[Paper Review] Nonadiabatic Quantum Annealing for One-Dimensional Trasverse-Field Ising Model

Hitoshi Katsuda, Hidetoshi Nishimori|arXiv (Cornell University)|Mar 25, 2013

Quantum Computing Algorithms and Architecture3 references3 citations

TL;DR

This paper proposes a nonadiabatic quantum annealing approach that replaces long adiabatic evolution with repeated short-time annealing trials, collecting final states to identify the ground state. For the one-dimensional transverse-field Ising model, it analytically and numerically demonstrates that the computational complexity remains O(N²), matching adiabatic quantum annealing, while offering potential robustness against external disturbances.

ABSTRACT

We propose a nonadiabatic approach to quantum annealing, in which we repeat quantum annealing in nonadiabatic time scales, and collect the final states of many realizations to find the ground state among them. In this way, we replace the diffculty of long annealing time in adiabatic quantum annealing by another problem of the number of nonsidabatic (short-time) trials. The one-dimensional transverse-field Ising model is used to test this idea, and it is shown that nonadiabatic quantum annealing has the same computational complexity to find the ground state as the conventional adiabatic annealing does. This result implies that the nonadiabatic method may be used to replace adiabatic annealing to avoid the effects of external disturbances, to which the adiabatic method is more prone than the nonadiabatic counterpart.

Motivation & Objective

• To address the challenge of long annealing times in adiabatic quantum annealing, especially for NP-hard problems with exponentially small energy gaps.
• To explore whether nonadiabatic quantum annealing—using short, non-adiabatic time scales—can achieve the same computational complexity as adiabatic annealing.
• To test the feasibility and efficiency of repeated nonadiabatic trials in finding the ground state of a quantum spin system.
• To evaluate the robustness of nonadiabatic annealing against external disturbances compared to adiabatic methods.

Proposed method

• The method employs repeated nonadiabatic quantum annealing cycles, each lasting a short time τ, with the final state collected after each run.
• The time evolution is modeled using the transverse-field Ising Hamiltonian with a time-dependent transverse field, decomposed into momentum-space modes.
• For small q-modes, the dynamics are approximated as Landau-Zener-type transitions around t = τ/2, where the energy gap is minimized.
• The probability of remaining in the ground state after a single nonadiabatic run is estimated using a path integral approach and parabolic cylinder functions.
• The average number of trials needed to find the ground state is approximated as 1/P_GS, and the total time is analyzed as τ/P_GS to determine computational complexity.
• Asymptotic approximations are applied to the integral expression for ln P_GS, leading to a simplified form involving the Riemann zeta function ζ(3/2).

Experimental results

Research questions

• RQ1 Can nonadiabatic quantum annealing achieve the same computational complexity as adiabatic quantum annealing for the one-dimensional transverse-field Ising model?
• RQ2 Does replacing long adiabatic evolution with repeated short-time nonadiabatic trials preserve the O(N²) scaling of the minimum annealing time?
• RQ3 Is the nonadiabatic approach more robust against external disturbances than the adiabatic counterpart?
• RQ4 What is the optimal annealing time τ that minimizes the total time τ/P_GS for nonadiabatic annealing?
• RQ5 How well do analytical approximations for the ground state probability P_GS match numerical solutions of the Schrödinger equation?

Key findings

• The computational complexity of nonadiabatic quantum annealing for the 1D transverse-field Ising model is O(N²), matching the complexity of adiabatic quantum annealing.
• The optimal annealing time τ that minimizes the total time τ/P_GS is τ = (N²/4π²) × ζ(3/2)², where ζ(3/2) ≈ 2.61248.
• The minimum total time to find the ground state scales as approximately 1.27732 × N², derived from (N/2π × ζ(3/2))² × e².
• Numerical solutions of the Schrödinger equation for individual q-modes confirm that τ/P_GS scales as O(N²), validating the analytical prediction.
• The standard deviation of the number of trials is shown to be subdominant in the asymptotic regime, preserving the O(N²) complexity.
• The nonadiabatic method is expected to be more robust against external disturbances due to shorter, less sensitive time scales compared to adiabatic evolution.