Related papers: A remark on the quantum complexity of the Kronecke…

Stirling numbers, which count partitions of a set and permutations in the symmetric group, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the…

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of…

We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The…

Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…

In this short note, by combining the work of Amiot-Iyama-Reiten and Thanhoffer de Volcsey-Van den Bergh on Cohen-Macaulay modules with the previous work of the author on orbit categories, we compute the (nonconnective) algebraic K-theory…

Assuming Calabi symmetry, we prove that a numerical condition ensures the solvability of the complex Hessian quotient equation, as conjectured by Sz\'ekelyhidi. We also propose a conjecture on the existence of a $k$-subharmonic…

We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible…

In this note, we provide a unifying framework to investigate the computational complexity of classical spin models and give the full classification on spin models in terms of system dimensions, randomness, external magnetic fields and types…

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in…

We improve on Gonek-Montgomery's quantitative version of Kronecker's approximation theorem.

We give explicit positive combinatorial interpretations for the plethysm coefficients $\langle s_\mu[s_\nu], s_\lambda\rangle$, when $\lambda$ has at most two rows, as counting certain marked trees. In the special case $\mu=(n)$, this also…

Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which…

The presence of symmetries, be they discrete or continuous, in a physical system typically leads to a reduction in the problem to be solved. Here we report that neither translational invariance nor rotational invariance reduce the…

We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether $S$ is a semigroup or has an identity element. If S is a monoid, we want to…

We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a p-summability condition and for integration of functions from Lebesgue…

The role of quantum groups and braid groups in the description of Standard Model particles is discussed. Some recent results on the use of the quantum group $SU_q(3)$ as a flavour symmetry are reviewed and a connection between two…

We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\sf g}-vector cone…

The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier…

We propose a taxonomy for quantum algorithms grounded in the fundamental symmetries, both continuous and discrete, underlying quantum state spaces, oracles, and circuit dynamics. By organizing algorithms according to their symmetry groups…

The implementation of physical symmetries into problem descriptions allows for the reduction of parameters and computational complexity. We show the integration of the permutation symmetry as the most restrictive discrete symmetry into…