arXiv subjects

Taehee Ko

Publications and source records attributed to Taehee Ko.

13 recordsLinked to original sources

arXiv2026

Exact and Efficient Circuit Construction for Block Encoding Matrix Polynomials

A recent interpolation-based Quantum Signal Processing (QSP) framework by Alase bypasses the phase-finding procedures required in conventional QSP, allowing for a direct encoding of the target polynomial into a quantum circuit. However, this approach assumes access to a diagonal block encoding of function values without providing an explicit circuit construction. In this work, we address this gap by developing an explicit circuit construction method for diagonal block encodings. The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods. Numerical results confirm this scaling, demonstrating that circuit parameters for polynomial degrees up to $10^7$ can be computed in about a minute on a standard CPU.

quant-ph↗

arXiv2025

Analytic and Stochastic Approach to Quantum Advantages in Ground State and Quantum State Preparation Problems

We study the problems of state preparation, ground state preparation and quantum state preparation. We propose an analytic approach to a stochastic quantum algorithm which prepares the ground state for $n$-qubit Hamiltonian that is represented by $\text{poly}(n)$ Pauli operators and has an inverse-polynomial gap, requiring only $\text{poly}(n)$ Pauli rotations, measurements, and classical time complexity when $n$ exceeds a threshold, to inverse-polynomial precision given the initial overlap being lower bounded by $\frac{1}{2^n}$. Extending this result, we prove that any $n$-qubit quantum state can be prepared in two regimes: (1) with a constant number of Pauli rotations to constant precision, or (2) with a polynomial number of rotations to inverse-polynomial precision. Our results improve over previous approaches to quantum state preparation in terms of gate complexity, thereby yielding quantum space advantage. As an application, we identify a practical condition under which quadratic unconstrained binary optimization (QUBO) problems can be solved with exponential quantum speedups.

quant-ph↗

arXiv2025

Fermionic-Adapted Shadow Tomography for dynamical correlation functions

Dynamical correlation functions are essential for characterizing the response of the quantum many-body systems to the external perturbation. As their calculation is classically intractible in general, quantum algorithms are promising in this aspect, but most rely on brute force measurement strategies that evaluate one body observable pair per circuit. In this work, we introduce Fermionic-Adapted Shadow Tomography (FAST) protocols, a new framework for the efficient calculation of multiple dynamical correlation functions. The key idea is to reformulate these functions into forms that are compatible with shadow tomography techniques. The circuits in our protocols require at most two-copy measurements with uncontrolled Hamiltonian simulation. We show that the proposed protocols enhance sample efficiency and/or reduce the number of measurement circuits by an order of one or two with respect to the number of qubits across a range of scenarios.

quant-ph↗

arXiv2025

Classical optimization algorithms for diagonalizing quantum Hamiltonians

Diagonalizing a Hamiltonian, which is essential for simulating its long-time dynamics, is a key primitive in quantum computing and has been proven to yield a quantum advantage for several specific families of Hamiltonians. Yet, despite its importance, only a handful of diagonalization algorithms exist, and correspondingly few families of fast-forwardable Hamiltonians have been identified. This paper introduces classical optimization algorithms for Hamiltonian diagonalization by formulating a cost function that penalizes off-diagonal terms and enforces unitarity via an orthogonality constraint, both expressed in the Pauli operator basis. We pinpoint a class of Hamiltonians that highlights severe drawbacks of existing methods, including exponential per-iteration cost, exponential circuit depth, or convergence to spurious optima. Our approach overcomes these shortcomings, achieving polynomial-time efficiency while provably avoiding suboptimal points. As a result, we broaden the known realm of fast-forwardable systems, showing that quantum-diagonalizable Hamiltonians extend to cases generated by exponentially large Lie algebras. On the practical side, we also present a randomized-coordinate variant that achieves a more efficient per-iteration cost than the deterministic counterpart. We demonstrate the effectiveness of these algorithms through explicit examples and numerical experiments.

quant-ph↗

arXiv2025

Quantum medical image encoding and compression using Fourier-based methods

Quantum image processing (QIMP) has recently emerged as a promising field for modern image processing applications. In QIMP algorithms, encoding classical image informaiton into quantum circuit is important as the first step. However, most of existing encoding methods use gates almost twice the number of pixels in an image, and simulating even a modest sized image is computationally demanding. In this work, we propose a quantum image encoding method that effectively reduces gates than the number of pixels by a factor at least 4. We demonstrate our method for various 1024 by 1024 high-quality medical images captured during the Bilateral Axillo-Breast Approach (BABA) robotic thyroidectomy surgery. Additionally, two compression techniques are proposed to further reduce the number of gates as well as pre-processing time with negligible loss of image quality. We suggest our image encoding strategy as a valuable option for large scale medical imaging.

quant-ph↗

arXiv2025

Interpolation-based coordinate descent method for parameterized quantum circuits

Parameterized quantum circuits (PQCs) are ubiquitous in the design of hybrid quantum-classical algorithms. In this work, we propose an interpolation-based coordinate descent (ICD) method to address the parameter optimization problem in PQCs. The ICD method provides a unified framework for existing structure optimization techniques such as Rotosolve, sequential minimal optimization, ExcitationSolve, and others. ICD employs interpolation to approximate the PQC cost function, effectively recovering its underlying trigonometric structure, and then performs an argmin update on a single parameter in each iteration. In contrast to previous studies on structure optimization, we determine the optimal interpolation nodes to mitigate statistical errors arising from quantum measurements. Moreover, in the common case of $r$ equidistant frequencies, we show that the optimal interpolation nodes are equidistant nodes with spacing $2\pi/(2r+1)$ (under constant variance assumption), and that our ICD method simultaneously minimizes the mean squared error, the condition number of the interpolation matrix, and the average variance of the approximated cost function. We perform numerical simulations and test on the MaxCut problem, the transverse field Ising model, and the XXZ model. Numerical results imply that our ICD method is more efficient than the commonly used gradient descent and random coordinate descent method.

quant-ph↗

arXiv2024

Quantum random power method for ground state computation

We present a quantum-classical hybrid random power method that approximates a ground state of a Hamiltonian. The quantum part of our method computes a fixed number of elements of a Hamiltonian-matrix polynomial via quantum polynomial filtering techniques with either Hamiltonian simulation or block encoding. The use of the techniques provides a computational advantage that may not be achieved classically in terms of the degree of the polynomial. The classical part of our method is a randomized iterative algorithm that takes as input the matrix elements computed from the quantum part and outputs an approximation of ground state of the Hamiltonian. We prove that with probability one, our method converges to an approximation of a ground state of the Hamiltonian, requiring a constant scaling of the per-iteration classical complexity. The required quantum circuit depth is independent of the initial overlap and has no or a square-root dependence on the spectral gap. The iteration complexity scales linearly as the dimension of the Hilbert space when the quantum polynomial filtering corresponds to a sparse matrix. We numerically validate this sparsity condition for well-known model Hamiltonians. We also present a lower bound of the fidelity, which depends on the magnitude of noise occurring from quantum computation regardless of its charateristics, if it is smaller than a critical value. Several numerical experiments demonstrate that our method provides a good approximation of ground state in the presence of systematic and/or sampling noise.

quant-ph↗

arXiv2023

Random coordinate descent: a simple alternative for optimizing parameterized quantum circuits

Variational quantum algorithms rely on the optimization of parameterized quantum circuits in noisy settings. The commonly used back-propagation procedure in classical machine learning is not directly applicable in this setting due to the collapse of quantum states after measurements. Thus, gradient estimations constitute a significant overhead in a gradient-based optimization of such quantum circuits. This paper introduces a random coordinate descent algorithm as a practical and easy-to-implement alternative to the full gradient descent algorithm. This algorithm only requires one partial derivative at each iteration. Motivated by the behavior of measurement noise in the practical optimization of parameterized quantum circuits, this paper presents an optimization problem setting that is amenable to analysis. Under this setting, the random coordinate descent algorithm exhibits the same level of stochastic stability as the full gradient approach, making it as resilient to noise. The complexity of the random coordinate descent method is generally no worse than that of the gradient descent and can be much better for various quantum optimization problems with anisotropic Lipschitz constants. Theoretical analysis and extensive numerical experiments validate our findings.

quant-ph↗

arXiv2023

Implementation of the Density-functional Theory on Quantum Computers with Linear Scaling with respect to the Number of Atoms

Density-functional theory (DFT) has revolutionized computer simulations in chemistry and material science. A faithful implementation of the theory requires self-consistent calculations. However, this effort involves repeatedly diagonalizing the Hamiltonian, for which a classical algorithm typically requires a computational complexity that scales cubically with respect to the number of electrons. This limits DFT's applicability to large-scale problems with complex chemical environments and microstructures. This article presents a quantum algorithm that has a linear scaling with respect to the number of atoms, which is much smaller than the number of electrons. Our algorithm leverages the quantum singular value transformation (QSVT) to generate a quantum circuit to encode the density-matrix, and an estimation method for computing the output electron density. In addition, we present a randomized block coordinate fixed-point method to accelerate the self-consistent field calculations by reducing the number of components of the electron density that needs to be estimated. The proposed framework is accompanied by a rigorous error analysis that quantifies the function approximation error, the statistical fluctuation, and the iteration complexity. In particular, the analysis of our self-consistent iterations takes into account the measurement noise from the quantum circuit. These advancements offer a promising avenue for tackling large-scale DFT problems, enabling simulations of complex systems that were previously computationally infeasible.

quant-ph↗

arXiv2023

Beyond potential energy surface benchmarking: a complete application of machine learning to chemical reactivity

We train an equivariant machine learning model to predict energies and forces for a real-world study of hydrogen combustion under conditions of finite temperature and pressure. This challenging case for reactive chemistry illustrates that ML learned potential energy surfaces (PESs) are always incomplete as they are overly reliant on chemical intuition of what data is important for training, i.e. stable or metastable energy states. Instead we show here that a negative design data acquisition strategy is necessary to create a more complete ML model of the PES, since it must also learn avoidance of unforeseen high energy intermediates or even unphysical energy configurations. Because this type of data is unintuitive to create, we introduce an active learning workflow based on metadynamics that samples a lower dimensional manifold within collective variables that efficiently creates highly variable energy configurations for further ML training. This strategy more rapidly completes the ML PES such that deviations among query by committee ML models helps to now signal occasional calls to the external ab initio data source to further molecular dynamics in time without need for retraining the ML model. With the hybrid ML-physics model we predict the change in transition state and/or reaction mechanism at finite temperature and pressure for hydrogen combustion, thereby delivering on the promise of real application work using ML trained models of an ab initio PES with two orders of magnitude reduction in cost.

physics.chem-ph↗

arXiv2023

Using Diffusion Maps to Analyze Reaction Dynamics for a Hydrogen Combustion Benchmark Dataset

We use local diffusion maps to assess the quality of two types of collective variables (CVs) for a recently published hydrogen combustion benchmark dataset~\cite{guan2022benchmark} that contains ab initio molecular dynamics trajectories and normal modes along minimum energy paths. This approach was recently advocated in~\cite{tlldiffmap20} for assessing CVs and analyzing reactions modeled by classical molecular dynamics simulations. We report the effectiveness of this approach to molecular systems modeled by quantum ab initio molecular dynamics. In addition to assessing the quality of CVs, we also use global diffusion maps to perform committor analysis as proposed in~\cite{tlldiffmap20}. We show that the committor function obtained from the global diffusion map allows us to identify transition regions of interest in several hydrogen combustion reaction channels.

physics.chem-ph↗

arXiv2022

A Local Convergence Theory for the Stochastic Gradient Descent Method in Non-Convex Optimization With Non-isolated Local Minima

Loss functions with non-isolated minima have emerged in several machine learning problems, creating a gap between theory and practice. In this paper, we formulate a new type of local convexity condition that is suitable to describe the behavior of loss functions near non-isolated minima. We show that such condition is general enough to encompass many existing conditions. In addition we study the local convergence of the SGD under this mild condition by adopting the notion of stochastic stability. The corresponding concentration inequalities from the convergence analysis help to interpret the empirical observation from some practical training results.

cs.LG↗

arXiv2021

Stochastic Algorithms for Self-consistent Calculations of Electronic Structures

The convergence property of a stochastic algorithm for the self-consistent field (SCF) calculations of electron structures is studied. The algorithm is formulated by rewriting the electron charges as a trace/diagonal of a matrix function, which is subsequently expressed as a statistical average. The function is further approximated by using a Krylov subspace approximation. As a result, each SCF iteration only samples one random vector without having to compute all the orbitals. We consider the common practice of SCF iterations with damping and mixing. We prove with appropriate assumptions that the iterations converge in the mean-square sense, when the stochastic error has an almost sure bound. We also consider the scenario when such an assumption is weakened to a second moment condition, and prove the convergence in probability.

math.NA↗