Quantum Algorithms for Matrix Multiplication and Product Verification

Years and Authors of Summarized Original Work

• 2006; Buhrman, Špalek

• 2012; Jeffery, Kothari, Magniez

Problem Definition

Let S be any algebraic structure over which matrix multiplication is defined, such as a field (e.g., real numbers), a ring (e.g., integers), or a semiring (e.g., the Boolean semiring). If we use + and ⋅ to denote the addition and multiplication operations over S, then the matrix product C of two n × n matrices A and B is defined as $C_{ij} :=\sum \nolimits_{ k=1}^{n}A_{ik} \cdot B_{kj}$ for all i, j ∈ { 1, 2, …, n}. Over the Boolean semiring, the addition and multiplication operations are the logical OR and logical AND operations, respectively, and thus, the matrix product C is defined as $C_{ij} :=\bigvee _{ k=1}^{n}(A_{ik} \wedge B_{kj})$. In this article we consider the following problems.

Problem 1 (Matrix multiplication)

Input: :

Two n × n matrices A and B with entries from S.

Output: :

The matrix C : = AB.

Problem 2 (Matrix product verification)

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