arXiv subjects

Bjorn K. Berntson

Publications and source records attributed to Bjorn K. Berntson.

At least 19 recordsLinked to original sources

arXiv2026

Optimal kernel functions for linear combination of Hamiltonian simulation

Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error $\varepsilon$, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion $\tfrac{2\mathrm{e}}π\big(\log\frac{1}{\varepsilon}-\log\log\frac{1}{\varepsilon}+o(1)\big)$ as ${\varepsilon\downarrow0}$. The provably optimal leading coefficient $2\mathrm{e}/π$ improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.

quant-ph↗

arXiv2026

An end-to-end quantum algorithm for weakly nonlinear plasma physics with superquadratic speedup

Nonlinear kinetic plasma simulation is high-dimensional and classically demanding, while quantum algorithms face different bottlenecks: embedding nonlinear dynamics into a linear computation, loading dense field-interaction data, and efficiently extracting information. We present an end-to-end quantum algorithm, with rigorous convergence guarantees, for a weakly nonlinear kinetic plasma model. The system describes a 3D electron-ion plasma with adiabatic electrons, kinetic ions, Debye screening, and Krook relaxation. After Fourier-Hermite truncation, the dynamics reduces to a high-dimensional quadratic ordinary differential equation. To tackle quantum bottlenecks we combine three key ingredients. First, we use a plasma free energy to identify a Lyapunov transform under which a Carleman linear embedding converges exponentially in the truncation order within a certified weakly nonlinear regime. Second, we develop a hierarchical block-encoding protocol for dense matrices, exploiting the spatial decay of the field to avoid polynomial overhead from sparse access encodings. Third, we introduce a subroutine for information extraction that exploits nonlinear components encoded in the full Carleman history state to improve the estimation of linear observables. We construct a quantum algorithm to estimate the spacetime-averaged kinetic energy using $\widetilde{O}\!\left( N_F N_H^{1/2} \operatorname{polylog}\!\left(\frac{T}ε\right)\frac{1}ε\right)$ gates and $\widetilde{O}\!\left(\log\!\left(N_F N_H^{1/2}T\right)\log\!\left(\frac{1}ε\right)\right)$ qubits, where $N_F$ and $N_H$ are the Fourier and Hermite cutoffs. Relative to a Fourier-Hermite spectral solver, this yields exponential memory savings and superquadratic improvements in time. Together, these results establish a controlled nonlinear plasma benchmark for quantum simulation.

quant-ph↗

arXiv2025

Matrix inversion polynomials for the quantum singular value transformation

Quantum matrix inversion with the quantum singular value transformation (QSVT) requires a polynomial approximation to $1/x$. Several methods from the literature construct polynomials that achieve the known degree complexity $\mathcal{O}(κ\log(κ/\varepsilon))$ with condition number $κ$ and uniform error $\varepsilon$. However, the \emph{optimal} polynomial with lowest degree for fixed error $\varepsilon$ can only be approximated numerically with the resource-intensive Remez method, leading to impractical preprocessing runtimes. Here, we derive an analytic shortcut to the optimal polynomial. Comparisons with other polynomials from the literature, based on Taylor expansion, Chebyshev iteration, and convex optimization, confirm that our result is optimal. Furthermore, for large $κ\log(κ/\varepsilon)$, our polynomial has the smallest maximum value on $[-1,1]$ of all approaches considered, leading to reduced circuit depth due to the normalization condition of QSVT. With the Python code provided, this paper will also be useful for practitioners in the field.

quant-ph↗

arXiv2025

Complementary polynomials in quantum signal processing

Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this problem employ numerical methods that are not amenable to explicit error analysis. We present a new approach to complementary polynomials using complex analysis. Our main mathematical result is a contour integral representation for a canonical complementary polynomial. On the unit circle, this representation has a particularly simple and efficacious Fourier analytic interpretation, which we use to develop a Fast Fourier Transform-based algorithm for the efficient calculation of $Q$ in the monomial basis with explicit error guarantees. Numerical evidence that our algorithm outperforms the state-of-the-art optimization-based method for computing complementary polynomials is provided.

quant-ph↗

arXiv2025

Quantum Computation of Electronic Structure with Projector Augmented-Wave Method and Plane Wave Basis Set

Quantum simulation of materials is a promising application area of quantum computers. To practically realize this promise, we must reduce quantum resources while maintaining accuracy. In electronic structure calculations on classical computers, resource reduction has been achieved by using the projector augmented-wave method (PAW) and plane wave basis sets. However, the PAW method generalized for many-body states introduces non-orthogonality effects which impede its direct application to quantum computing. In this work, we develop a unitary variant of the PAW (UPAW) that preserves the orthogonality constraints. We provide a linear-combination-of-unitaries decomposition of the UPAW Hamiltonian to enable ground state estimation using qubitized quantum phase estimation. Additionally, we further improve algorithmic efficiency by extending classical down-sampling techniques into the quantum setting. We then estimate quantum resources for crystalline solids to estimate the energy within chemical accuracy with respect to the full basis set limit, and also consider a supercell approach which is more suitable for calculations of defect states. We provide the quantum resources for energy estimation of a nitrogen-vacancy defect centre in diamond which is a challenging system for classical algorithms and a quintessential problem in the studies of quantum point defects.

quant-ph↗

arXiv2025

Two exact quantum signal processing results

Quantum signal processing (QSP) is a framework for implementing certain polynomial functions via quantum circuits. To construct a QSP circuit, one needs (i) a target polynomial $P(z)$, which must satisfy $\lvert P(z)\rvert\leq 1$ on the complex unit circle $\mathbb{T}$ and (ii) a complementary polynomial $Q(z)$, which satisfies $\lvert P(z)\rvert^2+\lvert Q(z)\rvert^2=1$ on $\mathbb{T}$. We present two exact mathematical results within this context. First, we obtain an exact expression for a certain uniform polynomial approximant of $1/x$, which is used to perform matrix inversion via quantum circuits. Second, given a generic target polynomial $P(z)$, we construct the complementary polynomial $Q(z)$ exactly via integral representations, valid throughout the entire complex plane.

quant-ph↗

arXiv2025

Quantum Simulations of Chemistry in First Quantization with any Basis Set

Quantum computation of the energy of molecules and materials is one of the most promising applications of fault-tolerant quantum computers. Practical applications require development of quantum algorithms with reduced resource requirements. Previous work has mainly focused on quantum algorithms where the Hamiltonian is represented in second quantization with compact basis sets while existing methods in first quantization are limited to a grid-based basis. In this work, we present a new method to solve the generic ground-state chemistry problem in first quantization using any basis set. We achieve asymptotic speedup in Toffoli count for molecular orbitals, and orders of magnitude improvement using dual plane waves as compared to the second quantization counterparts. In some instances, our approach provides similar or even lower resources compared to previous first quantization plane wave algorithms that, unlike our approach, avoids the loading of the classical data. The developed methodology can be applied to variety of applications, where the matrix elements of a first quantized Hamiltonian lack simple circuit representation.

quant-ph↗

arXiv2025

Pauli Decomposition via the Fast Walsh-Hadamard Transform

The decomposition of a square matrix into a sum of Pauli strings is a classical pre-processing step required to realize many quantum algorithms. Such a decomposition requires significant computational resources for large matrices. We present an exact and explicit formula for the Pauli string coefficients which inspires an efficient algorithm to compute them. More specifically, we show that up to a permutation of the matrix elements, the decomposition coefficients are related to the original matrix by a multiplication of a generalised Hadamard matrix. This allows one to use the Fast Walsh-Hadamard transform and calculate all Pauli decomposition coefficients in $\mathcal{O}(N^2\log N)$ time and using $\mathcal{O}(1)$ additional memory, for an $N\times N$ matrix. A numerical implementation of our equation outperforms currently available solutions.

quant-ph↗

arXiv2023

Conformal field theory, solitons, and elliptic Calogero--Sutherland models

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

math-ph↗

arXiv2022

A focusing-defocusing intermediate nonlinear Schrödinger system

We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing-defocusing Manakov system on one hand and a limiting case of the intermediate nonlinear Schrödinger equation on the other. We show that this new system, which we call the intermediate mixed Manakov (IMM) system, admits multi-soliton solutions governed by a complexification of the hyperbolic Calogero-Moser (CM) system. Furthermore, we introduce a spatially periodic version of the IMM system, for which our result is a class of exact solutions governed by a complexified elliptic CM system.

nlin.SI↗

arXiv2022

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic $A$-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin-Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

math-ph↗

arXiv2022

Consistency of the Bäcklund transformation for the spin Calogero-Moser system

We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero-Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ODEs; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency and show, under mild assumptions, that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.

nlin.SI↗

arXiv2022

Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics

We present a class of periodic solutions of the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation, which was recently introduced by the authors together with Langmann as a classical, continuum limit of an Inozemtsev-type spin chain. These exact analytic solutions are constructed via a spin-pole ansatz written in terms of certain elliptic functions. The dynamical parameters in our solutions solve an elliptic spin Calogero-Moser (CM) system subject to certain constraints. In the course of our construction, we establish a novel Bäcklund transformation for this constrained elliptic spin CM system.

math-ph↗

arXiv2022

Spin generalizations of the Benjamin-Ono equation

We present new soliton equations related to the $A$-type spin Calogero-Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin-Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

math-ph↗

arXiv2021

The non-chiral intermediate Heisenberg ferromagnet equation

We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter $δ>0$, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit $δ\to \infty$ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of a Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.

math-ph↗

arXiv2021

On the non-chiral intermediate long wave equation II: periodic case

We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) use the Hirota method to construct the periodic multi-soliton solutions, (d) derive a Bäcklund transformation, (e) use the Bäcklund transformation to obtain an infinite number of conservation laws.

nlin.SI↗

arXiv2021

On the non-chiral intermediate long wave equation

We study integrability properties of the non-chiral intermediate long wave equation recently introduced by the authors as a parity-invariant variant of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) derive a Bäcklund transformation, (d) use, separately, the Bäcklund transformation and the Lax representation to obtain an infinite number of conservation laws.

nlin.SI↗

arXiv2020

Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators

We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and 2-parameter rational potential is the best known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.

math-ph↗