Stretched Exponential Scaling of Parity-Restricted Energy Gaps in a Random Transverse-Field Ising Model

Abstract

The success of a quantum annealing algorithm requires a polynomial scaling of the energy gap. Recently it was shown that a two-dimensional transverse-field Ising model on a square lattice with nearest-neighbor $\pm J$ random coupling has a polynomial energy gap in the symmetric subspace of the parity operator [Nature 631, 749-754 (2024)], indicating the efficient preparation of its ground states by quantum annealing. However, it is not clear if this result can be generalized to other spin glass models with continuous or biased randomness. Here we prove that under general independent and identical distributions (i.i.d.) of the exchange energies, the energy gap of a one-dimensional random transverse-field Ising model follows a stretched exponential scaling even in the parity-restricted subspace. We discuss the implication of this result to quantum annealing problems.

Figures (3)

• Figure 1: Numerical results for the activated scaling of RTIM's energy gaps. (a) The standard energy gap $\Delta_1$ (blue dots), the parity-restricted energy gap $\Delta_p$ (red squares) and the derived upper bound $\Delta_b$ (black triangles) versus the system size $L$, when the random Ising couplings are independently drawn from a Gaussian distribution $\mathcal{N}(\mu,\sigma^2)$ with a mean value $\mu=1$ and a standard deviation $\sigma=0.5$. The transverse field $h$ is fixed at the critical point $h_c=\exp(\overline{\ln|J|})$ where the average over the random distribution is estimated from a sample size of ten times the largest system size. Here we plot the energy gaps in logarithmic scale and the system size in square root scale to visualize the activated scaling as a straight line. Each data point is the geometric average (typical gap PhysRevB.58.9131) over 500 random realizations, with the shaded area representing 99% confidence intervals. The dashed lines are linear fitting results, where we have dropped the data points for $\Delta_1<10^{-13}$ to suppress the numerical error. (b) The histogram of $C\equiv L^{-1/2}\ln\Delta_p$. The three curves for $L=50,\,637,\,2000$ collapse well with each other. (c) and (d) Similar plots when the random Ising couplings are independently drawn from a two-point distribution $\{0.5,\,1\}$ with equal probabilities.
• Figure 2: Quality of different fitting models versus level of random fluctuation. After obtaining the upper bound of the parity-restricted energy gap $\Delta_b$ versus the system size $L$ as in Fig. \ref{['fig:exponential_gap']}(a), we can linearly fit them by either the $\ln \Delta_b$ vs. $\sqrt{L}$ model or the $\ln \Delta_b$ vs. $\ln L$ model, so as to numerically distinguish between an activated scaling and a polynomial scaling. Here we choose three typical standard deviations of the Gaussian distribution in (a) $\sigma=0.01$, (b) $\sigma=0.05$ and (c) $\sigma=0.1$, and examine the quality of these two fitting models versus different system sizes $L$ up to which the fitting is performed. For small random fluctuation, initially the polynomial fitting can give higher $R^2$ values and the activated scaling outperforms only when the system size gets above certain value $L^*$ [indicated by the vertical dashed lines in (b) and (c)]. (d) The critical system size $L^*$ after which the activated scaling dominates versus the random fluctuation level $\sigma$. The energy gaps are evaluated at system sizes with an increment of $\Delta L=50$, so we estimated an error bar of $\pm \Delta L/2$ for the extracted $L^*$. A power-law relation $L^*\sim \sigma^{-2.46}$ is fitted.
• Figure 3: A deterministic sequence with a parity-restricted energy gap showing an activated scaling. Here we consider a deterministic sequence $J_{x,x+1}=\sin(x \sin x)$ ($x=1,\,2,\,\cdots,\,L-1$), and plot the upper bound $\Delta_b$ of the parity-restricted energy gap. Similar to a single random realization of the RTIM, multiple plateaus are observed along with the overall activated scaling as the system size $L$ increases. In comparison, we also plot $\Delta_b$ for an RTIM whose exchange energies are sampled from a uniform distribution between $[-1,\,1]$ (red dashed). The data points for the random distribution are the geometric mean over $500$ realizations, with the shaded area representing 99% confidence intervals.

Theorems & Definitions (2)

• Theorem
• proof : Proof sketch