Spin-Glass Bottlenecks in Quantum Annealing Sergey Knysh SGT Inc., NASA Ames Research Center Nature Communications 7, 12370 (2016).
Quantum Adiabatic Annealing H e u r i s t i c a l g o r i t h m f o r t a c k l i n g N P - c o m p l e t e p r o b l e m s . Kadowaki & Nishimori, PRE '98 H = − 1 2 ∑ z − ∑ z −Γ( t ) ∑ z σ k x J ik σ i h i σ i σ i Farhi et al. , Science '01 i ,k i i spin-flip dynamics objective function Transverse field slowly decreased to zero. Γ( t ) ∣ Ψ ( 0 ) 〉 = 1 N / 2 ∑ ∣ Ψ ( T ) 〉 = ∣ s m i n 〉 s ∈{± 1 } N ∣ s 〉 Ground state interpolates from to 2 Adiabatic condition d Γ/ d t ≪Δ E ⋅Δ Γ F o r L a n d a u - Z e n e r c r o s s i n g Δ E ∼Δ Γ Gap closes at QCP in thermodynamic limit. Finite-size scaling gives average-case complexity. − N / 2 Example: 1 s t order phase transition in REM Δ E c ∼ 2
Continuous Phase Transition Critical scaling at 2 nd order QCP Normalized GSE ( sing ) / N ∼ | γ a | E 0 (singular component): b E 1 − E 0 ∼γ Gap in PM phase: γ=Γ−Γ c Finite-size scaling: − b a − b Δ E c ∼ N 1 − a − b Δ Γ c ∼ N Polynomial annealing rate avoids QCP bottleneck. − 1 /ν ∼ N − 1 /( d ν) ΔΓ c ∼ξ a = 2 −α=( d + z )ν 1 / d ξ∼ L = N b = z ν − z ∼ N − z / d Δ E c ∼ξ
Exceptions to Polynomial Scaling Disorder J k,k + 1 J k + 1 , k + 2 ⋯⋯⋯ 1D chain with i.i.d. random J k,k + 1 1 Δ E c ∼ e − c √ N “Finite-size” critical field Γ c ≈ ( J 12 J 23 ⋯ J n − 1, n ) n − 1 Different parts of the system become critical at different times Slow dynamics as clusters of spins are flipped Not an issue with all-to-all connectivity “Fixable” by synchronizing phase transitions with local Γ i Frustration 1D loop with odd number of antiferromagnetic couplings “Competition” between solutions 2 < IK I < J < K J Develops exponentially small gap Polynomial gap at Γ c = K in the ordered phase, Γ<Γ c 2 − J 2 )( J 2 − I 2 ) ( K Γ * = 1 Exponential gap at I 2 + K 2 − 2 J 2 I
Spin-Glass Bottlenecks Spin-glass phase characterized by many valleys Santoro et al ., Science '02 Altshuler et al. , PNAS '10 Energy levels “reshuffled” as Γ changes Farhi et al ., PRE '12 But: Ground state is less sensitive (extreme value) Effect of the Transverse Field d spin flips “Smoothes out” energy landscapes on scales ~Γ Lowers energy of wide valleys −Γ d Deep-and-narrow and shallow-and-wide valleys can come into resonance tunneling Fractal Energy Landscapes N h.b. =α ln Γ c No intrinsic scale (Γ≪Γ c ) Γ min Expected # of hard bottlenecks Γ min ∼ 1 N h.b. [Γ 1, Γ 2 ]= f (Γ 2 /Γ 1 ) Γ c ∼ 1 δ N Additivity: N h.b. [Γ 1 ; Γ 2 ]= N h.b. [Γ 1 ; Γ ' ]+ N h.b. [Γ ' ; Γ 2 ]
Associative Memory: Hopfield Network Nishimori & Nonomura, JPSJ '96 ( 1 ) ={ 1, − 1, − 1, … , 1 } ξ i C r a f t H a m i l t o n i a n e n c o d i n g p ` p a t t e r n s ' ( 2 ) ={− 1,1, − 1, … , − 1 } ξ i ⋯⋯⋯⋯⋯ p J ik = 1 N ∑ (μ) ξ k (μ) ξ i (μ) min =±ξ i s i μ= 1 Small p : `project' onto patterns m = 1 N ∑ ⃗ z ⟩ ξ i ⟨ ^ ⃗ σ i i Barriers are O ( N ) Classical (Γ=0) gap is O (1) attractors, ± O ( 1 ) Δ Γ c ∼ N − 2 / 3 − 1 / 3 Δ E c ∼ N QCP is the only bottleneck: , Capacity limit: p = O ( N ) m i n = ± s g n ∑ (μ) μ α μ ξ i s i Spurious states become globally stable: Smaller barriers; classical gap vanishes asymptotically
Hopfield Model with Gaussian Patterns Spurious states appear for p ≥2 Classical gap is O ( 1 / N ) Barriers are O ( √ N ) Mean Field Theory Finite-temperature partition function β 1 2 ∫ ( ∑ 2 d t + ∑ ⃗ ξ i s i ( t )) K [ s i ( t )] Z (β)= ∑ degenerate to O ( N ) e 0 i i J ik = 1 [ { s i ( t ) } ] ξ i ⃗ ⃗ (# of kinks) × 1 ξ k 2 ln tanh (Γ Δ t ) N Rewrite as a path integral using Hubbard-Stratonovich β − N 2 ∫ 2 ( t ) d t + ∑ ⃗ m ln Z i Z (β)= ∫ [ d ⃗ 1 2 2 ( ∑ ⃗ m ( t )] e ξ i s i ) 0 i ∝ ∫ d ⃗ m ⃗ 2 / 2 + ⃗ −⃗ ξ i s i m e m e i Single-site partition function β β ∫ ∫ ( h i ( t ) ^ z +Γ ^ x ) d t h i ( t ) s ( t ) d t + K [ s ( t )] σ σ Z i = ∑ = Tr Τ e e h i ( t )=⃗ 0 0 ξ ⃗ m ( t ) [ s ( t )]
Mapping to Ordinary Quantum Mechanics ● Saddle-point solution is stationary s 1 h i ̂ σ z h i m = 1 s k N ∑ ⃗ h i =⃗ ⃗ ξ i ξ i ⃗ m √ Γ s i 2 + h i 2 Γ ̂ σ x i replace sum by s N disorder average m ( t )≈ m Γ ( − sin ϑ( t ) cos ϑ( t ) ) ● Finite- N corrections: path integral is dominated by ⃗ ● is slow-varying ϑ( t ) β ( √ Γ 2 ) ) d t ln Z i = ∫ 2 + h i 2 ( t )+ O ( ( d h i / d t ) 0 ● Disorder realization – dependent partition function non-adiabatic corrections β − ∫ 2 / 2 + V Γ (ϑ) ) d t ( M ( d ϑ/ d t ) − N β⟨ F ⟩ ∫ [ d ⃗ Z (β)= e ϑ( t )] e 0 ● Low energy spectrum is equivalent to that of a particle on a ring ξ i =ξ i ( cos θ i sin θ i ) ⃗ 2 ξ i i √ Γ 2 + m Γ V Γ (ϑ)=− ∑ 2 sin 2 (ϑ−θ i )+ N ⟨ √ ⋯⟩
Evolution of Random Potential i √ Γ V Γ (ϑ)=− ∑ 2 +[ m Γ ξ i sin (ϑ−θ i )] 2 + N ⟨ √ ⋯⟩ Scales as (central limit theorem) √ N Smooth near critical point 1 √ N V Γ (ϑ)= C + ∑ ( A k cos2 k ϑ+ B k sin2 k ϑ) 2 k A k ,B k = m k 2 k − 1 Γ Becomes increasingly rugged for small Γ Continuous Process Orthogonalize correlated 2D random process ∞ ∫ f Γ ⟨ζ n (θ)ζ n' (θ ' )⟩=δ nn' δ(θ−θ ' ) V Γ (ϑ)= ∑ ( n ) (ϑ−θ)ζ n (θ) d θ white noise n = 0 f n (ϑ) Choose to match covariance ⟨ V Γ (ϑ) V Γ ' (ϑ ' )⟩ ∞ √ Γ 2 +⋯×ξ 2 e −ξ ( n ) (ϑ)∝ ∫ 2 / 2 L n ( 1 ) (ξ 2 / 2 ) d ξ Use orthogonal polynomials (Laguerre) f Γ 0
Evolution of Random Potential (cont'd) ∞ 1 √ N V Γ (θ)∝( F Γ ∗χ)(ϑ)+ ∑ ( n ) ∗η n )(ϑ)+ const ( G Γ n = 1 smoothing kernel classical potential brownian motion of width Γ Convolution with raises energy of narrow valleys F Γ (ϑ) 2 nd term vanishes for Γ=0; comparable contribution for Γ>0 Classical potential Neglect near a global minimum 2 χ θ d √ N Γ 3 / 2 V 2 +χ=ζ 0 (ϑ) d ϑ Δθ∼Γ 3 / 2 Γ Δ ϕ∼Γ Condition on the fact that χ(ϑ)≥χ(θ * )=χ * ϕ θ Without losing generality ϑ * = 0, χ * = 0
Classical Potential near Global Minimum υ= d χ ● Markov process in `time' ( is the `velocity') (χ , υ) ϑ d ϑ 2 p ∂ p ∂ϑ +υ ∂ p ∂ ∂χ− 1 2 = 0 2 ∂υ χ→+ 0 p (θ ; χ , υ)= 0 for υ> 0 lim ● Only include paths with : χ≥ 0 ● Renormalize probability so that it is conserved q (ϑ ; χ , υ)∝ p (ϑ ; χ , υ) ∫ P (Θ ; Χ , Υ | ϑ ; χ , υ) d Χ d Υ Χ> 0 survival probability P Θ (χ , υ) ● Before: p (Δ υ> 0 )= p (Δ υ< 0 )= 1 / 2 ● After: (the process with more likely to survive) υ ' >υ p (Δ υ> 0 )> 1 / 2 > p (Δ υ< 0 ) ∂υ ( 1 ∂ υ q ) ∂ P Θ + ∂ ● Probability is conserved but adds repulsion: P Θ
“Stationary” Solution Green's function satisfies time-reversed PDE P (Θ ; Χ , Υ | ϑ ; χ , υ)= P (ϑ ; χ , −υ | Θ ; Χ , −Υ) Asymptotic form (independent of initial conditions): p (ϑ ; χ , υ)∼ A p * (χ , υ) α ϑ 3 / 2 , [υ]=[ϑ] 1 / 2 [χ]=[ϑ] Dimensional analysis: 2 α/ 3 p * (υ/χ 1 / 3 ) p * (χ , υ)=χ α= 1 4 + 3 n ODE for yields quantized eigenvalues p * (ν) 2 for n ≥ 0 Dimensionless `time' − 2 / 3 d ϑ d τ=χ Dimensionless `velocity' 1 / 3 ν=υ/χ Dimensionless `coordinate' μ= ln χ τ Regard as a Markov process in `time' (ϑ , χ , υ)
Langevin Process ● PDE after the change of variables ∂ ν ( ∂ U q ) − 1 2 ¯ ∂¯ q 2 μ / 3 ∂ ¯ ∂ϑ +ν ∂¯ q q ∂ q ∂μ− ∂ ∂ τ + e 2 = 0 ∂ ν ¯ 2 ∂ ν 3 U ∼ ν 9 − ln p * (−ν) ● Describes a solution to a stochastic differential equation d ν ± =− U ' (ν) sgn (τ)+ζ(τ) d τ − 2 U (ν) ρ ( ν( 0 ) ) ∼ e ● Further integrated twice d ( ln χ) d ϑ 2 / 3 d τ =±χ =±ν ± d τ ● To yield a parametric representation ( χ(τ) , ϑ(τ) ) of function χ(ϑ) 3 / 2 ● but can drop by arbitrarily large percentage χ∼ϑ
Results E n e r g y l a n d s c a p e i s a s e l f - s i m i l a r r a n d o m p r o c e s s (every realization happens on some scale) 3 / 2 × λ × λ There will be realizations where two minima compete Numerically integrate stochastic equations Typical gap V (ϑ)∼ K 2 2 (ϑ−ϑ * ) 3 / 4 Δ E ∼ √ K / M ∼ Γ 1 / 4 N K ∼ √ N /Γ Minimum gap 2 M ∼ N /Γ 3 / 2 √ N V ∼Γ 3 / 4 Δ E ∼ e − √ MV Δ ϑ ∼ e − c (Γ N ) Δϑ∼Γ
Discussion Bottlenecks progressively easier toward the end of the algorithm (problem solved for ) Γ< 1 / N Only become relevant for large problems N h.b. ≈α ln N > 1 Crossover from polynomial to α≈ 0.15 N c ∼ 1000 exponential complexity Cf. Sherrington-Kirkpatrick model: time to solution Classical gap scales as 1 / √ N Barrier heights scale as 1 / 3 N “easy” “hard” Stronger disorder fluctuations, J ik ∼ 1 / √ N N N c
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