Quantum and Classical Query Complexities of Functions of Matrices

Let A be an s -sparse Hermitian matrix, f ( x ) be a univariate function, and i , j be two indices. In this work, we investigate the query complexity of approximating i f ( A ) j . We show that for any continuous function f ( x ):[−1,1]→ [−1,1], the quantum query complexity of computing i f ( A ) j ± ε/4 is lower bounded by Ω(deg ε ( f )). The upper bound is at most quadratic in deg ε ( f ) and is linear in deg ε ( f ) under certain mild assumptions on A . Here the approximate degree deg ε ( f ) is the minimum degree such that there is a polynomial of that degree approximating f up to additive error ε in the interval [−1,1]. We also show that the classical query complexity is lower bounded by Ω(( s /2) (deg 2ε ( f )−1)/6 ) for any s ≥ 4. Our results show that the quantum and classical separation is exponential for any continuous function of sparse Hermitian matrices, and also imply the optimality of implementing smooth functions of sparse Hermitian matrices by quantum singular value transformation. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.

Authors

• Ashley Montanaro
• Changpeng Shao

Keywords

• Quantum computing
• functions of matrices
• polynomial method
• query complexity