7 research outputs found
Almost Optimal Solution of Initial-Value Problems by Randomized and Quantum Algorithms
We establish essentially optimal bounds on the complexity of initial-value
problems in the randomized and quantum settings. For this purpose we define a
sequence of new algorithms whose error/cost properties improve from step to
step. These algorithms yield new upper complexity bounds, which differ from
known lower bounds by only an arbitrarily small positive parameter in the
exponent, and a logarithmic factor. In both the randomized and quantum
settings, initial-value problems turn out to be essentially as difficult as
scalar integration.Comment: 16 pages, minor presentation change
Implicit Hybrid Quantum-Classical CFD Calculations using the HHL Algorithm
Implicit methods are attractive for hybrid quantum-classical CFD solvers as
the flow equations are combined into a single coupled matrix that is solved on
the quantum device, leaving only the CFD discretisation and matrix assembly on
the classical device. In this paper, an implicit hybrid solver is investigated
using emulated HHL circuits. The hybrid solutions are compared with classical
solutions including full eigen-system decompositions. A thorough analysis is
made of how the number of qubits in the HHL eigenvalue inversion circuit affect
the CFD solver's convergence rates. Loss of precision in the minimum and
maximum eigenvalues have different effects and are understood by relating the
corresponding eigenvectors to error waves in the CFD solver. An iterative
feed-forward mechanism is identified that allows loss of precision in the HHL
circuit to amplify the associated error waves. These results will be relevant
to early fault tolerant CFD applications where every (logical) qubit will
count. The importance of good classical estimators for the minimum and maximum
eigenvalues is also relevant to the calculation of condition number for Quantum
Singular Value Transformation approaches to matrix inversion
Complexity of randomized algorithms for underdamped Langevin dynamics
We establish an information complexity lower bound of randomized algorithms
for simulating underdamped Langevin dynamics. More specifically, we prove that
the worst strong error is of order , for
solving a family of -dimensional underdamped Langevin dynamics, by any
randomized algorithm with only queries to , the driving Brownian
motion and its weighted integration, respectively. The lower bound we establish
matches the upper bound for the randomized midpoint method recently proposed by
Shen and Lee [NIPS 2019], in terms of both parameters and .Comment: 27 pages; some revision (e.g., Sec 2.1), and new supplementary
materials in Appendice