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Ákos Nagy

Publications and source records attributed to Ákos Nagy.

16 recordsLinked to original sources
arXiv2026

Simulating high-accuracy nuclear motion Hamiltonians using discrete variable representation and Walsh-Hadamard QROM on fault-tolerant quantum computers

We present a quantum algorithm for simulating rovibrational Hamiltonians on fault-tolerant quantum computers. The method integrates exact curvilinear kinetic energy operators and general-form potential energy surfaces expressed in a hybrid finite-basis/discrete-variable representation. The Hamiltonian is encoded as a unitary quantum circuit using a quantum read-only memory construction based on the Walsh-Hadamard transform, enabling high-accuracy quantum phase estimation of rovibrational energy levels and dynamics simulations. Our technique provides asymptotic reductions in both logical qubit count and T-gate complexity that are exponential in the number of atoms and at least polynomial in the total Hilbert-space size, relative to existing block-encoding techniques based on linear combinations of unitaries and variational basis representation. Compared with classical variational methods, it offers exponential memory savings and polynomial reductions in time complexity. The quantum volume required for computing the rovibrational spectrum of water can be reduced by up to 100 000 times compared with other quantum methods, increasing to at least 1 million for a classically intractable 30-dimensional (12-atom) molecular system. For this case with a six-body coupled potential, estimating spectroscopic-accuracy energy levels would require about three months on a 1 MHz fault-tolerant quantum processor with fewer than 300 logical qubits, versus over 30 000 years on the fastest current classical supercomputer. These estimates are approximate and subject to technological uncertainties, and realizing the asymptotic advantage will require substantial quantum resources and continued algorithmic progress.

quant-ph↗
arXiv2025

Conjugate linear perturbations of Dirac operators and Majorana fermions

We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw-Rossi operator, which describes electronic excitations on topological superconductors. We also describe the low energy spectrum of these operators on complete surfaces, under mild hypotheses.

math.DG↗
arXiv2024

Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems

We study a Grover-type method for Quadratic Unconstrained Binary Optimization (QUBO) problems. For an $n$-dimensional QUBO problem with $m$ nonzero terms, we construct a marker oracle for such problems with a tuneable parameter, $Λ\in \left[ 1, m \right] \cap \mathbb{Z}$. At $d \in \mathbb{Z}_+$ precision, the oracle uses $O (n + Λd)$ qubits, has total depth of $O \left( \tfrac{m}Λ \log_2 (n) + \log_2 (d) \right)$, and non-Clifford depth of $O \left( \tfrac{m}Λ \right)$. Moreover, each qubit required to be connected to at most $O \left( \log_2 (Λ+ d) \right)$ other qubits. In the case of a maximum graph cuts, as $d = 2 \left\lceil \log_2 (n) \right\rceil$ always suffices, the depth of the marker oracle can be made as shallow as $O (\log_2 (n))$. For all values of $Λ$, the non-Clifford gate count of these oracles is strictly lower (at least by a factor of $\sim 2$) than previous constructions. Furthermore, we introduce a novel \textit{Fixed-point Grover Adaptive Search for QUBO Problems}, using our oracle design and a hybrid Fixed-point Grover Search, motivated by the works of Boyer et al. and Li et al. This method has better performance guarantees than previous Grover Adaptive Search methods. Some of our results are novel and useful for any method based on Fixed-point Grover Search. Finally, we give a heuristic argument that, with high probability and in $O \left( \tfrac{\log_2 (n)}{\sqrtε} \right)$ time, this adaptive method finds a configuration that is among the best $ε2^n$ ones.

quant-ph↗
arXiv2024

Novel oracle constructions for quantum random access memory

We present new designs for quantum random access memory. More precisely, for each function, $f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2^d$, we construct oracles, $\mathcal{O}_f$, with the property \begin{equation} \mathcal{O}_f \left| x \right\rangle_n \left| 0 \right\rangle_d = \left| x \right\rangle_n \left| f(x) \right\rangle_d. \end{equation} Our methods are based on the Walsh-Hadamard Transform of $f$, viewed as an integer valued function. In general, the complexity of our method scales with the sparsity of the Walsh-Hadamard Transform and not the sparsity of $f$, yielding more favorable constructions in cases such as binary optimization problems and function with low-degree Walsh-Hadamard Transforms. Furthermore, our design comes with a tuneable amount of ancillas that can trade depth for size. In the ancilla-free design, these oracles can be $ε$-approximated so that the Clifford + $T$ depth is $O \left( \left( n + \log_2 \left( \tfrac{d}ε \right) \right) \mathcal{W}_f \right)$, where $\mathcal{W}_f$ is the number of nonzero components in the Walsh-Hadamard Transform. The depth of the shallowest version is $O \left( n + \log_2 \left( \tfrac{d}ε \right) \right)$, using $n + d \mathcal{W}_f$ qubit. The connectivity of these circuits is also only logarithmic in $\mathcal{W}_f$. As an application, we show that for boolean functions with low approximate degrees (as in the case of read-once formulas) the complexities of the corresponding QRAM oracles scale only as $2^{\widetilde{O} \left( \sqrt{n} \log_2 \left( n \right) \right)}$.

quant-ph↗
arXiv2023

On the bifurcation theory of the Ginzburg-Landau equations

We construct nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles.

math.AP↗
arXiv2022

Construction of Nahm data and BPS monopoles with continuous symmetries

We study solutions to Nahm's equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data, and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry.

math-ph↗
arXiv2022

Nonminimal solutions to the Ginzburg-Landau equations on surfaces

We prove the existence of novel, nonminimal and irreducible solutions to the (self-dual) Ginzburg-Landau equations on closed surfaces. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. Our method works with the 2-dimensional, critically coupled Ginzburg-Landau theory and uses the topology of the moduli space. The method is nonconstructive, but works for all values of the remaining coupling constant. We also prove the instability of these solutions.

math.DG↗
arXiv2022

The asymptotic geometry of $\rm{G}_2$-monopoles

This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical $\rm{G}_2$-manifold.

math.DG↗
arXiv2022

On the hyperbolic Bloch transform

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case, that is when the Hilbert space is the regular representation of the Fuchsian group, $Γ$. Second, when $Γ\subset \mathrm{PSU} (1, 1)$ acts isometrically on the hyperbolic plane, $\mathbb{H}$, and the Hilbert space is $L^2 \left( \mathbb{H} \right)$, then we define a modified, geometric Bloch transform, that sends wave functions to sections of stable, flat bundles over $Σ= \mathbb{H} / Γ$ and transforms the hyperbolic Laplacian into the covariant Laplacian.

math-ph↗
arXiv2022

The Kapustin--Witten equations on ALE and ALF gravitational instantons

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.

math.DG↗
arXiv2022

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12].

math.DG↗
arXiv2022

The Berry connection of the Ginzburg-Landau vortices

We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.

math-ph↗
arXiv2022

Stationary solutions to the Keller-Segel equation on curved planes

We study stationary solutions to the Keller--Segel equation on curved planes. We prove the necessity of the mass being $8 π$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller--Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller--Segel equation (with any mass). Finally, as a complementary result, we prove a curved version of the logarithmic Hardy--Littlewood--Sobolev inequality and use it to show that the Keller--Segel free energy is bounded from below exactly when the mass is $8 π$, even in the curved case.

math.AP↗
arXiv2018

Irreducible Ginzburg-Landau fields in dimension 2

Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $α$ and $β$. We give conditions on $α$ and $β$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\mathbb{R}^2$, spheres, tori, etc.) with de Gennes-Neumann boundary conditions. We also prove that, for each such manifold and all positive $α$ and $β$, the Ginzburg-Landau free energy is a Palais-Smale function on the space of gauge equivalence classes, Ginzburg-Landau fields exist for only a finite set of energy values, and the moduli space of Ginzburg-Landau fields is compact.

math-ph↗
arXiv2018

From vortices to instantons on the Euclidean Schwarzschild manifold

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\mathrm{SU} (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\mathrm{SU} (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.

math.DG↗