Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Algorithm 288: solution of simultaneous linear Diophantine equations [F4]
Communications of the ACM
Quantum computation and quantum information
Quantum computation and quantum information
Quantum Factoring, Discrete Logarithms, and the Hidden Subgroup Problem
Computing in Science and Engineering
Quantum Cryptanalysis of Hidden Linear Functions (Extended Abstract)
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
Weitere zum Erfüllungsproblem polynomial äquivalente kombinatorische Aufgaben
Komplexität von Entscheidungsproblemen, Ein Seminar
Simulating Quantum Computation by Contracting Tensor Networks
SIAM Journal on Computing
Generalized clifford groups and simulation of associated quantum circuits
Quantum Information & Computation
A linearized stabilizer formalism for systems of finite dimension
Quantum Information & Computation
Classical simulations of Abelian-group normalizer circuits with intermediate measurements
Quantum Information & Computation
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The quantum Fourier transform (QFT) is an important ingredient in various quantum algorithms which achieve superpolynomial speed-ups over classical computers. In this paper we study under which conditions the QFT can be simulated efficiently classically. We introduce a class of quantum circuits, called normalizer circuits: a normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. In our main result we prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. Our result generalizes the Gottesman-Knill theorem: in particular, Clifford circuits for d-level qudits arise as normalizer circuits over the group Zdm. We also highlight connections between normalizer circuits and Shor's factoring algorithm, and the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.