Grover's Algorithm with Diffusion and Amplitude Steering

As featured on Cornell University Quantum Physics website:

Written by Robert L. Singleton JR, Michael L. Rogers, and David L. Ostby of the SavantX Research Center based in Santa Fe, NM

We review the basic theoretical underpinnings of the Grover algorithm, providing a rigorous and well motivated derivation. We then present a generalization of Grover’s algorithm that searches an arbitrary subspace of the multi-dimensional Hilbert space using a diffusion operation and an amplitude amplification procedure that has been biased by unitary {\em steering operators}. We also outline a generalized Grover’s algorithm that takes into account higher level correlations that could exist between database elements. In the traditional Grover algorithm, the Hadamard gate selects a uniform sample of computational basis elements when performing the phase selection and diffusion. In contrast, steered operators bias the selection process, thereby providing more flexibility in selecting the target state. Our method is a generalization of the recently proposal pattern matching algorithm of Hiroyuki et al.

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Floating-Point Calculations on a Quantum Annealer: Division and Matrix Inversion

As featured on the Frontiers in Physics Quantum Engineering and Technology website:

Written by Robert L. Singleton JR and Michael L. Rogers of the SavantX Research Center based in Santa Fe, NM

Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry, and statistics. Traditional solution methods such as Gaussian elimination are very time consuming for large matrices, and more efficient computational methods are desired. In the twilight of Moore’s Law, quantum computing is perhaps the most direct path out of the darkness. There are two complementary paradigms for quantum computing, namely, circuit-based systems and quantum annealers. In this paper, we express floating point operations, such as division and matrix inversion, in terms of a quadratic unconstrained binary optimization (QUBO) problem, a formulation that is ideal for a quantum annealer. We first address floating point division, and then move on to matrix inversion. We provide a general algorithm for any number of dimensions, and, as a proof-of-principle, we demonstrates results from the D-Wave quantum annealer for 2 × 2 and 3 × 3 general matrices. In principle, our algorithm scales to very large numbers of linear equations; however, in practice the number is limited by the connectivity and dynamic range of the machine.

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