Thermalization time bounds for Pauli stabilizer Hamiltonians Kristan Temme California Institute of Technology arXiv:1412.2858 QEC 2014, Zurich
t mix ≤ O ( N 2 e 2 �✏ )
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 Spectral gap bound: 3 e − 8 β J
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 c 1 d 1 Sn Co λ ≥ 1 Spectral gap bound: 3 e − 8 β J Implies mixing time bound: c 2 d 2
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 c 1 d 1 Sn Co λ ≥ 1 Spectral gap bound: 3 e − 8 β J Implies mixing time bound: c 2 t mix ≤ O ( Ne 8 β J ) d 2
The energy barrier | ψ 0 i | ψ 1 i
The energy barrier • Arrhenius law t mem ∼ e β E B Phenomenological law of the lifetime Bravyi, Sergey, and Barbara Terhal, J. Phys. 11 (2009) 043029 | ψ 0 i | ψ 1 i Olivier Landon-Cardinal, David Poulin Phys. Rev. Lett. 110, 090502 (2013)
The energy barrier • Arrhenius law t mem ∼ e β E B Phenomenological law of the lifetime Bravyi, Sergey, and Barbara Terhal, J. Phys. 11 (2009) 043029 | ψ 0 i | ψ 1 i Olivier Landon-Cardinal, David Poulin Phys. Rev. Lett. 110, 090502 (2013) • Question: Can we prove a connection between the energy barrier and thermalization ?
Stabilizer Hamiltonians Example : Toric Code A set of commuting Pauli matrices Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.
Stabilizer Hamiltonians Example : Toric Code A set of commuting Pauli matrices The Stabilizer Group Logical operators Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.
Stabilizer Hamiltonians Example : Toric Code A set of commuting Pauli matrices The Stabilizer Group Logical operators Stabilizer Hamiltonian Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.
Open system dynamics • Lindblad master equation k − 1 X L k ρ L † 2 { L † ρ = L ( ρ ) = − i [ H, ρ ] + ˙ k L k , ρ } + k
Open system dynamics • Lindblad master equation k − 1 X L k ρ L † 2 { L † ρ = L ( ρ ) = − i [ H, ρ ] + ˙ k L k , ρ } + k
Open system dynamics • Lindblad master equation k − 1 X L k ρ L † 2 { L † ρ = L ( ρ ) = − i [ H, ρ ] + ˙ k L k , ρ } + k • With a unique fixed point
Thermal noise model & Weak coupling limit
Thermal noise model & Weak coupling limit
Thermal noise model & Weak coupling limit The evolution : ρ S ( t + ∆ t ) = tr R [ e − iH ∆ t ( ρ ( t ) ⊗ ρ R ) e iH ∆ t ]
Thermal noise model & Weak coupling limit The evolution : ρ S ( t + ∆ t ) = tr R [ e − iH ∆ t ( ρ ( t ) ⊗ ρ R ) e iH ∆ t ] 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 For a single thermal bath: KMS conditions*: σ S α ( ω ) = e βω S α ( ω ) σ Ensures detail balance with: σ ∝ e − β H S Gibbs state as steady state * 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.
Davies generator for Pauli stabilizers Lindblad operators e iHt S α e − iHt = • X S α ( ω ) e i ω t ω
Davies generator for Pauli stabilizers Lindblad operators e iHt S α e − iHt = • X S α ( ω ) e i ω t ω Syndrome projectors X • S ↵ ( ω ) = σ ↵ i P ( a ) ! = ✏ a − ✏ a α
Davies generator for Pauli stabilizers Lindblad operators e iHt S α e − iHt = • X S α ( ω ) e i ω t ω Syndrome projectors X • S ↵ ( ω ) = σ ↵ i P ( a ) ! = ✏ a − ✏ a α The Lindblad operators are local! • (when the code is)
Davies generator for Pauli stabilizers Lindblad operators e iHt S α e − iHt = • X S α ( ω ) e i ω t ω Syndrome projectors X • S ↵ ( ω ) = σ ↵ i P ( a ) ! = ✏ a − ✏ a α The Lindblad operators are local! • (when the code is)
Convergence to the fixed point σ • For a unique fixed point: k e L t ( ⇢ 0 ) � � k tr ✏ t > t mix ( ✏ ) ⇒
Convergence to the fixed point σ • For a unique fixed point: k e L t ( ⇢ 0 ) � � k tr ✏ t > t mix ( ✏ ) ⇒ • Exponential convergence k e t L ( ρ 0 ) � σ k tr Ae − Bt
Convergence to the fixed point σ • For a unique fixed point: k e L t ( ⇢ 0 ) � � k tr ✏ t > t mix ( ✏ ) ⇒ • Exponential convergence Temme, K., et al. "The χ 2-divergence and mixing times of quantum Markov processes." Journal of Mathematical Physics 51.12 (2010): 122201.
Convergence to the fixed point σ • For a unique fixed point: k e L t ( ⇢ 0 ) � � k tr ✏ t > t mix ( ✏ ) ⇒ • Exponential convergence Temme, K., et al. "The χ 2-divergence and mixing times of quantum Markov processes." Journal of Mathematical Physics 51.12 (2010): 122201. • A thermal σ implies the bound t mix ∼ O ( β N λ − 1 ) k σ − 1 k ⇠ e c β N ⇒ =
Spectral gap bound Theorem 14 For any commuting Pauli Hamiltonian H , eqn. (1), the spectral gap λ of the Davies generator L β , c.f. eqn (15), with weight one Pauli couplings W 1 is bounded by λ ≥ h ∗ 4 η ∗ exp( − 2 β � ) , (81)
Spectral gap bound Theorem 14 For any commuting Pauli Hamiltonian H , eqn. (1), the spectral gap λ of the Davies generator L β , c.f. eqn (15), with weight one Pauli couplings W 1 is bounded by λ ≥ h ∗ 4 η ∗ exp( − 2 β � ) , (81) The constants are: η ∗ = O ( N ) The largest Pauli path: smallest transition rate: generalized energy barrier : ✏
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 Consider the toric code on an lattice X X H = − J A v − J B p v p
3D Toric Code Consider the toric code on an lattice
3D Toric Code Consider the toric code on an lattice leads to a bound
Fernando Pastawski 3D Toric Code Michael Kastoryano Consider the toric code on an lattice leads to a bound High temperature bound
Discussion of the bound Relationship to Arrhenius law • t mem ∼ e β E B
Discussion of the bound Relationship to Arrhenius law • t mem ∼ e β E B It would be nicer to have a bound that includes • “entropic contributions”
Discussion of the bound Relationship to Arrhenius law • t mem ∼ e β E B It would be nicer to have a bound that includes • “entropic contributions” Can we get rid of the 1/N factor? •
Proof sketch • The Poincare Inequality • Matrix pencils and the PI • The canonical paths bound • The spectral gap and the energy barrier
The Poincare Inequality λ Var σ ( f, f ) ≤ E ( f, f )
The Poincare Inequality ⇣ − tr [ σ f ] 2 ⌘ σ f † f σ f † L ( f ) ⇥ ⇤ ⇥ ⇤ tr ≤ − tr λ
The Poincare Inequality ⇣ − tr [ σ f ] 2 ⌘ σ f † f σ f † L ( f ) ⇥ ⇤ ⇥ ⇤ tr ≤ − tr λ For classical Markov processes Sampling the Permanent : • 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 • Cheeger’s bound Canonical paths
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