Box model of quantum annealing

Abstract: A particle-in-a-box model of continuous space quantum annealing is proposed and studied numerically by solving the Schrödinger wave equation directly. Three types of energy landscapes with multiple local minima are considered, namely a sinusoidal wave modulated by a concave, a convex, or a flat envelope. Both static (energy spectrum) and dynamical (residual energy) behaviors are analyzed in detail, paying particular attention to the effects of landscape roughness and annealing depth. Simulation results show that the residual energy as a function of annealing speed is largely independent of these two factors. The prevalence of diabatic transitions during annealing is observed, and the discrepancy between our numerical results and the Landau-Zener formula is discussed. An interesting feature in the energy gap spectrum, which we call flat gaps, is examined. Based on it, we propose a mechanism to explain the trapping of wave function in local minima during diabatic transitions, widely observed in our data.

• The paper develops a numerically efficient one-dimensional box model with analytic matrix elements, enabling quantum-annealing simulations across four decades of effective mass and basis sizes up to 1,500.
• The analysis finds that flat landscapes exhibit residual energy scaling as T⁻², while rugged concave and convex landscapes show exponential-to-polynomial crossovers, with performance largely insensitive to the number of local minima and annealing depth.
• The paper argues that flat spectral gaps generically promote diabatic trapping in local minima, making Landau–Zener theory quantitatively unreliable and motivating optimized schedules and new theories of finite-time annealing.

Motivation and model

This paper develops a one-dimensional "box" model for continuous-space quantum annealing (QA), in which a particle of tunable effective mass is confined to $0

$$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$

The parameter $\mu$ sets the number of local minima ($\mu/2+1$ total), while the envelope amplitude $a$ selects among three landscape classes: flat ($a=0$, all minima degenerate at zero energy), concave ($a>0$, global minima at the walls), and convex ($a<0$, unique global minimum at the center). The annealing Hamiltonian is $H(s)=p^2/(2ms)+V_{\mathrm{box}}(x)$, so that increasing $s$ increases the particle's effective mass and localizes its wave function.

The box construction is motivated by numerical practicality. In the coordinate representation used previously for the Rastrigin function (2605.07144), resolving many narrow minima and simultaneously accommodating a delocalized initial wave function requires prohibitively large grids; in the harmonic-oscillator energy basis, Hermite polynomials overflow numerically beyond roughly $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$0. The free-particle sine basis of the box avoids both problems: matrix elements of the cosine potentials are analytic Kronecker-delta expressions, and annealing can proceed over four decades of $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$1 in a single trajectory rather than in staged segments. The authors also justify their choice of $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$2 over the convex-combination form $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$3: the latter's ground state changes appreciably only when $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$4 is extremely close to 1, making it unwieldy.

Dynamics are obtained by direct integration of the time-dependent Schrödinger equation under a linear schedule $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$5, with performance measured by the residual energy $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$6. Statics are studied via exact diagonalization with basis sizes up to $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$7.

Flat envelope: degenerate minima and adiabatic scaling

For $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$8, all central minima are exactly degenerate. As $$V_{\mathrm{box}}(x) = \tfrac{1}{2}\left[1-\cos\left(\frac{\mu\pi x}{L}\right)\right] + a\left[1-\cos\left(\frac{2\pi x}{L}\right)\right].$$9 grows, the lowest $\mu$0 levels merge into an $\mu$1-fold degenerate band spanned by Gaussian peaks localized in each minimum. Two quantum effects absent from classical optimization are highlighted. First, degenerate minima are not treated equally: the ground-state density peaks higher at the center minimum than at off-center ones, reflecting the wholeness of the wave function and its memory of earlier delocalized profiles—implying that QA will find equal-energy minima with unequal probability. Second, wall minima are excluded from the ground-state band because the half-harmonic-oscillator boundary condition costs one extra quantum of zero-point energy; these "wall states" appear only in the next level.

The ground-state energy rises linearly with $\mu$2 despite all classical minima sitting at zero potential, an effect quantitatively explained by zero-point energy: $\mu$3 matches the numerical data closely.

Under linear annealing, $\mu$4 decays as $\mu$5, characteristic of the adiabatic regime. The paper derives this analytically via first-order adiabatic perturbation theory, obtaining

$\mu$6

whose upper bound $\mu$7 agrees well with simulation. The factor $\mu$8 implies cancellation between increased per-minimum curvature and the number of minima, so residual energies for $\mu$9 to 24 collapse onto a single curve—ruggedness does not degrade adiabatic performance here. Only $\mu/2+1$0 shows an exponential pre-asymptotic decay, traced to trapping in the large wall basins that overlap the initial wave function.

Concave envelope: first-order transition and diabatic prevalence

For $\mu/2+1$1, the two wall minima are global, but the ground state undergoes a first-order transition: at a critical $\mu/2+1$2 the gap closes and the density jumps from the two "adjacent" minima to the walls. The transition point is derived semi-classically by balancing the extra half-oscillator quantum at the wall against the excess potential of the adjacent minimum,

$\mu/2+1$3

which reproduces the observed gap closure at $\mu/2+1$4 for $\mu/2+1$5 essentially exactly. Because first-order transitions are notoriously hard for QA, the adjacent minima—not the true global minima—are adopted as the realistic target states, and $\mu/2+1$6 is set accordingly. The authors note plainly that strictly the "degenerate" levels carry exponentially small splittings invisible in double precision, consistent with the non-degeneracy theorem in one dimension.

The dynamical results are the most consequential claims of the paper. Residual-energy curves exhibit up to three regimes: a fast "shoulder," an exponential diabatic regime, and a polynomial $\mu/2+1$7 adiabatic regime. Two strong statements emerge:

• Diabatic transitions are prevalent. Across all concave-system curves, the diabatic regime dominates accessible annealing times; the authors argue that in practice one rarely reaches the adiabatic regime, so real QA runs will generally suffer diabatic excitation, leaving residual energy stored as population trapped in local minima.
• Landau–Zener theory fails quantitatively. Fitted exponential decays such as $\mu/2+1$8 cannot be matched to the Landau–Zener exponent. The authors attribute this to the gap structure: rather than a V-shaped avoided crossing, the spectrum exhibits a constant ("flat") gap over a wide interval of $\mu/2+1$9, violating a core assumption of Landau–Zener theory.

Two further results bear on computational complexity. Replotting $a$0 against annealing speed $a$1 collapses curves for all annealing depths onto a single master curve, so depth $a$2 mainly rescales time. The crossover time scales as $a$3—linearly, hence more favorably than simulated annealing's $a$4—but the spatial precision of the solution scales as $a$5, which the authors flag as expensive; they suggest defining $a$6 as the annealing parameter to recover SA-like $a$7 precision scaling. Increasing $a$8 from 8 to 28 leaves both shallow and deep annealing curves nearly unchanged apart from a $a$9 vertical shift in the adiabatic regime—the number of metastable traps does not increase computational cost, a result the authors themselves call surprising.

Convex envelope: flat gaps and a trapping mechanism

For $a=0$0 there is no gap closure. Instead, the gap spectrum $a=0$1 displays a terrace of plateaus—"flat gaps"—each corresponding to eigenfunctions localized as Gaussians in successive local minima, with the highest minima localizing first and the global minimum last. The authors propose this localization cascade as the continuous-space analogue of a quantum phase transition marker (in place of a vanishing gap) and, importantly, as the mechanism behind diabatic trapping: during annealing the system traverses the flat-gap interval where low-lying excited states are already localized in local minima, so diabatic excitations preferentially populate those traps—analogous to valley bifurcation at the critical temperature in simulated annealing.

A variational analysis with a Gaussian ansatz on the Rastrigin function explains the flatness. At the stationary variational solution, the gap gradient reduces to

$a=0$2

independent of the potential's detailed form. Flatness follows because (i) the $a=0$3 prefactor suppresses the gradient at large mass, and (ii) minima of similar curvature host Gaussians of similar width, $a=0$4. Since neither ingredient depends on the specific potential (boundary effects enter only as perturbations far from walls), the authors argue flat gaps should be generic in continuous systems with multi-minima landscapes. The plateau heights are captured by purely classical potential differences, $a=0$5, which agree closely with exact diagonalization.

Annealing dynamics mirror the concave case: exponential-to-polynomial crossovers, shoulders, collapse of $a=0$6 across depths, and insensitivity to $a=0$7 up to 28. One deviation is reported candidly: for $a=0$8 and 28 the adiabatic-regime curves bend upward because the ground state delocalizes over three adjacent minima, requiring "long-ranged" $a=0$9 matrix elements between distinct basins—a blurring of the intra-/inter-basin distinction that also softens the diabatic–adiabatic crossover.

Limitations and open questions

The paper concedes several boundaries of scope. All results are strictly one-dimensional with only three values of the envelope amplitude ($a>0$0); generalization to richer or realistic molecular landscapes is asserted plausible but untested. The physical mechanism of the fast "shoulder" regime remains unidentified, with intra-basin excitations offered only as a hypothesis and quenched dynamics suggested as a possible analytical route. The failure of Landau–Zener theory is diagnosed qualitatively (L-shaped versus V-shaped gaps) but no replacement quantitative theory of the diabatic regime is provided; likewise, the resemblance of the crossover time $a>0$1 to a second-order phase transition is noted without theoretical explanation. The claim that flat gaps are universal rests on a variational argument whose validity assumes comparable curvatures among minima and negligible boundary effects—conditions the authors themselves identify as potentially violated elsewhere. Finally, the proposed three-stage optimized schedule (rapid initial ramp, slow middle stage, adiabatic finish) is presented as an empirical prescription inferred from the data, not derived or benchmarked against alternatives.

Conclusion

The box model provides a numerically tractable platform—analytic matrix elements, single-trajectory annealing over four decades of mass—for systematic study of ruggedness and annealing depth in continuous-space QA. Its principal findings are that QA performance is largely insensitive to both the number of local minima and the annealing depth; that finite-time annealings are dominated by diabatic transitions not describable by Landau–Zener theory owing to flat-gap spectra; and that these flat gaps, explainable variationally and argued to be generic, account for wave-function trapping in local minima. The work also foregrounds the exponential-to-polynomial crossover structure of $a>0$2, which the authors contend has been underemphasized in the continuous-QA literature, and leaves open quantitative theories of the diabatic regime, the shoulder dynamics, and two-dimensional extensions where multidirectional tunneling may alter convergence behavior.