Post on 20-May-2020
Two dimensional quantum memories
David Poulin
Département de PhysiqueUniversité de Sherbrooke
Collaborators H. Bombin, S. Bravyi, G. Duclos-Cianci, O. Landon-Cardinal, and B. Terhal
Institut transdisciplinaire d’informatique quantique, Bromont, April 2013
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 1 / 31
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 2 / 31
Check operators & local codes
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 3 / 31
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX 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 PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost 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 codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost 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 codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost 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 codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost 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 codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost 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 codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
Check operators & local codes
Lattice
l
l
Two-dimensional square latticePeriodic boundary conditions
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 9 / 31
Check operators & local codes
Kitaev’s code
X
XX
ZZ
ZZ
X
As : Bp : H = !!
s
As !!
p
Bp
Site operator:As =
∏i∈v(s) σ
ix
Plaquette operator:Bp =
∏i∈v(p) σ
iz
H = −(∑
s As +∑
p Bp)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
Check operators & local codes
Kitaev’s code
X
XX
ZZ
ZZ
X
As : Bp : H = !!
s
As !!
p
Bp
Site operator:As =
∏i∈v(s) σ
ix
Plaquette operator:Bp =
∏i∈v(p) σ
iz
H = −(∑
s As +∑
p Bp)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
Check operators & local codes
Kitaev’s code
X
XX
ZZ
ZZ
X
As : Bp : H = !!
s
As !!
p
Bp
Site operator:As =
∏i∈v(s) σ
ix
Plaquette operator:Bp =
∏i∈v(p) σ
iz
H = −(∑
s As +∑
p Bp)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
Check operators & local codes
Desirable features
Let |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|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 |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|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 |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|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 |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|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 Rsuch that R(TrMρ) = ρ for all code states ρ.M correctable⇔ No order parameter on M ⇔ ΠOMΠ ∝ Π.
Minimum distanceThe minimum distance d is the size of the smallest non-correctableregion.
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 Rsuch that R(TrMρ) = ρ for all code states ρ.M correctable⇔ No order parameter on M ⇔ ΠOMΠ ∝ Π.
Minimum distanceThe minimum distance d is the size of the smallest non-correctableregion.
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 Rsuch that R(TrMρ) = ρ for all code states ρ.M correctable⇔ No order parameter on M ⇔ ΠOMΠ ∝ Π.
Minimum distanceThe minimum distance d is the size of the smallest non-correctableregion.
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
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
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 ∂Mis also correctable. Then, there exists a unitary operator U∂M actingonly on the boundary of M such that, for any code state |ψ〉,
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉
for some fixed state |φM〉 on M.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 15 / 31
Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe 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
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
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 theminimum distance |∂M| ≤ cd .
Bulky errors are not problematic: it’s the skinny ones we need toworry 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 theminimum distance |∂M| ≤ cd .
Bulky errors are not problematic: it’s the skinny ones we need toworry 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 theminimum distance |∂M| ≤ cd .
Bulky errors are not problematic: it’s the skinny ones we need toworry 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
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
ABCD
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
ABCD
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
ABCD
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.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
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
Capacity-Stability Tradeoff
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 20 / 31
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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 qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
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
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
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-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
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 atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
Thermal instability
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 24 / 31
Thermal instability
Classical memories are robust
0=
1=
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
Thermal instability
Thermal stability vs spectral stabilisy
Main result (Landon-Cardinal & DP)The minimum set of conditions required to prove spectral stability implythe existence of a sequence of local maps that corrupt the system atan energy cost bounded by a constant.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 28 / 31
Thermal instability
Noise model
1 2 k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31