Decoding problem for topological quantum codes Guillaume - - PowerPoint PPT Presentation

Decoding problem for topological quantum codes

Guillaume Duclos-Cianci

Département de Physique Université de Sherbrooke Joint work with: David Poulin and Hector Bombin

Second International Conference on Quantum Error Correction University of Southern California, Los Angeles, USA, December 2011

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 1 / 44

Motivation

Topological codes are promising candidates to implement fault-tolerant quantum computation From a physicist perspective

Physically realistic interactions : H = − Si with Si local Explicit and simple examples of system topologically ordered Explicit and simple examples of system having anyonic excitations Toolbox (e.g. twists, boundaries) have natural physical meaning (symmetry of the anyon model, particle condensation)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 2 / 44

Motivation

Topological codes are promising candidates to implement fault-tolerant quantum computation From a physicist perspective

Physically realistic interactions : H = − Si with Si local Explicit and simple examples of system topologically ordered Explicit and simple examples of system having anyonic excitations Toolbox (e.g. twists, boundaries) have natural physical meaning (symmetry of the anyon model, particle condensation)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 2 / 44

Motivation

Topological codes are promising candidates to implement fault-tolerant quantum computation From a physicist perspective

Physically realistic interactions : H = − Si with Si local Explicit and simple examples of system topologically ordered Explicit and simple examples of system having anyonic excitations Toolbox (e.g. twists, boundaries) have natural physical meaning (symmetry of the anyon model, particle condensation)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 2 / 44

Motivation

Topological codes are promising candidates to implement fault-tolerant quantum computation From a physicist perspective

Physically realistic interactions : H = − Si with Si local Explicit and simple examples of system topologically ordered Explicit and simple examples of system having anyonic excitations Toolbox (e.g. twists, boundaries) have natural physical meaning (symmetry of the anyon model, particle condensation)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 2 / 44

Motivation

Topological codes are promising candidates to implement fault-tolerant quantum computation From a physicist perspective

Physically realistic interactions : H = − Si with Si local Explicit and simple examples of system topologically ordered Explicit and simple examples of system having anyonic excitations Toolbox (e.g. twists, boundaries) have natural physical meaning (symmetry of the anyon model, particle condensation)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 2 / 44

Motivation

Topological codes are promising candidates to implement fault-tolerant quantum computation From a physicist perspective

Physically realistic interactions : H = − Si with Si local Explicit and simple examples of system topologically ordered Explicit and simple examples of system having anyonic excitations Toolbox (e.g. twists, boundaries) have natural physical meaning (symmetry of the anyon model, particle condensation)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 2 / 44

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 3 / 44

Kitaev’s toric code

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 4 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

X X X X Z Z Z Z

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

X X X X Z Z Z Z

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

X X X X Z Z Z Z

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

X X X X X X X X X X Z Z Z Z Z Z Z Z Z Z Z Z Z Z

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

Z1 X1

Z Z Z Z Z Z Z Z Z Z X X X X X X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

Z2 X2 γ

2

γ2

Z Z Z Z Z Z Z Z Z Z X X X X X X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Definition

H = −

s As − p Bp

Trivial action on ground space As|ψ = Bp|ψ = +1|ψ The As et Bp are trivial cycles Stabilizer elements are trivial loops. Non-trivial cycles commute with the stabilizers but are independent. Logical operators encoding information.

Z2 X2 γ

2

γ2

Z Z Z Z Z Z Z Z Z Z X X X X X X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 5 / 44

Kitaev’s toric code

Particle creation

σi

x anti-commutes with

adjacent plaquettes. This error has flipped a pair of syndrome bits ↔ created a pair of plaquette defects.

X 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 6 / 44

Kitaev’s toric code

Particle creation

σi

x anti-commutes with

adjacent plaquettes. This error has flipped a pair of syndrome bits ↔ created a pair of plaquette defects.

X 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 6 / 44

Kitaev’s toric code

Particle diffusion

New error occurs on neighboring qubit: Restores the sign of the middle plaquette Flips the sign of the right plaquette Number of particles is constant, but one has moved

X 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 7 / 44

Kitaev’s toric code

Particle diffusion

New error occurs on neighboring qubit: Restores the sign of the middle plaquette Flips the sign of the right plaquette Number of particles is constant, but one has moved

X X 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 7 / 44

Kitaev’s toric code

Particle diffusion

New error occurs on neighboring qubit: Restores the sign of the middle plaquette Flips the sign of the right plaquette Number of particles is constant, but one has moved

X X 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 7 / 44

Kitaev’s toric code

Error chains

Error chains are attached to a pair of particles. The syndrome configuration on the endpoint doesn’t depend on the geometry (path, length) of the string. Error chains can be stretched freely: constant energy cost.

X X 1 1 X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 8 / 44

Kitaev’s toric code

Error chains

Error chains are attached to a pair of particles. The syndrome configuration on the endpoint doesn’t depend on the geometry (path, length) of the string. Error chains can be stretched freely: constant energy cost.

X X 1 1 X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 8 / 44

Kitaev’s toric code

Error chains

Error chains are attached to a pair of particles. The syndrome configuration on the endpoint doesn’t depend on the geometry (path, length) of the string. Error chains can be stretched freely: constant energy cost.

X X 1 1 X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 8 / 44

Kitaev’s toric code

Particle annihilation

An error can annihilate two particles. The particle’s worldline is left behind after fusion. Particle fusion can leave behind a worldline corresponding to a logical

• peration.

Memory corruption

X X X X X X X X X X X X X X X 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 9 / 44

Kitaev’s toric code

Particle annihilation

An error can annihilate two particles. The particle’s worldline is left behind after fusion. Particle fusion can leave behind a worldline corresponding to a logical

• peration.

Memory corruption

X X X X X X X 1 1 X X X X X X X X X X X X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 9 / 44

Kitaev’s toric code

Particle annihilation

An error can annihilate two particles. The particle’s worldline is left behind after fusion. Particle fusion can leave behind a worldline corresponding to a logical

• peration.

Memory corruption

X X X X X X X 1 1 X X X X X X X X X X X X X X X X

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 9 / 44

Kitaev’s toric code

Site particles

The same story holds for σz errors These will create site particles located at the lattice’s vertices (plaquette of dual lattice).

Z Z 1 1 Z Z Z Z Z

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 10 / 44

Decoding problem

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 11 / 44

Decoding problem

Error model

Depolarizing error model Independent on every qubit. No error with probability 1 − p. Error X, Y, or Z with probability p/3. Bit-flip error model Independent on every qubit. No error with probability 1 − p. Error X with probability p.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 12 / 44

Decoding problem

Error syndrome & decoding

An error produces defects (error syndrome) Measure particle position, but not worldline. Many worldlines consistent with defects. Worldline with different homologies have different effect on ground space: MUST be distinguished. Decoding Infer worldline homology from particle location. 15 % Noise rate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Z Y X X Z Y Y Y Y Z Y X X Z X X X Y Y Y Z Z X Y Z Z Y

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

Decoding problem

Error syndrome & decoding

An error produces defects (error syndrome) Measure particle position, but not worldline. Many worldlines consistent with defects. Worldline with different homologies have different effect on ground space: MUST be distinguished. Decoding Infer worldline homology from particle location. 15 % Noise rate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

Decoding problem

Error syndrome & decoding

An error produces defects (error syndrome) Measure particle position, but not worldline. Many worldlines consistent with defects. Worldline with different homologies have different effect on ground space: MUST be distinguished. Decoding Infer worldline homology from particle location. 15 % Noise rate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

Decoding problem

Error syndrome & decoding

An error produces defects (error syndrome) Measure particle position, but not worldline. Many worldlines consistent with defects. Worldline with different homologies have different effect on ground space: MUST be distinguished. Decoding Infer worldline homology from particle location. 15 % Noise rate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

Decoding problem

Error syndrome & decoding

An error produces defects (error syndrome) Measure particle position, but not worldline. Many worldlines consistent with defects. Worldline with different homologies have different effect on ground space: MUST be distinguished. Decoding Infer worldline homology from particle location. 15 % Noise rate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

Decoding problem

Error syndrome & decoding

An error produces defects (error syndrome) Measure particle position, but not worldline. Many worldlines consistent with defects. Worldline with different homologies have different effect on ground space: MUST be distinguished. Decoding Infer worldline homology from particle location. 15 % Noise rate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

Decoding problem

Existing methods

Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O(ℓ6) Polynomial complexity, but still prohibitive, O(ℓ6). Recent progress : average O(1) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal:

Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

Decoding problem

Existing methods

Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O(ℓ6) Polynomial complexity, but still prohibitive, O(ℓ6). Recent progress : average O(1) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal:

Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

Decoding problem

Existing methods

Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O(ℓ6) Polynomial complexity, but still prohibitive, O(ℓ6). Recent progress : average O(1) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal:

Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

Decoding problem

Existing methods

Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O(ℓ6) Polynomial complexity, but still prohibitive, O(ℓ6). Recent progress : average O(1) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal:

Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

Decoding problem

Existing methods

Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O(ℓ6) Polynomial complexity, but still prohibitive, O(ℓ6). Recent progress : average O(1) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal:

Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

Decoding problem

Existing methods

Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O(ℓ6) Polynomial complexity, but still prohibitive, O(ℓ6). Recent progress : average O(1) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal:

Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

Decoding problem

Minimum distance vs Degeneracy

Two possible pairings with different homologies

First one has lower weight (Energy). Second one is highly degenerate (Entropy).

Optimal decoding Homology class with lowest free energy F = E − TS. Nishimori T −1 = ln 3(1−p)

p

. Sum over all equivalent errors.

1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

Decoding problem

Minimum distance vs Degeneracy

Two possible pairings with different homologies

First one has lower weight (Energy). Second one is highly degenerate (Entropy).

Optimal decoding Homology class with lowest free energy F = E − TS. Nishimori T −1 = ln 3(1−p)

p

. Sum over all equivalent errors.

1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

Decoding problem

Minimum distance vs Degeneracy

Two possible pairings with different homologies

First one has lower weight (Energy). Second one is highly degenerate (Entropy).

Optimal decoding Homology class with lowest free energy F = E − TS. Nishimori T −1 = ln 3(1−p)

p

. Sum over all equivalent errors.

1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

Decoding problem

Minimum distance vs Degeneracy

Two possible pairings with different homologies

First one has lower weight (Energy). Second one is highly degenerate (Entropy).

Optimal decoding Homology class with lowest free energy F = E − TS. Nishimori T −1 = ln 3(1−p)

p

. Sum over all equivalent errors.

1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

Decoding problem

Minimum distance vs Degeneracy

Two possible pairings with different homologies

First one has lower weight (Energy). Second one is highly degenerate (Entropy).

Optimal decoding Homology class with lowest free energy F = E − TS. Nishimori T −1 = ln 3(1−p)

p

. Sum over all equivalent errors.

1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

Decoding problem

Minimum distance vs Degeneracy

Two possible pairings with different homologies

First one has lower weight (Energy). Second one is highly degenerate (Entropy).

Optimal decoding Homology class with lowest free energy F = E − TS. Nishimori T −1 = ln 3(1−p)

p

. Sum over all equivalent errors.

1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

Decoding problem

Plaquette-Site string correlations

Two possible pairings with different homologies Both seemingly have same weight A Y error has same weight as X and Z: overcounting. Site Z and plaquette X errors are not independent.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

Decoding problem

Plaquette-Site string correlations

Two possible pairings with different homologies Both seemingly have same weight A Y error has same weight as X and Z: overcounting. Site Z and plaquette X errors are not independent.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

Decoding problem

Plaquette-Site string correlations

Two possible pairings with different homologies Both seemingly have same weight A Y error has same weight as X and Z: overcounting. Site Z and plaquette X errors are not independent.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

Decoding problem

Plaquette-Site string correlations

Two possible pairings with different homologies Both seemingly have same weight A Y error has same weight as X and Z: overcounting. Site Z and plaquette X errors are not independent.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

Decoding problem

Plaquette-Site string correlations

Two possible pairings with different homologies Both seemingly have same weight A Y error has same weight as X and Z: overcounting. Site Z and plaquette X errors are not independent.

1 1 1 1 1 1

Y Y

1 1 1

Y

1 1 1

Y

1 1 1

Y

1 1 1

Y Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

Decoding problem

Plaquette-Site string correlations

Two possible pairings with different homologies Both seemingly have same weight A Y error has same weight as X and Z: overcounting. Site Z and plaquette X errors are not independent.

1 1 1 1 1 1

Y Y

1 1 1

Y

1 1 1

Y

1 1 1

Y

1 1 1

Y Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

Renormalization Group Decoder

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 17 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Scale invariance

Original Bp checks Basis change (row operations on C) Obtain scale invariant generators Structure similar to a concatenated code.

Soft-decode each small block. Pass information to next encoding level.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

Renormalization Group Decoder

Concatenated code

Decoding P(L) = ′

E P(E) where

Sum over E equivalent to L and with right syndrome. P(E) given by error model. Decoding Compute error probability for each encoded qubit. Pass that probability to the next level up.

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

U

|0 |0

|ψ

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

Renormalization Group Decoder

Concatenated code

Decoding P(L) = ′

E P(E) where

Sum over E equivalent to L and with right syndrome. P(E) given by error model. Decoding Compute error probability for each encoded qubit. Pass that probability to the next level up.

U

|0 |0

|ψ

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

Renormalization Group Decoder

Concatenated code

Decoding P(L) = ′

E P(E) where

Sum over E equivalent to L and with right syndrome. P(E) given by error model. Decoding Compute error probability for each encoded qubit. Pass that probability to the next level up.

U L

†

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

Renormalization Group Decoder

Concatenated code

Decoding P(L) = ′

E P(E) where

Sum over E equivalent to L and with right syndrome. P(E) given by error model. Decoding Compute error probability for each encoded qubit. Pass that probability to the next level up.

U L

†

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

Renormalization Group Decoder

Concatenated code

Decoding P(L) = ′

E P(E) where

Sum over E equivalent to L and with right syndrome. P(E) given by error model. Decoding Compute error probability for each encoded qubit. Pass that probability to the next level up.

U U U U

U L

U U U U U U U U

†

† † † † † † † † † † † †

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

Renormalization Group Decoder

Renormalization: Information coarse graining

Think of Kitaev’s code as a concatenated code:

It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes.

Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to {I, X, Y, Z}2, 2 qubits. This is done by brute force: sum over all worldline configurations.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

Renormalization Group Decoder

Renormalization: Information coarse graining

Think of Kitaev’s code as a concatenated code:

It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes.

Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to {I, X, Y, Z}2, 2 qubits. This is done by brute force: sum over all worldline configurations.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

Renormalization Group Decoder

Renormalization: Information coarse graining

Think of Kitaev’s code as a concatenated code:

It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes.

Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to {I, X, Y, Z}2, 2 qubits. This is done by brute force: sum over all worldline configurations.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

Renormalization Group Decoder

Renormalization: Information coarse graining

Think of Kitaev’s code as a concatenated code:

It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes.

Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to {I, X, Y, Z}2, 2 qubits. This is done by brute force: sum over all worldline configurations.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

Renormalization Group Decoder

Overlaping cells

Topological codes are NOT concatenated codes. Cannot break lattices into constant-size cells in such a way that each stabilizer

• verlaps with a single

region. Use overlapping cells instead.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 21 / 44

Renormalization Group Decoder

Overlaping cells

Topological codes are NOT concatenated codes. Cannot break lattices into constant-size cells in such a way that each stabilizer

• verlaps with a single

region. Use overlapping cells instead.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 21 / 44

Renormalization Group Decoder

Overlaping cells

Topological codes are NOT concatenated codes. Cannot break lattices into constant-size cells in such a way that each stabilizer

• verlaps with a single

region. Use overlapping cells instead.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 21 / 44

Renormalization Group Decoder

Self-consistency

1 1

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Renormalization Group Decoder

Self-consistency

1 1

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Renormalization Group Decoder

Self-consistency

1 1

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Renormalization Group Decoder

Self-consistency

PL,i(X) = PR,i(X)

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Renormalization Group Decoder

Self-consistency

P

L,i(X) = P R,i(X)

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Renormalization Group Decoder

Self-consistency

P

L,i(X) = P R,i(X)

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Renormalization Group Decoder

Self-consistency

P

L,i(X) = P R,i(X)

Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O(ℓ2) parallelizable to constant time.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

Results for Kitaev’s code

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 23 / 44

Results for Kitaev’s code

RG decoder with 3 BP rounds

0.01 0.1 1 12 13 14 15 16 17

Probabilite d’erreur du decodeur Force du canal depolarizant, p (%)

l=8 l=16 l=32 l=64

Threshold ≈ 15%, compared to 15.5% for PMA. O(log ℓ) time complexity with marginal performance loss.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 24 / 44

Results for Kitaev’s code

Smaller unit cell, 2 × 1

0.0001 0.001 0.01 0.1 1 20 30 40 50 60 70 80 90 100

Probabilite d’erreur du decodeur Force du canal Bit-Flip, p (%)

l=8 l=16 l=32 l=64 l=128 l=256 l=512 l=1024

Failure probability Depolarization Strength % 2 4 6 8 10

Bit-flip threshold ≈ 8.2%, compared to 10.3% for PMA. Much faster even without parallelization (106 sites). Illustrates flexibility.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 25 / 44

Results for Kitaev’s code

Higher threshold

0.1 1 15 15.5 16 16.5 17 17.5 18

Probabilite d’erreur du decodeur Force du canal depolarizant, p (%)

l=8 l=16 l=32 l=64 l=128

PMA

Use of additional belief propagation. Threshold ≈ 16.5%, compared to 15.5% for PMA.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 26 / 44

Extension to other codes

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 27 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code

X X X X X X X X Z Z Z Z Z Z Z Z

Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code

X X X X X X X X Z Z Z Z Z Z Z Z

Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code

X X X X Z Z Z Z

Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code

X X X X Z Z Z Z

Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code

X X X X Z Z Z Z

Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Topological color code

Qubits located on vertices Plaquette stabilizers Sp =

j∈∂p σj,

for σ = σx and σz. Same particle types and statistics as 2 copies of Kitaev’s code

X X X X Z Z Z Z

Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

Extension to other codes

Equivalence of Topological codes

Every 2D, translationally invariant, non-chiral stabilizer code with local generators and macroscopic minimal distance is locally equivalent to a finite number of copies of Kitaev’s code. Topological Color code equivalent to two copies of Kitaev’s code : a local clifford map exists Do the mapping and operate decoding on KTCs instead. Local Clifford Map Noise model remains Pauli and local Map for syndrome bits (particles)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 29 / 44

Extension to other codes

Equivalence of Topological codes

Every 2D, translationally invariant, non-chiral stabilizer code with local generators and macroscopic minimal distance is locally equivalent to a finite number of copies of Kitaev’s code. Topological Color code equivalent to two copies of Kitaev’s code : a local clifford map exists Do the mapping and operate decoding on KTCs instead. Local Clifford Map Noise model remains Pauli and local Map for syndrome bits (particles)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 29 / 44

Extension to other codes

Equivalence of Topological codes

Every 2D, translationally invariant, non-chiral stabilizer code with local generators and macroscopic minimal distance is locally equivalent to a finite number of copies of Kitaev’s code. Topological Color code equivalent to two copies of Kitaev’s code : a local clifford map exists Do the mapping and operate decoding on KTCs instead. Local Clifford Map Noise model remains Pauli and local Map for syndrome bits (particles)

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 29 / 44

Extension to other codes

Local Clifford Map

TCC KTC 1 KTC 2 → → → → → → → → TCC KTC 1 KTC 2 → → Local : 1 body →≤ 3 body Preserves commutation relations

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 30 / 44

Extension to other codes

Charge Map

→ = ⇒ →

Z 1

→ = ⇒ →

1 Z Z Z Z Z

Map of hopping operators ⇒ map of the charges

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 31 / 44

Extension to other codes

Results for Topological color code

0.01 0.1 1 7 7.5 8 8.5 9 9.5 10 10.5 11

Decoding error probability Bit-Flip channel strength p%

l=16 l=32 l=64 l=128 l=256

Threshold ≈ 8.7% compared to ∼ 11%. First efficient decoder for this code

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 32 / 44

Extension to other codes

Topological subsystem color code

A B

Gauge generators: two-body operators Up to 24-body stabilizer generators Threshold? Suchara et al. 2010: 2% depolarizing threshold on the “five-squares" code.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 33 / 44

Extension to other codes

Results for Topological subsystem color code

0.0001 0.001 0.01 0.1 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Decoding error probability Depolarizing channel strength, p%

l=8 l=16 l=32 l=64 l=128

Threshold ∼ 2%

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 34 / 44

Fault-tolerance

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 35 / 44

Fault-tolerance

Noisy measurements

So far, we have assumed that we can perfectly monitor the presence of defects. Measurements will themselves be noisy. Performing error correction with noisy instruments can kill the computation. Can model erroneous measurements by ghost defects appearing with probability p. Can overcome measurement errors by repeating the measurements periodically. Model becomes 2+1 dimensional.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 36 / 44

Fault-tolerance

Noisy measurements

So far, we have assumed that we can perfectly monitor the presence of defects. Measurements will themselves be noisy. Performing error correction with noisy instruments can kill the computation. Can model erroneous measurements by ghost defects appearing with probability p. Can overcome measurement errors by repeating the measurements periodically. Model becomes 2+1 dimensional.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 36 / 44

Fault-tolerance

Noisy measurements

So far, we have assumed that we can perfectly monitor the presence of defects. Measurements will themselves be noisy. Performing error correction with noisy instruments can kill the computation. Can model erroneous measurements by ghost defects appearing with probability p. Can overcome measurement errors by repeating the measurements periodically. Model becomes 2+1 dimensional.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 36 / 44

Fault-tolerance

Noisy measurements

So far, we have assumed that we can perfectly monitor the presence of defects. Measurements will themselves be noisy. Performing error correction with noisy instruments can kill the computation. Can model erroneous measurements by ghost defects appearing with probability p. Can overcome measurement errors by repeating the measurements periodically. Model becomes 2+1 dimensional.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 36 / 44

Fault-tolerance

Noisy measurements

So far, we have assumed that we can perfectly monitor the presence of defects. Measurements will themselves be noisy. Performing error correction with noisy instruments can kill the computation. Can model erroneous measurements by ghost defects appearing with probability p. Can overcome measurement errors by repeating the measurements periodically. Model becomes 2+1 dimensional.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 36 / 44

Fault-tolerance

Noisy measurements

So far, we have assumed that we can perfectly monitor the presence of defects. Measurements will themselves be noisy. Performing error correction with noisy instruments can kill the computation. Can model erroneous measurements by ghost defects appearing with probability p. Can overcome measurement errors by repeating the measurements periodically. Model becomes 2+1 dimensional.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 36 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

2+1 world-lines

Some defects stay put. Some defects diffuse. Some charges can fuse. Some charges can nucleate. Some defects are missing, and assumed to be there. Some defects shouldn’t be there, and are ignored.

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 37 / 44

Fault-tolerance

Results in 3D using RG decoding

0.0001 0.001 0.01 0.1 1 1.4 1.6 1.8 2 2.2 2.4 2.6

Decoding error probability Bit-Flip channel strength p%

l=8 l=16 l=32 l=64

Fault tolerant threshold of roughly 1.8 % Comparable to the 2.9% reported by Harrington et al. using slow decoders

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 38 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Outline

1

Kitaev’s toric code

2

Decoding problem

3

Renormalization Group Decoder

4

Results for Kitaev’s code

5

Extension to other codes

6

Fault-tolerance

7

2D Fault-Tolerant Quantum Cellular Automaton

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 39 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Confinement

Current proposal Make millions of measurements. Send data to classical processor to be analyzed Feed information forward If particles were attracting each other, they would be confined and none of this would be necessary. String tension in 2D? (Hama et al. 2008, Chesi et al. 2010) Errors would be thermally suppressed (keep system cool).

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 40 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Confinement

Current proposal Make millions of measurements. Send data to classical processor to be analyzed Feed information forward If particles were attracting each other, they would be confined and none of this would be necessary. String tension in 2D? (Hama et al. 2008, Chesi et al. 2010) Errors would be thermally suppressed (keep system cool).

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 40 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Confinement

Current proposal Make millions of measurements. Send data to classical processor to be analyzed Feed information forward If particles were attracting each other, they would be confined and none of this would be necessary. String tension in 2D? (Hama et al. 2008, Chesi et al. 2010) Errors would be thermally suppressed (keep system cool).

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 40 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Confinement

Current proposal Make millions of measurements. Send data to classical processor to be analyzed Feed information forward If particles were attracting each other, they would be confined and none of this would be necessary. String tension in 2D? (Hama et al. 2008, Chesi et al. 2010) Errors would be thermally suppressed (keep system cool).

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 40 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Simulated confinement

Simulated confinement Take snap shot of particle locations (x1, x2, ...) Sent this data to an external computer. Computer simulates gravitational V field as if particles had a mass. Determine how particles would move if they were massed. Move particles accordingly Locally simulated confinement Control unit at each site location to

1

Perform syndrome measurements (determine local charge)

2

Simulate a confining potential

Exchange messages with neighboring control units

3

Control neighboring qubits

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 41 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Control unit

Control unit holds value of local potential V and ∇V Measures presence of defect (syndrome) Updates potential ∇2V − ∂2

∂t2 V = −ρ

Move particles according to force F = −∇V

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 42 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Control unit

Control unit holds value of local potential V and ∇V Measures presence of defect (syndrome) Updates potential ∇2V − ∂2

∂t2 V = −ρ

Move particles according to force F = −∇V

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 42 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Control unit

Control unit holds value of local potential V and ∇V Measures presence of defect (syndrome) Updates potential ∇2V − ∂2

∂t2 V = −ρ

Move particles according to force F = −∇V

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 42 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Control unit

Control unit holds value of local potential V and ∇V Measures presence of defect (syndrome) Updates potential ∇2V − ∂2

∂t2 V = −ρ

Move particles according to force F = −∇V

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 42 / 44

2D Fault-Tolerant Quantum Cellular Automaton

Preliminary result

0.001 0.01 0.1 1 7 7.5 8 8.5 9

Decoding error probability Bit-Flip channel strength p%

l=25 l=50 l=100 l=200

Problem with EM over Z2 Problem with proper lattice scaling

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 43 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44

Conclusion Summary

Decoding problem: infer defect worldline homology from "snapshots" of their configuration. RG decoding algorithm

Fast : log ℓ time. Versatile (other codes, time/performance tradeoff). Higher threshold. Heuristic (Bravyi has proved a threshold... 10−22) Extends beyond 2D (Fault-tolerance)

Local equivalence between codes

Defines topological phases Universality of decoding algorithms Enhanced fault-tolerance ? All 2D stabilizer codes topologically equivalent to Kitaev. (Chiral?) Can decode some subsystem codes as well.

Possible fault-tolerant 2D quantum cellular automaton

Application to non-Abelian models

Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 44 / 44