Abstract
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 become very time consuming for large matrices, and more efficient computational methods are desired. In the twilight of Moore\rq{}s Law, quantum computing is perhaps the most direct path out of the darkness. There are two complementary paradigms for quantum computing, namely, gated systems and quantum annealers. In this paper, we express floating point operations such as division and matrix inversion in terms of a {\em 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, but we provide results from the D-Wave quantum anneler for 2x2 and 3x3 general matrices. Our algorithm scales to very large numbers of linear equations. We should also mention that our algorithm provides the full solution the the matrix problem, while HHL provides only an expectation value.
Highlights
Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry, and statistics
Circuitbased systems exploit the deeper properties of quantum mechanics such coherence, entanglement, and non-locality, while quantum annealers mainly take advantage of tunneling between metastable states and the ground state
We express floating point operations such as division and matrix inversion as quadratic unconstrained binary optimization (QUBO) problems, which are ideal for a quantum annealer
Summary
Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry, and statistics. Quantum computers are physical systems that exploit the laws of quantum mechanics to perform arithmetic and logical operations much faster than a conventional computer. In [1], HHL introduces a circuit-based method by which the inverse of a matrix can be computed, and [2, 3] provide implementations of the algorithm to invert 2 × 2 matrices. We express floating point operations such as division and matrix inversion as quadratic unconstrained binary optimization (QUBO) problems, which are ideal for a quantum annealer. We have implemented our algorithms on the D-Wave 2000Q and 2X chips, illustrating that division and matrix inversion can be performed on an existing quantum annealer.
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