Related papers: Pauli Automorphisms, Clifford Groups, and Local Cl…
In the present paper, we examine in detail the method of "graph compactifications" of topological groups. The graph and Ellis methods of constructing proper compactifications of topological groups are applied for the investigation of…
The object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in…
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has…
This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the…
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves cryptography, the new generation of public key…
The groups of order 64p without a normal sylow p-subgroup are listed, and their automorphism groups are also determined. As a by-product of our original effort to get these groups, we needed to determine the automorphism groups of those…
When visualised as an operation on the Bloch sphere, the qubit "pi-over-eight" gate corresponds to one-eighth of a complete rotation about the vertical axis. This simple gate often plays an important role in quantum information theory,…
Efficient and effective compilation of quantum circuits remains an important aspect of executing quantum programs. In this paper, we propose a generic compilation framework particularly suitable for limited connectivity, that extends many…
The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and 'nearly diagonal' semi-Clifford gates are particularly important:…
This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived.…
Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that…
Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing…
Random quantum circuits continue to inspire a wide range of applications in quantum information science and many-body quantum physics, while remaining analytically tractable through probabilistic methods. Motivated by an interest in…
We use a certain rigid local system in order to prove the potential automorphy of certain Galois representations with values in $G_2,$ found by N. Katz and the author.
We study a scheme to implement an asymptotic unitary 3-design. The scheme implements a random Pauli once followed by the implementation of a random transvection Clifford by using state twirling. Thus the scheme is implemented in the form of…
In this paper, we study groups of automorphisms of algebraic systems over a set of $p$-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The main result…
The paper is devoted to projective Clifford groups of quantum $N$-dimensional systems. Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann-Knill theorem). However,…
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main…
We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentations of t.d.l.c.\ groups, and show their equivalence. In the process, we obtain an…
We describe the structure of the extended Clifford Group (defined to be the group consisting of all operators, unitary and anti-unitary, which normalize the generalized Pauli group (or Weyl-Heisenberg group as it is often called)). We also…