Quantum normalizer circuits were recently introduced as generalizations of
Clifford circuits [arXiv:1201.4867]: 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
[arXiv:1201.4867] 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 [quant-ph/9705052, quant-ph/9807006] 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
[quant-ph/9705052, quant-ph/9807006] 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.Comment: 26 pages+appendices. Title has changed in this second version. To
appear in Quantum Information and Computation, Vol.14 No.3&4, 201