arXiv subjects

Nhat A. Nghiem

Publications and source records attributed to Nhat A. Nghiem.

At least 37 records · Page 2Linked to original sources

arXiv2025

(Nearly) Optimal Time-dependent Hamiltonian Simulation

We describe a simple quantum algorithm to simulate time-dependent Hamiltonian, extending the methodology of quantum signal processing. The framework achieves optimal scaling up to some factor with respect to other parameters, and nearly optimal in inverse of error tolerance, which could be improved to optimal scaling under certain input models. As applications, we discuss the problem of simulating generalized lattice system and time-periodic, or Floquet system, showing that our framework provides a neater yet highly efficient solution, achieving optimal/nearly optimal scaling in all parameters. In particular, our method also paves a new way for studying phase transition on quantum computer, extending the reach of quantum simulation.

quant-ph↗

arXiv2025

Quantum Speedup in Dissecting Roots and Solving Nonlinear Algebraic Equations

It is shown that quantum computer can detect the existence of root of a function almost exponentially more efficient than the classical counterpart. It is also shown that a quantum computer can produce quantum state corresponding to the solution of nonlinear algebraic equations quadratically faster than the best known classical approach. Various applications and implications are discussed, including a quantum algorithm for solving dense linear systems with quadratic speedup, determining equilibrium states, simulating the dynamics of nonlinear coupled oscillators, estimating Lyapunov exponent, an improved quantum partial differential equation solver, and a quantum-enhanced collision detector for robotic motion planning. This provides further evidence of quantum advantage without requiring coherent quantum access to classical data, delivering meaningful real-world applications.

quant-ph↗

arXiv2025

Simple Quantum Gradient Descent Without Coherent Oracle Access

The gradient descent method aims at finding local minima of a given multivariate function by moving along the direction of its gradient, and hence, the algorithm typically involves computing all partial derivatives of a given function, before updating the solution iteratively. In the work of Rebentrost et al. [New Journal of Physics, 21(7):073023, 2019], the authors translated the iterative optimization algorithm into a quantum setting, with some assumptions regarding certain structure of the given function, with oracle or black-box access to some matrix that specifies the structure. Here, we develop an alternative quantum framework for the gradient descent problem, based on the seminal quantum singular value transformation framework. We show that given only classical information of function of interest, it is possible to construct a quantum gradient descent algorithm with a running time logarithmical in the number of variables. In particular, our framework consumes exponentially less qubits than the prior quantum gradient descent algorithm and removes the need for any coherent oracle access to classical information. Thus, our work provides another example demonstrating the power of quantum singular value transformation framework, and in particular, it adds another instance revealing that quantum coherent access is not necessary for quantum computational advantage.

quant-ph↗

arXiv2025

New Quantum Algorithm For Solving Linear System of Equations

Linear equations play a pivotal role in many areas of science and engineering, making efficient solutions to linear systems highly desirable. The development of quantum algorithms for solving linear systems has been a significant breakthrough, with such algorithms rapidly becoming some of the most influential in quantum computing. Subsequent advances have focused on improving the efficiency of quantum linear solvers and extending their core techniques to address various challenges, such as least-squares data fitting. In this article, we introduce a new quantum algorithm for solving linear systems based on the gradient descent method. Inspired by the vector/density state formalism, we represent a point, or vector, as a density state-like entity, enabling us to leverage recent advancements in quantum algorithmic frameworks, such as block encoding, to directly manipulate operators and carry out the gradient descent method in a quantum setting. The operator corresponding to the intermediate solution is updated iteratively, with a provable guarantee of convergence. The quantum state representing the final solution to the linear system can then be extracted by further measurement. We provide a detailed complexity analysis and compare our approach with existing methods, demonstrating significant improvement in certain aspects.

quant-ph↗

arXiv2025

Estimation of Nonlinear Physical Quantities By Measuring Ancillas

In this article, we present quantum algorithms for estimating von Neumann entropy and Renyi entropy, which are crucial physical and information-theoretical properties of a given quantum state $ρ$. Although there have been existing works that achieved the same goal, some prior developments assume the unitary that prepares the purification to the target state $ρ$. Here, we consider an alternative setting where only copies of $ρ$ are given and construct a quantum algorithm that estimates the desired entropy. Our framework can complete the given task by measuring a small number of ancilla qubits without directly measuring the system, and that it achieves significant improvement over prior relevant developments. For example, for the Renyi entropy of the order of non-integral $α$, our method achieves almost power-of-two improvement in sample complexity with respect to the rank of the given state and almost a power-of-two improvement in error tolerance compared with the work by Wang et al. [Phys. Rev. Applied 19, 044041 (2023)].

quant-ph↗

arXiv2025

New Quantum Algorithm for Principal Component Analysis

Quantum principal component analysis (QPCA) ignited a new development toward quantum machine learning algorithms. Initially showcasing as an active way for analyzing a quantum system using the quantum state itself, QPCA also found potential application in analyzing a large-scale dataset, represented by the so-called covariance matrix. Inspired by recent advancement in quantum algorithms, we give an alternatively new quantum framework for performing principal component analysis. By analyzing the performance in detail, we shall identify the regime in which our proposal performs better than the original QPCA. In addition, we also provide a new approach for preparing the covariance matrix, given classical dataset, on a quantum computer. Thus, our work provides an efficient complementary framework for revealing features of the quantum state, while keeping the philosophy of original QPCA, as the state could play an active role in analyzing itself.

quant-ph↗

arXiv2024

A Framework For Estimating Amplitudes of Quantum State With Single-Qubit Measurement

We propose and analyze a simple framework for estimating the amplitudes of a given $n$-qubit quantum state $\ketψ = \sum_{i=0}^{2^n-1} a_i \ket{i}$ in computational basis, utilizing a single-qubit measurement only. Previously, it was a common procedure that one could measure all qubits in order to collect measurement outcomes, from which one can estimate amplitudes of given quantum state. Here, we show that if restricting to single-qubit measurement, and one can perform measurement on arbitrary basis, then the measurement outcomes can be used to assist the finding of amplitudes in the usual computational, or Z basis. More concretely, such outcomes are capable of constructing a system of nonlinear algebraic equations, and by classically solving them, we obtain $\Tilde{a}_i$, which is the approximation to the corresponding amplitudes $a_i$, including both real and imaginary component. We then discuss our framework from a broader perspective. First, we show that estimating all (norms of) amplitudes to additive accuracy $δ$, i.e., $| |\Tilde{a}_i - |a_i| | \leq δ$ for all $i$, $\mathcal{O}(4^n/δ^4)$ single-qubit measurements is sufficient. Second, we show that to achieve total variation $\sum_{i=0}^{2^n-1} | |\Tilde{a}_i|^2 - |a_i|^2| \leq δ$, $\mathcal{O}(6^n/δ^4)$ a single bit measurement is required. Finally, in order to achieve an average $L_1$ norm error $ \sum_{i=0}^{2^n-1} | |\Tilde{a}_i| - |a_i| |/2^n \leq δ$, a single bit measurement $\mathcal{O}(2^n/ δ^4)$ is needed.

quant-ph↗

arXiv2024

Improved Time-independent Hamiltonian Simulation

We describe a simple method for simulating time-independent Hamiltonian $H$ that could be decomposed as $H = \sum_{i=1}^m H_i$ where each $H_i$ can be efficiently simulated. Approaches relying on product formula generally work by splitting the evolution time into segments, and approximate the evolution in each segment by the evolution of composing Hamiltonian $H_i$. This key step incur a constraint, that prohibits a (poly)logarithmic scaling on approximation error. We employ the recently introduced quantum singular value transformation framework to utilize the ability to simulate $H_i$ in an alternative way, which then allows us to construct and simulate the main Hamiltonian $H$ with polylogarithmical scaling on the inverse of desired error, which is a major improvement with respect to product formula approaches.

quant-ph↗

arXiv2024

Simulating Time-dependent Hamiltonian Based On High Order Runge-Kutta and Forward Euler Method

We propose a new method for simulating certain type of time-dependent Hamiltonian $H(t) = \sum_{i=1}^m γ_i(t) H_i$ where $γ_i(t)$ (and its higher order derivatives) is bounded, computable function of time $t$, and each $H_i$ is time-independent, and could be efficiently simulated. Our quantum algorithms are based on high-order Runge-Kutta method and forward Euler method, where the time interval is divided into subintervals. Then in an iterative manner, the evolution operator at given time step is built upon the evolution operator at previous time step, utilizing algorithmic operations from the recently introduced quantum singular value transformation framework.

quant-ph↗

arXiv2024

Quantum Algorithm For Testing Convexity of Function

Functions are a fundamental object in mathematics, with countless applications to different fields, and are usually classified based on certain properties, given their domains and images. An important property of a real-valued function is its convexity, which plays a very crucial role in many areas, such as thermodynamics and geometry. Motivated by recent advances in quantum computation as well as the quest for quantum advantage, we give a quantum algorithm for testing convexity of polynomial functions, which appears frequently in multiple contexts, such as optimization, machine learning, physics, etc. We show that quantum computers can reveal the convexity property superpolynomially faster than classical computers with respect to number of variables. As a corollary, we provide a significant improvement and extension on quantum Newton's method constructed in earlier work of Rebentrost et al [New J. Phys. \textbf{21} 073023 (2019)]. We further discuss our algorithm in a broader context, such as potential application in the study of geometric structure of manifold, testing training landscape of variational quantum algorithm and also gradient descent/Newton's method for optimization.

quant-ph↗

arXiv2024

Quantum Algorithm For Solving Nonlinear Algebraic Equations

Nonlinear equations are challenging to solve due to their inherently nonlinear nature. As analytical solutions typically do not exist, numerical methods have been developed to tackle their solutions. In this article, we give a quantum algorithm for solving a system of nonlinear algebraic equations, in which each equation is a multivariate polynomial of known coefficients. Building upon the classical Newton method and some recent works on quantum algorithm plus block encoding from the quantum singular value transformation, we show how to invert the Jacobian matrix to execute Newton's iterative method for solving nonlinear equations, where each contributing equation is a homogeneous polynomial of an even degree. A detailed analysis are then carried out to reveal that our method achieves polylogarithmic time in relative to the number of variables. Furthermore, the number of required qubits is logarithmic in the number of variables. In particular, we also show that our method can be modified with little effort to deal with polynomial of various types, thus implying the generality of our approach. Some examples coming from physics and algebraic geometry, such as Gross-Pitaevski equation, Lotka-Volterra equations, and intersection of algebraic varieties, involving nonlinear partial differential equations are provided to motivate the potential application, with a description on how to extend our algorithm with even less effort in such a scenario. Our work thus marks a further important step towards quantum advantage in nonlinear science, enabled by the framework of quantum singular value transformation.

quant-ph↗

arXiv2024

Improved Quantum Power Method and Numerical Integration Using Quantum Singular Value Transformation

Quantum singular value transformation (QSVT) is a framework that has been shown to unify many primitives in quantum algorithms. In this work, we leverage the QSVT framework in two directions. We first show that the QSVT framework can accelerate one recently introduced quantum power method, which substantially improves its running time. Additionally, we incorporate several elementary numerical integration techniques, such as the rectangular method, Monte Carlo method, and quadrature method, into the QSVT framework, which results in polynomial speedup with respect to the size or the number of points of the grid. Our results thus provide further examples to demonstrate the potential of the QSVT and how it may enhance quantum algorithmic tasks.

quant-ph↗

arXiv2024

Quantum Algorithm for Computing Distances Between Subspaces

Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given data collection. Computing distances between different subspaces is generally a computationally challenging problem with both theoretical and applicable consequences, as, for example, the results can be used to classify data from different categories. Fueled by the fast-growing development of quantum algorithms, we consider such problems in the quantum context and provide a quantum algorithm for estimating two kinds of distance: Grassmann distance and ellipsoid distance. Under appropriate assumptions and conditions, the speedup of our quantum algorithm is exponential with respect to both the dimension of the given data and the number of data points. Some extensions regarding estimating different kinds of distance are then discussed as a corollary of our main quantum algorithmic method.

quant-ph↗

arXiv2024

Alternative Method for Estimating Betti Numbers

Topological data analysis (TDA) is a fast-growing field that utilizes advanced tools from topology to analyze large-scale data. A central problem in topological data analysis is estimating the so-called Betti numbers of the underlying simplicial complex. While the difficulty of this problem has been established as NP-hard, previous works have showcased appealing quantum speedup. In this article, we provide an alternative method for estimating Betti numbers and normalized Betti numbers of given simplicial complex, based on some recent advances in quantum algorithm, specifically, quantum singular value transformation. Our method can be faster than the best-known classical method for finding Betti numbers, and interestingly, it can also find the Betti numbers of the complement graph to our original one. Comparing to the best known quantum algorithm, our method generally requires lower depth circuit, in trade-off for longer running time. Regarding normalized Betti numbers, our method could match the running time of best-known quantum method in the case of dense simplices.

quant-ph↗

arXiv2023

An Improved Method for Quantum Matrix Multiplication

Following the celebrated quantum algorithm for solving linear equations (so-called HHL algorithm), Childs, Kothari and Somma [SIAM Journal on Computing, {\bf 46}: 1920, (2017)] provided an approach to solve a linear system of equations with exponentially improved dependence on precision. In this note, we aim to complement such a result for applying a matrix to some quantum state, based upon their Chebyshev polynomial approach. A few examples that motivate this application are included and we further discuss an application of this improved matrix application algorithm explicitly with an efficient quantum procedure.

quant-ph↗

arXiv2023

Quantum Algorithm For Estimating Eigenvalue

A majority of numerical scientific computation relies heavily on handling and manipulating matrices, such as solving linear equations, finding eigenvalues and eigenvectors, and so on. Many quantum algorithms have been developed to advance these computational tasks, and in some cases, such as solving linear equations, can be shown to yield exponential speedup. Here, employing the techniques in the HHL algorithm and the ideas of the classical power method, we provide a simple quantum algorithm for estimating the largest eigenvalue in magnitude of a given Hermitian matrix. As in the case of the HHL algorithm, our quantum procedure can also yield exponential speedup compared to classical algorithms that solve the same problem. We also discuss a few possible extensions and applications of our quantum algorithm, such as a version of a hybrid quantum-classical Lanczos algorithm.

quant-ph↗

arXiv2023

Constant-time Quantum Algorithm for Homology Detection in Closed Curves

Given a loop or more generally 1-cycle $r$ of size L on a closed two-dimensional manifold or surface, represented by a triangulated mesh, a question in computational topology asks whether or not it is homologous to zero. We frame and tackle this problem in the quantum setting. Given an oracle that one can use to query the inclusion of edges on a closed curve, we design a quantum algorithm for such a homology detection with a constant running time, with respect to the size or the number of edges on the loop $r$, requiring only a single usage of oracle. In contrast, classical algorithm requires $Ω(L)$ oracle usage, followed by a linear time processing and can be improved to logarithmic by using a parallel algorithm. Our quantum algorithm can be extended to check whether two closed loops belong to the same homology class. Furthermore, it can be applied to a specific problem in the homotopy detection, namely, checking whether two curves are \textit{not} homotopically equivalent on a closed two-dimensional manifold.

quant-ph↗

arXiv2021

Subtleties in the trainability of quantum machine learning models

A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called Quantum Machine Learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called Variational Quantum Algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.

quant-ph↗