核心概念
A quantum algorithm is proposed to efficiently solve linear systems that can be expressed as a low-rank tensor sum, which commonly arise in discretized PDE problems. The algorithm achieves polylogarithmic time complexity in the problem dimension, providing exponential speedup over classical methods.
要約
The content describes a quantum algorithm for solving linear systems that can be expressed as a low-rank tensor sum. This type of linear system often arises in discretized PDE problems, where the problem size grows exponentially with the length of the tensor product chain.
The key highlights are:
The authors focus on linear systems that can be represented as a low-rank tensor sum, i.e., the coefficient matrix and right-hand-side vector can be expressed as a linear combination of a few tensor products of 2-by-2 matrices and 2-dimensional vectors, respectively.
Previous classical algorithms for such linear systems, such as modified Krylov subspace methods, have a polylogarithmic per-iteration complexity but no guarantees on the total convergence cost.
The authors propose a quantum algorithm based on recent advances in adiabatic-inspired quantum linear system algorithms (QLSAs). They provide a detailed analysis of the circuit depth for implementing this algorithm.
The authors show that the total complexity of their quantum algorithm implementation is polylogarithmic in the problem dimension, which is comparable to the per-iteration complexity of the classical heuristic methods, but with exponential speedup.
The key to the efficient implementation is the ability to decompose the Hamiltonian used in the QLSA into a sum of two types of structured Hamiltonians, which can be implemented efficiently using quantum circuits.