quantum error correction joshua kretchmer gautam wilkins eric zhou
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Quantum Error Correction
Joshua Kretchmer
Gautam Wilkins
Eric Zhou
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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
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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
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• Bit flip channel: bit is flipped with prob. = p < 1/2• Encoding:
• 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
Classical 3-Bit Code: Bit Flip Error
b b0 b0 b
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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
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Quantum 3-Bit Code: Bit Flip Error
• 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
|>=a|0>+b|1>|0>|0>
M
MX
|>
|0>
|0>Encoding Error
ChannelDiagnose and Correct
Decode
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• After channel 8 possible resultsState: Probability:
a|000>+b|111> (1-p)3
a|100>+b|011> p(1-p)2
a|010>+b|101> p(1-p)2
a|001>+b|110> p(1-p)2
a|110>+b|001> p2(1-p)
a|101>+b|010> p2(1-p)
a|011>+b|100> p2(1-p)
a|111>+b|000> p3
Quantum 3-Bit Code: Bit Flip Error
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Quantum 3-Bit Code: Bit Flip Error
• After CNOT’s 4 possible resultsState: Probability:
a|000>+b|111>|00> (1-p)3
a|100>+b|011>|10> p(1-p)2
a|010>+b|101>|01> p(1-p)2
a|001>+b|110>|11> p(1-p)2
a|110>+b|001>|01> p2(1-p)
a|101>+b|010>|10> p2(1-p)
a|011>+b|100>|11> p2(1-p)
a|111>+b|000>|00> p3
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• 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
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• 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: Bit Flip Error
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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
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• Apply H to each qubit at either end of the channel
• HIH = HH = I; HzH = x
HPH = cos()I +isin()x
• Same result from bit flip code– Fidelity = 1 - 3p2
– p = <sin2()> (2)2/3 for <<1
Quantum 3-Bit Code: Phase Error
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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>
01|0>|E1> + 02|1>|E2> +
13|1>|E3> + 14|0>|E4>
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• (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
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• (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
General Quantum Error
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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
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Shor’s Algorithm
• Each qubit is encoded as nine qubits
( )( )( )
( )( )( )11100011100011100022
11
11100011100011100022
10
−−−→
+++→
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Shor’s Algorithm
Assume decoherence on first bit of first triple, becomes:
( ) ( )[ ]
( )( )
( )( )
( )( )
( )( )01110022
1
01110022
1
11100022
1
11100022
1
111000102
1
21
21
30
30
3210
−++
+++
−−+
++
=
+++
αα
αα
αα
αα
αααα
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Shor’s Algorithm
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( ) ( )[ ]
( )( )
( )( )
( )( )
( )( )01110022
1
01110022
1
11100022
1
11100022
1
111000102
1
21
21
30
30
3210
−++
+++
−−+
++
=
+++
αα
αα
αα
αα
αααα
Shor’s Algorithm
• No error
• Z error
• X error
• ZX = Y error
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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)836p2
• Therefore probability that 9*k qubits can be decoded is (1-36p2)k
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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
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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
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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
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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
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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
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Steane’s Algorithm
• Let |a and |b be expressed as [7, 3, 4] in basis 1:
00101101000011010010111100000011001
100110001010101111111
11010010111100101101000011111100110
011001110101010000000
+++++
++=
+++++
++=
b
a
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
0000111
1111000
1100110
1010101
,
1111000
1100110
1010101
GH
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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
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Implications for Physical Realizations of Quantum
Computers
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Why Do We Need It?
• Quantum computers are very delicate.
• External interactions result in decoherence and introduction of errors.
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Fault-tolerance
• Especially important when considering physical implementations.
• Must consider errors introduced by all parts, including gates.
• Incorrect syndromes introduce errors.
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Impact on Physical Systems
• Increased size
• Level of coherence determines increase
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Alternative to Error Correction
• Topological Quantum Computing
• Involves particles called anyons that form braids, whose topology determines quantum state.
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Topological Quantum Computing
Slight perturbations to system cause braids to be deformed, but only large disturbances result in them being cut or joined.
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Summary
• Error correction is vital for physical realizations of trapped particle quantum computers.
• Allows reliable quantum computation without requiring extremely high levels of coherence.