15.4 Quantum Linear Solver
Harrow, Hassidim and Lloyd (HHL) [133] devised a quantum algorithm to solve linear systems, beating classical computation times. Linear systems are ubiquitous in applications, and many aspects of quantitative finance rely on being able to solve such (low- or high-dimensional) systems. We highlight below two key examples of fundamental importance in finance: solving Partial Differential Equations (PDEs) and portfolio optimisation.
15.4.1 Theoretical aspects
The problem can be stated as follows: given a matrix A ∈MN(ℂ) and a vector b ∈ℂN, find the vector x ∈ℂN such that
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In order for the algorithm to work, the matrix A needs to be Hermitian. If A is not Hermitian, we can nevertheless consider the augmented system
similarly to the Hamiltonian embedding in Section 7.6. We assume from now now that A is indeed Hermitian. The first step of the algorithm is to assume that the vector...
