Quantum algorithms offer significant speed-ups over their classical counterparts in various applications. In this paper, we develop quantum algorithms for the Kalman filter widely used in classical control engineering using the block encoding method. The entire calculation process is achieved by performing matrix operations on Hamiltonians based on the block encoding framework, including addition, multiplication, and inversion, which can be completed in a unified framework compared to previous quantum algorithms for solving control problems. We demonstrate that the quantum algorithm exponentially accelerates the computation of the Kalman filter compared to traditional methods. The time complexity can be reduced from $O(n^3)$ to $O(\kappa poly\log(n/\epsilon)\log(1/\epsilon'))$, where $n$ represents the matrix dimension, $\kappa$ denotes the condition number for the matrix to be inverted, $\epsilon$ indicates desired precision in block encoding, $\epsilon'$ signifies desired precision in matrix inversion. This paper provides a comprehensive quantum solution for implementing the Kalman filter and serves as an attempt to broaden the scope of quantum computation applications. Finally, we present an illustrative example implemented in Qiskit (a Python-based open-source toolkit) as a proof-of-concept.

We consider a finite-dimensional oscillatory integral which provides a "finite-dimensional model" for analytically continued $SU(2)$ Chern-Simons theory on closed 3-manifolds that are described by plumbing trees. This model allows an efficient description of Stokes phenomenon for perturbative expansions in Chern-Simons theory around classical solutions - $SL(2,\mathbb{C})$ flat connections. Moreover, the Stokes coefficients can be categorified, i.e. promoted to graded vector spaces, in terms of this finite-dimensional model. At least naively, the categorification gives BPS spectrum of 5d maximally supersymmetric Yang-Mills theory on the 3-manifold times a line with appropriate boundary conditions. We also comment on necessity of taking into account "flat connections at infinity" to capture Stokes phenomenon for certain 3-manifolds.

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SU$_q(1,1)$. We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SU$_q(1,1)$, whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the $q$-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.

What does it mean for a boundary condition to be symmetric with respect to a non-invertible global symmetry? We discuss two possible definitions in 1+1d. On the one hand, we call a boundary weakly symmetric if the symmetry defects can terminate topologically on it, leading to conserved operators for the Hamiltonian on an interval (in the open string channel). On the other hand, we call a boundary strongly symmetric if the corresponding boundary state is an eigenstate of the symmetry operators (in the closed string channel). These two notions of symmetric boundaries are equivalent for invertible symmetries, but bifurcate for non-invertible symmetries. We discuss the relation to anomalies, where we observe that it is sometimes possible to gauge a non-invertible symmetry in a generalized sense even though it is incompatible with a trivially gapped phase. The analysis of symmetric boundaries further leads to constraints on bulk and boundary renormalization group flows. In 2+1d, we study the action of non-invertible condensation defects on the boundaries of $U(1)$ gauge theory and several TQFTs. Starting from the Dirichlet boundary of free Maxwell theory, the non-invertible symmetries generate infinitely many boundary conditions that are neither Dirichlet nor Neumann.

The physical scalar product between spin-networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNN), which are quantum neural networks previously introduced by the authors in the context of quantum machine learning. However, the effective evaluation of the scalar product remains a bottleneck for the applicability of the theory. We introduce an algorithm for the evaluation of the physical scalar product defined by Noui and Perez between spin-network with hexagonal shape. By means of recoupling theory and the properties of the Haar integration we obtain an efficient algorithm, and provide several proofs regarding the main steps. We investigate the behavior of the TQNN evaluations on certain classes of spin-networks with the classical and quantum recoupling. All results can be independently reproduced through the "this http URL" framework~\href{this https URL}{\nolinkurl{this https URL}}

We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and prove that it satisfies codescent with respect to the target S, which implies the locality property. We apply this result to construct a classifying space for concordance classes of functorial field theories with geometric data, solving a conjecture of Stolz and Teichner about the existence of such a space. We use our classifying space construction to develop a geometric theory of power operations, following the recent work of Barthel, Berwick-Evans, and Stapleton.

Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, which is difficult in general, and not trivial even for well-structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well-structured sparse matrices, and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

Given a codimension one Riemannian embedding of Riemannian spin$^c$-manifolds $\imath:X \to Y$ we construct a family $\{\imath_!^ \epsilon\}_{0< \epsilon< \epsilon_0}$of unbounded$KK$-cycles from$C(X)$to$C_0(Y)$, each equipped with a connection $\nabla^\epsilon$ and each representing the shriek class $\imath_! \in KK(C(X), C_0(Y))$. We compute the unbounded product of $\imath_!^\epsilon$ with the Dirac operator $D_Y$ on $Y$ and show that this represents the $KK$-theoretic factorization of the fundamental class $[X] = \imath_! \otimes [Y]$for all$\epsilon$. In the limit$\epsilon \to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}{\epsilon} T + D_X + \mathcal{O}(\epsilon)$where the ``divergent'' part$T$ is an index cycle representing the unit in $KK(\mathbb{C}, \mathbb{C})$ and the constant ``renormalized'' term is the Dirac operator $D_X$ on $X$. The curvature of $(\imath_!^\epsilon, \nabla^\epsilon)$ is further shown to converge to the square of the mean curvature of $\imath$ as $\epsilon \to 0$.

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum $\mathfrak{sl}_2$ at a root of unity. These are generalized quantum invariants depend both on a knot $K$ and a representation of the fundamental group of its complement into $\mathrm{SL}_2(\mathbb{C})$; equivalently, we can think of $\mathrm{KR}(K)$ as associating to each knot a function on (a slight generalization of) its character variety. In this paper we clarify some details of their construction. In particular, we show that for $K$ a hyperbolic knot $\mathrm{KaRe}(K)$ can be viewed as a function on the geometric component of the $A$-polynomial curve of $K$. We compute some examples at a third root of unity.

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user friendly resource for beginners and experts alike.

We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category $\mathcal{C}$ is represented by a Frobenius algebra in $\mathcal{C}$. Several directions for further investigation are also proposed.

We prove the equivalence of VOA tensor categories and conformal net tensor categories for the following examples: all WZW models; all lattice VOAs; all unitary parafermion VOAs; type $ADE$ discrete series $W$-algebras; their tensor products; their regular cosets. A new proof of the complete rationality of conformal nets is also given.

In this paper we prove the isomorphism of the positive half of the quantum toroidal algebra and the positive half of the Maulik-Okounkov quantum affine algebra of affine type $A$ via the monodromy representation for the Dubrovin connection. The main tool is on the proof of the fact that the degeneration limit of the algebraic quantum difference equation is the same as that of the Okounkov-Smirnov geometric quantum difference equation.

We study the Factorization Paradox from the bottom up by adapting methods from perturbative renormalization. Just as quantum field theories are plagued with loop divergences that need to be cancelled systematically by introducing counterterms, gravitational path integrals are plagued by wormhole contributions that spoil the factorization of the holographic dual. These wormholes must be cancelled by some stringy effects in a UV complete, holographic theory of quantum gravity. In a simple model of two-dimensional topological gravity, we outline a gravitational analog of the recursive BPHZ procedure in order to systematically introduce ``counter-wormholes" which parametrize the unknown stringy effects that lead to factorization. Underlying this procedure is a Hopf algebra of symmetries which is analogous to the Connes--Kreimer Hopf algebra underlying perturbative renormalization. The group dual to this Hopf algebra acts to reorganize contributions from spacetimes with distinct topology, and can be seen as a gauge group relating various equivalent ways of constructing a factorizing gravitational path integral.

This paper addresses the question of how categorical symmetries act on extended operators in quantum field theory. Building on recent results in two dimensions, we introduce higher tube categories and algebras associated to higher fusion category symmetries. We show that twisted sector extended operators transform in higher representations of higher tube algebras and interpret this result from the perspective of the sandwich construction of finite symmetries via the Drinfeld center. Focusing on three dimensions, we discuss a variety of examples to illustrate the general constructions. In the case of invertible symmetries, we show that higher tube algebras are higher analogues of twisted Drinfeld doubles of finite groups, generalising known constructions in two dimensions. Building on this foundation, we discuss non-invertible Ising-like symmetry categories obtained by gauging finite subgroups. We also consider non-invertible topological symmetry lines described by braided fusion categories and discuss connections to the M\"uger center and braided module categories.

Let $\Bbbk$ be an algebraically closed field of characteristic 0 and $H$ a finite-dimensional Hopf algebra over $\Bbbk$ with the dual Chevalley property. In this paper, we show that $\operatorname{gr}^c(H)$ is of tame corepresentation type if and only if $\operatorname{gr}^c(H)\cong (\Bbbk\langle x,y\rangle/I)^* \times H^\prime$ for some finite-dimensional semisimple Hopf algebra $H^\prime$ and some special ideals $I$. Then, by the method of link quiver and bosonization, we discuss which of the above ideals will occur when $(\Bbbk\langle x,y\rangle/I)^* \times H_0$ is a Hopf algebra of tame corepresentation type under some assumptions.

We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of $\mathcal{\ddot{B}}$ we produce automorphisms and anti-involutions of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements $C$ and $k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$ up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type $A$ due to Miki which have been instrumental in the study of the structure and representation theory of $U_{q}(\mathfrak{sl}_{n+1,\mathrm{tor}})$.

A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of $\operatorname{Spec}(Z)$. In this article we consider a Poisson order $A$ over a complex affine Poisson algebra $Z$. We stratify the primitive spectrum $\operatorname{Prim}(A)$ into symplectic cores, which should be thought of as families of non-commutative symplectic leaves. We then introduce a category $A$-$\mathcal{P}$-Mod of $A$-modules adapted to the Poisson structure on $Z$, and we show that when $\operatorname{Spec}(Z)$ is smooth with locally closed symplectic leaves, there is a natural homeomorphism from the spectrum of annihilators of simple objects in $A$-$\mathcal{P}$-Mod to the set of symplectic cores in $\operatorname{Prim}(A)$ with its quotient topology. Several application are given to Poisson representation theory. Our main tool is the Poisson enveloping algebra $A^e$ of a Poisson order $A$, which captures the Poisson representation theory of $A$. For $Z$ regular and affine we prove a PBW theorem for $A^e$ and use this to characterise the annihilators of simple Poisson modules: they coincide with the Poisson weakly locally closed, the Poisson primitive and the Poisson rational ideals. We view this as a generalised weak Poisson Dixmier--Moeglin equivalence.

A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of $\operatorname{Spec}(Z)$. In this article we consider a Poisson order $A$ over a complex affine Poisson algebra $Z$. We stratify the primitive spectrum $\operatorname{Prim}(A)$ into symplectic cores, which should be thought of as families of non-commutative symplectic leaves. We then introduce a category $A$-$\mathcal{P}$-Mod of $A$-modules adapted to the Poisson structure on $Z$, and we show that when $\operatorname{Spec}(Z)$ is smooth with locally closed symplectic leaves, there is a natural homeomorphism from the spectrum of annihilators of simple objects in $A$-$\mathcal{P}$-Mod to the set of symplectic cores in $\operatorname{Prim}(A)$ with its quotient topology. Several application are given to Poisson representation theory. Our main tool is the Poisson enveloping algebra $A^e$ of a Poisson order $A$, which captures the Poisson representation theory of $A$. For $Z$ regular and affine we prove a PBW theorem for $A^e$ and use this to characterise the annihilators of simple Poisson modules: they coincide with the Poisson weakly locally closed, the Poisson primitive and the Poisson rational ideals. We view this as a generalised weak Poisson Dixmier--Moeglin equivalence.