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Spatial search by quantum walk
Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that aExpand
Exponential algorithmic speedup by a quantum walk
TLDR
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. Expand
Simulating Hamiltonian dynamics with a truncated Taylor series.
TLDR
A simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification. Expand
Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters
TLDR
An algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest is presented, and a new lower bound is proved showing that no algorithm can have sub linear dependence on tau. Expand
Constructing elliptic curve isogenies in quantum subexponential time
TLDR
This work gives a new subexponential-time quantum algorithm for constructing nonzero isogenies between two such elliptic curves, assuming the Generalized Riemann Hypothesis (but with no other assumptions). Expand
On the Relationship Between Continuous- and Discrete-Time Quantum Walk
Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantumExpand
Universal computation by quantum walk.
TLDR
It is shown that quantum walk can be regarded as a universal computational primitive, with any quantum computation encoded in some graph, even if the Hamiltonian is restricted to be the adjacency matrix of a low-degree graph. Expand
Exponential improvement in precision for simulating sparse Hamiltonians
TLDR
The algorithm is based on a significantly improved simulation of the continuous- and fractional- query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. Expand
Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision
TLDR
The algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation, and allows the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive, to be bypassed. Expand
An Example of the Difference Between Quantum and Classical Random Walks
TLDR
A general definition of quantum random walks on graphs is discussed and with a simple graph the possibility of very different behavior between a classical random walk and its quantum analog is illustrated. Expand
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