Equivalence of 2D color codes (without translationalsymmetry) to surface codes

Arjun Bhagoji1 Pradeep Sarvepalli1

1Department of Electrical EngineeringIndian Institute of Technology, Madras

IEEE International Symposium on Information Theory 2015, HongKong

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 1 / 20

Motivation

Surface codes, a class of topological codes, have many advantages(local stabilizer generators, low complexity decoders etc.)

Limited in terms of transversal gates that can be implemented

Color codes, another class of topological codes, can implement entireClifford group transversally

So, they are inequivalent?No!

Bombin, Duclos-Cianci and Poulin (2012) showed that translationallyinvariant 2D color codes can be mapped to a finite number of copies of

Kitaev’s toric code.Question: What about translation variant codes?

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 2 / 20

Surface codes

Topological codes are a class of quantum codes where information isstored in topological degrees of freedomSurface codes are topological codes with qubits attached to the edgesof a lattice embedded in a given closed surfaceEach face (plaquette) and vertex of the lattice defines a stabilizeracting on the surrounding qubits

XX

XX

Z

Z

Z Z

E1

E2

Figure: Left: Stabilizers on a copy of Kitaev’s toric code. Right: E1 is a homologicallytrivial error, can be detected and corrected. E2 is homologically non-trivial, cannot bedetected, interacts with the encoded qubits.

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 3 / 20

Color codes

Different class of topological codes, embedded on 2-colexes (trivalent,3-face-colorable complexes)Qubits attached to every vertexEach face of the lattice defines a stabilizer acting on the qubits in itsboundary

X

X

X

X

X

X

Z

ZZ

Z

Z

Z

Figure: Hexagonal color code

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 4 / 20

Charges and syndromes

Think of each violated check (stabilizer anti-commuting with error) asa charge

4 types of charges on a surface code: i) electric (ε) on vertices, ii)magnetic (µ) on the plaquettes, iii) composite electric and magneticcharge (εµ) on both plaquettes and vertices, and iv) the vacuum (ι)

Only two of these are independent, which we choose to be ε (due toZ -type) and µ (due to X -type)

Color code charges also have color associated with them.

16 types of charges on a color code (Bombin et.al., 2012) with 2independent pairs

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 5 / 20

Hopping operators

Any element of the Pauli group that moves chargesStabilizers can be viewed as a combination of hopping operatorsmoving charges back to their initial position

Figure: Hopping operators on the surface code (left) and color code (right)

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 6 / 20

Mapping between color codes and surface codes

Observation:1 4 types of charges on a surface code and 16 types of charges on a color

code.2 2 pairs of independent charges on the color code

⇒ Decompose the color code into 2 surface codes, mapping eachcharge pair from the color code to one of the surface codes

Electric charges on the surface codes live on the vertices while themagnetic charges live on the plaquettes.

Consider the charges on the color code Γ: all live on plaquettes

How do we relate the two?

Notation: For what follows, we denote the faces, edges and vertices of the embedding of a graph Γ by F(Γ), E(Γ) and V(Γ).Bσ

f =∏

v∈fσv denotes the stabilizers on the color code, where σ ∈ X , Z .

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 7 / 20

Desirable constraints on map

The map π between the color code and the surface codes should have thefollowing desirable properties:

1 Linear2 Invertible (i.e. map should be bijective)3 Local4 Efficiently computable5 Preserve the commutation relations between the Pauli error operators

on V(Γ), i.e. PV(Γ)6 Consistent in the description of the movement of charges on the color

code and surface codes.

Focus will be on the consistency of charge movement and the preservationof commutation relations. Time permitting, other constraints will bediscussed.

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 8 / 20

Surface codes obtained from color codes

Contract all the c-colored plaquettes in the embedding of Γ. This will giverise to a new graph τc(Γ).

Figure: Illustrating the contraction of a color code via τc and the resultant surface code.Only portions of the codes are shown.

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 9 / 20

Charge Mapping

Each c-colored face in Γ can host εc and µc .Both these charges live on the vertices of a contracted latticeIdentify εc ≡ ε1 and µc ≡ ε2.

Now choose the magnetic charges on Γ1 and Γ2 from εc′ , εc′′ and µc′ , µc′′ .How?

Lemma (Charge mapping)

Let c, c ′, c ′′ be three distinct colors. Then, {εc , µc′} and {εc′ , µc} arepermissible pairings of the charges so that the color code on Γ can bemapped to a pair of surface codes on Γi = τc(Γ). In other words, ε1 ≡ εc ,µ1 ≡ µc′ , ε2 ≡ µc and µ2 ≡ εc′ , where εi and µi are the electric andmagnetic charges of the surface code on Γi .

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 10 / 20

Elementary hopping operators (electric)

Figure: Mapping the electric charge hopping operators from the color code to thesurface codes

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 11 / 20

Mapping elementary hopping operators

Lemma (Elementary hopping operators)

Let f , f ′ ∈ Fc(Γ) where the edge (u, v) is incident on f and f ′. Then, thefollowing choices reflect the charge movement on Γ onto the surface codeson Γi .

π(Hεcu,v ) =

[Zτ(u)

]1

=[Zτ(v)

]1

(1)

π(Hµcu,v ) =

[Zτ(u)

]2

=[Zτ(v)

]2, (2)

where [T ]i indicates the instance of the surface code on which T acts.Now if f , f ′ ∈ Fc′(Γ) and (u, v) ∈ Ec′(Γ) such that u ∈ f and v ∈ f ′ andHεc′

u,v and Hµc′u,v are chosen to be independent hopping operators of f , then

π(Hεc′u,v ) =

[Xτ(u)Xτ(v)

]2

; π(Hµc′u,v ) =

[Xτ(u)Xτ(v)

]1. (3)

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 12 / 20

Dependent hopping operators

Not all the hopping operators on a particular face are independent.

Lemma (Dependent hopping operators)Let f ∈ Fc′′(Γ) and 1, . . . , 2`f be the vertices in its boundary so that(2i − 1, 2i) ∈ Ec(Γ), (2i , 2i + 1) ∈ Ec′(Γ) for 1 ≤ i ≤ `f and 2`f + 1 ≡ 1.If π is invertible, then π(Bσ

f ) 6= I and there are 4`f − 2 independentelementary hopping operators along the edges of f .

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 13 / 20

Mapping the single qubit errorsLet the operators Hεc′

1,2`f and Hµc′2m,2m+1 be the dependent ones.

Z1Z2Z3Z4

...Z2`−1Z2`

X1X2X3X4

...X2`−1X2`

π7→

ZZ

. . .Z

ZZ

. . .Z

1∪2

(4)

Z2Z3Z4Z5

...Z2`−2Z2`−1

π7→

X X

X X. . .

X X

2

(5)

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 14 / 20

Single qubit error splitting

X2X3...

X2m−2X2m−1X2m+2X2m+3

...X2`X1

π7→

X X. . .X X

X X. . .

X X

1

(6)

Lemma (Splitting)

The following choices lead to an invertible π while respecting thecommutation relations with the independent hopping operators.

π(gX1) = [Xτ(1)]1 where g ∈ {I,BXf ,BY

f ,BZf } (7)

π(gZ2m) = [Xτ(2m)]2 where g ∈ {I,BXf } (8)

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 15 / 20

Projection theorem

Theorem

Any 2D color code (on a 2-colex Γ without parallel edges) is equivalent toa pair of surface codes τ(Γ) under the map π.

Mapping can be done locally for each c ′′-colored face

Stabilizers on the color code are mapped to stabilizers on the surfacecode, so code capabilities are preserved

Decoding of color codes is a possible application

Mapped errors can be decoded on the surface code, and uniquelylifted back to the color code, as the map is bijective

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 16 / 20

Map for X -type errors on the hexagonal color code

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 17 / 20

Map for X -type errors on the hexagonal color code

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 18 / 20

Comparison with previous work

Authors Translationinvariancereqd.

No. ofcopies

Ancillaqubitsreqd.

Injective Conseq. fordecoding

Bombinet. al.

Yes Finite,could bemany

Sometimes Yes High com-plexity

Delfosseet. al.

No 3 Not sure Yes No pre-image forsome errors

Presentwork

No 2 No Yes Non-CSSnature, upto 2 de-coders persurface code

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 19 / 20

That’s all folks!

Bhagoji, Sarvepalli Equivalence of 2D color codes (without translational symmetry) to surface codes ISIT 2015 20 / 20