7,454 research outputs found
Adiabatic optimization without local minima
Several previous works have investigated the circumstances under which
quantum adiabatic optimization algorithms can tunnel out of local energy minima
that trap simulated annealing or other classical local search algorithms. Here
we investigate the even more basic question of whether adiabatic optimization
algorithms always succeed in polynomial time for trivial optimization problems
in which there are no local energy minima other than the global minimum.
Surprisingly, we find a counterexample in which the potential is a single basin
on a graph, but the eigenvalue gap is exponentially small as a function of the
number of vertices. In this counterexample, the ground state wavefunction
consists of two "lobes" separated by a region of exponentially small amplitude.
Conversely, we prove if the ground state wavefunction is single-peaked then the
eigenvalue gap scales at worst as one over the square of the number of
vertices.Comment: 20 pages, 1 figure. Journal versio
The Short Path Algorithm Applied to a Toy Model
We numerically investigate the performance of the short path optimization
algorithm on a toy problem, with the potential chosen to depend only on the
total Hamming weight to allow simulation of larger systems. We consider classes
of potentials with multiple minima which cause the adiabatic algorithm to
experience difficulties with small gaps. The numerical investigation allows us
to consider a broader range of parameters than was studied in previous rigorous
work on the short path algorithm, and to show that the algorithm can continue
to lead to speedups for more general objective functions than those considered
before. We find in many cases a polynomial speedup over Grover search. We
present a heuristic analytic treatment of choices of these parameters and of
scaling of phase transitions in this model.Comment: 11 pages, 9 figures; v2 final version published in Quantu
Algorithmic approach to adiabatic quantum optimization
It is believed that the presence of anticrossings with exponentially small
gaps between the lowest two energy levels of the system Hamiltonian, can render
adiabatic quantum optimization inefficient. Here, we present a simple adiabatic
quantum algorithm designed to eliminate exponentially small gaps caused by
anticrossings between eigenstates that correspond with the local and global
minima of the problem Hamiltonian. In each iteration of the algorithm,
information is gathered about the local minima that are reached after passing
the anticrossing non-adiabatically. This information is then used to penalize
pathways to the corresponding local minima, by adjusting the initial
Hamiltonian. This is repeated for multiple clusters of local minima as needed.
We generate 64-qubit random instances of the maximum independent set problem,
skewed to be extremely hard, with between 10^5 and 10^6 highly-degenerate local
minima. Using quantum Monte Carlo simulations, it is found that the algorithm
can trivially solve all the instances in ~10 iterations.Comment: 7 pages, 3 figure
Adiabatic Quantum Optimization for Associative Memory Recall
Hopfield networks are a variant of associative memory that recall information
stored in the couplings of an Ising model. Stored memories are fixed points for
the network dynamics that correspond to energetic minima of the spin state. We
formulate the recall of memories stored in a Hopfield network using energy
minimization by adiabatic quantum optimization (AQO). Numerical simulations of
the quantum dynamics allow us to quantify the AQO recall accuracy with respect
to the number of stored memories and the noise in the input key. We also
investigate AQO performance with respect to how memories are stored in the
Ising model using different learning rules. Our results indicate that AQO
performance varies strongly with learning rule due to the changes in the energy
landscape. Consequently, learning rules offer indirect methods for
investigating change to the computational complexity of the recall task and the
computational efficiency of AQO.Comment: 22 pages, 11 figures. Updated for clarity and figures, to appear in
Frontiers of Physic
Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians
We investigate the relationship between the energy spectrum of a local
Hamiltonian and the geometric properties of its ground state. By generalizing a
standard framework from the analysis of Markov chains to arbitrary
(non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap
can always be upper bounded by an isoperimetric ratio that depends only on the
ground state probability distribution and the range of the terms in the
Hamiltonian, but not on any other details of the interaction couplings. This
means that for a given probability distribution the inequality constrains the
spectral gap of any local Hamiltonian with this distribution as its ground
state probability distribution in some basis (Eldar and Harrow derived a
similar result in order to characterize the output of low-depth quantum
circuits). Going further, we relate the Hilbert space localization properties
of the ground state to higher energy eigenvalues by showing that the presence
of k strongly localized ground state modes (i.e. clusters of probability, or
subsets with small expansion) in Hilbert space implies the presence of k energy
eigenvalues that are close to the ground state energy. Our results suggest that
quantum adiabatic optimization using local Hamiltonians will inevitably
encounter small spectral gaps when attempting to prepare ground states
corresponding to multi-modal probability distributions with strongly localized
modes, and this problem cannot necessarily be alleviated with the inclusion of
non-stoquastic couplings
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
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