Title: Introduction to Quantum Error Correction
1Introduction to Quantum Error Correction
Fault-Tolerant Quantum Logic
2Why Quantum Error Correction?6
- Cause circuit interacts with the surroundings
- decoherence
- decay of the quantum information stored in the
device - Solution Quantum Error Correcting Codes
- protect quantum information against errors.
- perform operations fault-tolerantly on encoded
states.
3Major Difficulties 9
- 1. No cloning theorem impossible to duplicate an
arbitrary unknown qubit - Solution Fight Entanglement with
entanglement(encode the information that we want
to protect in entanglement).
impossible
possible
Explain why no cloning is important in this
context
4Major Difficulties 9
- 2. Errors are continuous continuous
errors?requires infinite resources and infinite
precision - Solution Digitalize the errors that circuit
makes. - 3. Measurement destroys quantum information
recovery is impossible if quantum information
state is destroyed. - Solution Measure the errors without measuring
the data.
5Central Idea of QEC
- A small subspace of the Hilbert space of the
device is designated as the code subspace. - This space is carefully chosen so that all of the
errors that we want to correct move the code
space to mutually orthogonal error subspaces. - We can make a measurement after our system has
interacted with the environment that tells us in
which of these mutually orthogonal spaces the
system resides, and hence infer exactly what type
of error occurred. - The error can then be repaired by applying an
appropriate unitary transformation.
6Key Ideas of QEC
- Encode the message with redundant information
- Redundancy in the encoded message allows to
recover the information in the original message.
- Measure the errors, not the data.
7General Model of QEC 1
- Deal with errors
- Error detection
- Error correction
- Errors also in encoding and recovery (they are
themselves complex quantum computations) ?But,
fault-tolerant recovery possible if error rate is
not high (Peter Shor, 1996). - Problems to store an unknown quantum state with
high fidelity for an indefinitely long time and
problems to do quantum computation?But, possible
if error rate is below threshold (Manny Knill and
Raymond Laflamme, 1996).
8Application of what?
- Storage CDs, DVDs, hard drives
- Wireless Cell phones, wireless links
- Satelite and Space TV, Mars rover
- Digital Television DVD, MPEGS layover
- High Speed Modems ADSL, DSL
9Classical Error Correction
linear
cyclic
BCH
Bose-Chaudhuri-Hochquenghem
Hamming
Reed-Solomon
10Classical Repetition Code
- Transmission Sending one bit of information
across the channel. - Noise flips the bit with the probability p
- Encoding triple each bit 0?000, 1?111,
C000,111 - Decoding majority voting
- Example 10?111000 101001? 10
- Limitation not possible to recover the
information correctly if more than one bit is
flipped.
11Classical Repetition Code
Transmission method 1. Non-encoded transmission 3-bit repetition code
Probability of error p Probability that two or more of the bits are flipped 3p2(1-p)p3
When p0.25 0.25 0.15625 (when plt0.5, this method is better.)
What is this? Explain better this and the whole
table
12QEC The Three Qubit Bit Flip Code
- Example sending one qubit through a channel.
- Noise flips the qubit with the probability p.
- In other words, the state ?gt is taken to state
X\?gt with the probability p, where X
(bit flip matrix) - Encoding
- Decoding Majority Logic
- Limitation may be unable to recover the
information correctly if more than one bit is
flipped in some cases.
13QEC The Three Qubit Bit Flip Code
- Encoding
- ?0gt?1gt ? ?000gt?111gt
- Encoding Circuit
- Measuring the ancilla bits reveals the error but
not the information qubit.2
Explain Why?
14QEC The Three Qubit Bit Flip Code
Pre-assumption of the errors One or none error
occurs
- Transfer the stored information to the output
qubit. - Limited if more than one error.
- We dont have enough info of the location of
errors.
15Analysis OF WHAT?
Add more detailed captions to the table
The error probability can depend significantly on
the initial state.
0gt?000gt 0gt?000gt 0gt?000gt ? ? ?
Encode Decode Prob. (P0.25) Encode Decode Prob OF WHAT.
000gt 0gt (1-0.25)3 0.4219 0.4219
100gt 0gt 0.1406 0.1406
010gt 0gt 0.1406 0.1406
001gt 0gt 0.1406 0.1406
110gt 1gt 0.0469 0.0469
101gt 1gt 0.0469 0.0469
011gt 1gt 0.0469 0.0469
111gt 1gt 0.0156 0.0156
Error Sum Error Sum 0.01563 Error Sum Error Sum 0
16Fault-Tolerant Computation
Preparation/ Encode
Verify
Computation of Error Syndrome
Recovery
Explain better what each block does, especially
verify
17Fault-Tolerant Computation6
- Rules
- Implement gates that can process encoded
information. - Control propagation of errors.
- Ensure that recovery from errors is performed
reliably.
18Fault-Tolerant Computation6
- 1st Law Dont use the same bit twice.
Bad Error propagates, so infection spreads.
Good!
19Fault-Tolerant Computation6
- 2nd Law Copy/measure the errors, not the data.
- Copy the information from the data to the
ancilla. - Measure the ancilla to find an error syndrome.
- Based on the error syndrome, we perform the
required recovery.
20Fault-Tolerant Computation6
- 3rd Law Verify when you encode a known quantum
state.
A nondestructive measurement is performed (twice
performed above) to verify that the encoding was
successful.
More explanation needed
21Fault-Tolerant Computation
- 4th Law Repeat the operations
More explanation needed
22Fault-Tolerant Computation
- 5th Law Use the right code
More explanation needed
23Error Correction in The Three Qubit Code
- 0gt?000gt, 1gt?111gt
- Error Correction
More explanation needed
Measurement(M1, M2) Action
(0,0) None
(1,0) Flip the second bit
(0,1) Flip the third bit
(1,1) None
24Example The Shor Code
- Also known as the 9-qubit code
- Combination of the three qubit phase flip codes
and bit flip codes. - Seen as a two-level concatenated code.3
- One qubit is encoded into 9 qubits
- The data is no longer stored in a single qubit,
but instead spread out among nine of them.8 - Correction of bit flips majority voting.
25Assumptions of the Shor Code
- For simplicity, we assume that any qubit error
consists in the application of bit flip error,
phase flip error, and/or combination of these
two. - X (bit flip error)
- Z (phase flip error)
- Y iXZ (combination of bit flip and
phase flip error)
26Preparation in The Shor Code
Block 1
27Majority Logic in The Shor Code3
Explain decoding and recovery, how majority
works, may be you need more slides for this
28Bit Flip Correction
- Bit flip switch 0gt and 1gt
- Describe the error as bit flip matrix X
- Correction
- For a block, compare the first two qubits, and
compare the first with the third. - If the first was flipped, it will disagree with
the third. - If the second was flipped, the first and third
will agree.
29Phase Flip Correction
- Example
- Describe the error as phase flip matrix Z
- Correction
- By comparing the sign of the first block of three
with the second block of three, we can see that a
sign error has occurred in one of those blocks. - Then, by comparing the signs of the first and
third blocks of three, we can narrow down the
location of phase error and flip it back.
30Simultaneous Bit and Phase Flip Error
- Describe the error as YiXZ
- Correction We can fix the bit flip first, and
then fix the phase flip for the simultaneous bit
and phase flip error, even if they are on
different qubits.
31Stabilizer Coding in The Shor Code8
- Bit Flip Error
- Equivalent to measure the eigenvalues of Z1Z2 and
Z1Z3. - For example, if the first two qubits are the
same, the eigenvalue of Z1Z2 is 1 otherwise,
the value is 1. - Phase Flip Error
- Equivalent to measure the eigenvalues of
X1X2X3X4X5X6 and X1X2X3X7X8X9. - If the signs agree, the eigenvalues will be 1
otherwise, the values is 1.
Remind on an example what are eigenvalues ,
define them
32Stabilizer Coding in The Shor Code
- In order to totally correct the code, we must
measure the eigenvalues of a total of eight
operators.
M1 Z Z I I I I I I I
M2 Z I Z I I I I I I
M3 I I I Z Z I I I I
M4 I I I Z I Z I I I
M5 I I I I I I Z Z I
M6 I I I I I I Z I Z
M7 X X X X X X I I I
M8 X X X I I I X X X
Explain what we see here
33Another Phase Error Correction in The Shor Code
- Hadamard Transformation on each qubit.
- The qubits taken 1, 4, and 7 (or 2,5,8 or
3,6,9).
34Stabilizer Code
- Many quantum states can be more easily described
by working with the operators that stabilize them
than by working explicitly with the state itself.
- ?gt
- ?X1X2?gt ?gt and Z1Z2?gt ?gt
- ??gt is stabilized by the operators X1X2 and
Z1Z2. - ??gt is the unique quantum state which is
stabilized by these operators X1X2 and Z1X2.
35Stabilizer Code
- In making continuous weak measurements on our
system, we would like to choose the measurements
in such a manner that we gather as much
information about the errors as possible while
disturbing the logical qubits as little as
possible?quantum error correcting code. - Stabilizer formalism provides a way to easily
characterize many of the error correcting codes. - Pauli group Pn ?1, ?i?I,X,Y,Z?n
Give examples of Pauli group operators
36Stabilizer Code4
- There exist a set of operators in Pn, called the
stabilizer generators and denoted by g1, g2, ...,
gr. - They are such that every state in C is an
eigenstate with eigenvalue 1 of all the
stabilizer generators. - That is, gi?gt ?gt for all i and for all states
?gt in C. - Moreover, these stabilizer generators are all
mutually commuting.
37Stabilizer Code4a
- The stabilizer code error correction procedure
involves - 1) simultaneously measuring all the stabilizer
generators and then - 2) inferring what correction to apply from the
measurement results. - The formalism states that the stabilizer
measurement results indicate a unique correction
operation.
38ExampleThe Three Qubit Bit Flip Code
ZZI IZZ Error Correction Unitary Action
1 1 None None None
-1 1 Bit 1 flipped XII Flip bit 1
-1 -1 Bit 2 flipped IXI Flip bit 2
1 -1 Bit 3 flipped IIX Flip bit 3
39The Classical 7,4,3 Hamming Code
0
0
1
1
Parity bits comes from the rule that the total
number of 1s contained in each circle should be
even.
0
1
1
0
1
1
0
40Steanes Code
- One qubit is encoded into seven qubits.
- Logic 0 those with even number of 1s
- Logic 1 those with odd number of 1s
41Encoder for the Steanes Code
42Computation of Bit Flip Syndrome
43Computation of Phase Errors
R Hadamard Rotation
44Fault-Tolerant Logic Gate
- The fault-tolerant quantum gates and measurements
must prevent a single error from propagating to
more than one error in any code block. - Therefore the small correctable errors will not
grow to exceed the correction capability of the
code. 7
45One-Bit Teleportation
- One-teleportation is based on Swap gate
I do not understand . This is not swap Explain
why we need one-bit teleportation
46Several Facts to derive one-bit teleportation
47Several Facts to derive one-bit teleportation
- Fact 2 When the control qubit is measured, a
quantum-controlled gate can be replaced by a
classical controlled operation.
U is performed if the measurement result is 1.
Why it is so?
48Z-teleportation
- The two bits are disentangled before the second
Hadamard gate. - Therefore the second qubit can be measured
before the second Hadamard gate without affecting
the unknown state in the first qubit.
H?gt
Why it is so?
49X-teleportation
Why it is so?
50Fault-Tolerant Toffoli Using One-bit Teleportation
Non-FT Gate
51Fault-Tolerant Toffoli Gate
0gt
x gt
H
X
0gt
y gt
X
H
Z
0gt
Z
x gt
y gt
z gt
H
Mistake
52Alternative FT Toffoli Gate
53Anticommute Commute
- Anticommute A, BABBA0
- Commute A,BAB-BA0
- Two commuting matrices can be simultaneously
diagonalized. - This means that we can measure the eigenvalue of
one of them without disturbing the eigenvalues of
the other. - Conversely, if two operators do not commute,
measuring one will disturb the eigenvectors of
the other, so we cannot simultaneously measure
non-commuting operators.
54References
- 1 Web resouce from http//www-2.cs.cmu.edu/afs/c
s/project/pscico-guyb/realworld/www/ - 2 Quantum Codes, Class slides from
http//math.uwyo.edu/moorhous/quantum - 3 Quantum Physics, abstractquant-ph/0211071
- From Jumpei NIWA view email Date Wed, 13 Nov
2002 084346 GMT (584kb) Simulating the Effects
of Quantum Error-correction Schemes - Authors Jumpei Niwa, Keiji Matsumoto, Hiroshi
ImaiComments 13 pages, 25 figures - 4 A Practical Scheme for Error Control using
Feedback, Mohan Sarovar,1, . Charlene Ahn,2,
Kurt Jacobs,3, and Gerard J. Milburn1, - 1Centre for Quantum Computer Technology, and
School of Physical Sciences, - The University of Queensland, St Lucia, QLD 4072,
Australia - 2Institute for Quantum Information, California
Institute of Technology, Pasadena, CA 91125, USA - 3Centre for Quantum Computer Technology, Centre
for Quantum Dynamics, - School of Science, Grith University, Nathan, QLD
4111, Australia - 6 Reliable Quantum Computers, J. Preskill,
California Institute of Technology - 7 Towards Robust Quantum Computation,
dissertation, Debbie W. Leung, 2000 - 8 Stabilizer Codes and Quantum Error
Correction, Thesis, Daniel Gottesman,
California,Institute of Technology, 2001
55References
- 9 Quantum Computation and Quantum Information,
M.A. Nielsen I. L. Chuang, 2000