Matrix-Vector vs. Matrix-Matrix Multiplication: Potential in DD-based Simulation of Quantum Computations

Abstract

The simulation of quantum computations basically boils down to the multiplication of vectors (describing the respective quantum state) and matrices (describing the respective quantum operations). However, since those matrices/vectors are exponential in size, most of the existing solutions (relying on arrays for their representation) are either limited to rather small quantum systems or require substantial hardware resources. To overcome these shortcomings, solutions based on decision diagrams (DD-based simulation) have been proposed recently. They exploit redundancies in quantum states as well as matrices and, by this, allow for a compact representation and manipulation. This offers further (unexpected) potential. In fact, simulation has been conducted thus far by applying one operation (i.e. one matrix-vector multiplication) after another. Besides that, there is the possibility to combine several operations (requiring a matrix-matrix multiplication) before applying them to a vector. But since, from a theoretical perspective, matrix-vector multiplication is significantly cheaper than matrix-matrix multiplication, the potential of this direction was rather limited thus far. In this work, we show that this changes when decision diagrams are employed. In fact, their more compact representation frequently makes matrix-matrix multiplication more beneficial—leading to substantial improvements by exploiting the combination of operations. Experimental results confirm the proposed strategies for combining operations lead to speed-ups of several factors or—when additionally exploiting further knowledge about the considered instance—even of several orders of magnitudes.