a theory of single-shot error correction for …a theory of single-shot error correction for...
TRANSCRIPT
![Page 1: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/1.jpg)
Earl CampbellEarl Campbell
A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE
ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY
Sheff ield
![Page 2: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/2.jpg)
Earl Campbell
Sheffield my group works on: Quantum error correction Magic state distillation Circuit compilation Resource estimation Classical simulation methods Magic theory
![Page 3: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/3.jpg)
Earl Campbell
“Old” South Wales my place of birth!
![Page 4: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/4.jpg)
Earl Campbell
what?
Deal with noisy measurements without measurement repetition.
Fault tolerance in a single shot Bombin Phys. Rev. X 5, 031043 2015 / QIP2015
why?
faster: no need for d repeated measurements
more reliable: handles time correlated noise / fabrication faults Bombin Phys. Rev. X 6, 041034 (2016) / QIP2017
MOTIVATION
![Page 5: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/5.jpg)
Earl Campbell
4D topological codes
3D gauge codes
Topological single shot codes
Trade off bounds Bravyi-Poulin-Terhal PRL (2010) e.g. in 2D constrained by kd
2 = O(n)
MOTIVATION
![Page 6: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/6.jpg)
Earl Campbell
4D topological codes
3D gauge codes
Topological single shot codes
All single shot codes
????? Codes with
better [[n,k,d]] parameters
????
MOTIVATION
![Page 7: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/7.jpg)
Earl CampbellRESULTS
Key results
Given any classical code use homological product
to construct quantum “single shot” code
classical LDPC -> quantum LDPC
+ 4 mini-results on foundations of single-shot error correction
![Page 8: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/8.jpg)
Earl Campbell
Review single-shot
3 mini-results
OUTLINE
1
2
3 The homological product codes
![Page 9: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/9.jpg)
Earl Campbell
Noisy measurement problem in Toric code problem: two measurement errors
looks same as long qubit error
point-like syndromes
string-like errors
![Page 10: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/10.jpg)
Earl Campbell
Solution for toric code: repeat measurements
space
space
time
2D Toric code: with history of measurement date leads to 3D space-time.
Perform minimum weight decoding in space-time picture.
2D cross section from [Dennis Kitaev Preskill ’01]
![Page 11: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/11.jpg)
Earl Campbell
Single shot: the basic ideaMeasure code stabilisers once and try to infer error,
but measurement outcomes are noisy!
Possible for 3D gauge colour code or 4D topological code (e.g. 4D toric code)
[Kubica, Preskill arXiv:1809.10145] [Breuckmann PhD thesis ’18] [Breuckmann, Duivenvoorden, Michels, Terhal QIC ’17] [Brown, Nickerson, Browne, Nat Comms ’16] [Pastawski, Clemente, Circa PRA ’11] [Dennis Kitaev Preskill J. Math. Phys. ’01]
See e.g.
![Page 12: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/12.jpg)
Earl Campbell
A bitParity check measure parity of connected bits e.g. ZZ
ISING MODEL: CLASSICAL WARM UP
Classical single-shot: possible in 2D! warm up example
![Page 13: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/13.jpg)
Earl Campbell
A bitParity check measure parity of connected bits e.g. ZZ
X A physical error
−1 “-1” measurement result
X
−1
−1 −1
−1
ISING MODEL: CLASSICAL WARM UP
![Page 14: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/14.jpg)
Earl Campbell
A bitParity check measure parity of connected bits e.g. ZZ
X A physical error
−1 “-1” measurement result
Meta-check: “a check on checks” calculates parity of all connected checks
−1 “-1” meta-check result
X
−1
−1
−1
−1
ISING MODEL: CLASSICAL WARM UP
![Page 15: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/15.jpg)
Earl Campbell
A bitParity check measure parity of connected bits e.g. ZZ
X A physical error
−1 “-1” measurement result
Meta-check: “a check on checks” calculates parity of all connected checks
−1 “-1” meta-check result
X
−1
−1
−1 −1
−1
Measurement error!!! should be “-1” but experiment reports “+1”
ISING MODEL: CLASSICAL WARM UP
![Page 16: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/16.jpg)
Earl Campbell
Consider a set of Pauli checks that stabiliser codespace:
M = {M1,M2, . . .}
allow to be overcomplete
A redundancy is a sets of check multiplying to identity
M
M1M2M3 = Ie.g.
Leads to consistency condition on parity checks. Metacheck are a subset of redundancies that we choose to enforce.
Formal definition: Redundancy & metacheck:
![Page 17: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/17.jpg)
Earl Campbell
Two stage decoder
Stage 1: Correct measurement error Look at metachecks, find low weight measurement correction.
Stage 2: Correct physical bits Look at repaired checks find low weight physical correction.
X
−1
−1
−1 −1
−1
Measurement error!!! should be “-1” but experiment reports “+1”
ISING MODEL: CLASSICAL WARM UP
![Page 18: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/18.jpg)
Earl Campbell
Two stage decoder
Stage 1: Correct measurement error Look at metachecks, find low weight measurement correction.
Stage 2: Correct physical bits Look at repaired checks find low weight physical correction.
−1
−1
−1 −1
−1
Same signature caused by triple measurement error and no physical error Correction leads to a residual error
ISING MODEL: CLASSICAL WARM UP
![Page 19: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/19.jpg)
Earl Campbell
Area-law / soundness property
string-like syndromesforming trivial cycles
area-like errors
string-like syndromesforming trivial cycles
area-like errors E E
boundary size = syndrome weight
then
= x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
wt[E?] f(x) = x2/2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
![Page 20: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/20.jpg)
Earl Campbell
Adversarial quantum error correctionError model
x = num. of of measurement errors y= num. of single-qubit Pauli errors
Standard QEC
if x=0 and y < d/2 then
can correct perfectly Residual error weight = 0
Single-shot QEC
if x, y < some upperbound then
there is a decoder such that Residual error weight <= f(x)
where f is some function s.t. f(0)=0
![Page 21: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/21.jpg)
Earl Campbell
Adversarial quantum error correction
Good family of SS codes
a family of n-qubit codes has
good single shot QEC if
1) x,y bounds grow
2)
Single-shot QEC
⌦(na)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
f 2 poly(x)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
if x, y < some upperbound then
there is a decoder such that Residual error weight <= f(x)
where f is some function s.t. f(0)=0
Error model x = num. of of measurement errors y= num. of single-qubit Pauli errors
![Page 22: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/22.jpg)
Earl Campbell
Area-law / soundness property
Given any consistent syndrome of weight = x t<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
some constant
A code & check set is sound if (t, f)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
There exists an Pauli correction of weight
A code & check family has good soundness if
f(x) 2 poly(x)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
All members are where grows as (tn, f)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
tn<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌦(na)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Mini result 1: Good soundness good single-shot EC proof uses two-stage decoder
�!
![Page 23: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/23.jpg)
Earl Campbell
Toric code problem: has poor soundness
syndrome weight
then
point-like syndromes
string-like errors
wt[E] f(2) =?unbounded?<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= 2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
![Page 24: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/24.jpg)
Earl Campbell
soundness vs. energy barrier
![Page 25: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/25.jpg)
Earl Campbell
Is soundness just the same thing as a large energy barrier?
![Page 26: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/26.jpg)
Earl Campbell
Energy barrier for one check set
Consider walks from one logical state to an orthogonal logical state, A walk is made of small steps
|0Li |1Li
| j+1i = Pj | ji
1 2
3
4
The energy of a state is the number of violated stabiliser checks. The energy penalty of a walk is the “peak” energy during the walk.
The energy barrier of a check set is the min energy penalty over all walks.
0 5
a check family of n-qubit codes has a
Macroscopic Energy barrierif the energy barrier grows as ⌦(nc)
![Page 27: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/27.jpg)
Earl Campbell
LDPC + good soundness macroscopic energy barrier
LDPC + macroscopic energy
barriergood soundness
????
![Page 28: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/28.jpg)
Earl Campbell
No 2D topological code can have a macroscopic energy barrier, [Bravyi, Terhal NJP ‘09 ]
![Page 29: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/29.jpg)
Earl Campbell
Choice of check set
![Page 30: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/30.jpg)
Earl Campbell
Consider a set of Pauli checks that stabilise the codespace: M = {M1,M2, . . .}
Recall we:
Soundness is a property of the set of stabiliser checks. sooo….
We can ask: “Given a code family does these exist a check-family with
good soundness?”
![Page 31: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/31.jpg)
Earl Campbell
Thm: for all codes checks w/ good soundness!9
Proof sketch.
S1 = X1 Z2 I3 Z4 . . .S2 = Z1 X2 I3 Z4 . . .S3 = Z1 I2 X3 Z4 . . .
For every code we can find stabiliser generators that upto local Cliffords are of “diagonalised” form
Then every 1-bit syndrome can be corrected by a weight 1 error
ZjSjZj = �Sj
So residual error is less than syndrome, so f(x) = x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
![Page 32: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/32.jpg)
Earl Campbell
Mini result 1: Good soundness good single-shot EC
Mini result 2*: Good soundness + LDPC + distance macroscopic energy barrier
Mini result 3**: 2D topological code + distance bad soundness
Mini result 4: for all codes checks w/ good soundness!
�!
�!
* similar result (w/ stronger soundness def) is used in study of PCP the D. Aharonov and L. Eldar, SIAM Journal on Computing 44, 1230 (2015).
⌦(na)
** corollary following from S. Bravyi and B. Terhal,
⌦(na)�!
9
![Page 33: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/33.jpg)
Earl Campbell
Moral:
Easy to find codes/checks with good soundness
Tricky to find LDPC codes/checks with good soundness
![Page 34: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/34.jpg)
Earl Campbell
1 big result: homological
product constructions
![Page 35: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/35.jpg)
Earl CampbellTHE HYPER GRAPH / HOMOLOGICAL PRODUCT
Take two Tanner graphs representing classical code
Blue code
Pink code
Squares: bits
Circles: parity checks
Squares: bits
Circles: parity checks⇥H
Essentially same as hyper graph product: [Tillich & Zemor IEEE ’09]
Warmup: Revise hyper graph product
![Page 36: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/36.jpg)
Earl CampbellTHE HYPER GRAPH / HOMOLOGICAL PRODUCT
Clone the codes
![Page 37: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/37.jpg)
Earl CampbellTHE HYPER GRAPH / HOMOLOGICAL PRODUCT
Superimpose codes
![Page 38: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/38.jpg)
Earl CampbellTHE HYPER GRAPH / HOMOLOGICAL PRODUCT
Superimpose codes
= +
= += +
+=
where
![Page 39: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/39.jpg)
Earl CampbellTHE HYPER GRAPH / HOMOLOGICAL PRODUCT
Relabel vertices
= +
= += +
+=
where
→→→→ X
Z checkschecks
qubits
→
Finished! A quantum code
![Page 40: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/40.jpg)
Earl Campbell
⇥H =
Always yields valid quantum code for any pair of initial classical codes
Unlike CCS construction where duality is required.
![Page 41: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/41.jpg)
Earl Campbell
Why valid quantum code: because a chain complex!
C0
C1
δ0
Bipartite graph
[�0]i,j =
(1 if vertex i in C0 is adjacent to vertex j in C1
0 otherwise.
Tripartite graph
[�0]i,j =
(1 if vertex i in C0 is adjacent to vertex j in C1
0 otherwise.
C0
C1
C2
δ0
δ1
[�1]i,j =
(1 if vertex i in C1 is adjacent to vertex j in C2
0 otherwise.
We say such a graph forms a chain complex if and only if
for all i�i�i�1 = 0
![Page 42: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/42.jpg)
Earl Campbell
C0
C1
δ0
hypergraph product
�̃0 =
✓I⌦ �T0�0 ⌦ I
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
�̃1 =��0 ⌦ I I⌦ �T0
�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
�̃1�̃0 = 2�0 ⌦ �T0 = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
since
the new object is indeed a chain complex!
![Page 43: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/43.jpg)
Earl Campbell
If the graph has or more parts then we can define a quantum code
commutative
We say such a graph forms a chain complex if and only if
for all i�i�i�1 = 03
Cj�1 =
Cj =
Cj+1 =
the qubits
the X checks
the Z checks
HX = �j
HZ = �Tj�1
HXHTZ = �j(�
Tj�1)
T = �j�j�1 = 0
![Page 44: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/44.jpg)
Earl Campbell
Double homological product
⇥H ⇥H ⇥H
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
✓
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
✓
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
=<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>chain complex graph with 5 levels of structure!
give a quantum code with metachecks!
�̆1�̆0 = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
�̆2�̆1 = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
. . .<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
classical LDPC -> quantum LDPC
![Page 45: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/45.jpg)
Earl Campbell
If the graph has or more parts then we can define a quantum code
Cj�1 =
Cj =
Cj+1 =
the qubits
the X checks
the Z checks
HX = �j
HZ = �Tj�1
We say such a graph forms a chain complex if and only if
for all i�i�i�1 = 05
Cj�2 =
Cj+2 = the X metachecks
the Z metachecks
�i�i�1 = 0 entails that valid metacheck codes!
�j+1
�Tj�2
![Page 46: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/46.jpg)
Earl Campbell
Given any classical code with
n =k =d =
no. physical bits no. logical bits distance
Double homological
product nQ =kQ =dQ =
no. physical qubits no. logical bits distance
where…nQ = n
4 + 4n2(n� k)2 + (n� k)4 ⇠ O(n4)
kQ = k4
dQ = d2
Distance recently improved from bound, by Zeng and Pryadko arXiv:1810.01519
dQ � d
A quantum code
![Page 47: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/47.jpg)
Earl Campbell
Given any classical code with
n =k =d =
no. physical bits no. logical bits distance
Double homological
product nQ =kQ =dQ =
no. physical qubits no. logical bits distance
A quantum code
where…nQ = n
4 + 4n2(n� k)2 + (n� k)4 ⇠ O(n4)
kQ = k4
dQ = d2
Theorem: Furthermore, the above code is always (d-1, f) sound with or better!f(x) = x3/4
![Page 48: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/48.jpg)
Earl Campbell
Any classical code
Step one: First homological product
A quantum code with meta checks & good soundness
Step two: Second homological product
A quantum code without metachecks or a classical code with metachecks
⌘or
![Page 49: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/49.jpg)
Earl Campbell
Any classical code
Step one: First homological product
A quantum code without metachecks or a classical code with metachecks
⌘or
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
![Page 50: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/50.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
Initial code with parity check matrix Hypergraph classical codes with bits arranged in 2D:
checks of applied to every row
[n, k, d]<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
H<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
H<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
checks of applied to every columnHT
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
![Page 51: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/51.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
Consider some error pattern: which we then minimise weight of!
![Page 52: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/52.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
01110 is codeword!
Look for codewords for original code in rows/columns
![Page 53: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/53.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
01110 is codeword!
01110 is codeword!
Look for codewords for original code in rows/columns
![Page 54: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/54.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
01110 is codeword!
01110 is codeword!
Look for codewords for original code in rows/columns
confirm intersection is red
![Page 55: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/55.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
Form “product of code words”
has trivial syndrome
01110⇥ 01110
![Page 56: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/56.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
Add “product of code words”
reduced weight with same syndrome
01110⇥ 01110
+ =
![Page 57: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/57.jpg)
Earl Campbell
Lemma: Furthermore, the new classical code is always (d, f) sound with or better!f(x) = x2/4
Proof Sketch. by explicit decoder design
Repeat until
�
�
Not codewords r = no. of rows
Every row/column that is not a codeword leads to violated check
Not codewords c = no. of columns
Therefore, syndrome weight
but wt(E) cr
combined wt(E) x2/4
x � c+ r
![Page 58: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/58.jpg)
Earl Campbell
Redundancy, overheads
& talking heads
![Page 59: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/59.jpg)
Earl Campbell
⌫ =<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Quantify redundancynumber of checks measurements
num. stabiliser generators
no redundancy⌫ = 1 !<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
so
Analysis of double homological product shows…. ⌫ < 2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
“Interesting” examples of single-shot have : some constant ⌫ <latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Formalism allows “trivial/uninteresting” single shot achieved using repeated measurements, but then ⌫ ⇠ d
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
C0
C1
C2δ1
δ0bits
checks
metacheckstrivial
example
![Page 60: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/60.jpg)
Earl Campbell
space
space
space
Raussendorf 3D cluster state e.g. [Raussendorf, et al NJP ’07]
Like 2D toric code, but 3 “space” dimensions.
Prof D Browne
Surely “single-shot” just means you measure every qubit once. And you do this in the 3D Raussendorf cluster state model
Fiolated codes: extends idea to other CCS codes [Bolt, Duclos-Cianci, Poulin, Stace, PRL ’16 ]
Basically repeated teleportation
![Page 61: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/61.jpg)
Earl Campbell
Prof D Browne
Surely “single-shot” just means you measure every qubit once. And you do this in the 3D Raussendorf cluster state model
Hmmm…. maybe, maybe not…
But you have to pay an extra factor d in space cost in lieu of a factor d in measurement redundancy.
So if permitted, surely they would be uninteresting examples
![Page 62: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/62.jpg)
Earl Campbell
Prof D Gottesman (at QIP2018)
But, in the Steane model of error correction you don’t need repeated measurements, so isn’t that single shot!
Refresher on Steane (a.k.a oold skool) EC: Don’t measure checks individually, instead teleport through an ancilla which you prepare offline to very high fidelity
|+Li<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
![Page 63: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/63.jpg)
Earl Campbell
Prof D Gottesman (at QIP2018)
But, in the Steane model of error correction you don’t need repeated measurements, so isn’t that single shot!
Hmmm…. this sounds a lot like foliated codes again.
This offline state preparation doesn’t this normally use a factor d of extra resource.
Surely this is “uninteresting” again
![Page 64: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/64.jpg)
Earl Campbell
What matters is resource cost
n⌫<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
redundancy in measurementsnumber of qubits including ancilla used
(e.g. foliation / Steane EC cost)
but we also want lots of logical qubits…. son⌫
k<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
number of logical qubits
![Page 65: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/65.jpg)
Earl Campbell
How does this scale with the code distance
nc
kc= O(1)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
for double homological product codes
where [nc, kc, dc]<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
are initial
classical parameters
there are classical codes where
so possible to achieve
n⌫
k⇠ n4
c
k4c<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
n⌫
k= O(1)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
constant overhead
n⌫
k= O(d3)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
for 4D Toric code:
n⌫
k= O(d2)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
k = O(1), n = O(d2), ⌫ = O(d)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
k = O(1), n = O(d2), ⌫ = O(1)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
for 2D Toric code:
![Page 66: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/66.jpg)
Earl Campbell
Future work
![Page 67: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/67.jpg)
Earl Campbell
Closing remarks
Constant overhead quantum fault-tolerance with quantum expander codesAuthors: Omar Fawzi, Antoine Grospellier, Anthony Leverrier
Soundness is sufficient condition for single-shot error correction
Uses single-shot codes with bad soundness!
Need to revisit problem to include also these examples of single-shot.
![Page 68: A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR …A THEORY OF SINGLE-SHOT ERROR CORRECTION FOR ADVERSARIAL NOISE ARXIV:1805.09271 ACCEPTED TO QUANTUM SCIENCE & TECHNOLOGY Sheffield](https://reader030.vdocument.in/reader030/viewer/2022040509/5e4f1b5121d3a22a7b7cfd15/html5/thumbnails/68.jpg)
THANK YOU!
reference more papers,give examples of single shot codes/decoders,discuss redundancy,