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Hyperbolic Color Codes on Densest Tessellations

Hyperbolic Color Codes on Densest Tessellations

Clarice Dias de Albuquerque (UFCA) Reginaldo Palazzo Junior (UNICAMP) 27/07/2018

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Topological Quantum Codes

The main characteristic of topological codes is the contrast between the local character of the stabilizer generators and the overall nondetectable

• errors. This implies the simplicity of the stabilizer generators and the

code distance increases as the cardinality of the set of the stabilizer generators increases. Fault-tolerant quantum computation: the code space is realized in a quantum system ordered topologically The stabilizer operators constitute a Hamiltonian with local interactions, whose ground state coincides with the code subspace

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Kitaev’s Toric Code

◮ Square lattice l × l. ◮ qubits → edges of the lattice. ◮ n = |E| = 2l2 and k = 2.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Kitaev’s Toric Code

C C’ c’ c c c c’

◮ qubits → edges of the lattice. ◮ n = |E| = 2l2 and k = 2. ◮ dmin = l.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Kitaev’s Toric Code

F V

◮ Stabilizer operators:

Av =

• j∈Ev

σj

x

Bf =

• j∈Ef

σj

z.

◮ C = {|ψ : Av|ψ = |ψ, Bf |ψ = |ψ ∀ v, f }.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Hyperbolic Toric Codes

Conjecture: The performance of a quantum error-correcting codes on surfaces with genus g ≥ 2 can be better, in terms of the error probability, than that codes on surfaces with genus g < 2.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Hyperbolic Toric Codes

Conjecture: The performance of a quantum error-correcting codes on surfaces with genus g ≥ 2 can be better, in terms of the error probability, than that codes on surfaces with genus g < 2. In addition, these surfaces can be tilling by various tessellations, generating different codes that may provide more applications.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Plane Models of Surfaces

A compact topological surface M may be obtained of a polygon P′ by “pasting” edges in pairs, since length and angle conditions are satisfied. (i) For each edge s of P′ there exists a unique edge s′ of P′ such that s′ = γ(s), where γ is an isometry of an isometry group Γ. (ii) Given edge-pairings of P′, for each vertex cycle (a chain of identifications with vertices), the sum of the angles has to be equal to 2π.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Plane Models of Surfaces

A compact topological surface M may be obtained of a polygon P′ by “pasting” edges in pairs, since length and angle conditions are satisfied. (i) For each edge s of P′ there exists a unique edge s′ of P′ such that s′ = γ(s), where γ is an isometry of an isometry group Γ. (ii) Given edge-pairings of P′, for each vertex cycle (a chain of identifications with vertices), the sum of the angles has to be equal to 2π. We use polygons P′ of the type 4g-gon with the pairings of opposite edges of P′ as the plane models of the surfaces.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Tessellations of P′

◮ The solution of the following equation results in all possible tessellations {p, q} of P′ µ(P′) = nf µ(P), (1) where µ(P′) is the area of P′, µ(P) is the area of the polygon associated to {p, q}, nf is a positive integer and (p − 2)(q − 2) > 4. ◮ Dual tessellation {q, p} has to satisfy the same conditions. ◮ From the Gauss-Bonnet Theorem, (1) may be written as:

4π(g − 1) = nf

• (p − 2)π − 2pπ

q )

• .

(2) nf = 4q(g − 1) pq − 2p − 2q . (3)

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Toric Codes on Surfaces with g ≥ 2

◮ nf =

4q(g−1) pq−2p−2q.

◮ n = |E| = nf · p

2 and k = 2g.

◮ Example: 14 − gon surface with g = 3, with tessellation {7, 3}. We have: nf = 24, nv = 56 and n = 84.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Toric Codes on Surfaces with g ≥ 2

◮ Pairings of opposite edges. ◮ d =

• 2 arccosh[

cos(π/4g) sin(π/4g) ]

arccosh

• cos2(π/q)+cos(2π/p)

sin2(π/q)

• .

◮ Example: s2i+1 → s2i+6. In this case, dmin = 4.

Hyperbolic Color Codes on Densest Tessellations Topological Quantum Codes

Toric Codes on Surfaces with g ≥ 2

◮ Stabilizer operators:

Av =

• j∈Ev

σj

x

Bf =

• j∈Ef

σj

z.

◮ C = {|ψ : Av|ψ = |ψ, Bf |ψ = |ψ ∀ v, f }.

Hyperbolic Color Codes on Densest Tessellations Color Codes

Color Codes - H. Bombin and M. Delgado

◮ Color codes allow the transverse implementation of logic gates X = 1 1

• Z =

1 −1

• CNot =

    1 1 1 1     . ◮ And the gates H = 1 √ 2 1 1 1 −1

• K =

1 i

• .

◮ These gates generate the group of Clifford, composed by the

• perators that leave invariant by conjugation the group of Pauli,

that is sufficient to perform teleportation and distillation protocols, and hence to perform universal computation.

Hyperbolic Color Codes on Densest Tessellations Color Codes

Lattice

Bidimensional lattice embedded in a torus of arbitrary genus such that let the graph be: (i) trivalent (ii) 3-colorable

Hyperbolic Color Codes on Densest Tessellations Color Codes

Lattice

Bidimensional lattice embedded in a torus of arbitrary genus such that let the graph be: (i) trivalent (ii) 3-colorable Hexagonal Lattice {6, 3}, Square-octagon Lattice {4, 8, 8}, Truncated Trihexagonal Lattice {4, 6, 12}.

Hyperbolic Color Codes on Densest Tessellations Color Codes

Lattice

Bidimensional lattice embedded in a torus of arbitrary genus such that let the graph be: (i) trivalent (ii) 3-colorable Hexagonal Lattice {6, 3}, Square-octagon Lattice {4, 8, 8}, Truncated Trihexagonal Lattice {4, 6, 12}.

Hyperbolic Color Codes on Densest Tessellations Color Codes

Parameters

◮ Qubits ← → vertices of the bidimensional lattice of the torus. ◮ Stabilizer operators associated with the lattice faces. BX

f =

• i∈f

Xi BZ

f =

• i∈f

Zi ◮ C = {|ψ : BX

f |ψ = |ψ, BZ f |ψ = |ψ ∀ f }.

◮ n = |V | ◮ k = 4g

Hyperbolic Color Codes on Densest Tessellations Color Codes

Parameters

◮ d is the minimum length between paths with non-trivial homology. ◮ Consider, for example, red faces, placing a site at each face and connecting them through red edges. In this new (triangular) lattice, known as shrunk lattice, each edge corresponds to two vertices in the initial colored lattice and that the blue and green faces correspond to faces in the new lattice. There is a shrunk lattice for each color.

Hyperbolic Color Codes on Densest Tessellations Color Codes

Example - P. Sarvepalli and R. Raussendorf

◮ Hexagonal Lattice - the color code on this lattice has to satisfy certain restrictions on its dimensions in order to be embedded on the torus. ◮ [[18x2m, 4, 4xm]] = [[2(3xm)2, 4, 4xm]], where m can be any nonnegative integer and x ≥ 2. ◮ [[72, 4, 8]] code on the dual tessellation:

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Hyperbolic Color Codes

◮ Tessellations: {2p′, 3} ◮ nf = 6(g−1)

p′−3 .

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Operators and Parameters

◮ Operators: Bσ

f = i∈f σi

σ = X, Z

◮ C = {|ψ : BX

f |ψ = |ψ, BZ f |ψ = |ψ ∀ f }.

◮ n = |V | = nf

p q = nf p 3.

◮ k = 4 − 2χ = 4g.

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Operators and Parameters

d is the minimum length between paths with non-trivial homology, considering the shrunk lattice.

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Hyperbolic Color Codes on Densest Tessellations

Consider {12i − 6, 3}, then nf = g−1

i−1 .

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Hyperbolic Color Codes on Densest Tessellations

Consider {12i − 6, 3}, then nf = g−1

i−1 .

{18, 3} tessellation: ◮ [[6(g − 1), 4g, d]] and k

n → 2 3

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Hyperbolic Color Codes on Densest Tessellations

Consider {12i − 6, 3}, then nf = g−1

i−1 .

{18, 3} tessellation: ◮ [[6(g − 1), 4g, d]] and k

n → 2 3

{30, 3} tessellation: ◮ [[5(g − 1), 4g, d]] and k

n → 4 5

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Hyperbolic Color Codes on Densest Tessellations

Consider {12i − 6, 3}, then nf = g−1

i−1 .

{18, 3} tessellation: ◮ [[6(g − 1), 4g, d]] and k

n → 2 3

{30, 3} tessellation: ◮ [[5(g − 1), 4g, d]] and k

n → 4 5

{42, 3} tessellation: ◮ [[ 14

3 (g − 1), 4g, d]] and k n → 6 7

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Hyperbolic Color Codes on Densest Tessellations

Consider {12i − 6, 3}, then nf = g−1

i−1 .

{18, 3} tessellation: ◮ [[6(g − 1), 4g, d]] and k

n → 2 3

{30, 3} tessellation: ◮ [[5(g − 1), 4g, d]] and k

n → 4 5

{42, 3} tessellation: ◮ [[ 14

3 (g − 1), 4g, d]] and k n → 6 7

In general: [[ 2+4i

i−1 (g − 1), 4g, d]] with k n → 1

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Color codes on {18, 3} tessellation

◮ Parameters: [[6(g − 1), 4g, d]] ◮ Encoding rate:

k n → 2 3

g n k

k n

4 18 16 0.889 5 24 20 0.833 6 30 24 0.8 7 36 28 0.778

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Color codes on {30, 3} tessellation

◮ Parameters: [[5(g − 1), 4g, d]] ◮ Encoding rate:

k n → 4 5

g n k

k n

7 30 28 0.933 9 40 36 0.9 11 50 44 0.88 13 60 52 0.8667

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

Color codes on {42, 3} tessellation

◮ Parameters: [[ 14(g−1)

3

, 4g, d]] ◮ Encoding rate:

k n → 6 7

g n k

k n

10 42 40 0, 95 13 56 52 0, 93 16 70 64 0, 914 19 84 76 0, 905

Hyperbolic Color Codes on Densest Tessellations Hyperbolic Color Codes

◮ The classes associated with tessellations {12i − 6, 3} exist for genus g ≥ 2 since that g−1

i−1 is a positive integer. For example: for {18, 3},

g = 4, 5, 6, 7, . . .; for {30, 3}, g = 5, 7, 9, . . . is odd; for {42, 3}, g ≡ 1mod3. ◮ Such tessellations are the most dense, and have coding rate asymptotically 1. ◮ Comparing with the example [[18x2m, 4, 4xm]] in the torus, we find that the encoding rates of the codes obtained on surfaces of genus g ≥ 2 are much better.

Hyperbolic Color Codes on Densest Tessellations Bibliography

Bibliography

◮ A. Yu. Kitaev (2003), Fault-tolerant quantum computation by anyons, Annals of Physics, 303, pp. 2-30. ◮ C. D. de Albuquerque, R. Palazzo Jr. and E. B. da Silva (2009), Topological quantum codes on compact surfaces with genus g ≥ 2,

• J. Math. Phys., 50, pp. 023513.

◮ H. Bombin and M. A. Martin-Delgado (2006), Topological quantum distillation, Physical Review Letters, vol. 97, pp. 180501. ◮ P. Sarvepalli and R. Raussendorf (2012), Efficient decoding of topological color codes, Physical Review A, vol. 85, pp. 022317.

Hyperbolic Color Codes on Densest Tessellations Acknowledgements

Acknowledgements

CNPq FAPESP

Hyperbolic Color Codes on Densest Tessellations Acknowledgements

Acknowledgements

CNPq FAPESP

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