Stark-Coleman Invariants and Quantum Lower Bounds: An Integrated Framework for Real Quadratic Fields

• URL: http://arxiv.org/abs/2506.07640v1
• Date: Mon, 09 Jun 2025 11:06:17 GMT
• Title: Stark-Coleman Invariants and Quantum Lower Bounds: An Integrated Framework for Real Quadratic Fields
• Authors: Ruopengyu Xu, Chenglian Liu,
• Abstract summary: Stark-Coleman invariants $kappa_p(K) = log_p left( fracvarepsilon_mathrmSt,psigma(varepsilon_mathrmSt,p) mod pmathrmord_p(Delta_K)$ through a synthesis of $p$-adic Hodge theory and extended Coleman integration.<n>We show that Stark units constrain the geometric organization of class groups, providing theoretical insight into computational complexity barriers.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. While Stark's conjecture establishes theoretical connections between special units and class group structures, explicit constructions have remained elusive, and precise quantum complexity bounds for class group computations are lacking. Here we establish an integrated framework defining Stark-Coleman invariants $\kappa_p(K) = \log_p \left( \frac{\varepsilon_{\mathrm{St},p}}{\sigma(\varepsilon_{\mathrm{St},p})} \right) \mod p^{\mathrm{ord}_p(\Delta_K)}$ through a synthesis of $p$-adic Hodge theory and extended Coleman integration. We prove these invariants classify class groups under the Generalized Riemann Hypothesis (GRH), resolving the isomorphism problem for discriminants $D > 10^{32}$. Furthermore, we demonstrate that this approach yields the quantum lower bound $\exp\left(\Omega\left(\frac{\log D}{(\log \log D)^2}\right)\right)$ for the class group discrete logarithm problem, improving upon previous bounds lacking explicit constants. Our results indicate that Stark units constrain the geometric organization of class groups, providing theoretical insight into computational complexity barriers.

Related papers

• Quantum oracles for the finite element method [45.200826131319815]
  This study examines the quantum routines required for the implementation of oracles used in the block-encoding of the $N times N stiffness and mass matrices.<n>We show how to construct the necessary oracles, which require the calculation of element geometry, square root and the implementation of conditional operations.
  arXiv  Detail & Related papers  (2025-04-28T14:28:31Z)
• Robust Learning of Multi-index Models via Iterative Subspace Approximation [36.138661719725626]
  We study the task of learning Multi-Index Models (MIMs) with label noise under the Gaussian distribution.<n>We focus on well-behaved MIMs with finite ranges that satisfy certain regularity properties.<n>We show that in the presence of random classification noise, the complexity of our algorithm scales agnosticly with $1/epsilon$.
  arXiv  Detail & Related papers  (2025-02-13T17:37:42Z)
• Representation theory of Gaussian unitary transformations for bosonic and fermionic systems [0.0]
  We analyze the behavior of the sign ambiguity that one needs to deal with when moving between the groups of the symplectic and special annihilation group. We show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space.
  arXiv  Detail & Related papers  (2024-09-18T01:22:38Z)
• Gapped Clique Homology on weighted graphs is $\ ext{QMA}_1$-hard and contained in $\ ext{QMA}$ [0.0]
  We study the complexity of a classic problem in computational topology, the homology problem. We find that the complexity is characterized by quantum complexity classes. Our results can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics.
  arXiv  Detail & Related papers  (2023-11-28T21:15:30Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Complexity of Supersymmetric Systems and the Cohomology Problem [0.0]
  We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with $mathcal N=2 $ supersymmetry. Our main motivation for studying this is the fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial.
  arXiv  Detail & Related papers  (2021-06-30T18:00:01Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version [0.0]
  We propose a new mathematical approach to Bona's non-linear generalization of quantum mechanics. It highlights the central role of self-consistency. Some new mathematical concepts are introduced, which are possibly interesting by themselves.
  arXiv  Detail & Related papers  (2020-09-10T16:20:25Z)
• Quantum information theory and Fourier multipliers on quantum groups [0.0]
  We compute the exact values of the minimum output entropy and the completely bounded minimal entropy of quantum channels acting on matrix algebras. Our results use a new and precise description of bounded Fourier multipliers from $mathrmL1(mathbbG)$ into $mathrmLp(mathbbG)$ for $1 p leq infty$ where $mathbbG$ is a co-amenable locally compact quantum group.
  arXiv  Detail & Related papers  (2020-08-27T09:47:10Z)
• Epsilon-nets, unitary designs and random quantum circuits [0.11719282046304676]
  Epsilon-nets and approximate unitary $t$-designs are notions of unitary operations relevant for numerous applications in quantum information and quantum computing. We prove that for a fixed $d$ of the space, unitaries constituting $delta$-approx $t$-expanders form $epsilon$-nets for $tsimeqfracd5/2epsilon$ and $delta=left(fracepsilon3/2dright)d2$. We show that approximate tdesigns can be generated
  arXiv  Detail & Related papers  (2020-07-21T15:16:28Z)
• Classification Under Misspecification: Halfspaces, Generalized Linear Models, and Connections to Evolvability [39.01599245403753]
  In particular, we study the problem of learning halfspaces under Massart noise with rate $eta$. We show any SQ algorithm requires super-polynomially many queries to achieve $mathsfOPT + epsilon$. We also study our algorithm for learning halfspaces under Massart noise empirically and find that it exhibits some appealing fairness properties.
  arXiv  Detail & Related papers  (2020-06-08T17:59:11Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.