# Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

@article{Berry2007EfficientQA,
title={Efficient Quantum Algorithms for Simulating Sparse Hamiltonians},
author={Dominic W. Berry and Graeme Ahokas and Richard Cleve and Barry C. Sanders},
journal={Communications in Mathematical Physics},
year={2007},
volume={270},
pages={359-371}
}
• Published 18 August 2005
• Computer Science
• Communications in Mathematical Physics
We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and ||H|| is bounded by a constant, we may select any positive integer k such that the simulation requires O((log*n)t1+1/2k) accesses to matrix entries of H. We also show that the temporal…
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## References

SHOWING 1-10 OF 22 REFERENCES
• Physics
• 2000
We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that
Quantum information processing in continuous time
• Physics
• 2004
Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not
Quantum walk algorithm for element distinctness
• A. Ambainis
• Computer Science
45th Annual IEEE Symposium on Foundations of Computer Science
• 2004
An O(N/sup k/(k+1)/) query quantum algorithm is given for the generalization of element distinctness in which the authors have to find k equal items among N items.
Adiabatic quantum state generation and statistical zero knowledge
• Physics, Computer Science
STOC '03
• 2003
The ASG approach to quantum algorithms provides intriguing links between quantum computation and many different areas: the analysis of spectral gaps and groundstates of Hamiltonians in physics, rapidly mixing Markov chains, statistical zero knowledge, and quantum random walks.
Quantum lower bounds by polynomials
• Computer Science
Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
• 1998
This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.
The Complexity of the Local Hamiltonian Problem
• Mathematics
SIAM J. Comput.
• 2006
This paper settles the question and shows that the 2-LOCAL HAMILTONIAN problem is QMA-complete, and demonstrates that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.
Limit on the Speed of Quantum Computation in Determining Parity
• Computer Science
• 1998
It is shown that any quantum algorithm capable of determining the parity of f contains at least N/2 applications of the unitary operator which evaluates f and quantum computers cannot outperform classical computers.
Quantum computation and quantum information
• T. Paul
• Physics
Mathematical Structures in Computer Science
• 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal
Quantum random-walk search algorithm
• Computer Science
• 2003
It will be shown that this algorithm performs an oracle search on a database of N items with $O(\sqrt{N})$ calls to the oracle, yielding a speedup similar to other quantum search algorithms.
Exponential algorithmic speedup by a quantum walk
• Computer Science
STOC '03
• 2003
A black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer is constructed and it is proved that no classical algorithm can solve the problem in subexponential time.