Computing Quantum Lecture 14a (notes on QEC)
Michele Mosca
Classical Error Correcting Codes
Suppose errors in our physical system for storing 0 and 1 cause each physical bit to be toggled independently with probability p We can reduce the probability of error to be in O(p2 ) by using a repetition code e.g. : encode a logical 0 with the state 000 and a logical 1 with the state 111
Reversible networks for encoding and decoding
b
0 0
b b b
b b b
b 0 0
Classical Error Correcting Codes
After the errors occur, decode the logical bits by taking the majority answer of the three bits and correct the encoded bits So 000 ! 000 111 ! 111
001 ! 000 010 ! 000 100 ! 000
011 ! 111 101 ! 111 110 ! 111
Classical Error Correcting Codes
As long as less than 2 errors occurred, we will keep the correct value of the logical bit The probability of 2 or more errors is
3p (1 " p) + p = 3p " 2p ! O(p )
2 3 2 3 2
1 (which is less than p if p < ) 2
Reversible network for error correction
b ! e3 b ! e2 b ! e1
0 0
Assume that e3 + e2 + e1 ! 1
ei ! {0,1} b b b s1
s2
syndrome
If s1s2 = 00 then no error occurred Otherwise, the error occurred in bit j where j = 2s1 + s2
Equivalently
b e3 ! b ! e2 b ! e1
0 0
s1
b b b
s2
Stabilizer measurement??
0
This is implementing a Z1 measurement (interpreting 0 as +1, and 1 as 1)
Stabilizer measurement??
0
This is implementing a Z1Z 2 measurement
Stabilizer measurement??
0
H
H
This is implementing a X 1 X 2 measurement
Notation clarification
For an n-qubit system Z j denotes
I " 4 44 I2 3 I "L" 14 " L " I " Z " 1 24 I 4 3
j !1 n! j
E.g. n=3, then
Z1Z 2 = (Z ! I ! I )(I ! Z ! I ) = (Z ! Z ! I )
Perform operations on logical bits
e.g. NOT gate
b b b
X X X
b b b
Perform operations on logical bits
e.g. c-NOT gate
Quantum Error Correcting Codes
e.g. : encode a logical 0 with t...