Quantum error correction thresholds for the universal

Quantum error correction thresholds for the universalFibonacci Turaev-Viro code

arXiv:2012.04610

Alexis Schotte 1 Guanyu Zhu 2 Lander Burgelman 1 Frank Verstraete 1

1 Ghent University

2 IBM Quantum

Outline

Non-Abelian topological quantum error correction

The extended string-net code

Classical simulation

Summary & Outlook

Alexis Schotte arXiv:2012.04610 Non-Abelian topological quantum error correction 2 / 25

Topological quantum computationUse anyons to perform fault-tolerant quantum computation [Kitaev ’97]

Passive TQC Majorana wires, FQH states, . . .

I Information encoded in groundspace of physical systemI Protected by energy gap at T = 0I Need error correction for thermal noiseI non-Abelian anyons are possible

Active QEC surface code, color code, . . .

I Information encoded in codespace of system of qubitsI Actively measure stabilizersI Error correction when stabilizers are violated


Active Non-Abelian quantum error correctionI Codespace of system of qubitsI Actively measure set of commuting projectorsI Noise → Non-abelian anyons

Our goals: 1. Construction of a non-Abelian code of qubits2. Classical simulation of this code

Necessary ingredients:

QEC

TQFT TN

You are here


Abelian vs non-Abelian codes

Abelian Non-Abelian

Low-weight local stabilizer codesWell studiedHigh thresholds

Native universal gate set in 2DBeyond stabilizer codes

No native non-Clifford gate in 2DI Magic state distillationI Code-switching to higher

dimensional codes→ high overhead

ComplicatedHard to simulate classically


Previous work

I König, Kuperberg, Reichardt: Turaev-Viro codesarXiv:1002.2816

I Bonesteel, DiVincenzo: Measurement circuits for Levin-Wen OperatorsarXiv:1206.6048

I Feng, Bonesteel: Fixing violated vertices in Fibonacci Levin-Wen codePhD Thesis W. Feng

I Burton, Brell, Flammia: Threshold for phenomenological model of Fib anyonsarXiv:1506.03815


Outline




Summary & Outlook

Alexis Schotte arXiv:2012.04610 The extended string-net code 7 / 25

The Levin-Wen model

HΛ = −∑

v

Qv −∑

p

Bp

Qv

∣∣i jk⟩

= δijk

∣∣i jk⟩

Bp = 1D2

∑s

dsOsp

v

p

Defined by a Unitary Fusion Category C:I string types {1, i2, . . . , iN} → qudit states |0〉 , |1〉 , . . . , |N〉I branching rules δijk → determine Qv

I numerical data {di, F} → determine Bp


The Levin-Wen model

HΛ = −∑

v

Qv −∑

p

Bp

Qv

∣∣i jk⟩

= δijk

∣∣i jk⟩

Bp = 1D2

∑s

dsOsp

v

p

surface code: C = Z2{1, e} e× e = 1

C = FIB{1, τ} τ × τ = 1 + τ


The Levin-Wen model

HΛ = −∑

v

Qv −∑

p

Bp

Qv

∣∣i jk⟩

= δijk

∣∣i jk⟩

Bp = 1D2

∑s

dsOsp

v

p

a

b

e

c

d=∑n

Fabecdf

a df

cb

=1φ

+1√φ

=1√φ

−1φ


The extended Levin-Wen model

HΛ = −∑

v

Qv −∑

p

Bp

Qv

∣∣i jk⟩

= δijk

∣∣i jk⟩

Bp = 1D2

∑s

dsOsp

v

p

I Localized excitations follow DFIB anyonic statistics: {11, 1τ, τ1, ττ}

I Inside Hs.n. = {|ψ〉 |∏

v Qv |ψ〉 = |ψ〉}, the anyon charge of a plaquette is awell-defined quantum-number.


The tube algebra

Oxαyβ

∣∣∣∣m1

m2

m3

jr

j1j2

mr

jr+1

mr−1

j3

x′⟩

= δx,x′

∣∣∣∣m1

m2

m3

jr

j1j2

mr

jr+1

mr−1

j3

x yα

β

⟩

= δx,x′vαvβ

vy

∑k1,...,kr+1

Fj1jr+1xαβkr+1

( r∏ν=1

Fmνjνjν+1αkν+1kν

)Fkr+1k1yαj1β

∣∣∣∣m1

m2

m3

kr

k1k2

mr

kr+1

mr−1

k3

y

⟩

Central idempotents project on anyonic sectors

Bp = P11 ∼1 1

1

1

+ φ1

τ

1τ

P1τ ∼τ

τ

τ1

+√φe−3πi/5

τ

τ

ττ+ e4πi/5

τ

1

ττPτ1 = P1τ

Pττ ∼ φ21 1

1

1

− φ1

τ

1τ+ φ

τ

τ

1τ+

1√φ τ

τ

ττ+ (1 +

1φ

)τ

1

ττ


Measuring anyon charges

. . .

F-move(2-2 Pachner move)

0

CNOT0

0

(a) (b) (c)

(d) (e) (f)

(g) (h)

(i) (j)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

a

b

c

e

a

b

c

d

e a

b

c

d

e a

b

c

d

e a

b

c

de a

b

c

de

a

b

c

de a

b

c

d

e

a

b

c

de

12

11 109

8 7

6

543

21

13 14 15

1617

U Ua

b

c

e

a

b

c

d

ea

b

c

d

ea

b

c

d

ea

b

c

dea

b

c

de

a

b

c

dea

b

c

d

e

a

b

c

de

‘grow’ circuit ‘resolve’ circuit

abc

d

e

=

F-move

fe


Vertex correctionSome errors σe can take the initial state out of the string-net subspace:

[σe,∏

v

Qv] 6= 0 ⇒ σe |Ψ0〉 /∈ Hs.n.

After measuring all Qv projectors withoutcomes V the state is returned toHs.n. by unitary UV conditioned on V

UV |ΨV 〉 ∈ Hs.n.

UV : 7→


Error correction protocol

Apply noise

Measure vertex projectors + tails

Apply vertex correction

Measure anyon charge in all plaquetteswhile syndrome is nonempty1. Fuse pairs of anyons as dictated by decoding algorithm2. Measure anyon charges


The clustering decoderDecoding graph:

A

B

C

D

E

F

(A)

(A)

(B)

(C)

(D)

(E)

(F)

(A)(G) (H) (I) (J) (K)

G H I J KA

B

C

D

E

F

(A)

(A)

(B)

(C)

(D)

(E)

(F)

(A)

G H I J K

(G) (H) (I) (J) (K)

A

B

C

D

E

F

(A)

(A)

(B)

(C)

(D)

(E)

(F)

(A)

G H I J K

(G) (H) (I) (J) (K)

A

B

C

D

E

F

(A)

(A)

(B)

(C)

(D)

(E)

(F)

(A)

G H I J K

(G) (H) (I) (J) (K)

A

B

C

D

E

F

(A)

(A)

(B)

(C)

(D)

(E)

(F)

(A)

G H I J K

(G) (H) (I) (J) (K)


Outline




Summary & Outlook

Alexis Schotte arXiv:2012.04610 Classical simulation 17 / 25

Translating qubit errors to anyonic processesI Non-stabilizer code → Pauli noise is not natural here!I Fusion states of anyons in plaquettes ↔ unique lattice qubit states:

|~a,~b〉 =

a2

a4a3

a1

b

I The states {|~a,~b〉} form an orthonormal basis of Hs.n.

I Express joint action of noisevertex measurementvertex correction

in fusion basis:

〈~a′,~b′| UV PV σe |~a,~b〉

〈~a′,~b′| σe PV σe |~a,~b〉


Computing the matrix elements

I UV PV σe |~a,~b〉 and σePV σ

e |~a,~b〉 onlydepend on local properties of |~a,~b〉

σx

a1 a2

a4a3b1

b3

b2

I Tensor network representation of anyonic fusion states:

|~a,~b〉 =

a1 a2

a3 a4

a5

a4a3 b

1 b2

a2 a

1 b1

a4 b

2 a5


Computing the matrix elements

|~a,~b〉 =

. . .

. . .

. . . . . .

a1

a2

a3

a4

b1 b2

a5

〈~a′,~b′|σexPV σ

ex|~a,~b〉 =

where =σex

σexPA

a2 and =σex

σexPB

a4


Assumptions for the classical simulation

I Noise = random Pauli errorsI Total number of errors T ∼ Poisson(|E| p)I Perfect measurementsI Perfect recovery operations

We simulate the dynamics of the system for T timesteps

Individual time step in the simulation1. An edge e is selected at random

2. A random Pauli operator σei is applied to the state

3. All vertex projectors {Qv} and tail qubits are measured

4. Depending on the outcome V , vertex correction UV is applied

5. The anyon charge of all plaquettes is measured.


Results

Clustering decoder, depolarizing noise

pc ≈ 0.0475

Clustering decoder, dephasing noise

pc ≈ 0.0725

Comparable to surface code thresholds under similar assumptions!


Outline




Summary & Outlook

Alexis Schotte arXiv:2012.04610 Summary & Outlook 23 / 25

Summary

I Active Non-Abelian QEC → system of qubits

I Code space defined using modified Levin-Wen Hamiltonian

I 3 ingredients:I Quantum error correctionI Topological quantum field theory → tube algebra + anyonic fusion basisI Tensor networks → translating qubit errors to anyons

I First threshold for 2D code which natively supports a universal gate set

I Threshold comparable to surface code threshold under similar assumptions


Outlook

I More natural noise model?

I Simulating fully fault-tolerant setup

I Finding better decoders

I Applying our techniques to different non-Abelian codes (with lower-depthmeasurement circuits)

I Planar string-net codes

I Road to experimental implementation→ hardware-efficient multi-qubit gates


Documents

Quantum error correction thresholds for the universal