583 research outputs found
Efficient discrete-time simulations of continuous-time quantum query algorithms
The continuous-time query model is a variant of the discrete query model in
which queries can be interleaved with known operations (called "driving
operations") continuously in time. Interesting algorithms have been discovered
in this model, such as an algorithm for evaluating nand trees more efficiently
than any classical algorithm. Subsequent work has shown that there also exists
an efficient algorithm for nand trees in the discrete query model; however,
there is no efficient conversion known for continuous-time query algorithms for
arbitrary problems.
We show that any quantum algorithm in the continuous-time query model whose
total query time is T can be simulated by a quantum algorithm in the discrete
query model that makes O[T log(T) / log(log(T))] queries. This is the first
upper bound that is independent of the driving operations (i.e., it holds even
if the norm of the driving Hamiltonian is very large). A corollary is that any
lower bound of T queries for a problem in the discrete-time query model
immediately carries over to a lower bound of \Omega[T log(log(T))/log (T)] in
the continuous-time query model.Comment: 12 pages, 6 fig
Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
We consider the manifold of all quantum many-body states that can be
generated by arbitrary time-dependent local Hamiltonians in a time that scales
polynomially in the system size, and show that it occupies an exponentially
small volume in Hilbert space. This implies that the overwhelming majority of
states in Hilbert space are not physical as they can only be produced after an
exponentially long time. We establish this fact by making use of a
time-dependent generalization of the Suzuki-Trotter expansion, followed by a
counting argument. This also demonstrates that a computational model based on
arbitrarily rapidly changing Hamiltonians is no more powerful than the standard
quantum circuit model.Comment: Presented at QIP 201
Spectral Gap Amplification
A large number of problems in science can be solved by preparing a specific
eigenstate of some Hamiltonian H. The generic cost of quantum algorithms for
these problems is determined by the inverse spectral gap of H for that
eigenstate and the cost of evolving with H for some fixed time. The goal of
spectral gap amplification is to construct a Hamiltonian H' with the same
eigenstate as H but a bigger spectral gap, requiring that constant-time
evolutions with H' and H are implemented with nearly the same cost. We show
that a quadratic spectral gap amplification is possible when H satisfies a
frustration-free property and give H' for these cases. This results in quantum
speedups for optimization problems. It also yields improved constructions for
adiabatic simulations of quantum circuits and for the preparation of projected
entangled pair states (PEPS), which play an important role in quantum many-body
physics. Defining a suitable black-box model, we establish that the quadratic
amplification is optimal for frustration-free Hamiltonians and that no spectral
gap amplification is possible, in general, if the frustration-free property is
removed. A corollary is that finding a similarity transformation between a
stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the
black-box model, setting limits to the power of some classical methods that
simulate quantum adiabatic evolutions.Comment: 14 pages. New version has an improved section on adiabatic
simulations of quantum circuit
A Quantum Approach to Classical Statistical Mechanics
We present a new approach to study the thermodynamic properties of
-dimensional classical systems by reducing the problem to the computation of
ground state properties of a -dimensional quantum model. This
classical-to-quantum mapping allows us to deal with standard optimization
methods, such as simulated and quantum annealing, on an equal basis.
Consequently, we extend the quantum annealing method to simulate classical
systems at finite temperatures. Using the adiabatic theorem of quantum
mechanics, we derive the rates to assure convergence to the optimal
thermodynamic state. For simulated and quantum annealing, we obtain the
asymptotic rates of and , for the temperature and magnetic field, respectively. Other
annealing strategies, as well as their potential speed-up, are also discussed.Comment: 4 pages, no figure
Necessary Condition for the Quantum Adiabatic Approximation
A gapped quantum system that is adiabatically perturbed remains approximately
in its eigenstate after the evolution. We prove that, for constant gap, general
quantum processes that approximately prepare the final eigenstate require a
minimum time proportional to the ratio of the length of the eigenstate path to
the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We
also give a necessary condition for the adiabatic approximation that depends on
local properties of the path, which is appropriate when the gap varies.Comment: 5 pages, 1 figur
Quantum Speedup by Quantum Annealing
We study the glued-trees problem of Childs et. al. 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.Comment: 7 page
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