The paper studies 4-qubit Clifford states and their properties.
problem Understanding the set and properties of 4-qubit Clifford states.
method Analyzing the 293760 4-qubit Clifford states, splitting them into 18 groups, and studying the action of CNOT gates and local gates.
result There are 293760 4-qubit Clifford states with specific entanglement entropies, and any pair can be connected with local gates and at most 3 CNOT gates.
The Clifford group for 2 qubits is divided into 20 orbits, each with 4608 matrices.
problem Understanding the structure of the Clifford group for 2 qubits.
method Equivalence relation based on local Clifford gates and analysis of orbits.
result The Clifford group for 2 qubits is divided into 20 orbits, each with 4608 matrices.
GKP codes connect quantum gates to algebraic curves, enabling fault-tolerant quantum computation.
problem Implementing fault-tolerant quantum computation in quantum harmonic oscillator systems.
method Exploring the topological and algebraic structure of GKP codes, showing how gates correspond to symplectic automorphisms and mapping class groups of surfaces.
result GKP Clifford gates are identified with symplectic automorphisms of GKP lattices and mapping class groups of surfaces, providing a topological interpretation of fault tolerance.
Topological theory for qLDPC codes enables non-Clifford gates and magic state injection.
problem Fault-tolerant quantum computation in qLDPC codes with non-Clifford gates and magic state resources.
method Developed a topological theory using simplicial or CW complex structures and deformation retraction.
result Achieved non-Clifford gates and magic state injection in qLDPC codes with constant rate and polynomial distance.
New non-semisimple Ising anyons enable robust universal quantum computation.
problem Limitation of semisimple theories in universal topological quantum computation.
method Developed non-semisimple Ising anyon model with new anyon types indexed by α. result Robust universality of braiding persists over an open interval of α. New quantum code breaks distance barrier with transversal non-Clifford gates.
problem Breaking the sqrt(N) distance barrier for quantum LDPC codes.
method Combining three qLDPC codes, Freedman-Hastings mapping, and triple cup product.
result Achieves Ω(N^(2/3)) distance and Θ(N^(2/3)) dimension, enabling fault-tolerant magic state preparation.
Single T-gate makes distribution learning hard for deep circuits.
problem Learning probability distributions from quantum circuits.
method Characterization of learnability and simulatability of quantum circuit outputs.
result Injection of a single T-gate into depth n^Ω(1) circuits makes distribution learning hard.
New fault-tolerant quantum gates for homological LDPC codes with constant or almost-constant rate.
problem Fault-tolerant quantum computing for homological LDPC codes with constant or almost-constant encoding rate.
method Derive generic formula for transversal and logical gates acting on 3-manifolds, using higher symmetries and cup product cohomology.
result Parallelizable logical gates for homological LDPC codes with constant or almost-constant rate.
Quantum circuits reveal pathways to dequantization in machine learning models.
problem Navigating the complex landscape of quantum machine learning models and algorithms.
method Introducing a framework connecting quantum circuit structure to function representability.
result Fundamental properties of quantum circuits determine classical simulability of models.
Quantum codes linked to abelian varieties, providing mathematical rigor.
problem Quantum error correction through complex abelian varieties.
method Mathematical formulation of Gottesman-Kitaev-Preskill codes using abelian varieties.
result Asymptotic isometry of encoding, precise gate realizations, and failure probability optimization.
MBQC linked to CQCA, yielding efficient Ansätze.
problem Quantum computation efficiency and Ansatz adaptation.
method Relating MBQC to CQCA and constructing Ansätze.
result MBQC Ansätze can lead to different performances on learning tasks.
Study shows limitations and possibilities of learning quantum circuit output distributions.
problem Learnability of output distributions of local quantum circuits.
method Investigated within two oracle models: statistical query model and direct sample access model.
result Output distributions of super-logarithmic depth Clifford circuits are not efficiently learnable in the statistical query model.
The study examines how quantum resources enhance the complexity of quantum circuits.
problem Quantum resource enhancement on circuit complexity.
method Utilizing quantum resource theories, the study analyzes statistical complexities of quantum circuits with limited quantum resources.
result Bounds for statistical complexities of quantum circuits are derived and applied to specific cases.
Study on topological order on fractal geometries, proving no-go theorem and fault-tolerant gates.
problem Investigating topological order on fractal geometries embedded in n dimensions.
method Using quantum error-correcting codes and systolic geometry to diagnose topological order.
result Proves no-go theorem for topological order on 2D fractals, survival on higher dimensions, and construction of fault-tolerant gates.
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
Study relates Finsler structures to Clifford bundles for flat metrics.
problem Relating Finsler structures to Clifford bundles for flat metrics.
method Examines extensions of Clifford bundles and Finsler type structures for flat metrics.
result Triangle map exists between Finsler structures constructed from metrics and 1-forms.
Extends Kostant's results to symmetric pairs in Clifford algebras.
problem Analyzing k-invariants in Clifford algebras of symmetric pairs. method Proves Cartan theorem, transgression theorem, Harish-Chandra isomorphism, and Clifford algebra conjecture for relative case.
result Establishes a relative transgression theorem and Harish-Chandra isomorphism for Clifford algebras.
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the same distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous if for any two points x1,x2∈M there is a Clifford-Wolf translation ρ such that ρ(x1)=x2. In this paper, we give a complete classifi…
We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for som…
New symmetric Willmore tori emerge from Clifford torus in Berger spheres.
problem Finding new symmetric Willmore surfaces from Clifford torus.
method Applying bifurcation theory to estimate Morse index of Willmore surfaces.
result New symmetric Willmore tori emerge from Clifford torus.
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous if for any two point x1,x2∈M there is a Clifford-Wolf translation ρ such that ρ(x1)=x2. In this paper, we study Clifford-Wolf transl…
Quantum algorithm for multi-asset option pricing under different volatility models.
problem Efficiently pricing multi-asset options under various volatility models using quantum computing.
method Developed an end-to-end quantum PDE framework for European option pricing, solving PDEs after discretization on spatial grids.
result Quantum framework provides polynomial improvement in resource usage compared to classical methods.
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). Aft…
Classifies compact Clifford-Klein forms for specific Lie algebras.
problem Classifying compact Clifford-Klein forms for given Lie algebra structures.
method Using Onishchik's results on semisimple Lie algebras, the paper classifies forms for triples (g,h,l).
result New examples of reductive homogeneous spaces with non-standard compact Clifford-Klein forms.
In this paper, we study Clifford-Wolf translations of Finsler spaces. We first give a characterization of Clifford-Wolf translations of Finsler spaces in terms of Killing vector fields. In particular, we show that there is a natural correspondence between Clifford-Wolf translations and the Killing vector fields of cons…
We consider pseudo-Riemannian generalizations of Osserman, Clifford, and the duality principle properties for algebraic curvature tensors and investigate relations between them. We introduce quasi-Clifford curvature tensors using a generalized Clifford family and show that they are Osserman. This allows us to discover …
Constructs a model for differential KO-theory using Clifford modules.
problem Refining Atiyah and Singer's families index with differential structure.
method Builds a model using families of Clifford modules with superconnection.
result Affords a differential refinement of Atiyah and Singer's families index.
The paper explores connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
problem Exploring connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
method Starting from collections of 'Kähler 2-forms', the paper constructs canonical 4-forms and calibrated 4-planes in dimensions 8 and 16.
result Explicit formulas for canonical 4-forms ΦSpin(8) and ΦSpin(7)U(1) are derived, and their calibrated 4-planes are characterized. A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein…
The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-S…
Unified study of surfaces using Clifford algebras.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
We provide a necessary condition for the existence of a compact Clifford-Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi-Ono with an elementary fact that certain two different Clifford-Klein forms have the same cohomology ring. We give some examples, S…
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…
Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out L indpendent Clifford cones in R2N+2 via the multi-screw motion, we construct minimal submanifolds in RL(2N+2)+1. Also, we sweep out the L-rays Clifford cone (introduced in Sectio…
Improved logistic MoE with sigmoid gate shows better sample efficiency.
problem Improving sample efficiency in logistic MoE models.
method Comprehensive analysis of multinomial logistic MoE with modified sigmoid gate, incorporating temperature parameter and using Euclidean score.
result The sigmoid gate leads to lower sample complexity than softmax gate for both parameter and expert estimation.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
Study pin manifolds using Clifford linear Dirac operator and KO-theory.
problem Index theory on Pin manifolds.
method Clifford linear Dirac operator and differential KO-theory.
result Systematic treatment of index theory on Pin manifolds.
New method prepares 3-qubit states using local gates and controlled-Z gates.
problem Preparation of 3-qubit states using quantum gates.
method Uses Ry(θ) gates and controlled-Z gates, with an optimal number of controlled-Z gates. result Optimal number of controlled-Z gates for preparing 3-qubit states is four. In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus S1(1)×S1(1) is the unique compact orientable Lagrangian self-shrinker in C2 with ∣A∣2≤2, which gives an affirmative answer to Ca…
The Clifford torus minimizes Willmore energy closely for small perturbations.
problem Finding the closest shape to the Clifford torus under small perturbations of Willmore energy.
method Analyzing integral 2-varifolds with specific properties and showing quantitative closeness to the Clifford torus.
result The support of the varifold is quantitatively close to the Clifford torus after a conformal transformation.
In this article we prove that under certain assumptions, a reductive homogeneous space G/H does not admit a solvable compact Clifford-Klein form. This generalizes the well known non-existence theorem of Benoist for nilpotent Clifford-Klein forms. This generalization works for a particular class of homogeneous spaces de…
We present a generalization of the Clifford action for other representations spaces of Spin(n), which is called the Clifford homomorphism. Their properties extend to the ones for the higher spin Dirac operators on spin manifolds. In particular, we have general Bochner identities for them, and an eigenvalue estimate o…
In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing pr…
This dissertation explores Clifford bundles and spinor fields in geometric and algebraic contexts.
problem Understanding spinor fields and their classification in geometric frameworks.
method Combines algebraic and geometric approaches to study Clifford structures on bundles and spinor fields.
result Identifies new spinor field classes in warped flux compactifications.
Classifies 4D spaces with compact Clifford-Klein forms.
problem Classifying 4D symmetric spaces with compact Clifford-Klein forms.
method Developed a method for 1-connected solvable symmetric spaces.
result Classification of 4D spaces with compact Clifford-Klein forms.
Sigmoid gating is more sample efficient than softmax in mixture of experts.
problem Softmax gating leads to unnecessary competition among experts, causing representation collapse.
method Theoretical analysis of a regression framework with mixture of experts, identifying identifiability conditions and convergence rates.
result Sigmoid gating requires fewer samples to achieve the same expert estimation error as softmax gating.
In quantum computation, series of quantum gates have to be arranged in a predefined sequence that led to a quantum circuit in order to solve a particular problem. What if the sequence of quantum gates is known but both the problem to be solved and the outcome of the so defined quantum circuit remain in the shadow? This…