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X Z X Z Z X X X Z X Z Z X X Z X Z X Z X X Z X Z Z X Z Z Z Z X Z X Z X
Z
X Z X Z X Z X
Recent Schemes for Increasing the Quantum Fault-tolerance Threshold
Ben Reichardt
Berkeley UL
X X Z Z
Recent Schemes for X X Increasing the Quantum Z Z - - PowerPoint PPT Presentation
Recent Schemes for X X Increasing the Quantum Z Z - - PowerPoint PPT Presentation
Recent Schemes for X X Increasing the Quantum Z Z Fault-tolerance U L Threshold Ben Reichardt Berkeley Z Z X X X X Z Z Z X X X X Z X X Z Z Z Z Z X X X X Z X X Z Z Z X X Z Z Z X X X Z Z Z Z
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[Von Neumann ‘56]
Classical fault-tolerance
Make fault-tolerant a circuit consisting of a universal set of operations, some faulty: 1 ,
,
Perfect op’s: Faulty op’s:
AND, NOT
1 1 1 1 Transverse gate application
r a n d
- m
p e r m u t a t i
- n
r a n d
- m
p e r m u t a t i
- n
1 1 1 Error correction
0L = 1L =
Encoding 1
1 1
fraction of 1’s
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Quantum bit (qubit)
Continuous quantum states
Classical bit
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- Errors inevitable
- Continuous quantum states
- Tradeoff between controllability and stability
- No cloning: local duplication undefined
Qu disadvantages Qu advantage!
- Can defer exposure of data to operations
?
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Universality result
– Theorem: Stabilizer operations plus the ability to prepare any pure state not a stabilizer state gives quantum universality.
Overview
Improved threshold result
– Modification of standard error correction scheme increases estimated threshold 3x, to almost 1%.
Knill’s threshold result
– Theorem: Threshold for erasure errors is for Bell measurements. – Estimated 5-10% threshold for independent depolarizing errors.
Postselection Teleportation Magic states distillation
[R ‘04] [R ‘04]
Defer exposure of data to operations
?
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0 / 1 Larger, fault-tolerant
U
, , ,
Quantum fault-tolerance problem
a b a a b
00 01 10 11 00 01 10 11
CNOT:
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Discrete errors
Duality: bit flips: phase flips:
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Quantum codes
- Classical codewords in the 0/1 basis
Correct bit flip X errors
- Classical codewords in the +/- basis
Correct phase flip Z errors
- E.g., quantum [[7,1,3]] code corrects arbitrary error on one qubit
Based on classical Hamming [7,4,3] code
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X
X X X
X
[Steane,…]
Standard fault-tolerance scheme
X Z a b a a b Def: CNOT data ancilla Fact 1: Fact 2: 1-a b 1-a 1-a+b
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. 5 . 1 . 1 5 . 2 . 2 5 . 5 . 1 . 1 5 . 2 . 2 5
Physical gate error rate Logical gate error rate
Threshold from concatenation
~p22 p23 p c p2 Probability
- f error
1 Physical bits per logical bit 7 72 73 O(log log N) concatenations O(log N) physical bits / logical
- N gate circuit
Want error 1/N 1/c
- [[7, 1, 3]] code only
corrects 1 error
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[R ‘04]
Universality result
– Theorem: Stabilizer operations plus the ability to prepare any pure state not a stabilizer state gives quantum universality.
Improved threshold result
– Modification of standard error correction scheme increases estimated threshold 3x, to almost 1%.
Knill’s threshold result
– Theorem: Threshold for erasure errors is for Bell measurements. – Estimated 5-10% threshold for independent depolarizing errors.
- Postselection
- Teleportation
- Magic states distillation
[R ‘04]
Overview
Defer exposure of data to operations
!
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X Z X Z X Z X Z X Z X Z Z X Z X Z X Z X Z X Z X Z X Z X Z X Z X Z X Z X error correction
d d e e t t e e c c t t i i
- n
n ! !
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Time to prepare encoded ancilla
5 10 15 20 25 .001 .002 .003 .004 .005 .006 .007 .008 .009 .010 Physical error rate Steane avg Reject avg
Logical ancilla errors
.000001 .00001 .0001 .001 .01 .1 .001 .002 .003 .004 .005 .006 .007 .008 .009 .010 Physical error rate Logical ancilla error rate Steane Z Steane X Reject Z Reject X
Effect on threshold
.0001 .001 .01 .1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 1 . 1 1 Physical error rate Crash rate Steane Reject 3/4
Effect of postselection in ancilla preparation
[R ‘04]
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Teleportation: Knill’s erasure threshold of
Theorem [Knill ‘03]: Threshold for erasure error is for Bell measurements. X Z
Teleportation
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Teleportation
Alice Bob
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Teleportation
+ Computation
U Alice Bob
Assume no errors!
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Teleportation: Knill’s erasure threshold of
Theorem [Knill ‘03]: Threshold for erasure error is for Bell measurements. [Knill ‘04]: Estimated threshold of 5-10%. X Z UL
X X Z Z + Computation + Fault-tolerance
U
Teleportation
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- Errors inevitable in quantum computers
- Fault-tolerance schemes can tolerate physically plausible error rates
Open questions
Efficient? Provable? UL
X X Z Z
?
[R ‘04] [Knill ‘04]
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