Matrices, Exponential Operators and Physical Applications

Willi-Hans Steeb, Yorick Hardy

Pauli spin matrices, Pauli group, commutators, anti-commutators and the Kronecker product are studied. Applications to eigenvalue problems, exponential functions of such matrices, spin Hamilton operators, mutually unbiased bases, Fermi operators and Bose operators are provided.

Xabier Gutiérrez, Nicola Mariella, Javier González-Conde +2 more

We present quantum algorithms to approximate the exponential of banded Toeplitz matrices. These matrices have central importance in many PDE related problems, but their quantum implementation is hindered by their possibly large norms. Constructing directly the exponential we can circumvent this limitation. By relating the lower/upper shift operators to circulant and skewcirculant generators, which are diagonalised by the QFT, we construct (i) an LCU-based block encoding of banded Toeplitz matrices, (ii) an efficient, controllably truncated Pauli-string decomposition of the circulant eigenphases with a closed-form error bound, and (iii) a specialized QFT-Trotter product formula. As an application we build a block encoding of the propagator of the discretised heat equation with periodic, Dirichlet and Neumann boundary conditions, using a frequency cutoff.

Ravinder Rupchand Puri

R. Aldrovandi

Quynh T. Nguyen, Bobak T. Kiani, Seth Lloyd

Many quantum algorithms for numerical linear algebra assume black-box access to a block-encoding of the matrix of interest, which is a strong assumption when the matrix is not sparse. Kernel matrices, which arise from discretizing a kernel function k(x,x′) , have a variety of applications in mathematics and engineering. They are generally dense and full-rank. Classically, the celebrated fast multipole method performs matrix multiplication on kernel matrices of dimension N in time almost linear in N by using the linear algebraic framework of hierarchical matrices. In light of this success, we propose a block-encoding scheme of the hierarchical matrix structure on a quantum computer. When applied to many physical kernel matrices, our method can improve the runtime of solving quantum linear systems of dimension N to O(κpolylog⁡(Nε)) , where κ and ε are the condition number and error bound of the matrix operation. This runtime is near-optimal and, in terms of N , exponentially improves over prior quantum linear systems algorithms in the case of dense and full-rank kernel matrices. We discuss possible applications of our methodology in solving integral equations and accelerating computations in N-body problems.

G. Dattoli, G. Quattromini, Alessandra della Torre +1 more