A brief review of quantum annealing Hidetoshi Nishimori Tokyo - PowerPoint PPT Presentation

A brief review of quantum annealing Hidetoshi Nishimori Tokyo Institute of Technology

Combinatorial optimization Traveling salesman problem Minimize the cost function (tour length)

Combinatorial optimization Form ulation Minimize the cost function (a function of discrete variables) Ising model Cost function,, (classical) Hamiltonian = ∑ − σ σ σ = ± H J ( 1 ) 0 ij i j i

Simulated Annealing (SA) Search by thermal fluctuations − ∆ E / T e ↓↓ ↑↓ ↓↑ ↓↑

Quantum Annealing (QA) • Search by quantum fluctuations Quantum probability ↑↓ ↓↓ ↓↑ ↓↑

Questions  Does quantum annealing w ork? Yes.  Is it better than sim ulated annealing? Yes, in some sense … . 6 6

T vs Γ : Hopfield model ∑ p ∑ = − σ σ µ ξ µ = ξ J H J (Finite T ) ij i j ij i j µ = 1 p α = N Am it, Gutfreunt, Som polinsky (1985)

T vs Γ : Hopfield model ∑ ∑ = − σ σ − Γ σ = z z x H J ( T 0 ) ij i j i i Nishim ori & Nonom ura (1996)

Numerical evidence

Master eqn vs. Schrödinger eqn Random J ij , 8 spins 3 Γ = ( t ) t Schrödinger eqn. 3 = T ( t ) t Master eqn. Kadow aki & Nishim ori (1998) 10 10 10 10 10 10 10 10 10

Monte Carlo for TSP (1002 cities)  −  t t = +   = ⇒ τ = H ( t ) H 1 H H ( 0 ) H H ( ) H τ τ 0 quantum   quantum 0 Residual energy τ − H ( ) E true 11 11 11 11 11 11 Martonak, Santoro &Tosatti (2004)

Monte Carlo for 3SAT QA QA SA Battaglia, Santoro, Tosatti (2005) 12 12 12 12 12 12 12 12 12

Theoretical background 13

Convergence theorem Control parameter cN = T ( t ) ( Gem an-Gem an for SA) log t Γ = − c ' / N Morita & Nishim ori ( t ) t t ∑ ∑ = + = − σ σ − Γ σ z z x H H H J ( t ) 0 quantum ij i j i 14 14 14 14 14 14 14 14 14 14 14

Adiabatic computation 15 15

Quantum adiabatic computation E Farhi et al (2001) ∆ t τ 0 Trivial initial state Non-trivial final state  −  t t ∑ ∑ = − σ − σ σ   x z z H ( t ) 1 J τ τ i ij i j   16 16 16 16 16 16

Computational complexity E Finite-size analysis − τ ∝ ∆ 2 Adiabatic theorem τ 0  − aN t e 1 st order phase transition ∆ ∝ Gap scaling  − b  N 2 nd order phase transition  2 aN e (hard) Complexity τ ∝  2 b  N (easy) 17 17 17 17 17 17 17 17 17 17 17

Summary so far  QA works and is better than SA.  1 st order quantum transitions is problematic.  Question: What happens when there exists no classical phase transition but T there is a quantum transition? Γ c Γ 0 18 18 18 18 18 18 18 18 18 18 18 18 18

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Classical dynamics and quantum Hamiltonian Hidetoshi Nishimori Tokyo Institute of Technology 23 June 2014 Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 1 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Classical dynamics to quantum Hamiltonian ( ) (Classical) Ising model H 0 ( σ ) , σ = { σ 1 , σ 2 , · · · , σ N } Master equation (fixed T , single-spin flip) dP σ ( t ) ∑ = W σσ ′ P σ ′ ( t ) dt σ ′ Transverse-field Ising model 2 βH 0 ( σ ) W σσ ′ e − 1 1 2 βH 0 ( σ ′ ) H σσ ′ := − e Eigenvalue spectrum W, − H : λ 0 = 0 > − λ 1 > − λ 2 > · · · W : λ 1 = τ − 1 ( inverse relaxation time ; P ( t ) ∼ P eq + a e − λ 1 t ) H : λ 1 = ∆ cf. Castelnovo, Chamon, Mudry, and Pujol, Ann. Phys. (2005) Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 2 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Example 1d ferromagnetic Ising model N ∑ H 0 ( σ ) = − J σ j σ j +1 j =1 W of heat-bath dynamics (at fixed T ) is equivalent to: N H = − 1 ˆ ∑ σ z j σ z 2 tanh 2 K j +1 j =1 N 1 cosh 2 K − sinh 2 K σ z ∑ j − 1 σ z σ x ( ) − j +1 j 2 cosh 2 K j =1 Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 3 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Quantize it! Quantum Hamiltonian: Real symmetric (Hermitian) ( ˆ 1 2 βH 0 ( σ ) W σσ ′ e − 1 2 βH 0 ( σ ′ ) H σσ ′ = ) H σσ ′ = − e ⇒ H σσ ′ = H σ ′ σ ( ← detailed balance) Eigenvector and eigenvalue ψ ( R,n ) = − λ n ˆ 1 2 β ˆ We − 1 2 β ˆ W ˆ ˆ ˆ H 0 ˆ ψ ( R,n ) , H 0 H = − e φ ( n ) := e 2 β ˆ 1 H 0 ˆ ˆ ψ ( R,n ) φ ( n ) = λ n ˆ ⇒ ˆ H ˆ φ ( n ) = Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 6 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Matrix elements of ˆ H Off-diagonal 1 2 βH 0 ( σ ) W σσ ′ e − 1 2 βH 0 ( σ ′ ) = − w σσ ′ ( < 0) H σσ ′ = − e W σσ ′ = w σσ ′ e − 1 ( 2 β ( H 0( σ ) − H 0( σ ′ )) ) Diagonal w σσ ′ e − 1 ∑ 2 β ( H 0 ( σ ′ ) − H 0 ( σ )) H σσ = − W σσ = σ ′ ∈N ( σ ) Combined: operator representation ( ) w σσ ′ e − 1 ˆ ∑ ∑ 2 β ( H 0 ( σ ′ ) − H 0 ( σ )) | σ ⟩⟨ σ | − w σσ ′ | σ ′ ⟩⟨ σ | H = σ σ ′ ∈N ( σ ) Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 7 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Locality + single-spin flip Assume H 0 ( σ ) is local. ∑ ∑ ( ) H 0 ( σ ) = H j , H j = − h j σ j − σ j J jk σ k − · · · j k ∈N ( j ) Assume σ → σ ′ : σ j → − σ j (single-spin flip) H 0 ( σ ) − H 0 ( σ ′ ) = H j − ( − H j ) = 2 H j (local) Operator representation ( 1 ) ˆ ∑ ∑ 2 β ( H 0 ( σ ) − H 0 ( σ ′ )) | σ ⟩⟨ σ | − w σσ ′ | σ ′ ⟩⟨ σ | H = w σσ ′ e σ σ ′ ∈N ( σ ) ∑ w ( σ z j → − σ z e βH j I − σ x ( ) = j ) j j Local Hamiltonian! Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 8 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Example: 1d ferromagnetic Ising model Heat-bath dynamics N H = (const) − 1 ˆ ∑ σ z j σ z 2 tanh 2 K j +1 j =1 N 1 cosh 2 K − sinh 2 K σ z ∑ j − 1 σ z σ x ( ) − j +1 j 2 cosh 2 K j =1 Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 9 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Adaptive change of local transverse fields Transverse-field term N 1 cosh 2 K − sinh 2 K σ z ∑ ( j − 1 σ z ) σ x − j +1 j 2 cosh 2 K j =1 cosh 2 K − sinh 2 K σ z j − 1 σ z j +1 = 1 : Weak field j +1 = − 1 : cosh 2 K + sinh 2 K σ z j − 1 σ z Strong field Weak/strong field for desirable/undesirable configuration Adaptive transverse field → no phase transition (no transition in dynamics in 1d) cf. Uniform transverse field → quantum phase transition Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 10 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Simulated annealing with β as a function of t Master equation with ˆ W ( t ) d ˆ P ( t ) = ˆ W ( t ) ˆ P ( t ) dt 1 2 β ( t ) ˆ H 0 ˆ 1 2 β ( t ) ˆ H 0 ˆ W ( t ) e − 1 2 β ( t ) ˆ ˆ ˆ H 0 φ ( t ) := e P ( t ) , H ( t ) = − e Rewrite the master equation in terms of ˆ φ ( t ) and ˆ H ( t ) d ˆ φ ( t ) φ ( t ) + 1 = − ˆ H ( t )ˆ β ( t ) ˆ ˙ H 0 ˆ φ ( t ) dt 2 t → it : Schr¨ odinger equation id ˆ φ ( t ) H ( t ) − 1 ( ) ˆ β ( t ) ˆ ˙ ˆ = H 0 φ ( t ) dt 2 Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 11 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Quantum to classical: Construction of transition matrix Given ˆ ( ˆ H ) σσ ′ ≤ 0 ( σ ̸ = σ ′ ) H : ) 2 ˆ ∑ ∑ ( ∑ J ij σ z i σ z σ x σ x H = − j − Γ 1 i + Γ 2 excluded i ij i i φ (0) = 0 (ground state) H ˆ ˆ Shift the energy: φ (0) Perron-Frobenius: > 0 ( ∀ σ ) σ H 0 ( σ ) := − 2 ln φ (0) Define the Ising model: σ √ 2 β ˆ φ (0) = e − 1 cf. Classical to quantum: ˆ H 0 / Z Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 12 / 16

Overview Classical to quantum Time dependent ˆ W Quantum to classical Summary Quantum to classical (2) H 0 ( σ ) := − 2 ln φ (0) Define the Ising model: σ √ 2 β ˆ φ (0) = e − 1 cf. Classical to quantum: ˆ H 0 / Z Non-local ∑ ∑ H 0 ( σ ) = h j σ j + J ij σ i σ j + · · · + Jσ 1 σ 2 · · · σ N j ij W := − e − 1 2 ˆ 1 2 ˆ H 0 ˆ ˆ H 0 Transition matrix: He Hidetoshi Nishimori Classical dynamics and quantum Hamiltonians 13 / 16