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Thermalization time bounds for Pauli stabilizer Hamiltonians

Kristan Temme California Institute of Technology

QEC 2014, Zurich arXiv:1412.2858

tmix ≤ O(N 2e2✏)

Overview

• Motivation & previous results
• Mixing and thermalization
• The spectral gap bound
• Proof sketch

Thermalization in Kitaev’s 2D model

• Spectral gap bound for the 2D toric code and 1D Ising
• R. Alicki, M. Fannes, M. Horodecki J. Phys. A: Math. Theor. 42 (2009) 065303

Thermalization in Kitaev’s 2D model

• Spectral gap bound for the 2D toric code and 1D Ising
• R. Alicki, M. Fannes, M. Horodecki J. Phys. A: Math. Theor. 42 (2009) 065303

λ ≥ 1 3e−8βJ

Spectral gap bound:

Thermalization in Kitaev’s 2D model

• Spectral gap bound for the 2D toric code and 1D Ising
• R. Alicki, M. Fannes, M. Horodecki J. Phys. A: Math. Theor. 42 (2009) 065303

Sn Co d1 d2 c1 c2

λ ≥ 1 3e−8βJ

Spectral gap bound: Implies mixing time bound:

Thermalization in Kitaev’s 2D model

• Spectral gap bound for the 2D toric code and 1D Ising
• R. Alicki, M. Fannes, M. Horodecki J. Phys. A: Math. Theor. 42 (2009) 065303

Sn Co d1 d2 c1 c2

λ ≥ 1 3e−8βJ

Spectral gap bound: Implies mixing time bound:

tmix ≤ O(Ne8βJ)

The energy barrier

|ψ1i |ψ0i

The energy barrier

tmem ∼ eβEB

• Arrhenius law

|ψ1i |ψ0i

Phenomenological law of the lifetime

Bravyi, Sergey, and Barbara Terhal, J. Phys. 11 (2009) 043029 Olivier Landon-Cardinal, David Poulin Phys. Rev. Lett. 110, 090502 (2013)

The energy barrier

tmem ∼ eβEB

• Arrhenius law

|ψ1i |ψ0i

• Question:

Can we prove a connection between the energy barrier and thermalization ? Phenomenological law of the lifetime

Bravyi, Sergey, and Barbara Terhal, J. Phys. 11 (2009) 043029 Olivier Landon-Cardinal, David Poulin Phys. Rev. Lett. 110, 090502 (2013)

Stabilizer Hamiltonians

Kitaev, A.

• Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.

Example : Toric Code

A set of commuting Pauli matrices

Stabilizer Hamiltonians

Kitaev, A.

• Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.

Example : Toric Code

A set of commuting Pauli matrices The Stabilizer Group Logical operators

Stabilizer Hamiltonians

Kitaev, A.

• Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.

Example : Toric Code

A set of commuting Pauli matrices The Stabilizer Group Logical operators Stabilizer Hamiltonian

Open system dynamics

• Lindblad master equation

˙ ρ = L(ρ) = −i[H, ρ] + X

k

LkρL†

k − 1

2{L†

kLk, ρ}+

Open system dynamics

• Lindblad master equation

˙ ρ = L(ρ) = −i[H, ρ] + X

k

LkρL†

k − 1

2{L†

kLk, ρ}+

Open system dynamics

• Lindblad master equation

˙ ρ = L(ρ) = −i[H, ρ] + X

k

LkρL†

k − 1

2{L†

kLk, ρ}+

• With a unique fixed point

Thermal noise model & Weak coupling limit

Thermal noise model & Weak coupling limit

Thermal noise model & Weak coupling limit

ρS(t + ∆t) = trR[e−iH∆t(ρ(t) ⊗ ρR)eiH∆t]

The evolution :

Thermal noise model & Weak coupling limit

ρS(t + ∆t) = trR[e−iH∆t(ρ(t) ⊗ ρR)eiH∆t]

The evolution : Weak coupling limit & Markovian approximation:

Davies, E. B. (1974). Markovian master equations. Communications in Mathematical Physics, 39(2), 91–110.

The Davies generator

The Davies generator

* Kubo, R. (1957). Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. Journal of the Physical Society of Japan, 12(6), 570–586. Martin, P ., & Schwinger, J. (1959). Theory of Many-Particle Systems. I. Physical Review, 115(6), 1342–1373.

For a single thermal bath:

KMS conditions*: Ensures detail balance with: Gibbs state as steady state

σSα(ω) = eβωSα(ω)σ σ ∝ e−βHS

Davies generator for Pauli stabilizers

eiHtSαe−iHt = X

ω

Sα(ω)eiωt

• Lindblad operators

Davies generator for Pauli stabilizers

S↵(ω) = X

!=✏a−✏aα

σ↵

i P(a)

eiHtSαe−iHt = X

ω

Sα(ω)eiωt

• Lindblad operators
• Syndrome projectors

Davies generator for Pauli stabilizers

• The Lindblad operators are local!

(when the code is)

S↵(ω) = X

!=✏a−✏aα

σ↵

i P(a)

eiHtSαe−iHt = X

ω

Sα(ω)eiωt

• Lindblad operators
• Syndrome projectors

Davies generator for Pauli stabilizers

• The Lindblad operators are local!

(when the code is)

S↵(ω) = X

!=✏a−✏aα

σ↵

i P(a)

eiHtSαe−iHt = X

ω

Sα(ω)eiωt

• Lindblad operators
• Syndrome projectors

Convergence to the fixed point σ

t > tmix(✏) ⇒

keLt(⇢0) ktr  ✏

• For a unique fixed point:

Convergence to the fixed point σ

ketL(ρ0) σktr  Ae−Bt

t > tmix(✏) ⇒

keLt(⇢0) ktr  ✏

• For a unique fixed point:
• Exponential convergence

Convergence to the fixed point σ

t > tmix(✏) ⇒

keLt(⇢0) ktr  ✏

• For a unique fixed point:
• Exponential convergence

Temme, K., et al. "The χ2-divergence and mixing times of quantum Markov processes." Journal of Mathematical Physics 51.12 (2010): 122201.

Convergence to the fixed point σ

t > tmix(✏) ⇒

keLt(⇢0) ktr  ✏

tmix ∼ O(βNλ−1)

• For a unique fixed point:
• Exponential convergence
• A thermal σ implies the bound

kσ−1k ⇠ ecβN

= ⇒

Temme, K., et al. "The χ2-divergence and mixing times of quantum Markov processes." Journal of Mathematical Physics 51.12 (2010): 122201.

Spectral gap bound

Theorem 14 For any commuting Pauli Hamiltonian H, eqn. (1), the spectral gap λ of the Davies generator Lβ, c.f. eqn (15), with weight one Pauli couplings W1 is bounded by λ ≥ h∗ 4η∗ exp(−2β ), (81)

Spectral gap bound

The constants are:

Theorem 14 For any commuting Pauli Hamiltonian H, eqn. (1), the spectral gap λ of the Davies generator Lβ, c.f. eqn (15), with weight one Pauli couplings W1 is bounded by λ ≥ h∗ 4η∗ exp(−2β ), (81)

generalized energy barrier :

✏

smallest transition rate: The largest Pauli path:

η∗ = O(N)

Generalized energy barrier

Paths on the Pauli Group

Generalized energy barrier

Paths on the Pauli Group

Generalized energy barrier

Paths on the Pauli Group

Generalized energy barrier

Paths on the Pauli Group

Generalized energy barrier

Paths on the Pauli Group Reduced set of generators

Generalized energy barrier

Paths on the Pauli Group Reduced set of generators Energy barrier of the Pauli

Generalized energy barrier

Paths on the Pauli Group Reduced set of generators Energy barrier of the Pauli

Generalized energy barrier

Paths on the Pauli Group Reduced set of generators Energy barrier of the Pauli The generalized energy barrier

Generalized energy barrier

Paths on the Pauli Group Reduced set of generators Energy barrier of the Pauli The generalized energy barrier

Example: 2D Toric Code

3D Toric Code

H = −J X

v

Av − J X

p

Bp

Consider the toric code on an lattice

3D Toric Code

Consider the toric code on an lattice

3D Toric Code

Consider the toric code on an lattice leads to a bound

3D Toric Code

Consider the toric code on an lattice leads to a bound

High temperature bound

Fernando Pastawski Michael Kastoryano

Discussion of the bound

tmem ∼ eβEB

• Relationship to Arrhenius law

Discussion of the bound

tmem ∼ eβEB

• Relationship to Arrhenius law
• It would be nicer to have a bound that includes

“entropic contributions”

Discussion of the bound

tmem ∼ eβEB

• Relationship to Arrhenius law
• Can we get rid of the 1/N factor?
• It would be nicer to have a bound that includes

“entropic contributions”

Proof sketch

• The Poincare Inequality
• Matrix pencils and the PI
• The canonical paths bound
• The spectral gap and the energy barrier

The Poincare Inequality

λVarσ(f, f) ≤ E(f, f)

The Poincare Inequality

λ ⇣ tr ⇥ σf †f ⇤ − tr [σf]2⌘ ≤ −tr ⇥ σf †L(f) ⇤

The Poincare Inequality

• Sampling the Permanent :

For classical Markov processes

Cheeger’s bound Canonical paths

• M. Jerrum, A. Sinclair. "Approximating the permanent."

SIAM journal on computing 18.6 (1989): 1149-1178.

• Powerful because it can lead to a geometric interpretation

λ ⇣ tr ⇥ σf †f ⇤ − tr [σf]2⌘ ≤ −tr ⇥ σf †L(f) ⇤

The Poincare Inequality

• Sampling the Permanent :

For classical Markov processes Challenges in the quantum setting

Cheeger’s bound Canonical paths

• We are missing a general geometric picture
• M. Jerrum, A. Sinclair. "Approximating the permanent."

SIAM journal on computing 18.6 (1989): 1149-1178.

• Powerful because it can lead to a geometric interpretation

λ ⇣ tr ⇥ σf †f ⇤ − tr [σf]2⌘ ≤ −tr ⇥ σf †L(f) ⇤

Poincare and a Matrix pencil

λ−1 = τ

Varσ(f, f) = (f|ˆ V|f) E(f, f) = (f| ˆ E|f)

τ ˆ E − ˆ V ≥ 0

τ

minimize subject to

λVarσ(f, f) ≤ E(f, f)

Equivalent formulation for and where

Poincare and a Matrix pencil

λ−1 = τ

Varσ(f, f) = (f|ˆ V|f) E(f, f) = (f| ˆ E|f)

τ ˆ E − ˆ V ≥ 0

τ

minimize subject to

λVarσ(f, f) ≤ E(f, f)

Equivalent formulation for and where

AW = B

τ = min kWk2

ˆ E = AA†

ˆ V = BB†

Lemma: Let

subject to and

Boman, Erik G., and Bruce Hendrickson. "Support theory for preconditioning." SIAM Journal on Matrix Analysis and Applications 25.3 (2003): 694-717.

Suitable matrix factorization

L(f) ∼ X

i:αi

(σαi

i fσαi i

− f)

β → 0

E(f, f)

Some intuition from

Suitable matrix factorization

L(f) ∼ X

i:αi

(σαi

i fσαi i

− f)

β → 0

V(f) ∼ 1 4N X

γ

(σγ1

1 . . . σγN N fσγ1 1 . . . σγN N − f)

E(f, f)

Var(f, f)

Some intuition from

Suitable matrix factorization

L(f) ∼ X

i:αi

(σαi

i fσαi i

− f)

β → 0

V(f) ∼ 1 4N X

γ

(σγ1

1 . . . σγN N fσγ1 1 . . . σγN N − f)

E(f, f)

Var(f, f) (σx

1fσx 1 − f) + (σz 2fσz 2 − f) + (σx 3fσx 3 − f)

(σx

1σz 2σx 3 f σx 1σz 2σx 3 − f)

Choosing a decomposition in terms of

∼

Some intuition from

Suitable matrix factorization

L(f) ∼ X

i:αi

(σαi

i fσαi i

− f)

β → 0

V(f) ∼ 1 4N X

γ

(σγ1

1 . . . σγN N fσγ1 1 . . . σγN N − f)

E(f, f)

Var(f, f) (σx

1σz 2σx 3 f σx 1σz 2σx 3 − f)

(σx

1σz 2 f σx 1σz 2 − f) + (σx 3fσx 3 − f)

Choosing a decomposition in terms of

∼

Some intuition from

Suitable matrix factorization

L(f) ∼ X

i:αi

(σαi

i fσαi i

− f)

β → 0

V(f) ∼ 1 4N X

γ

(σγ1

1 . . . σγN N fσγ1 1 . . . σγN N − f)

E(f, f)

Var(f, f) (σx

1σz 2σx 3 f σx 1σz 2σx 3 − f)

(σx

1σz 2 f σx 1σz 2 − f) + (σx 3fσx 3 − f)

Choosing a decomposition in terms of

∼

Some intuition from A generalization yields to the matrix triple [A, B, W]

kWk2

can be bounded by suitable norm bounds

Canonical paths bound

Dressed Pauli paths :

• The norm bound on can be evaluated

in the following picture

kWk2

Canonical paths bound

Dressed Pauli paths :

• The norm bound on can be evaluated

in the following picture

kWk2

Canonical paths bound

Dressed Pauli paths :

• The norm bound on can be evaluated

in the following picture

kWk2

Canonical paths bound

τ ≤ max

ξ

4η∗ 2Nh(ωα(b))ρb X

ˆ ηa∈Γ(ξ)

ρaρaη

Dressed Pauli paths : The matrix norm bound yields

• The norm bound on can be evaluated

in the following picture

kWk2

The spectral gap and the energy barrier

• The only challenge is the maximum in the definition of τ

The spectral gap and the energy barrier

Φξ : Γ(ξ) → PN

Injective map (Jerrum & Sinclair)

• The only challenge is the maximum in the definition of τ

The spectral gap and the energy barrier

Φξ : Γ(ξ) → PN

[Φξ(ˆ ηa)]k = ⇢ (0, 0)k : k ≤ ξ ηk : k > ξ

Injective map (Jerrum & Sinclair)

• The only challenge is the maximum in the definition of τ

The spectral gap and the energy barrier

Φξ : Γ(ξ) → PN

[Φξ(ˆ ηa)]k = ⇢ (0, 0)k : k ≤ ξ ηk : k > ξ

Injective map (Jerrum & Sinclair) Bounding

✏bη⊕ξ + ✏bξ − ✏b − ✏bη ≤ 2✏

τ

• The only challenge is the maximum in the definition of τ

The spectral gap and the energy barrier

Φξ : Γ(ξ) → PN

[Φξ(ˆ ηa)]k = ⇢ (0, 0)k : k ≤ ξ ηk : k > ξ

Injective map (Jerrum & Sinclair) Bounding

h↵(ω↵(a))ρaρbΦξ(ˆ

ηb) ≥ h∗e−2✏ρbρbη

τ

• The only challenge is the maximum in the definition of τ

The spectral gap and the energy barrier

Φξ : Γ(ξ) → PN

[Φξ(ˆ ηa)]k = ⇢ (0, 0)k : k ≤ ξ ηk : k > ξ

Injective map (Jerrum & Sinclair) Bounding

h↵(ω↵(a))ρaρbΦξ(ˆ

ηb) ≥ h∗e−2✏ρbρbη

τ0 ≤ 4η∗ h∗ e2✏ max

ˆ ⇠

X

ˆ ⌘b∈Γ(ˆ ⇠)

1 2N ρbΦξ(ˆ

ηb)

τ

• The only challenge is the maximum in the definition of τ

The spectral gap and the energy barrier

Φξ : Γ(ξ) → PN

[Φξ(ˆ ηa)]k = ⇢ (0, 0)k : k ≤ ξ ηk : k > ξ

Injective map (Jerrum & Sinclair) Bounding

h↵(ω↵(a))ρaρbΦξ(ˆ

ηb) ≥ h∗e−2✏ρbρbη

τ0 ≤ 4η∗ h∗ e2✏ max

ˆ ⇠

X

ˆ ⌘b∈Γ(ˆ ⇠)

1 2N ρbΦξ(ˆ

ηb)

| {z }

≤ 1

τ

• The only challenge is the maximum in the definition of τ

Conclusion and Open Questions

• It would be great if one could extend the results

to more general quantum memory models.

• This only provides a converse to the lifetime of the classical
• memory. It would be great if one could find a converse for the

quantum memory time

• Can we get rid of the prefactor?
• Is it possible to find a bound that also takes the “entropic”

contributions into account?