arXiv subjects

Ivica Turkalj

Publications and source records attributed to Ivica Turkalj.

7 recordsLinked to original sources

arXiv2026

Classical Tensor Network and Quantum Fourier Transform Approaches for Large-Scale Carr-Madan Option Pricing

Fourier-based methods are among the most widely used techniques for pricing European options when the characteristic function of the underlying asset process is available. Their applicability to increasingly fine discretizations, however, is limited by the rapidly growing memory requirements of classical Fourier transforms, which become a computational bottleneck for large-scale pricing problems. In this work, we overcome this limitation by reformulating the Carr-Madan pricing framework using tensor networks. Specifically, we employ the Superfast Fourier Transform (SFFT), a compressed Tensor Train representation of the Quantum Fourier Transform (QFT), and apply it directly to tensorized option pricing without ever explicitly constructing exponentially large vectors or Fourier operators. This formulation also enables a direct comparison between the classical tensor network algorithm and its quantum counterpart through QFT-based option pricing on quantum simulators and quantum hardware. Numerical experiments for European call options demonstrate that the proposed SFFT method maintains pricing accuracy while substantially reducing memory requirements and achieving subexponential computational scaling compared with conventional FFT-based pricing. The accompanying quantum simulations and hardware executions enable a direct comparison between the classical tensor network formulation and its QFT-based quantum counterpart, showing that both approaches avoid the exponential scaling of conventional Fourier implementations and provide complementary perspectives on large-scale option pricing. Together, these results establish a unified framework connecting classical Fourier pricing, tensor network algorithms, and quantum computing approaches, demonstrating how tensorized Fourier methods can provide scalable alternatives for high-dimensional financial computations.

quant-ph↗

arXiv2025

Enhancing Variational Quantum Algorithms for Multicriteria Optimization

This paper presents methodological improvements to variational quantum algorithms (VQAs) for solving multicriteria optimization problems. We introduce two key contributions. First, we reformulate the parameter optimization task of VQAs as a multicriteria problem, enabling the direct use of classical algorithms from various multicriteria metaheuristics. This hybrid framework outperforms the corresponding single-criteria VQAs in both average and worst-case performance across diverse benchmark problems. Second, we propose a method that augments the hypervolume-based cost function with coverage-oriented indicators, allowing explicit control over the diversity of the resulting Pareto front approximations. Experimental results show that our method can improve coverage by up to 40\% with minimal loss in hypervolume. Our findings highlight the potential of combining quantum variational methods with classical population-based search to advance practical quantum optimization.

quant-ph↗

arXiv2024

Application of ZX-calculus to Quantum Architecture Search

This paper presents a novel approach to quantum architecture search by integrating the techniques of ZX-calculus with Genetic Programming (GP) to optimize the structure of parameterized quantum circuits employed in Quantum Machine Learning (QML). Recognizing the challenges in designing efficient quantum circuits for QML, we propose a GP framework that utilizes mutations defined via ZX-calculus, a graphical language that can simplify visualizing and working with quantum circuits. Our methodology focuses on evolving quantum circuits with the aim of enhancing their capability to approximate functions relevant in various machine learning tasks. We introduce several mutation operators inspired by the transformation rules of ZX-calculus and investigate their impact on the learning efficiency and accuracy of quantum circuits. The empirical analysis involves a comparative study where these mutations are applied to a diverse set of quantum regression problems, measuring performance metrics such as the percentage of valid circuits after the mutation, improvement of the objective, as well as circuit depth and width. Our results indicate that certain ZX-calculus-based mutations perform significantly better than others for Quantum Architecture Search (QAS) in all metrics considered. They suggest that ZX-diagram based QAS results in shallower circuits and more uniformly allocated gates than crude genetic optimization based on the circuit model.

quant-ph↗

arXiv2024

Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks

We analyze the frequency spectrum of Quantum Neural Networks (QNNs) using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area $A:=R\cdot L$ that preserves the frequency spectrum, where $R$ denotes the number of qubits and $L$ the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in $R$ and $L$ in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area $A=RL$ and not on the individual values of $R$ and $L$. Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area $A$, thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on $A = RL$, but also on the choice of $(R,L)$, so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability.

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arXiv2023

Quantum Architecture Search for Quantum Monte Carlo Integration via Conditional Parameterized Circuits with Application to Finance

Classical Monte Carlo algorithms can theoretically be sped up on a quantum computer by employing amplitude estimation (AE). To realize this, an efficient implementation of state-dependent functions is crucial. We develop a straightforward approach based on pretraining parameterized quantum circuits, and show how they can be transformed into their conditional variant, making them usable as a subroutine in an AE algorithm. To identify a suitable circuit, we propose a genetic optimization approach that combines variable ansatzes and data encoding. We apply our algorithm to the problem of pricing financial derivatives. At the expense of a costly pretraining process, this results in a quantum circuit implementing the derivatives' payoff function more efficiently than previously existing quantum algorithms. In particular, we compare the performance for European vanilla and basket options.

quant-ph↗

arXiv2022

Quantum Risk Analysis: Beyond (Conditional) Value-at-Risk

Risk measures are important key figures to measure the adequacy of the reserves of a company. The most common risk measures in practice are Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). Recently, quantum-based algorithms are introduced to calculate them. These procedures are based on the so-called quantum amplitude estimation algorithm which lead to a quadratic speed up compared to classical Monte-Carlo based methods. Based on these ideas, we construct quantum-based algorithms to calculate alternatives for VaR and CVaR, namely the Expectile Value-at-Risk (EVaR) and the Range Value-at-Risk (RVaR). We construct quantum algorithms to calculate them. These algorithms are based on quantum amplitude estimation. In a case study, we compare their performance with the quantum-based algorithms for VaR and CVaR. We find that all of the algorithms perform sufficiently well on a quantum simulator. Further, the calculations of EVaR and VaR are robust against noise on a real quantum device. This is not the case for CVaR and RVaR.

quant-ph↗

arXiv2016

Totally-Reflective Genera of Integral Lattices

In this paper we give a complete classification of totally-reflective, primitive genera in dimension 3 and 4. Our method breaks up into two parts. The first part consists of classifying the square free, totally-reflective, primitive genera by calculating strong bounds on the prime factors of the determinant of genera of positive definite quadratic forms (lattices) with this property. We achieve these bounds by combining the Minkowski-Siegel mass formula with the combinatorial classification of reflective lattices accomplished by Scharlau \& Blaschke. In a second part, we use a lattice transformation that goes back to Watson, to generate all totally-reflective, primitive genera when starting with the square-free case.

math.NT↗