Normalizer circuits [1, 2] are generalized Clifford circuits that act on arbitrary finitedimensional systems Hd1 ⊗· · ·⊗Hdn with a standard basis labeled by the elements of a finite Abelian group G = Zd1 × · · · × Zdn . Normalizer gates implement operations associated with the group G and can be of three types: quantum Fourier transforms, group automorphism gates and quadratic phase gates. In t...

Normalizer circuits [3, 4] are a family of quantum circuits which generalize Cli ord circuits [5 8] to Hilbert spaces associated with arbitrary nite abelian groups G = Zd1 × · · · × Zdn . Normalizer circuits are composed of normalizer gates. Important examples are quantum Fourier transforms (QFTs), which play a central role in quantum algorithms, such as Shor's [9]. Refs. [3, 4] showed that nor...

In this study, we deal with the conjecture given in [R. Keskin, Suborbital graph for the normalizer of Γ0(m), European Journal of Combinatorics 27 (2006) 193206.], that when the normalizer of Γ0(N) acts transitively on Q ∪ {∞}, any circuit in the suborbital graph G(∞, u/n) for the normalizer of Γ0(N), is of the form v → T (v) → T (v) → · · · → T (v) → v, where n > 1, v ∈ Q ∪ {∞} and T is an ell...

This work presents a precise connection between Clifford circuits, Shor’s factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup problems. We show that all these different forms of quantum computation belong to a common new restricted model of quantum operations that we call black-box normalizer circuits. To define ...

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 ...

Normalizers and p-normalizers of maximal tori in p-compact groups can be characterized by the Euler characteristic of the associated homogeneous spaces. Applied to centralizers of elementary abelian p-groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus; i.e. that “norm...

A fundamental problem for network intrusion detection systems is the ability of a skilled attacker to evade detection by exploiting ambiguities in the traffic stream as seen by the monitor. We discuss the viability of addressing this problem by introducing a new network forwarding element called a traffic normalizer. The normalizer sits directly in the path of traffic into a site and patches up...

We present an Agda formalization of a normalization proof for simply-typed lambda terms. The normalizer consists of two coinductively defined functions in the delay monad: One is a standard evaluator of lambda terms to closures, the other a type-directed reifier from values to η-long β-normal forms. Their composition, normalization-by-evaluation, is shown to be a total function a posteriori, us...

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 n...