A taxonomy of problems with fast parallel algorithms
Information and Control
Journal of Computer and System Sciences - Special issue on the 36th IEEE symposium on the foundations of computer science
Space-bounded Quantum complexity
Journal of Computer and System Sciences
Quantum computation and quantum information
Quantum computation and quantum information
Classical and Quantum Computation
Classical and Quantum Computation
On the complexity of simulating space-bounded quantum computations
Computational Complexity
Adiabatic Quantum State Generation
SIAM Journal on Computing
Algorithms for quantum computation: discrete logarithms and factoring
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Unbounded-error quantum computation with small space bounds
Information and Computation
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We show that quantum computers improve on the best known classical algorithms for matrix inversion (and singular value decomposition) as far as space is concerned. This adds to the (still short) list of important problems where quantum computers are of help. Specifically, we show that the inverse of a well conditioned matrix can be approximated in quantum logspace with intermediate measurements. This should be compared with the best known classical algorithm for the problem that requires Ω(log2 n) space. We also show how to approximate the spectrum of a normal matrix, or the singular values of an arbitrary matrix, with ε additive accuracy, and how to approximate the singular value decomposition (SVD) of a matrix whose singular values are well separated. The technique builds on ideas from several previous works, including simulating Hamiltonians in small quantum space (building on [2] and [10]), treating a Hermitian matrix as a Hamiltonian and running the quantum phase estimation procedure on it (building on [5]) and making small space probabilistic (and quantum) computation consistent through the use of offline randomness and the shift and truncate method (building on [8]).