Check operators & local codes Quantum codes Set of states that obey a bunch of check conditions C = {| ψ � : P j | ψ � = | ψ � , ∀ j } There must be more than one state in C for the code to be interesting. We measure the check operators, eigenvalue � = + 1 indicates an error. Locality Because coherent measurement of checks requires coupling the qubits, we restrict the P j to couple only neighbouring qubits in some geometry. In 2D, this leads to topological codes. C = degenerate ground space of Hamiltonian H = − � j P j . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes Quantum codes Set of states that obey a bunch of check conditions C = {| ψ � : P j | ψ � = | ψ � , ∀ j } There must be more than one state in C for the code to be interesting. We measure the check operators, eigenvalue � = + 1 indicates an error. Locality Because coherent measurement of checks requires coupling the qubits, we restrict the P j to couple only neighbouring qubits in some geometry. In 2D, this leads to topological codes. C = degenerate ground space of Hamiltonian H = − � j P j . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes Quantum codes Set of states that obey a bunch of check conditions C = {| ψ � : P j | ψ � = | ψ � , ∀ j } There must be more than one state in C for the code to be interesting. We measure the check operators, eigenvalue � = + 1 indicates an error. Locality Because coherent measurement of checks requires coupling the qubits, we restrict the P j to couple only neighbouring qubits in some geometry. In 2D, this leads to topological codes. C = degenerate ground space of Hamiltonian H = − � j P j . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes Quantum codes Set of states that obey a bunch of check conditions C = {| ψ � : P j | ψ � = | ψ � , ∀ j } There must be more than one state in C for the code to be interesting. We measure the check operators, eigenvalue � = + 1 indicates an error. Locality Because coherent measurement of checks requires coupling the qubits, we restrict the P j to couple only neighbouring qubits in some geometry. In 2D, this leads to topological codes. C = degenerate ground space of Hamiltonian H = − � j P j . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes Quantum codes Set of states that obey a bunch of check conditions C = {| ψ � : P j | ψ � = | ψ � , ∀ j } There must be more than one state in C for the code to be interesting. We measure the check operators, eigenvalue � = + 1 indicates an error. Locality Because coherent measurement of checks requires coupling the qubits, we restrict the P j to couple only neighbouring qubits in some geometry. In 2D, this leads to topological codes. C = degenerate ground space of Hamiltonian H = − � j P j . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes Quantum codes Set of states that obey a bunch of check conditions C = {| ψ � : P j | ψ � = | ψ � , ∀ j } There must be more than one state in C for the code to be interesting. We measure the check operators, eigenvalue � = + 1 indicates an error. Locality Because coherent measurement of checks requires coupling the qubits, we restrict the P j to couple only neighbouring qubits in some geometry. In 2D, this leads to topological codes. C = degenerate ground space of Hamiltonian H = − � j P j . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Definitions Λ is a 2D lattice. Each vertex occupied by d -level quantum particle. Hamiltonian H = − � X ⊂ Λ P X with P X = 0 if radius( X ) ≥ w . [ P X , P Y ] = 0. P X are projectors (optional). Code C = { ψ : P X | ψ � = | ψ �} = ground space of H = image of code projector Π = � X P X With proper coarse graining, we can assume that Λ is a regular square lattice. Each P X acts on 2 × 2 cell. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes Well known examples Kitaev’s toric code Bombin’s topological color codes Levin & Wen’s string-net models Turaev-Viro models Kitaev’s quantum double models Most known models with topological quantum order David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes Well known examples Kitaev’s toric code Bombin’s topological color codes Levin & Wen’s string-net models Turaev-Viro models Kitaev’s quantum double models Most known models with topological quantum order David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes Well known examples Kitaev’s toric code Bombin’s topological color codes Levin & Wen’s string-net models Turaev-Viro models Kitaev’s quantum double models Most known models with topological quantum order David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes Well known examples Kitaev’s toric code Bombin’s topological color codes Levin & Wen’s string-net models Turaev-Viro models Kitaev’s quantum double models Most known models with topological quantum order David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes Well known examples Kitaev’s toric code Bombin’s topological color codes Levin & Wen’s string-net models Turaev-Viro models Kitaev’s quantum double models Most known models with topological quantum order David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes Well known examples Kitaev’s toric code Bombin’s topological color codes Levin & Wen’s string-net models Turaev-Viro models Kitaev’s quantum double models Most known models with topological quantum order David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes Lattice l Two-dimensional square lattice Periodic boundary conditions l David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 9 / 31
Check operators & local codes Kitaev’s code X X X X Site operator: i ∈ v ( s ) σ i A s = � Z x Z Z Plaquette operator: Z i ∈ v ( p ) σ i B p = � z H = − ( � s A s + � p B p ) � � A s : B p : H = − A s − B p s p David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
Check operators & local codes Kitaev’s code X X X X Site operator: i ∈ v ( s ) σ i A s = � Z x Z Z Plaquette operator: Z i ∈ v ( p ) σ i B p = � z H = − ( � s A s + � p B p ) � � A s : B p : H = − A s − B p s p David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
Check operators & local codes Kitaev’s code X X X X Site operator: i ∈ v ( s ) σ i A s = � Z x Z Z Plaquette operator: Z i ∈ v ( p ) σ i B p = � z H = − ( � s A s + � p B p ) � � A s : B p : H = − A s − B p s p David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Other codes Motivation Aharonov & Eldar ’11: Topological order requires 4-qubit commuting checks. Low-weight non-commuting checks possible? Less error-prone Bombin ’10, Topological subsystem colour codes Weight=2. Low threshold. Bravyi, Duclos-Cianci, DP , Suchara Weight = 3. High threshold. Surface with boundaries. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes Desirable features Let | ψ 1 � and | ψ 2 � be two code states (ground states). Suppose there exists a local (e.g. single spin) measurement σ that distinguishes them. Then the environment can also learn which state is encoded by “looking" at a single spin. � | ψ 1 � with prob . | α | 2 α | ψ 1 � + β | ψ 2 � → with prob . | β | 2 | ψ 2 � So a code should not have such local “order parameter" : all codes states should look identical locally. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
Check operators & local codes Desirable features Let | ψ 1 � and | ψ 2 � be two code states (ground states). Suppose there exists a local (e.g. single spin) measurement σ that distinguishes them. Then the environment can also learn which state is encoded by “looking" at a single spin. � | ψ 1 � with prob . | α | 2 α | ψ 1 � + β | ψ 2 � → with prob . | β | 2 | ψ 2 � So a code should not have such local “order parameter" : all codes states should look identical locally. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
Check operators & local codes Desirable features Let | ψ 1 � and | ψ 2 � be two code states (ground states). Suppose there exists a local (e.g. single spin) measurement σ that distinguishes them. Then the environment can also learn which state is encoded by “looking" at a single spin. � | ψ 1 � with prob . | α | 2 α | ψ 1 � + β | ψ 2 � → with prob . | β | 2 | ψ 2 � So a code should not have such local “order parameter" : all codes states should look identical locally. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
Check operators & local codes Desirable features Let | ψ 1 � and | ψ 2 � be two code states (ground states). Suppose there exists a local (e.g. single spin) measurement σ that distinguishes them. Then the environment can also learn which state is encoded by “looking" at a single spin. � | ψ 1 � with prob . | α | 2 α | ψ 1 � + β | ψ 2 � → with prob . | β | 2 | ψ 2 � So a code should not have such local “order parameter" : all codes states should look identical locally. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
Check operators & local codes Standard definitions Correctable region A region M ⊂ Λ is correctable if there exists a recovery operation R such that R ( Tr M ρ ) = ρ for all code states ρ . M correctable ⇔ No order parameter on M ⇔ Π O M Π ∝ Π . Minimum distance The minimum distance d is the size of the smallest non-correctable region. Logical operator Operator L such that L | ψ � is a code state for any code state | ψ � . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 13 / 31
Check operators & local codes Standard definitions Correctable region A region M ⊂ Λ is correctable if there exists a recovery operation R such that R ( Tr M ρ ) = ρ for all code states ρ . M correctable ⇔ No order parameter on M ⇔ Π O M Π ∝ Π . Minimum distance The minimum distance d is the size of the smallest non-correctable region. Logical operator Operator L such that L | ψ � is a code state for any code state | ψ � . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 13 / 31
Check operators & local codes Standard definitions Correctable region A region M ⊂ Λ is correctable if there exists a recovery operation R such that R ( Tr M ρ ) = ρ for all code states ρ . M correctable ⇔ No order parameter on M ⇔ Π O M Π ∝ Π . Minimum distance The minimum distance d is the size of the smallest non-correctable region. Logical operator Operator L such that L | ψ � is a code state for any code state | ψ � . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 13 / 31
Holographic Disentangling Lemma Outline Check operators & local codes 1 Holographic Disentangling Lemma 2 Holographic Minimum Distance 3 Capacity-Stability Tradeoff 4 String-Like Logical Operators 5 Thermal instability 6 David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 14 / 31
Holographic Disentangling Lemma Statement of the lemma Holographic disentangling lemma (Bravyi, DP , Terhal) Let M ⊂ Λ be a correctable region and suppose that its boundary ∂ M is also correctable. Then, there exists a unitary operator U ∂ M acting only on the boundary of M such that, for any code state | ψ � , U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � for some fixed state | φ M � on M . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 15 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. A B C Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . D M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. A B C Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . D M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. A B C Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . D M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. A B C Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . D M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Disentangling Lemma With pictures Let M be correctable. M Assume ∂ M is correctable. A B C Let M = A ∪ B , M = C ∪ D , and ∂ M = B ∪ C . D M = Λ \M There exists a unitary transformation U ∂ M such that, for any | ψ � ∈ C U ∂ M | ψ � = | φ M � ⊗ | ψ ′ M � where | φ M � is the same for all | ψ � . Remark For a trivial code Tr Π = 1, every region is correctable, so we recover the area law S ( M ) ≤ | ∂ M | for commuting Hamiltonians of Wolf, Verstraete, Hastings, and Cirac. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
Holographic Minimum Distance Outline Check operators & local codes 1 Holographic Disentangling Lemma 2 Holographic Minimum Distance 3 Capacity-Stability Tradeoff 4 String-Like Logical Operators 5 Thermal instability 6 David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 17 / 31
Holographic Minimum Distance Statement of the result Holographic minimum distance (Bravyi, DP , Terhal) Region M ⊂ Λ is correctable if its boundary is smaller than the minimum distance | ∂ M | ≤ cd . Bulky errors are not problematic: it’s the skinny ones we need to worry about. This hints at our next result: string-like logical operators. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 18 / 31
Holographic Minimum Distance Statement of the result Holographic minimum distance (Bravyi, DP , Terhal) Region M ⊂ Λ is correctable if its boundary is smaller than the minimum distance | ∂ M | ≤ cd . Bulky errors are not problematic: it’s the skinny ones we need to worry about. This hints at our next result: string-like logical operators. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 18 / 31
Holographic Minimum Distance Statement of the result Holographic minimum distance (Bravyi, DP , Terhal) Region M ⊂ Λ is correctable if its boundary is smaller than the minimum distance | ∂ M | ≤ cd . Bulky errors are not problematic: it’s the skinny ones we need to worry about. This hints at our next result: string-like logical operators. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 18 / 31
Holographic Minimum Distance Proof M Let M ⊂ Λ be a correctable region. If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M Let M ⊂ Λ be a correctable region. If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M A Let M ⊂ Λ be a correctable region. B C D If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M A Let M ⊂ Λ be a correctable region. B C D If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M A Let M ⊂ Λ be a correctable region. B C D If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M Let M ⊂ Λ be a correctable region. If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M Let M ⊂ Λ be a correctable region. If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Holographic Minimum Distance Proof M Let M ⊂ Λ be a correctable region. If | ∂ M | ≤ d , then ∂ M is also correctable. M = Λ \M Thus, we can reconstruct any code state ρ from ρ AD = Tr ∂ M ρ . But from the Holographic disentangling lemma, ρ AD = η A ⊗ ρ D with η A independent of the encoded state ρ . Thus, we can reconstruct ρ from ρ D = Tr M ∪ ∂ M ρ , so M ∪ ∂ M is correctable. We can continue to grow M this way until | ∂ M | ≥ d . David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Capacity-Stability Tradeoff Outline Check operators & local codes 1 Holographic Disentangling Lemma 2 Holographic Minimum Distance 3 Capacity-Stability Tradeoff 4 String-Like Logical Operators 5 Thermal instability 6 David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 20 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
Capacity-Stability Tradeoff Statement of the result n = number of qubits k = number of encoded qubits d = minimum distance Capacity-Stability Tradeoff k ≤ c n d 2 Singleton’s bound: k ≤ n − 2 ( d − 1 ) . � � 1 − d 2 n log 3 − H ( d Hamming bound: k ≤ n 2 n ) . Kitaev’s codes (with punctures) saturate this bound, so it is tight. No “good codes" in 2D, i.e. k ∝ n and d ∝ n . For 2D classical codes, k ≤ c n d . √ David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
String-Like Logical Operators Outline Check operators & local codes 1 Holographic Disentangling Lemma 2 Holographic Minimum Distance 3 Capacity-Stability Tradeoff 4 String-Like Logical Operators 5 Thermal instability 6 David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 22 / 31
String-Like Logical Operators Statement of the result String-like logical operators (Haah, Preskill) There exists a non-trivial logical operator supported on a string-like region. Exists U M such that U M | ψ � = | ψ ′ � . M | ψ � � = | ψ ′ � . | ψ � , | ψ ′ � ∈ C . Λ Well known for Kitaev’s toric code. Intuitive for known models that support anyons: The ground state can be changed by dragging an anyon around a topologically non-trivial loop. This process is realized on a string, and generated a logical operation. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
String-Like Logical Operators Statement of the result String-like logical operators (Haah, Preskill) There exists a non-trivial logical operator supported on a string-like region. Exists U M such that U M | ψ � = | ψ ′ � . M | ψ � � = | ψ ′ � . | ψ � , | ψ ′ � ∈ C . Λ Well known for Kitaev’s toric code. Intuitive for known models that support anyons: The ground state can be changed by dragging an anyon around a topologically non-trivial loop. This process is realized on a string, and generated a logical operation. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
String-Like Logical Operators Statement of the result String-like logical operators (Haah, Preskill) There exists a non-trivial logical operator supported on a string-like region. Exists U M such that U M | ψ � = | ψ ′ � . M | ψ � � = | ψ ′ � . | ψ � , | ψ ′ � ∈ C . Λ Well known for Kitaev’s toric code. Intuitive for known models that support anyons: The ground state can be changed by dragging an anyon around a topologically non-trivial loop. This process is realized on a string, and generated a logical operation. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
String-Like Logical Operators Statement of the result String-like logical operators (Haah, Preskill) There exists a non-trivial logical operator supported on a string-like region. Exists U M such that U M | ψ � = | ψ ′ � . M | ψ � � = | ψ ′ � . | ψ � , | ψ ′ � ∈ C . Λ Well known for Kitaev’s toric code. Intuitive for known models that support anyons: The ground state can be changed by dragging an anyon around a topologically non-trivial loop. This process is realized on a string, and generated a logical operation. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
String-Like Logical Operators Statement of the result String-like logical operators (Haah, Preskill) There exists a non-trivial logical operator supported on a string-like region. Exists U M such that U M | ψ � = | ψ ′ � . M | ψ � � = | ψ ′ � . | ψ � , | ψ ′ � ∈ C . Λ Well known for Kitaev’s toric code. Intuitive for known models that support anyons: The ground state can be changed by dragging an anyon around a topologically non-trivial loop. This process is realized on a string, and generated a logical operation. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
String-Like Logical Operators Statement of the result String-like logical operators (Haah, Preskill) There exists a non-trivial logical operator supported on a string-like region. Exists U M such that U M | ψ � = | ψ ′ � . M | ψ � � = | ψ ′ � . | ψ � , | ψ ′ � ∈ C . Λ Well known for Kitaev’s toric code. Intuitive for known models that support anyons: The ground state can be changed by dragging an anyon around a topologically non-trivial loop. This process is realized on a string, and generated a logical operation. David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
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