Transcript Introduction to & Quantum Error Correction Fault-Tolerant Quantum Logic

Introduction to
Quantum Error Correction
& Fault-Tolerant Quantum Logic
Cherrie Huang
05/05/04
1
Why 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.
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Major Difficulties [9]
1. No cloning theorem: impossible to
duplicate an arbitrary unknown qubit
impossible
Solution: Fight Entanglement with
entanglement(encode the information
that we want to protect in
entanglement).
possible
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Explain why “no cloning”
is important in this context
3
Major 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.
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Central 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.
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Key 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.
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General Model of
QEC [1]
Deal with errors:
1. Error detection
2. Error correction
1. 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).
2. 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).
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Application
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
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Classical Error Correction
Hierarchy
linear
cyclic
BCH
Hamming
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Bose-Chaudhuri-Hochquenghem
Reed-Solomon
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Classical 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,
C={000,111}
Decoding: majority voting
Example: 10111000
101001 10
Limitation: not possible to recover the
information correctly if more than one bit is
flipped.
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Classical Repetition Code
Analysis
Transmission
method
Probability of
error
When p=0.25
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1. Nonencoded
transmission
p
0.25
3-bit repetition
code
Probability that two
or more of the bits
are flipped:
3p2(1-p)+p3
0.15625
(when p<0.5, this
method is better.)
What is this? Explain better this and the whole table
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QEC: 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 |ψ> is taken to state
X\ψ> with the probability p, where X  01 10


(bit flip matrix)
Encoding:
|0  |0 L |000 
|1|  |1 L |111 
Decoding: Majority Logic
Limitation: may be unable to recover the
information correctly if more than one bit is
flipped in some cases.
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QEC: The Three Qubit Bit Flip Code
Encoding :
|0>+|1>  |000>+|111>
Encoding Circuit
Measuring the ancilla bits reveals the
error but not the information qubit.[2]
Explain Why?
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QEC: The Three Qubit Bit Flip Code
Pre-assumption
of the errors:
One or none
error occurs
1.
2.
3.
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Transfer the
stored information
to the output
qubit.
Limited if more
than one error.
We don’t have
enough info of the
location of errors.
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Analysis OF WHAT?
Add more
detailed captions
The error probability can depend significantly on the initial state. to the table
|0>|000>
1
(|0  |1 )
2
Encode Decode Prob.
Encode
(P=0.25)

1
(|000  |111 )
2
Decode
Prob OF
WHAT.
|000>
|0>
(1-0.25)3
=0.4219
1
(|000  |111 )
2
|100>
|0>
0.1406
|0>
0.1406
1
(|0  |1 )
2
1
(|0  |1 )
2
0.1406
|010>
1
(|100  |011 )
2
1
(|010  |101 )
2
|001>
|0>
0.1406
|1>
0.0469
|101>
|1>
0.0469
1
(|0  |1 )
2
1
(|1  |0 )
2
1
(|1  |0 )
2
0.1406
|110>
1
(|001  |110 )
2
1
(|110  |001 )
2
1
(|101  |010 )
2
|011>
|1>
0.0469
|1>
0.0156
1
(|1  |0 )
2
1
(|1  |0 )
2
0.0469
|111>
1
(|011  |100 )
2
1
(|111  |000 )
2
0.01563
Error Sum
Error Sum
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1
(|0  |1 )
2
0.4219
0.1406
0.0469
0.0469
0.0156
0
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Fault-Tolerant Computation
General Stages:
Preparation/
Encode
Verify
Computation of
Error Syndrome
Recovery
Explain better what each
block does, especially
verify
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Fault-Tolerant Computation[6]
Rules:
Implement gates that can process
encoded information.
Control propagation of errors.
Ensure that recovery from errors is
performed reliably.
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Fault-Tolerant Computation[6]
1st Law: Don’t use the same bit twice.
Bad: Error propagates, so
infection spreads.
Good!
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Fault-Tolerant Computation[6]
2nd Law: Copy/measure the errors, not the
data.
Copy the information from the data to the
ancilla.
2. Measure the ancilla to find an error
syndrome.
3. Based on the error syndrome, we perform
the required recovery.
1.
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Fault-Tolerant Computation[6]
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
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Fault-Tolerant Computation
4th Law: Repeat the operations
More explanation needed
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Fault-Tolerant Computation
5th Law: Use the right code
More explanation needed
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Error Correction in The Three Qubit Code
|0>|000>, |1>|111>
Error Correction:
More explanation needed
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Measurement(M1, M2)
Action
(0,0)
None
(1,0)
Flip the second bit
(0,1)
Flip the third bit
(1,1)
None
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Example: 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:
1
(|000  |111 )(|000  |111 )(|000  |111 )
8
1
|1| |1L 
(|000  |111 )(|000  |111 )(|000  |111 )
8
|0 |0 L 
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.
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Assumptions 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.
1
0

X 1 0 (bit flip error)
Z
(phase flip error)
0  i 

Y  i 0  = iXZ (combination of bit flip
and phase flip error)
1
 
0
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0 
 1

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Preparation in The Shor Code
Block #1
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Majority Logic in The Shor Code[3]
Explain decoding and recovery, how majority works, may be you
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need more
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Bit Flip Correction
Bit flip : switch |0> and |1>
0

Describe the error as bit flip matrix X 1
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.
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1
0

Phase Flip Correction
Example :
1
(|000  |111 )(|000  |111 )(|000  |111 )
8
1
|1L 
(|000  |111 )(|000  |111 )(|000  |111 )
8
|0 L 
Describe the error as phase flip matrix Z
Correction :
1
 
0
0 
 1

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.
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Simultaneous Bit and Phase Flip Error
0
 
i
 i
0 

Describe the error as Y=iXZ
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.
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Stabilizer Coding in The Shor Code[8]
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
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Stabilizer Coding in The Shor Code
In order to totally correct the code, we
must measure the eigenvalues of a total
of eight operators.
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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
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Another Phase Error Correction in
The Shor Code
1.
2.
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Hadamard Transformation on each qubit.
The qubits taken : 1, 4, and 7 (or 2,5,8 or 3,6,9).
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Stabilizer 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.
|00  |11 

|ψ>
2
X1X2|ψ> = |ψ> and Z1Z2|ψ> = |ψ>
|ψ> is stabilized by the operators X1X2 and
Z1Z2.
|ψ> is the unique quantum state which is
stabilized by these operators X1X2 and Z1X2.
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Stabilizer 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
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Stabilizer Code[4]
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|ψ> = |ψ> for all i and for all states
|ψ> in C.
Moreover, these stabilizer generators are all
mutually commuting.
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Stabilizer Code[4a]
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.
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Example:The Three Qubit Bit Flip Code
ZZI
IZZ
Error
Correction
Unitary
Action
+1
+1
None
None
None
-1
+1
XII
-1
-1
+1
-1
Bit 1
flipped
Bit 2
flipped
Bit 3
flipped
Flip bit
1
Flip bit
2
Flip bit
3
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IXI
IIX
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The Classical [7,4,3] Hamming Code
Transmit the block 0011
0
1 0 1
Parity bits = comes from the rule that the total number of 1’s
contained in each circle should be even.
1
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0 1
0
1
1
0
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Steane’s Code
One qubit is encoded into seven qubits.
1
(|0000000  |0001111  |0110011  |0111100  |1010101  |1011010  |1100110  |1101001 )
8
1
|1| |1L 
(|1111111  |1110000  |1001100  |1000011  |0101010  |0100101  |0011001  |0010110 )
8
|0 |0 L 
Logic 0= those with even number of 1’s
Logic 1= those with odd number of 1’s
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Encoder for the Steane’s Code
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Computation of Bit Flip Syndrome
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Computation of Phase Errors
R = Hadamard Rotation =
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1
2
1 1 
1  1


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Fault-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]
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One-Bit Teleportation
One-teleportation is based on Swap
gate
I do not understand . This is not
swap
Explain why we need one-bit
teleportation
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Several Facts to derive one-bit teleportation
Fact 1 : X = HZH where
0 1
1 0 
1 1 1 
X 
, Z  0  1, H 
1  1
1
0
2






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Several 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
U is performed if the
operation.
measurement result is 1.
Why it is so?
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Z-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|>
Why it is so?
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X-teleportation
Why it is so?
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Fault-Tolerant Toffoli Using One-bit
Teleportation
|0>
H
|0>
H
X
X
|0>
Z
|x >
|y >
|z + xy >
|x >
|y >
|z >
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H
Non-FT
Gate
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Fault-Tolerant Toffoli Gate
|0>
H
|0>
H
X
Z
|0>
Z
X
|x >
|y >
|z + xy >
|x >
|y >
|z >
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H
Mistake
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Alternative FT Toffoli Gate
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Anticommute & Commute
Anticommute : {A, B}=AB+BA=0
Commute : [A,B]=AB-BA=0
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.
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References
[1] Web resouce from http://www-2.cs.cmu.edu/afs/cs/project/pscicoguyb/realworld/www/
[2] Quantum Codes, Class slides from http://math.uwyo.edu/~moorhous/quantum
[3] Quantum Physics, abstract
quant-ph/0211071
From: Jumpei NIWA [view email] Date: Wed, 13 Nov 2002 08:43:46 GMT (584kb)
Simulating the Effects of Quantum Error-correction Schemes
Authors: Jumpei Niwa, Keiji Matsumoto, Hiroshi Imai
Comments: 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
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References
[9] Quantum Computation and
Quantum Information, M.A. Nielsen & I.
L. Chuang, 2000
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