Quantum Error Correction
Joshua Kretchmer
Gautam Wilkins
Eric Zhou
Error Correction
• Physical devices are imperfect
• Interactions with the environment
• Error must be controlled or compensated for
– One step has probability to succeed = p
– t steps has probability to succeed = pt
Classical Error Correction
• Error Model
– Channels provide description of the type of error
• Encoding
– Extra bits added to protect logical bit
– String of bits  codeword
– Redundancy
• Error Recovery
– Recovery operation
– Measure bits and re-set all values to majority vote
Classical 3-Bit Code: Bit Flip Error
• Bit flip channel: bit is flipped with prob. = p < 1/2
• Encoding:
b
0
0
b
b
b
• Error Recovery:
000  { (000, (1-p)3),
(001, p(1-p)2), (010, p(1-p)2), (100, p(1-p)2)
(011, p2(1-p)), (110, p2(1-p)), (101, p2(1-p))
(111, p3) }
– Prob(unrecoverable error) = 3p2(1-p)+p3 = 3p2-2p3
Problems with QEC
• No cloning theorem
– Can’t copy an arbitrary quantum state
– Entanglement
• Measurement
– Cannot directly measure a qubit
– Error syndrome
• Quantum evolution is continuous
Quantum 3-Bit Code: Bit Flip Error
|>=a|0>+b|1>
|0>
|0>
Encoding
|>
Error
Channel
|0>
|0>
M
M
X
Diagnose and
Correct
• Encoding  a|000>+b|111>
• Error channel
– Noise acts on each qubit independently
– Probability noise does nothing = 1 - p
– Probability noise applies x = p < 1/2
Decode
Quantum 3-Bit Code: Bit Flip Error
• After channel  8 possible results
State:
a|000>+b|111>
a|100>+b|011>
a|010>+b|101>
a|001>+b|110>
a|110>+b|001>
a|101>+b|010>
a|011>+b|100>
a|111>+b|000>
Probability:
(1-p)3
p(1-p)2
p(1-p)2
p(1-p)2
p2(1-p)
p2(1-p)
p2(1-p)
p3
Quantum 3-Bit Code: Bit Flip Error
• After CNOT’s  4 possible results
State:
a|000>+b|111>|00>
a|100>+b|011>|10>
a|010>+b|101>|01>
a|001>+b|110>|11>
a|110>+b|001>|01>
a|101>+b|010>|10>
a|011>+b|100>|11>
a|111>+b|000>|00>
Probability:
(1-p)3
p(1-p)2
p(1-p)2
p(1-p)2
p2(1-p)
p2(1-p)
p2(1-p)
p3
Quantum 3-Bit Code: Bit Flip Error
• Measure 2 ancilla qubits  error syndrome
Measured syndrome
action
00
do nothing
01
apply x to 3rd qubit
10
apply x to 2nd qubit
11
apply x to 1st qubit
• Designed to correct if there’s an error in 1 or
no qubits
• Error in 2 or 3 qubits is an uncontrollable error
Quantum 3-Bit Code: Bit Flip Error
• Failing probability  pu = 3p2(1-p)+p3
= 3p2-2p3 = O(p2)
• Fidelity  success probability = 1- pu = 1- 3p2
• Without error correction pu = O(p)
Quantum 3-Bit Code: Phase Error
• Random rotation of qubits about z-axis
• Continuous error
• P() = ei 0 = cos()I +isin()z
0
e-i
•  - fixed quantity stating typical size of rotation
•  - random angle
Quantum 3-Bit Code: Phase Error
• Apply H to each qubit at either end of the
channel
• HIH = HH = I; HzH = x
•  HPH = cos()I +isin()x
• Same result from bit flip code
– Fidelity = 1 - 3p2
– p = <sin2()>  (2)2/3 for <<1
General Quantum Error
• Errors occur due to interaction with
environment
• |0>|E>  1|0>|E1> + 2|1>|E2>
• |1>|E>  3|1>|E3> + 4|0>|E4>
• (0|0> + 1|1>)|E> 
01|0>|E1> + 02|1>|E2> +
13|1>|E3> + 14|0>|E4>
General Quantum Error
• (0|0> + 1|1>)|E> 
1/2(0|0> + 1|1>)(1|E1> + 3|E3>)
+ 1/2(0|0> - 1|1>)(1|E1> - 3|E3>)
+ 1/2(0|1> + 1|0>)(2|E2> + 4|E4>)
+ 1/2(0|1> - 1|0>)(2|E1> - 4|E4>)
• 0|0> + 1|1> = |>
0|0> - 1|1> = Z|>
0|1> + 1|0> = X|>
0|1> - 1|0> = XZ|>
General Quantum Error
• (0|0> + 1|1>)|E> 
1/2(|>)(1|E1> + 3|E3>)
+ 1/2(Z|>)(1|E1> - 3|E3>)
+ 1/2(X|>)(2|E2> + 4|E4>)
+ 1/2(XZ|>)(2|E1> - 4|E4>)
•
•
•
•
Error basis = I, X, Z, XZ
|>L|>e  (i|>L)|i>e
|>L  general superposition of quantum codewords
i  error operator = tensor product of pauli operators
Correction of General Errors
• |>L|>e  (i|>L)|i>e
• |>L - orthonormal set of n qubit states
• To extract syndrome attach an n-k qubit ancilla
“a” to system  perform operations to get
syndrome  |si>a
 |0>a(i|>L)|i>e  |si>a(i|>L)|i>e
• Measure si to determine i-1  correct for error
 |si>a(i|>L)|i>e  |si>a(|>L)|i>e
Shor’s Algorithm
• Each qubit is encoded as nine qubits
0 
1 
1
2 2
1
2 2
 000
 111  000  111  000  111 
 000
 111  000  111  000  111 
Shor’s Algorithm
Assume decoherence on first bit of first triple,
becomes:
1
2
 
0   1 1  00    2 0   3 1  11
0

1

0
 3
 000
2 2
1
 0  3

2 2
1
 1   2

2 2
1
 1   2

2 2
 111

 000
 111

 100
 011

 100
 011


Shor’s Algorithm
Shor’s Algorithm
1
  0 0  1 1  00    2 0
2

1
  0   3  000  111 
2 2
1
  0   3  000  111 

2 2
1
 1   2  100  011 

2 2
1
 1   2  100  011 

2 2
  3 1  11 
•
No error
•
Z error
•
X error
•
ZX = Y error
Shor’s Algorithm
• Success Rate:
– Works if only one qubit decoheres
– If probability of a qubit decohering is p
• Probability of 2 or more out of 9 decohering is1(1+8p)(1-p)836p2
• Therefore probability that 9*k qubits can be decoded is
(1-36p2)k
Shor’s Algorithm
• More on decoherence
– Decoherence probability increases with time
– Use watchdog effect to periodically reset quantum
state
– Unfortunately, each reset introduces small
amount of extra error
– Therefore cannot store indefinitely
Steane’s Algorithm
• Basis 1 is |0 , |1 
– Also called basis F, or “flip” basis
• Basis 2 is |0 + |1 , |0 - |1 
– Also called basis P, or “phase” basis
Steane’s Algorithm
• The word |000…0  consisting of all zeroes in basis 1 is equal
to a superposition of all 2n possible words in basis 2, with
equal coefficients.
• If the jth bit of each word is complemented in basis 1, then all
words in basis 2 in which the jth bit is a 1 change sign.
• Hamming Distance
– The number of places two words of the same length differ
• Minimum Distance
– Smallest Hamming distance between any two code words in a code
Steane’s Algorithm
• A code of minimum distance d allows [(d-1)/2] to
be corrected
– If less than d/2 errors occur, the correct original code
word that gave rise of the erroneous word can be
identified as the only code word at a distance of less
than d/2 from the received word.
• [n,k,d] is a linear set of 2k code words each of
length n, with minimum distance d
Steane’s Algorithm
• Parity Check Matrix
– Matrix H of dimensions (n-k) by n, where Hv = 0 iff v is in
the code C
• Generator Matrix
– Matrix G of dimensions n by k, basis for a linear code
– w = cG, where w is a unique codeword of linear code C,
and c is a unique row vector
• For a linear code C in basis 1, a superposition with
equal coefficients, then in basis 2 the words of the
superposition form the dual code of C
• The Parity Check Matrix of C is the Generator Matrix
for its dual code
Steane’s Algorithm
•
Let |a  and |b  be expressed as [7, 3, 4] in
basis 1:
a  0000000  1010101  0110011
 1100110  0001111  1011010  0111100  1101001
b  1111111  0101010  1001100
 0011001  1110000  0100101  1000011  0010110
1

1
0
1
0
1
0
1




0
H   0 1 1 0 0 1 1, G  
0
 0 0 0 1 1 1 1



1

0 1 0 1 0 1

1 1 0 0 1 1
0 0 1 1 1 1

1 1 0 0 0 0 
Steane’s Algorithm
• |a  and |b  are non-overlapping, and have
distance of 3
• Find bit flip with parity check
• Switch to basis 2:
– |c  =|a  +|b 
• Contains only even parity words of a [7,4,3] code
– |d  =|a  -|b 
• Contains only odd parity words
• Distance between |c  and |d  is at least 3
• Phase error can be found with a parity check
Implications for Physical
Realizations of Quantum
Computers
Why Do We Need It?
• Quantum computers are very delicate.
• External interactions result in decoherence
and introduction of errors.
Fault-tolerance
• Especially important when considering
physical implementations.
• Must consider errors introduced by all parts,
including gates.
• Incorrect syndromes introduce errors.
Impact on Physical Systems
• Increased size
• Level of coherence determines increase
Alternative to Error Correction
• Topological Quantum Computing
• Involves particles called anyons that form
braids, whose topology determines quantum
state.
Topological Quantum Computing
Slight perturbations to system cause
braids to be deformed, but only large
disturbances result in them being cut or
joined.
Summary
• Error correction is vital for physical
realizations of trapped particle quantum
computers.
• Allows reliable quantum computation without
requiring extremely high levels of coherence.