[PPT] - Fault-tolerant quantum computing with color codes Andrew J. Landahl PowerPoint Presentation

SLIDE 1

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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Fault-tolerant quantum computing with color codes

Andrew J. Landahl

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This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories.

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with Jonas T. Anderson and Patrick R. Rice. arXiv:1108.5738

SLIDE 2

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

Photos placed in horizontal posi1on with even amount of white space between photos and header

Fault-tolerant quantum computing with color codes

Andrew J. Landahl

12/8/11

QEC11: Second Interna1onal Conference on Quantum Error Correc1on

This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories.

                                                       

  

with Jonas T. Anderson and Patrick R. Rice. arXiv:1108.5738

SLIDE 3

Color codes

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 3 The three semiregular 2D topological color codes 4.8.8 6.6.6 4.6.12 3.4.6.4 The 2D topological subsystem color code S is transversal Fewest qubits/distance S is transversal Two‐body checks suffice Planar color codes: 3m corners These codes are naturally suited to 2D quantum technologies in which long‐distance quantum transport is imprac1cal. The 2D surface code has many promising features for fault‐tolerant quantum compu1ng, including a high accuracy threshold and no need for syndrome ancilla dis4lla4on. How do 2D color codes compare? Checks: Xf, Zf

[Bombin & Mar1n‐Delgado, PRL 97, 180501 (2006)] [Bombin, PRA 80, 032301 (2010)]

SLIDE 4

Noise model:

Standard assumpDons: No leakage, reliable classical computa1on.

1. Circuit‐based noise model

Each prepara1on and one‐qubit gate followed by BP(p).
Each CNOT gate followed by DP(p).
Each measurement preceded by BP(p) and result flipped with probability p.

2. Phenomenological noise model

Same, except each syndrome‐bit extrac1on circuit modeled “phenomenologically” as a measurement that

fails with probability p; ignores noise propaga1on between data and ancilla. Gates only appear in encoded computa1on. 3. Code‐capacity noise model

Same as phenomenological model, except syndrome measurements are perfect.

Control model:

(Faulty) gate basis:
Standard assumpDons: Parallel opera1on, refreshable ancilla, fast classical computa1on, equal‐1me gates.
Locality assumpDons: 2D layout, local quantum processing.

Control & noise models

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 4 BP channel: Bit‐flip channel B(p) followed by phase‐flip channel Φ(p). DP channel: Applies each two‐qubit (“double Pauli”) product with probability p/16. {I, X, Z, H, S, S†, CNOT, MZ, MX, |0, |+, |π/4}

SLIDE 5

Decoders & thresholds

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 5 OpDmal decoder: Returns recovery most likely to succeed given the syndrome. MLE decoder: Returns most likely error that occurred given the syndrome.

[A: Sarvepalli & Raussendorf, arXiv:1111.0831] [B: Fowler, Whiteside, and Hollenberg, arXiv:1110.5133] [See our paper 1108.5738 for other references.]

7.8% [A] 0.9% [B]

SLIDE 6

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 6

Syndrome extraction

                                                       

                                                                               

   

XZ sequenDal schedule XZ interleaved schedule

|0

MZ
|+ •
MX
Example of error propagaDon
X error on X‐check bit (red circle) between 1me

steps 5 and 6.

Propagates to 3 data errors; detected correctly by

3 Z‐check bits (yellow)

SLIDE 7

Decoding

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 7 Code‐capacity MLE decoder: (Works for all CSS codes.) OpDmizaDon problem Integer program over GF(2) Integer program over the reals Phenomenological MLE decoder: (Works for all CSS codes.) min cT y sto Ay = z y ∈ Bn z := s + 2t1 + 4t2 + 8t3 y := (xT , tT

1 , tT 2 , tT 3 )T

c :=

1T , 0T , 0T , 0T T

A := (H| − 2I| − 4I| − 8I) Measurement error Data error Integer program over the reals ∆z := ∆s + 2t1 + 4t2 + 8t3 y := (xT

data, xT synd, tT 1 , tT 2 , tT 3 )T

c :=

1T , 1T , 0T , 0T , 0T T

A :=      H H ... H I I I ... I I −2I −4I −8I      . min 1T x sto Hx = s mod 2 x ∈ Bn min

v

xv sto

v∈f

xv = sf ∀f xv ∈ B := {0, 1} min cT y sto Ay = ∆z y ∈ Bn

SLIDE 8

Code-capacity threshold

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 8 Exact curves found up to d = 7. pfail =

failing patterns E

p|E|(1 − p)n−|E|, p(est)

fail

= Nfail N (σ2

fail)(est) =

p(est)

fail

1 − p(est)

fail

N
Theory:
Fit:

ξ ∼ |p − pc|−ν0 pfail = (p − pc)d1/ν0 pfail = A + B(p − pc)d1/ν0 Threshold by scaling Ansatz fit, not curve crossing esDmate. Monte‐Carlo esDmate for d = 9.

[Wang, Harrington, & Preskill,

Ann. Phys. 303, 31 (2003)]

pth = 10.56(1)%

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.05 0.1 0.15 0.2

PFailure PError

d=3 d=5 d=7 d=9 0.125 0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.101 0.103 0.105 0.107 0.109 0.111 (a) 0.141 0.1412 0.1414 0.1416 0.1418 0.142 0.1422 0.1054 0.1055 0.1056 0.1057 (b)

N.B. Finite size effects may ma:er.

SLIDE 9

Phenomenological threshold

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 9

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.024 0.026 0.028 0.03 0.032 0.034 0.036

PFailure PError

d=5 d=7 d=9 0.05 0.1 0.15 0.2 0.02 0.025 0.03 0.035 0.04

PFailure PError

d=5 d=7 d=9

pth = 3.05(4)%

Algorithm 1 : pfail(p) by Monte Carlo

1: nfaces ← 1

4(d + 1)2 − 1.

2: for i = 1 to N do 3:

// Generate data and syndrome errors for d time slices.

4:

for t = 1 to d do

5:

for j = 1 to n do

6:

E[t, j] ← 1 with probability p. // Data errors.

7:

end for

8:

for j = n + 1 to n + 1 + nfaces do

9:

E[t, j] ← 1 with probability p. // Synd. errors.

10:

end for

11:

end for

12:

Emin ← Decode(Syndrome(E)). // 3D error volume.

13:

E′ ← L

t E[t] ⊕ Emin[t]. // 2D error plane.

14:

E′

min ← Decode(Syndrome(E′)). // Ideal decoding.

15:

if (L

i E′[i] ⊕ E′ min[i] = 1) then

16:

Nfail ← Nfail + 1.

17:

end if

18: end for 19: return p(est)

fail

= Nfail/N.

SLIDE 10

Circuit threshold

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 10

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

PFailure PError

d=5 d=7 d=9 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

PFailure PError

d=5 d=7 d=9 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.0006 0.0008 0.001 0.0012 0.0014

p(XZ interleaved)

th

= 8.2(3) × 10−4 p(XZ sequential)

th

= 8.0(4) × 10−4

Circuit generates correlated errors up to weight four (“hooks”).
Can accommodate by modifying fi^ng funcDon:

pfail = A + B(p − pc)d1/ν0 + C(p − pc)2d2/ν0.

SLIDE 11

Self-avoiding walk bound

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 11

[Dennis, Kitaev, Landahl, & Preskill, JMP 43, 4452 (2002)]

Code‐capacity noise model

pfail =

O∈log ops

p(O ∈ E + Emin) =

O∈string-net

p(O ∈ E + Emin) +

O∈string

p(O ∈ E + Emin) ≤ 2

O∈SAW(≥ d)

p(O ∈ E + Emin) ≤ 2 · n

L≥d

nSAW(L) · 2L · [p(1 − p)]L/2

# star1ng points for SAW # ways to choose flips

n E + Emin
Prob. flips at

those loca1ons

µ4.8.8 ≈ 1.808 830 01(6) 1.804 596 ≤ µ4.8.8 ≤ 1.829 254 ∆max = 10 nSAW(L) ≤ P(L) µL pth(1 − pth) ≤ 1 4µ2

Phenomenological noise model

nSAW(L) ≤ ∆max(∆max − 1)L−1 pth ≥ 0.502 5% pth ≥ 8.335 745(1)%

cf. 10.56(1)%
cf. 3.05(4)%

SLIDE 12

Architectures & computation

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 12

     

Transversal

Push noise channels through gates to get effec1ve noise channels for FTQEC
Magic states for T gate injected by teleporta1on.

pI = pX = pZ = p|0 = p|+ = 3 2pCNOT = 3 2p(nondestructive)

MX

= 3 2p(nondestructive)

MZ

= 2pH = 2p(bit-flip)

S

= 2p(bit-flip)

T

= 3p(phase-flip)

S

= 3p(bit-flip)

T

= 8.2(3) × 10−4 p(destructive)

MX

= p(destructive)

MZ

= 10.56(1)% p|π/4 = ( √ 2 − 1)/2 √ 2 ≈ 14.6% Code deformaDon

Local ops to grow, shrink, & move “defects.”
H, S, CNOT gate by code deforma1on.
Magic states for T gate injected by code deforma1on.

|M

MX
/

H

Unencode
MZ
/
X

Z

M

SLIDE 13

Summary

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 13 Color codes vs. surface codes

Code‐capacity & phenomenological thresholds comparable to surface codes. (11%, 3%)
Circuit‐model threshold about 10 1mes smaller than surface‐code’s.
Rigorous bounds on threshold very weak at this point.
Qubit overhead comparable to surface codes.

Open quesDons:

TSCCs: Do they have beper thresholds?
Leakage errors: How well are they tolerated?
Efficient decoders: TCC = 2 surface codes‐‐‐run efficient matching on these. Threshold?
Rigorous lower bounds: Tighter analysis techniques?
Magic‐state injec1on: How much of 14.6% threshold is consumed by imperfect injec1on?
Topological quantum computa1on: How to formulate a color‐code quantum double model?