arXiv subjects

Mohsen Bagherimehrab

Publications and source records attributed to Mohsen Bagherimehrab.

10 recordsLinked to original sources

arXiv2026

Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas

Hamiltonian simulation using product formulas is arguably the most straightforward and practical approach for algorithmic simulation of a quantum system's dynamics on a quantum computer. Here we present corrected product formulas (CPFs), a variation of product formulas achieved by injecting auxiliary terms called correctors into standard product formulas. We establish several correctors that improve the accuracy of standard product formulas by orders of magnitude when simulating Hamiltonians comprised of two exactly simulatable partitions, a common structure of lattice Hamiltonians. Importantly, injecting these correctors increases the overall simulation cost by only a small additive or multiplicative factor. We show that correctors are particularly advantageous for perturbed systems, where one partition has a relatively small norm compared to the other, as they allow the small norm to be utilized as an additional parameter for controlling the simulation error. We demonstrate the performance of CPFs by numerical simulations for several lattice Hamiltonians. Numerical results show that our theoretical error bounds for CPFs match or outperform the empirical error scaling of standard product formulas for these systems. We also demonstrate improvements offered by CPFs by implementing small-size systems on actual quantum hardware, as well as on noisy and noiseless quantum simulators. CPFs could be a valuable algorithmic tool for early fault-tolerant quantum computers with limited computing resources. As for standard product formulas, CPFs could also be used for simulations on a classical computer.

quant-ph↗

arXiv2025

The generative quantum eigensolver (GQE) and its application for ground state search

We introduce the generative quantum eigensolver (GQE), a new quantum computational framework that operates outside the variational quantum algorithm paradigm by applying classical generative models to quantum simulation. The GQE algorithm optimizes a classical generative model to produce quantum circuits with desired properties. Here, we develop a transformer-based implementation, which we name the generative pre-trained transformer-based (GPT) quantum eigensolver (GPT-QE). We show a proof-of-concept of training and pretraining of GPT-QE applied to electronic structure Hamiltonians, and demonstrate its ability illustrated by surpassing coupled cluster singles and doubles (CCSD) for the strong bond dissociation of the nitrogen molecule and approaching chemical accuracy. We also demonstrate the method on real quantum hardware.

quant-ph↗

arXiv2025

Fast quantum algorithm for differential equations

Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at least linearly with the condition number $κ$ of the matrices involved in the computation. For many practical applications, $κ$ scales polynomially with the size $N$ of the matrices, rendering a polynomial complexity in $N$ for these algorithms. Here we present a quantum algorithm with a complexity that is polylogarithmic in $N$ but is independent of $κ$ for a large class of PDEs. Our algorithm generates a quantum state from which features of the solution can be extracted. Central to our methodology is using a wavelet basis as an auxiliary system of coordinates in which the condition number of associated matrices becomes independent of $N$ by a simple diagonal preconditioner. We present numerical simulations showing the effect of the wavelet preconditioner for several differential equations. Our work could provide a practical way to boost the performance of quantum simulation algorithms where standard methods are used for discretization.

quant-ph↗

arXiv2025

Chemically Motivated Simulation Problems are Efficiently Solvable by a Quantum Computer

Simulating chemical systems is highly sought after and computationally challenging, as the number of degrees of freedom increases exponentially with the size of the system. Quantum computers have been proposed as a computational means to overcome this bottleneck , thanks to their capability of representing this amount of information efficiently. Most efforts so far have been centered around determining the ground states of chemical systems. However, hardness results and the lack of theoretical guarantees for efficient heuristics for initial-state generation shed doubt on the feasibility. Here, we propose a heuristically guided approach that is based on inherently efficient routines to solve chemical simulation problems, requiring quantum circuits of size scaling polynomially in relevant system parameters. If a set of assumptions can be satisfied, our approach finds good initial states for dynamics simulation by assembling them in a scattering tree. In particular, we investigate a scattering-based state preparation approach within the context of mergo-association. We discuss a variety of quantities of chemical interest that can be measured after the quantum simulation of a process, e.g., a reaction, following its corresponding initial state preparation.

quant-ph↗

arXiv2025

Quantum-Assisted Support Vector Regression

A popular machine-learning model for regression tasks, including stock-market prediction, weather forecasting and real-estate pricing, is the classical support vector regression (SVR). However, a practically realisable quantum SVR remains to be formulated. We devise annealing-based algorithms, namely simulated and quantum-classical hybrid, for training two SVR models and compare their empirical performances against the SVR implementation of Python's scikit-learn package for facial-landmark detection (FLD), a particular use case for SVR. Our method is to derive a quadratic-unconstrained-binary formulation for the optimisation problem used for training a SVR model and solve this problem using annealing. Using D-Wave's hybrid solver, we construct a quantum-assisted SVR model, thereby demonstrating a slight advantage over classical models regarding FLD accuracy. Furthermore, we observe that annealing-based SVR models predict landmarks with lower variances compared to the SVR models trained by gradient-based methods. Our work is a proof-of-concept example for applying quantum-assisted SVR to a supervised-learning task with a small training dataset.

quant-ph↗

arXiv2024

qSWIFT: High-order randomized compiler for Hamiltonian simulation

Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work, we present qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. In qSWIFT, the required number of gates for a given precision is independent of the number of terms in Hamiltonian, while the systematic error is exponentially reduced with regards to the order parameter. In this respect, our qSWIFT is a higher-order counterpart of the previously proposed quantum stochastic drift protocol (qDRIFT), in which the number of gates scales linearly with the inverse of the precision required. We construct the qSWIFT channel and establish a rigorous bound for the systematic error quantified by the diamond norm. qSWIFT provides an algorithm to estimate given physical quantities using a system with one ancilla qubit, which is as simple as other product-formula-based approaches such as regular Trotter-Suzuki decompositions and qDRIFT. Our numerical experiment reveals that the required number of gates in qSWIFT is significantly reduced compared to qDRIFT. Particularly, the advantage is significant for problems where high precision is required; for example, to achieve a systematic relative propagation error of $10^{-6}$, the required number of gates in third-order qSWIFT is 1000 times smaller than that of qDRIFT.

quant-ph↗

arXiv2024

Efficient Quantum Algorithm for All Quantum Wavelet Transforms

Wavelet transforms are widely used in various fields of science and engineering as a mathematical tool with features that reveal information ignored by the Fourier transform. Unlike the Fourier transform, which is unique, a wavelet transform is specified by a sequence of numbers associated with the type of wavelet used and an order parameter specifying the length of the sequence. While the quantum Fourier transform, a quantum analog of the classical Fourier transform, has been pivotal in quantum computing, prior works on quantum wavelet transforms~(QWTs) were limited to the second and fourth order of a particular wavelet, the Daubechies wavelet. Here we develop a simple yet efficient quantum algorithm for executing any wavelet transform on a quantum computer. Our approach is to decompose the kernel matrix of a wavelet transform as a linear combination of unitaries (LCU) that are compilable by easy-to-implement modular quantum arithmetic operations and use the LCU technique to construct a probabilistic procedure to implement a QWT with a \textit{known} success probability. We then use properties of wavelets to make this approach deterministic by a few executions of the amplitude amplification strategy. We extend our approach to a multilevel wavelet transform and a generalized version, the packet wavelet transform, establishing computational complexities in terms of three parameters: the wavelet order $M$, the dimension $N$ of the transformation matrix, and the transformation level $d$. We show the cost is logarithmic in $N$, linear in $d$ and superlinear in $M$. Moreover, we show the cost is independent of $M$ for practical applications. Our proposed quantum wavelet transforms could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.

quant-ph↗

arXiv2022

Entanglement in quantum field theory via wavelet representations

Quantum field theory (QFT) describes nature using continuous fields, but physical properties of QFT are usually revealed in terms of measurements of observables at a finite resolution. We describe a multiscale representation of a free scalar bosonic and Ising model fermionic QFTs using wavelets. Making use of the orthogonality and self similarity of the wavelet basis functions, we demonstrate some well known relations such as scale dependent subsystem entanglement entropy and renormalization of correlations in the ground state. We also find some new applications of the wavelet transform as a compressed representation of ground states of QFTs which can be used to illustrate quantum phase transitions via fidelity overlap and holographic entanglement of purification.

quant-ph↗

arXiv2022

Tight Bound for Estimating Expectation Values from a System of Linear Equations

The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$, the condition number $κ$ of $A$, a vector $b$, a Hermitian matrix $M$ and an accuracy $ε$, and the task is to estimate $x^\dagger Mx$, where $x$ is the solution vector to the equation $Ax = b$. We aim to establish a lower bound on the complexity of the end-to-end quantum algorithms for SLEP with respect to $ε$, and devise a quantum algorithm that saturates this bound. To make lower bounds attainable, we consider query complexity in the setting in which a block encoding of $M$ is given, i.e., a unitary black box $U_M$ that contains $M/α$ as a block for some $α\in \mathbb R^+$. We show that the quantum query complexity for SLEP in this setting is $Θ(α/ε)$. Our lower bound is established by reducing the problem of estimating the mean of a black box function to SLEP. Our $Θ(α/ε)$ result tightens and proves the common assertion of polynomial accuracy dependence (poly$(1/ε)$) for SLEP, and shows that improvement beyond linear dependence on accuracy is not possible if $M$ is provided via block encoding.

quant-ph↗

arXiv2022

Nearly optimal quantum algorithm for generating the ground state of a free quantum field theory

We devise a quasilinear quantum algorithm for generating an approximation for the ground state of a quantum field theory (QFT). Our quantum algorithm delivers a super-quadratic speedup over the state-of-the-art quantum algorithm for ground-state generation, overcomes the ground-state-generation bottleneck of the prior approach and is optimal up to a polylogarithmic factor. Specifically, we establish two quantum algorithms -- Fourier-based and wavelet-based -- to generate the ground state of a free massive scalar bosonic QFT with gate complexity quasilinear in the number of discretized-QFT modes. The Fourier-based algorithm is limited to translationally invariant QFTs. Numerical simulations show that the wavelet-based algorithm successfully yields the ground state for a QFT with broken translational invariance. Furthermore, the cost of preparing particle excitations in the wavelet approach is independent of the energy scale. Our algorithms require a routine for generating one-dimensional Gaussian (1DG) states. We replace the standard method for 1DG-state generation, which requires the quantum computer to perform lots of costly arithmetic, with a novel method based on inequality testing that significantly reduces the need for arithmetic. Our method for 1DG-state generation is generic and could be extended to preparing states whose amplitudes can be computed on the fly by a quantum computer.

quant-ph↗