Quantum Adiabatic Algorithm and Scaling of Gaps at First-Order Quantum Phase Transitions

Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first-order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first-order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbor interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of essentially topological origin. This establishes that (i) the QAA can be successful even across first-order transitions but also that (ii) it can fail on exceedingly simple problems readily solved by inspection, or by classical annealing.

Figure 1

Spectrum of dimer model on even length periodic ladder with dimer configurations overlaid. The two staggered $w=\pm 1$ states are at $E=0$, while the columnar sector $w=0$ splits into a band of resonating dimer states under resonance $\Gamma$.

Figure 2

The fully frustrated Ising ladder. Location of Ising spins are denoted by filled (empty) circles, subject to fields of strength $-K$ ($U/2$), respectively. Black solid (dashed) lines represent (anti)ferromagnetic interactions of strength $K$. Dimers live on the quasidual lattice (fat-dashed lines), with a dimer on the top row denoting an unsatisfied field term. The ground states of the $K$ terms are the dimer states, while the $U$ term selects the staggered ones among them.

Figure 3

Phase diagram of frustrated two-leg ladder. Points indicate position of minimum gap in $k=0$ sector as a function of $\Gamma$ at fixed $K$, $L=12$. Dotted line $\Gamma =K$ indicates second-order condensation predicted to leading order in $K/U$; dashed line indicates first-order transition predicted to second order in $U/K$ (see text). (inset) Exponential scaling of the gap for $K$ in the first-order transition regime. Dashed lines are best fit exponentials to minimum gap data. From top to bottom, $K/U=2.5–5.0$ in steps of 0.5.