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Recent theoretical results in quantum machine learning have demonstrated a general trade-off between the expressive power of quantum neural networks (QNNs) and their trainability; as a corollary of these results, practical exponential…
We consider a bipartite quantum object, composed of a quantum system and a quantum actuator which is periodically reset. We show that the reduced dynamics of the system approaches unitarity as the reset frequency of the actuator is…
One of the main achievements of LQG is the consistent quantization of the Wheeler-DeWitt equation which is free of UV problems. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set…
Integrals for the product of unitary-matrix elements over the U(n) group will be discussed. A group-theoretical formula is available to convert them into a multiple sum, but unfortunately the sums are often tedious to compute. In this…
Polynomial invariants of a group action often appear only in high degree, and in many representations the invariant ring imposes severe degree constraints before any nontrivial invariants can occur. In contrast, the larger class of unitary…
We revisit the calculation of quantum-gravitational corrections to the power spectra of scalar and tensor perturbations in the Born-Oppenheimer approach to quantum gravity. We focus on the issue of the definition of the inner product of the…
We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity $O(\exp(C\sqrt{\log N}))$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a…
The Howe duality between quantum general linear supergroups was firstly established by Y. Zhang via quantum coordinate superalgebras. In this paper, we provide two other approaches to this Howe duality. One is constructed by quantum…
We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups $U_{N}^{+}$. In other words, we classify all discrete quantum subgroups of $\widehat{U}_{N}^{+}$, thereby proving a quantum variant…
We construct the HNN extension of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is K-amenable.
Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups…
The Verma modules over the quantum groups $\mathrm U_q(\mathfrak{gl}_{l + 1})$ for arbitrary values of $l$ are analysed. The explicit expressions for the action of the generators on the elements of the natural basis are obtained. The…
The problem of distinguishing two unitary transformations, or quantum gates, is analyzed and a function reflecting their statistical distinguishability is found. Given two unitary operations, $U_1$ and $U_2$, it is proved that there always…
We introduce a combinatorial criterion for verifying whether a formula is not the conjunction of an equation and a co-equation. Using this, we give a proof for the nonequationality of the free group. Furthermore, we generalize the latter…
We study $\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\epsilon$-represendations are uniformly close to…
Quantum groups in general and the quantum Anti-de Sitter group $U_q(so(2,3))$ in particular are studied from the point of view of quantum field theory. We show that if $q$ is a suitable root of unity, there exist finite-dimensional, unitary…
The study of representations invariant to common transformations of the data is important to learning. Most techniques have focused on local approximate invariance implemented within expensive optimization frameworks lacking explicit…
We construct families of irreducible representations for a class of quantum groups $U_{q}(f_{m}(K,H)$. First, we realize these quantum groups as Hyperbolic algebras. Such a realization yields natural families of irreducible weight…
Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define $N(G,m)$ to be the number of conjugacy classes of elements of finite order $m$ in a Lie group $G$, and $N(G,m,s)$ to be the number of such…
We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of…