Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-$s$ chain with nearest-neighbor interactions can be for small values of $s$. While FF spin-$1/2$ chains are known to have unentangled ground states, the case $s=1$ remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right brackets separated by empty spaces. Entanglement entropy of one half of the chain scales as $\frac{1}{2}\log n+O(1)$, where $n$ is the number of spins. We prove that the energy gap above the ground state is polynomial in $1/n$. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.

We study the glued-trees problem from A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. Spielman, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing (ACM, San Diego, CA, 2003), p. 59. in the adiabatic model of quantum computing and provide an annealing schedule to solve an oracular problem exponentially faster than classically possible. The Hamiltonians involved in the quantum annealing do not suffer from the so-called sign problem. Unlike the typical scenario, our schedule is efficient even though the minimum energy gap of the Hamiltonians is exponentially small in the problem size. We discuss generalizations based on initial-state randomization to avoid some slowdowns in adiabatic quantum computing due to small gaps.

Systematic easing via efficiently computable measures can make quantum matter with a sign problem amenable to Monte Carlo methods.

Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the `yes' and `no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous. Our construction is natively quantum, based on the analogy of a product of two reflections and a quantum walk. Second, in some special cases we show how to amplify the acceptance probability for good witnesses to 1, making a step towards the proof that QMA with one-sided error QMA_1 is equal to QMA. Finally, we simplify the filter-state method to search for QMA witnesses by Poulin and Wocjan.

We give a generalization to an infinite tree geometry of Vidal’s infinite time-evolving block decimation (iTEBD) algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] for simulating an infinite line of quantum spins. We numerically investigate the quantum Ising model in a transverse field on the Bethe lattice using the matrix product state ansatz. We observe a second order phase transition, with certain key differences from the transverse field Ising model on an infinite spin chain. We also investigate a transverse field Ising model with a specific longitudinal field. When the transverse field is turned off, this model has a highly degenerate ground state as opposed to the pure Ising model whose ground state is only doubly degenerate.