Quantum Computing
Lecture 22
Michele Mosca
Correcting Phase Errors

Suppose the environment effects error
e
x1 x2 x3  Z  Z
1
e2
Z
e3
x1 x2 x3
on our quantum computer, where
H(e1 , e2, e3 )  1
This is a description of errors
in phase because we use
powers of operator Z
Quantum Error Correction

We can encode
0L 
1L 

1
2 2
1
2 2
0
1
 0
1
 0
1

0
1
 0
1
 0
1

Z error in
upper bit
Consider error term Z  I  I acting on the
logical 0 gives
0
1
 0
 1
 0
 1

Such error arriving in
decoder is shown next slide
Quantum Error Correction
error
0  1 H
0  1
0  1
H
H
H
H
H
H
H
H
0
Please observe
repetitions of these
patterns
H
H
H
H
H
Equivalently , cancelling pairs of
H inside the diagram we get
0  1 H
0  1 H
0  1
0
H
H
H
H
Final circuit for correcting
phase errors
Quantum Error Correction

If the error effected on the system in state
is of the form 
( )


H ( e1e2e3 ) 1


pe1e2e3 Ee1e2e3 
Ee1e2e3  Z  Z  Z
e1
e2
e3
pe1e2e3 Ee1e2e3

t
Quantum Error Correction
and if the state  only consists of mixtures
of superpositions of codewords
 0  1  0  1  0  1  and 0  1  0  1  0  1 

then the correction procedure (call it
will map
  ()  
)
Correcting both phase
errors and bit flip errors

Consider the codewords of Shor’s code
0L   000  111
1L   000
 000  111  000  111 
 111  000  111  000  111 
We can easily correct any single X- error in
one of the 3 three-bit parts
 We can then also correct a single Z- error on
one of the 9 qubits.
 This means we can also correct Y-errors on
one of the 9 qubits

Quantum Error Correction

Theorem 10.2: Suppose C is a quantum code
and  is the error-correction operation
constructed in the proof of Theorem 10.1 to
recover from a noise process  with
operation elements {Ei } . Suppose F is a
quantum operation with elements {Fj } which
are linear combinations of the {Ei } . Then the
error correction operation  also corrects
the effects of the noise process F on the
code C.
Correcting any error
Since any error operator Ek can be written as
a linear combination of I,X,Z and Y, then the
same procedure will correct ANY error acting
on just 1 of the 9 qubits.
 If  ()  (1  p2 )()  p2()
T
where  is a quantum operator whose
operator terms are correctable with
correction operator  , then

  T ()  (1  p )  p   ()
2
2

Correcting any error
Theorem 10.1 (Quantum Error Correction
Conditions) Let C be a quantum code, and let P
be the projector onto C. Suppose  is a
quantum operation with operation elements {Ei }
A necessary and sufficient condition for the
existence of an error-correction operation 
correcting  on C is that
t
for some Hermitian matrix
numbers.
 (no mention of efficiency)
PEi EjP  ijP
 of complex
Degenerate Codes
Consider the 9-qubit code.
 A single Z-error on the first qubit of a
codeword produces the same outcome as a
single Z-error on either the 2nd or 3rd qubit.
 The correction procedure will correct these
errors regardless
 A degenerate code is one where two
correctable errors produce the same
effect on the codewords (this is impossible
with classical codes).

Quantum Hamming Bound

Any non-degenerate quantum error
correcting code that encodes k logical
qubits into n qubits and can correct errors
on up to t qubits must have
n  j k
n
 3 2  2

j1  j 
 If t=k=1, we get n  5 (there exists a 5t
qubit code that accomplishes this)