IA Scholar Query: Limitations of quantum coset states for graph isomorphism.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 22 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Another Round of Breaking and Making Quantum Money: How to Not Build It from Lattices, and More
https://scholar.archive.org/work/ghlrkjfs2nef5fwmkalccul75i
Public verification of quantum money has been one of the central objects in quantum cryptography ever since Wiesner's pioneering idea of using quantum mechanics to construct banknotes against counterfeiting. So far, we do not know any publicly-verifiable quantum money scheme that is provably secure from standard assumptions. In this work, we provide both negative and positive results for publicly verifiable quantum money. **In the first part, we give a general theorem, showing that a certain natural class of quantum money schemes from lattices cannot be secure. We use this theorem to break the recent quantum money scheme of Khesin, Lu, and Shor. **In the second part, we propose a framework for building quantum money and quantum lightning we call invariant money which abstracts some of the ideas of quantum money from knots by Farhi et al.(ITCS'12). In addition to formalizing this framework, we provide concrete hard computational problems loosely inspired by classical knowledge-of-exponent assumptions, whose hardness would imply the security of quantum lightning, a strengthening of quantum money where not even the bank can duplicate banknotes. **We discuss potential instantiations of our framework, including an oracle construction using cryptographic group actions and instantiations from rerandomizable functional encryption, isogenies over elliptic curves, and knots.Hart Montgomery, Jiahui Liu, Mark Zhandrywork_ghlrkjfs2nef5fwmkalccul75iTue, 22 Nov 2022 00:00:00 GMTElliptic Trace Map on Chiral Algebras
https://scholar.archive.org/work/v2b2dtlbovflrbh2bn6hbfafmu
Trace map on deformation quantized algebra leads to the algebraic index theorem. In this paper, we investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras developed by Beilinson and Drinfeld. We construct a trace map on the elliptic chiral homology of the free beta gamma-bc system using the BV quantization framework. As an example, we compute the trace evaluated on the unit constant chiral chain and obtain the formal Witten genus in the Lie algebra cohomology. We also construct a family of elliptic trace maps on coset models.Zhengping Gui, Si Liwork_v2b2dtlbovflrbh2bn6hbfafmuTue, 22 Nov 2022 00:00:00 GMTTopological Holography: Towards a Unification of Landau and Beyond-Landau Physics
https://scholar.archive.org/work/p6ncec5ppjao5eqex2pjxnxkzy
We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry G in one lower dimension. This exposes a vast web of dualities which act on the space of G-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to 1+1d quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in 2+1d. We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.Heidar Moradi, Seyed Faroogh Moosavian, Apoorv Tiwariwork_p6ncec5ppjao5eqex2pjxnxkzyMon, 21 Nov 2022 00:00:00 GMTLectures on modular forms and strings
https://scholar.archive.org/work/rra4lqiauzezpnt4kq5ag3axnu
The goal of these lectures is to present an informal but precise introduction to a body of concepts and methods of interest in number theory and string theory revolving around modular forms and their generalizations. Modular invariance lies at the heart of conformal field theory, string perturbation theory, Montonen-Olive duality, Seiberg-Witten theory, and S-duality in Type IIB superstring theory. Automorphic forms with respect to higher arithmetic groups as well as mock modular forms enter in toroidal string compactifications and the counting of black hole microstates. After introducing the basic mathematical concepts including elliptic functions, modular forms, Maass forms, modular forms for congruence subgroups, vector-valued modular forms, and modular graph forms, we describe a small subset of the countless applications to problems in Mathematics and Physics, including those mentioned above.Eric D'Hoker, Justin Kaidiwork_rra4lqiauzezpnt4kq5ag3axnuFri, 18 Nov 2022 00:00:00 GMTQuantum permutation groups
https://scholar.archive.org/work/pzyvi36aive4fei5crwurpwmry
The permutation group S_N has a quantum analogue S_N^+, which is infinite at N≥4. We review the known facts regarding S_N^+, and its versions S_F^+, with F being a finite quantum space. We discuss then the structure of the closed subgroups G⊂ S_N^+ and G⊂ S_F^+, with particular attention to the quantum reflection groups.Teo Banicawork_pzyvi36aive4fei5crwurpwmryMon, 14 Nov 2022 00:00:00 GMTPieri and Murnaghan–Nakayama type Rules for Chern classes of Schubert Cells
https://scholar.archive.org/work/ncrs7sqm5fdjjfsjoh5n6hghtq
We develop Pieri type as well as Murnaghan--Nakayama type formulas for equivariant Chern--Schwartz--MacPherson classes of Schubert cells in the classical flag variety. These formulas include as special cases many previously known multiplication formulas for Chern--Schwartz--MacPherson classes or Schubert classes. We apply the equivariant Murnaghan--Nakayama formula to the enumeration of rim hook tableaux.Neil J.Y. Fan, Peter L. Guo, Rui Xiongwork_ncrs7sqm5fdjjfsjoh5n6hghtqSun, 13 Nov 2022 00:00:00 GMTA Panorama Of Physical Mathematics c. 2022
https://scholar.archive.org/work/5ic2soxwonefdcg6gr2bi7r7gy
What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].Ibrahima Bah, Daniel S. Freed, Gregory W. Moore, Nikita Nekrasov, Shlomo S. Razamat, Sakura Schafer-Namekiwork_5ic2soxwonefdcg6gr2bi7r7gyWed, 09 Nov 2022 00:00:00 GMT3-Manifolds and VOA Characters
https://scholar.archive.org/work/z4gm3d4snfb4ved5z4excltsnm
By studying the properties of q-series Z-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for Z-invariants leads to many infinite families of new fermionic formulae for VOA characters.Miranda C. N. Cheng, Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah M. Harrison, Davide Passarowork_z4gm3d4snfb4ved5z4excltsnmFri, 04 Nov 2022 00:00:00 GMTCoherent sheaves and quantum Coulomb branches II: quiver gauge theories and knot homology
https://scholar.archive.org/work/63huosyatzbptnt2egq6kbbhme
We continue our study of noncommutative resolutions of Coulomb branches in the case of quiver gauge theories. These include the Slodowy slices in type A and symmetric powers in ℂ^2 as special cases. These resolutions are based on vortex line defects in quantum field theory, but have a precise mathematical description, which in the quiver case is a modification of the formalism of KLRW algebras. While best understood in a context which depends on the geometry of the affine Grassmannian and representation theory in characteristic p, we give a description of the Coulomb branches and their commutative and non-commutative resolutions which can be understood purely in terms of algebra. This allows us to construct a purely algebraic version of the knot homology theory defined using string theory by Aganagić, categorifying the Reshetikhin-Turaev invariants for minuscule representations of type ADE Lie algebras. We show that this homological invariant agrees with the categorification of these invariants previously defined by the author, and thus with Khovanov-Rozansky homology in type A.Ben Websterwork_63huosyatzbptnt2egq6kbbhmeThu, 03 Nov 2022 00:00:00 GMTReflected entropy in random tensor networks II: a topological index from the canonical purification
https://scholar.archive.org/work/gnfwsbymxnh5leyyq4tb7aqyh4
In arXiv:2112.09122, we analyzed the reflected entropy (S_R) in random tensor networks motivated by its proposed duality to the entanglement wedge cross section (EW) in holographic theories, S_R=2 EW/4G. In this paper, we discover further details of this duality by analyzing a simple network consisting of a chain of two random tensors. This setup models a multiboundary wormhole. We show that the reflected entanglement spectrum is controlled by representation theory of the Temperley-Lieb (TL) algebra. In the semiclassical limit motivated by holography, the spectrum takes the form of a sum over superselection sectors associated to different irreducible representations of the TL algebra and labelled by a topological index k∈ℤ_≥ 0. Each sector contributes to the reflected entropy an amount 2k EW/4G weighted by its probability. We provide a gravitational interpretation in terms of fixed-area, higher-genus multiboundary wormholes with genus 2k-1 initial value slices. These wormholes appear in the gravitational description of the canonical purification. We confirm the reflected entropy holographic duality away from phase transitions. We also find important non-perturbative contributions from the novel geometries with k≥ 2 near phase transitions, resolving the discontinuous transition in S_R. Along with analytic arguments, we provide numerical evidence for our results. We comment on the connection between TL algebras, Type II_1 von Neumann algebras and gravity.Chris Akers, Thomas Faulkner, Simon Lin, Pratik Rathwork_gnfwsbymxnh5leyyq4tb7aqyh4Wed, 02 Nov 2022 00:00:00 GMTGeometrical aspects of Amplitudes and Correlators in N=4 SYM
https://scholar.archive.org/work/4odqeqb3yzdlvdeqevearx5ria
This thesis describes progresses made by the author and collaborators in the positive geometry description of superamplitudes and supercorrelators in planar N = 4 SYM.Gabriele Dianwork_4odqeqb3yzdlvdeqevearx5riaSat, 29 Oct 2022 00:00:00 GMTTree-level amplitudes from the pure spinor superstring
https://scholar.archive.org/work/fd6gq6z7gbecpdzzadjoaki4uq
We give a comprehensive review of recent developments on using the pure spinor formalism to compute massless superstring scattering amplitudes at tree level. The main results of the pure spinor computations are placed into the context of related topics including the color-kinematics duality in field theory and the mathematical structure of α'-corrections.Carlos R. Mafra, Oliver Schlottererwork_fd6gq6z7gbecpdzzadjoaki4uqTue, 25 Oct 2022 00:00:00 GMTTransformations of Stabilizer States in Quantum Networks
https://scholar.archive.org/work/gonzl4y2ajhjrktaanvyupatuq
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47, 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2, which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.Matthias Englbrecht, Tristan Kraft, Barbara Krauswork_gonzl4y2ajhjrktaanvyupatuqTue, 25 Oct 2022 00:00:00 GMTEfficient decoding up to a constant fraction of the code length for asymptotically good quantum codes
https://scholar.archive.org/work/oqreepes5rclxd3h3q5jrciyji
We introduce and analyse an efficient decoder for the quantum Tanner codes of that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight O(√(n log n)). We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.Anthony Leverrier, Gilles Zémorwork_oqreepes5rclxd3h3q5jrciyjiTue, 25 Oct 2022 00:00:00 GMTFlat bands and band-touching from real-space topology in hyperbolic lattices
https://scholar.archive.org/work/wtdj3tcht5fohkaoqsgr7irtui
Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetries. We show that two characteristic features of the energy spectrum of those models, namely, the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be captured exactly by a combination of real-space topology arguments and a reciprocal-space description via the formalism of hyperbolic band theory. Furthermore, using real-space numerical diagonalization on finite lattices with periodic boundary conditions, we obtain insights into higher-dimensional irreducible representations of the non-Euclidean (Fuchsian) translation group of hyperbolic lattices. First, we find that the fraction of states in the flat band is the same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that only Abelian states participate in the formation of touching points between the flat and dispersive bands.Tomáš Bzdušek, Joseph Maciejkowork_wtdj3tcht5fohkaoqsgr7irtuiMon, 24 Oct 2022 00:00:00 GMTOn Vacuum Structures and Quantum Corrections in String Theory
https://scholar.archive.org/work/26gtp4n6enfwlklbnlwofhzady
A key target for fundamental physics remains developing a clear understanding of ultra-violet (UV) limits of Effective Field Theories (EFTs) coupled to gravity. In this context, string theory has emerged as a viable candidate for a UV complete theory of quantum gravity. Its compactifications result in a landscape of string vacua encompassing an immensely rich and diverse structure of EFTs. Extracting reliable low energy information from string compactifications notoriously requires a systematic derivation of corrections to the tree level actions which remains a key challenge. Further, despite astonishing progress in constructing string solutions, locating realistic string vacua with desirable properties in the landscape proves to be a delicate task. The objectives of this thesis are threefold: I) first, to perform an extensive analysis of quantum corrections in string theory; II) second, to create a systematic framework to study geometries and backgrounds for viable string compactifications; and III) third, to assess the attainable EFTs in the context of moduli stabilisa- tion. The synergy of these strategies constitutes an innovative approach towards addressing phenomenological questions in string theory. The first part of this thesis describes progress in deriving corrections to classi- cal string effective actions from multi-dimensional investigations by employing the powerful machinery of string dualities and symmetries. Initially, we investigate the structure of higher derivative terms involving the 3-form G3 in the α′ and string-loop expansion of the ten-dimensional Type IIB effective action. Subsequently, we ex- plore α′ corrections in F-theory compactifications to four dimensions in Ch. 6. Here, we focus on the moduli dependence of perturbative corrections to scalar potentials by performing a dimensional analysis. The second part concerns the development of new techniques to examine large classes of Calabi-Yau (CY) geometries and realising the Standard Model in string compactifications. In a first step, w [...]Andreas Schachner, Apollo-University Of Cambridge Repository, Fernando Quevedowork_26gtp4n6enfwlklbnlwofhzadyFri, 21 Oct 2022 00:00:00 GMTTransformations of Stabilizer States in Quantum Networks
https://scholar.archive.org/work/rpuoixzjqrhevnebqhzjzoci6i
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47, 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2, which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.Matthias Englbrecht, Tristan Kraft, Barbara Krauswork_rpuoixzjqrhevnebqhzjzoci6iThu, 20 Oct 2022 00:00:00 GMTFailing to hash into supersingular isogeny graphs
https://scholar.archive.org/work/47spzax6rbdjnefzo53nisufqy
An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular ℓ-isogeny graph which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd's of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces; and (v) using quantum random walks.Jeremy Booher, Ross Bowden, Javad Doliskani, Tako Boris Fouotsa, Steven D. Galbraith, Sabrina Kunzweiler, Simon-Philipp Merz, Christophe Petit, Benjamin Smith, Katherine E. Stange, Yan Bo Ti, Christelle Vincent, José Felipe Voloch, Charlotte Weitkämper, Lukas Zobernigwork_47spzax6rbdjnefzo53nisufqyWed, 19 Oct 2022 00:00:00 GMTVu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points
https://scholar.archive.org/work/jvxfz4ccm5h5vpxmmyl53pbupu
We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing k > 1 focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and k formal power series, up to a (ℤ_2 × D_k)-action, where D_k is the k-th dihedral group. The k formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold (M,ω) near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.Álvaro Pelayo, Xiudi Tangwork_jvxfz4ccm5h5vpxmmyl53pbupuTue, 18 Oct 2022 00:00:00 GMTStable envelopes for slices of the affine Grassmannian
https://scholar.archive.org/work/aikxk2iaanbdtekomxhwnro2we
The affine Grassmannian associated to a reductive group 𝐆 is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of D. Maulik and A. Okounkov [arXiv:1211.1287] in this family. We construct an explicit recursive relation for the stable envelopes in the 𝐆 = 𝐏𝐒𝐋_2 case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.Ivan Danilenkowork_aikxk2iaanbdtekomxhwnro2weTue, 18 Oct 2022 00:00:00 GMT