[Paper Review] Tunneling through high energy barriers in simulated quantum annealing

Elizabeth Crosson, Mingkai Deng|arXiv (Cornell University)|Oct 30, 2014

Quantum Computing Algorithms and Architecture8 references6 citations

TL;DR

This paper demonstrates that Simulated Quantum Annealing (SQA) can efficiently tunnel through high energy barriers in optimization problems—specifically the 'Hamming weight with a spike' problem—where classical Simulated Annealing (SA) fails exponentially. Using Path-Integral Monte Carlo with sufficiently fine imaginary-time discretization (L ∝ n), SQA achieves polynomial convergence time, inheriting quantum annealing's advantage over classical methods.

ABSTRACT

We analyze the performance of simulated quantum annealing (SQA) on an optimization problem for which simulated classical annealing (SA) is provably inefficient because of a high energy barrier. We present evidence that SQA can pass through this barrier to find the global minimum efficiently. This demonstrates the potential for SQA to inherit some of the advantages of quantum annealing (QA), since this problem has been previously shown to be efficiently solvable by quantum adiabatic optimization.

Motivation & Objective

• To investigate whether Simulated Quantum Annealing (SQA) can overcome high energy barriers that impede classical Simulated Annealing (SA).
• To determine if SQA inherits the quantum advantage of quantum annealing (QA) on problems with exponential barriers.
• To analyze the dependence of SQA convergence time on system size n and imaginary-time discretization L.
• To establish conditions under which SQA remains efficient despite topological obstructions present in classical annealing.

Proposed method

• SQA is implemented using Path-Integral Monte Carlo with a Suzuki-Trotter approximation to map the quantum annealing process onto a classical statistical mechanical system.
• The effective classical energy function E_C includes both the problem Hamiltonian f(z) and ferromagnetic couplings J along the imaginary-time direction, with J ∝ 1/Γ as Γ → 0.
• Markov chain Monte Carlo with local single-spin flips and Metropolis acceptance rules is used to sample the equilibrium distribution π(𝐳) ∝ exp(−βE_C(𝐳)).
• Worldline updates are considered but not required; local updates alone suffice to show polynomial convergence.
• The number of sweeps τ_s needed to sample the global minimum is measured as a convergence metric at each Γ_i.
• The transverse field Γ is decreased geometrically from 1 to 10⁻¹², with β = 32 to ensure the true minimum dominates the stationary distribution.

Experimental results

Research questions

• RQ1 Can SQA efficiently tunnel through a high energy barrier that causes classical SA to fail exponentially?
• RQ2 Does the equilibration time of SQA scale polynomially with system size n when the energy barrier is present?
• RQ3 What is the required scaling of the imaginary-time discretization L to prevent SQA from reverting to classical behavior?
• RQ4 How does the dynamical critical exponent z of SQA compare to that of the 1D kinetic Ising model?
• RQ5 Is the performance of SQA on this problem consistent with quantum annealing’s polynomial runtime advantage?

Key findings

• SQA achieves a convergence time τ_s ≈ O(n^1.98), indicating polynomial scaling with system size n.
• The dynamical critical exponent z ≈ 1.98 is consistent with the expected z = 2 for the 1D kinetic Ising model in the β → ∞ limit.
• The equilibration time remains polynomial even in the presence of the high energy barrier at Hamming weight n/4.
• SQA’s efficiency is preserved only when the number of Trotter slices L scales linearly with n, ensuring non-exponentially small acceptance probabilities for barrier-crossing moves.
• When L is constant, SQA’s convergence time becomes exponential, indicating a crossover to classical behavior.
• These results suggest SQA can inherit the quantum advantage of tunneling through high barriers, provided the imaginary-time discretization is sufficiently fine.