Jeffrey Goldstone

Spatial search by quantum walk

Grovers 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 a database of N items laid out in d spatial dimensions can be searched in time of order {radical}(N) for d>2, and in time of order {radical}(N) poly(log N) for d=2. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that {radical}(N) speedup can also be achieved on the hypercube. We show that full {radical}(N) speedup can be achieved on a d-dimensional periodic lattice for d>4. In d=4, the quantum walk search algorithm takes time of order {radical}(N) poly(log N), and in d<4, the algorithm does not provide substantial speedup.

A Quantum Algorithm for the Hamiltonian NAND Tree

We give a quantum algorithm for the binary NAND tree problem in the Hamil- tonian oracle model. The algorithm uses a continuous time quantum walk with a running time proportional to p N. We also show a lower bound of W( p N) for the NAND tree problem in the Hamiltonian oracle model.

Limit on the Speed of Quantum Computation in Determining Parity

Consider a function f which is defined on the integers from 1 to N and takes the values {minus}1 and +1 . The parity of f is the product over all x from 1 to N of f(x) . With no further information about f , to classically determine the parity of f requires N calls of the function f . We show that any quantum algorithm capable of determining the parity of f contains at least N/2 applications of the unitary operator which evaluates f . Thus, for this problem, quantum computers cannot outperform classical computers. {copyright} {ital 1998} {ital The American Physical Society}

Spatial search and the Dirac equation

We consider the problem of searching a d-dimensional lattice of N sites for a single marked location. We present a Hamiltonian that solves this problem in time of order {radical}(N) for d>2 and of order {radical}(N) log N in the critical dimension d=2. This improves upon the performance of our previous quantum walk search algorithm (which has a critical dimension of d=4), and matches the performance of a corresponding discrete-time quantum walk algorithm. The improvement uses a lattice version of the Dirac Hamiltonian, and thus requires the introduction of spin (or coin) degrees of freedom.

How probability arises in quantum mechanics

Abstract A version of the postulates of quantum mechanics is presented in which no reference is made to probability. Instead we rely on a weaker postulate referring to eigenvalues and eigenstates. The modulus squared of the inner product of two state vectors is shown to be an eigenvalue of the operator representing a frequency measurement on the system of an infinite number of copies of the original system. The argument makes essential use of the Strong Law of Large Numbers.

HOW TO MAKE THE QUANTUM ADIABATIC ALGORITHM FAIL

The quantum adiabatic algorithm is a Hamiltonian based quantum algorithm designed to find the minimum of a classical cost function whose domain has size N. We show that poor choices for the Hamiltonian can guarantee that the algorithm will not find the minimum if the run time grows more slowly than . These poor choices are nonlocal and wash out any structure in the cost function to be minimized, and the best that can be hoped for is Grover speedup. These failures tell us what not to do when designing quantum adiabatic algorithms.

Quantum search by measurement

We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar in spirit to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grovers unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms.

Bound on the number of functions that can be distinguished with k quantum queries

Suppose an oracle is known to hold one of a given set of D two-valued functions. To successfully identify which function the oracle holds with k classical queries, it must be the case that D is at most

Quantum transverse-field Ising model on an infinite tree from matrix product states

{2}^{k}.

Spin content of the bosonic string

In this paper we derive a bound for how many functions can be distinguished with k quantum queries.