Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
A new universal and fault-tolerant quantum basis
Information Processing Letters
Quantum computation and quantum information
Quantum computation and quantum information
Fault-Tolerant Quantum Computation with Higher-Dimensional Systems
QCQC '98 Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A Computational Introduction to Number Theory and Algebra
A Computational Introduction to Number Theory and Algebra
Modern Computer Arithmetic
Decomposing finite Abelian groups
Quantum Information & Computation
Cluster states, algorithms and graphs
Quantum Information & Computation
Simulating quantum computers with probabilistic methods
Quantum Information & Computation
A linearized stabilizer formalism for systems of finite dimension
Quantum Information & Computation
Quantum Information & Computation
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Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits [1]: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In [1] it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations [2, 3]) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism [2, 3] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.