𝐼𝑋𝑋
𝑔5 𝐼𝑍𝑍𝐼𝑍𝐼𝑍
𝑔4 𝑍𝐼𝑍𝑍𝐼𝐼𝑍
𝑔3 𝐼𝐼𝑋𝑋𝑋𝑋𝐼
![Page 21: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/21.jpg)
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:
![Page 22: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/22.jpg)
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
![Page 23: Quantum Error Correction - LTH · iiixxxx g 2 ixxiixx g 3 xixixix g 4 iiizzzz g 5 izziizz g 6 ziziziz 1 0001111 1011010 0111100 1101001 0000000 1010101 0110011 1100110 8 0 l 1 1110000](https://reader031.fdocuments.in/reader031/viewer/2022022806/5cccb96b88c9932b558c74f5/html5/thumbnails/23.jpg)
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