Parallel computation for well-endowed rings and space-bounded probabilistic machines
Information and Control
Random generation of combinatorial structures from a uniform
Theoretical Computer Science
NP is as easy as detecting unique solutions
Theoretical Computer Science
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
The complexity of very simple Boolean formulas with applications
SIAM Journal on Computing
Machine models and simulations
Handbook of theoretical computer science (vol. A)
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Graph orientations with no sink and an approximation for a hard case of #SAT
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Theoretical Computer Science
Quantum designer and network simulator
Proceedings of the 1st conference on Computing frontiers
Journal of Symbolic Computation
On the complexity of mixed discriminants and related problems
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
A dichotomy theorem for homomorphism polynomials
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Classical simulation of dissipative fermionic linear optics
Quantum Information & Computation
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A model of quantum computation based on unitary matrix operations was introduced by Feynman and Deutsch. It has been asked whether the power of this model exceeds that of classical Turing machines. We show here that a significant class of these quantum computations can be simulated classically in polynomial time. In particular we show that two-bit operations characterized by 4 \times 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time. This contrasts with the known universality of two-bit operations, and demonstrates that efficient quantum computation of restricted classes is reconcilable with the Polynomial Time Turing Hypothesis. In other words it is possible that quantum phenomena can be used in a scalable fashion to make computers but that they do not have superpolynomial speedups compared to Turing machines for any problem. The techniques introduced bring the quantum computational model within the realm of algebraic complexity theory. In a manner consistent will one view of quantum physics, the wave function is simulated deterministically, and randomization arises only in the course of making measurements. The results generalize the quantum model in that they do not require the matrices to be unitary. In a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read-twice Boolean formulae. All our results are based on the use of gates that are defined in terms of their graph matching properties.