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a digital image for back-up. However, quantum me- ing and quantum information processing.
chanics does not allow unknown quantum states to Here is how quantum error-correction codes work.
be distinguished or copied exactly. For example, we We consider the single qubitcase. First assume that
cannot reliably distinguish between quantum states a qubit α 0 +α 1 is passed through a bit flip
0 1
| i | i
0 and (0 + 1 )/√2. If we perform measurement channel which flips the state of a qubit from 0
| i | i | i | i
for quantum state 0 , the measurement result will to 1 and from 1 to 0 , each with probability p,
| i | i | i | i
be 0 with probability 1, while measuring quantum and leaves each of states 0 and 1 untouched with
| i | i
state (0 + 1 )/√2yields measurements 0or 1with probability 1 p. We describe a bit flip code that
| i | i −
equal probability. A measurement result of 0 can- protects the qubit against quantum noise from the
not tell the identity of the quantum state being bit flip channel.
measured. A theorem known as a no-cloning theo- We encode states 0 and 1 in three qubits, with
| i | i
rem states that unknown quantum states cannot be 0 encodedas 000 and 1 as 111 . Thusthequbit
| i | i | i | i
copied exactly (Wootters and Zurek (1982); Nielsen state α 0 + α 1 is encoded in three qubits as
0 1
| i | i
and Chuang (2000)). α 000 +α 111 . We pass each of the three qubits
0 1
| i | i
As we discussed in Section 3.3, quantum entan- through an independent copy of the bit flip chan-
glement plays a crucial role in strange quantum ef- nel, and assume that at most one qubit is flipped.
fects such as quantum teleportation, violation of The following simple two-step error-correction pro-
Bell’s inequality, and superdense coding (Hayashi cedure can be used to recover the correct quantum
(2006); Nielsen and Chuang (2000)). Entanglement state.
is a new type of resource that differs vastly from Step1.Performameasurementonaspeciallycon-
thetraditionalresourcesinclassicalinformationthe- structed observableand call the measurementresult
ory. We are far from having a general theory to an error syndrome. The error syndrome can inform
understandquantum entanglement butencouraging uswhaterror,ifany,occurredonthequantumstate.
progress made so far reveals the amazing property The observable has eigenvalues 0, 1, 2 and 3, with
19
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
corresponding projection operators, α 1 , and with probability 1 p, leaves alone the
1
| i −
qubit. The following scheme is to turn the phase
Q = 000 000 + 111 111 no error,
0
| ih | | ih | flip channel into a bit flip channel. Let + =(0 +
| i | i
Q 1= 100 100 + 011 011 1 )/√2 and =(0 1 )/√2 be a qubit basis.
| ih | | ih | | i |−i | i−| i
The phase flip channel leaves alone states + and
bit flip on the first qubit, | i
with probability 1 p and changes + to
Q 2= 010 010 + 101 101 a|− ndi viceversawithprob− abilityp.Inother| woi rds,| t− hei
| ih | | ih |
phase flip channel with respect to the basis + and
bit flip on the second qubit,
| i
acts just like a bit flip channel with respect to
Q 3= 001 001 + 110 110 | t− hei basis 0 and 1 . Thus we encode 0 as +++
| ih | | ih | | i | i | i | i
and 1 as for protection against phase flip
bit flip on the third qubit.
| i |−−−i
errors. The operations for encoding, error-detection
Ifoneof three qubitshasone ornobitflip,the error
and recovery are the same as for the bit flip channel
syndrome will be one of 0, 1, 2 and 3, with 0 corre-
but with respect to the + and basis instead of
sponding to no flip, and 1, 2 and 3 to a bit flip on | i |−i
the 0 and 1 basis.
the first, second and third qubit, respectively. For | i | i
Last we describe Shor error-correction code. It is
example, if the first qubit is flipped, the corrupted
a combination of the three-qubit phase flip and bit
state is ψ =α 100 +α 011 . Since ψ Q ψ =1
0 1 1 flip codes. First use the phase flip code to encode
| i | i | i h | | i
and ψ Q ψ =0 for j =1, in this case the error
j states 0 and 1 inthreequbits,with 0 encodedas
h | | i 6
syndrome is 1. Although performing measurements | i | i | i
+++ and 1 as ;next,usethethree-qubit
usually causes change to the quantum state, the | i | i |−−−i
bitflipcodetoencodeeach of thesequbits,with +
speciality of the constructed observable is that syn- encoded as (000 + 111 )/√2 and encoded| asi
drome measurement does not perturb the quantum | )/√i 2.T| heri esultednine| -− qui
(000 111 bitcodehas
state: it is easy to check that the state is ψ both | i−| i
codeworks as follows:
| i
before and after the syndrome measurement. While
000 + 111 000 + 111 000 + 111
the syndrome provides information about what flip 0 | i | i| i | i| i | i,
| i→ √2 √2 √2
error has occurred, it does not contain any infor-
mation about the state being protected, that is, it 000 111 000 111 000 111
1 | i−| i| i−| i| i−| i.
does not allow us to deduce anything about the am- | i→ √2 √2 √2
plitudes α and α . Such a special property is the
0 1
With the mixture of both phase flip and bit flip