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 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.
The quantum adiabatic algorithm is a Hamiltonian based quantum algorithm designed to find the minimum of a classical cost function whose domain has size N . We show that poor choices for the Hamiltonian can guarantee that the algorithm will not find the minimum if the run time grows more slowly than √ N . These poor choices are nonlocal and wash out any structure in the cost function to be minimized, and the best that can be hoped for is Grover speedup. These failures tell us what not to do when designing quantum adiabatic algorithms.
We study the glued-trees problem of Childs, et. al. [1] 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 socalled 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.PACS numbers: 03.67. Ac, 03.67.Lx, 42.50.Lc Quantum annealing is a powerful heuristic to solve problems in optimization [2,3]. In quantum computing, the method consists of preparing a low-energy or ground state |ψ of a quantum system such that, after a simple measurement, the optimal solution is obtained with large probability. |ψ is prepared by following a particular annealing schedule, with a parametrized Hamiltonian path subject to initial and final conditions. A ground state of the initial Hamiltonian is then transformed to |ψ by varying the parameter adiabatically. In contrast to more general quantum adiabatic state transformations, the Hamiltonians along the path in quantum annealing are termed stoquastic and do not suffer from the so-called numerical sign problem [4]: for a specified basis, the offdiagonal Hamiltonian-matrix entries are nonpositive [5]. This property is useful for classical simulations [3].A sufficient condition for convergence of the quantum method is given by the quantum adiabatic approximation. It asserts that, if the rate of change of the Hamiltonian scales with the energy gap ∆ between their two lowest-energy states, |ψ can be prepared with controlled accuracy [6,7]. Such an approximation may also be necessary [8]. However, it could result in undesired overheads if ∆ is small but transitions between the lowestenergy states are forbidden due to selection rules, or if transitions between lowest-energy states can be exploited to prepare |ψ . The latter case corresponds to the annealing schedule in this Letter. It turns out that the relevant energy gap for the adiabatic approximation in these cases is not ∆ and can be much bigger.Because of the properties of the Hamiltonians, the annealing can also be simulated using probabilistic classical methods such as quantum Monte-Carlo (QMC) [9]. The goal in QMC is to sample according to the distribution of the ground state, i.e. with probabilities coming from amplitudes squared. While we lack of necessary conditions that guarantee convergence, the power of QMC is widely recognized [3,9,10]. In fact, if the Hamiltonians satisfy an additional frustration-free property, efficient QMC simulations for quantum annealing exist [11,12]. This places a doubt on whether a quantum-computer simulation of general quantum annealing processes can ever be done using substantially less resources than QMC or any other classical simulation.Towards answering this question, we...
We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of 10dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state which encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis.This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless BPP=BQP). * Electronic address: daniel.nagaj@savba.sk † Electronic address:
We give a generalization to an infinite tree geometry of Vidal's infinite time-evolving block decimation ͑iTEBD͒ algorithm ͓G. Vidal, Phys. Rev. Lett. 98, 070201 ͑2007͔͒ 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.
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