116 research outputs found
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
Efficient Algorithms for Universal Quantum Simulation
A universal quantum simulator would enable efficient simulation of quantum
dynamics by implementing quantum-simulation algorithms on a quantum computer.
Specifically the quantum simulator would efficiently generate qubit-string
states that closely approximate physical states obtained from a broad class of
dynamical evolutions. I provide an overview of theoretical research into
universal quantum simulators and the strategies for minimizing computational
space and time costs. Applications to simulating many-body quantum simulation
and solving linear equations are discussed
Simulating sparse Hamiltonians with star decompositions
We present an efficient algorithm for simulating the time evolution due to a
sparse Hamiltonian. In terms of the maximum degree d and dimension N of the
space on which the Hamiltonian H acts for time t, this algorithm uses
(d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the
sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders,
which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a
general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of
non-zero entries have the property that every connected component is a star,
and efficiently simulate each of these pieces.Comment: 11 pages. v2: minor correction
Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer
We present a quantum algorithm to prepare the thermal Gibbs state of
interacting quantum systems. This algorithm sets a universal upper bound
D^alpha on the thermalization time of a quantum system, where D is the system's
Hilbert space dimension and alpha < 1/2 is proportional to the Helmholtz free
energy density of the system. We also derive an algorithm to evaluate the
partition function of a quantum system in a time proportional to the system's
thermalization time and inversely proportional to the targeted accuracy
squared
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