Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin...

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

Check operators & local codes

Outline

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

Classical codes

0→ 1 & 1→ 0

Encoding :0→ 0001→ 111

Receive 001→ 000

Classical codes

0→ 1 & 1→ 0

Encoding :0→ 0001→ 111

Receive 001→ 000

Classical codes

0→ 1 & 1→ 0

Encoding :0→ 0001→ 111

Receive 001→ 000

Classical codes

0→ 1 & 1→ 0

Encoding :0→ 0001→ 111

Receive 001→ 000

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

PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|

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 .

Quantum 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.

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

Definitions

∏X PX

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

Well known examples

Lattice

Two-dimensional square latticePeriodic boundary conditions

Kitaev’s code

As : Bp : H = !!

Site operator:As =

∏i∈v(s) σ

Plaquette operator:Bp =

∏i∈v(p) σ

H = −(∑

s As +∑

Kitaev’s code

As : Bp : H = !!

Site operator:As =

∏i∈v(s) σ

∏i∈v(p) σ

H = −(∑

s As +∑

Kitaev’s code

As : Bp : H = !!

Site operator:As =

∏i∈v(s) σ

∏i∈v(p) σ

H = −(∑

s As +∑

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.

Other 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.

Desirable features

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〉.

Correctable region

Logical operator

Correctable region

Logical operator

Holographic Disentangling Lemma

Outline

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.

With pictures

Let M be correctable.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〉 = |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.

With pictures

M = Λ\M

With pictures

M = Λ\M

With pictures

M = Λ\M

With pictures

M = Λ\M

With pictures

M = Λ\M

With pictures

M = Λ\M

Holographic Minimum Distance

Outline

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.

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 .

M = Λ\M

Capacity-Stability Tradeoff

Outline

n = number of qubitsk = number of encoded qubitsd = minimum distance

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√

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

k ≤ c nd2

2n log 3− H( d2n )].

String-Like Logical Operators

Outline

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.

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.

Thermal instability

Outline

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.

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.

Thermal instability

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

Thermal instability

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

Thermal instability

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

Thermal instability

| "" . . . "i

| ## . . . #i2B

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

Thermal instability

| "" . . . "i

| ## . . . #i2B

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

Thermal instability

| "" . . . "i

| ## . . . #i2B

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

Thermal instability

| "" . . . "i

| ## . . . #i2B

α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉

Unknown B:(

dB−−−→(|α|2 00 |β|2

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 = −∑

σzi σ

zi+1 + σz

The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?

Thermal instability

H = −∑

σzi σ

zi+1 + σz

Thermal instability

H = −∑

σzi σ

zi+1 + σz

Thermal instability

H = −∑

σzi σ

zi+1 + σz

Thermal instability

H = −∑

σzi σ

zi+1 + σz

Thermal instability

H = −∑

σzi σ

zi+1 + σz

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.

Thermal instability

Noise model

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)

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

Thermal instability

Noise model

Pk�1,k

If P23 = 0 go to 3.

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.

Conclusion

Take home messages

Conclusion

Take home messages

Conclusion

Take home messages

Conclusion

Take home messages

Conclusion

Take home messages

Conclusion

Take home messages

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).

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

Conclusion

Open questions

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