arXiv subjects

Sachin S. Bharadwaj

Publications and source records attributed to Sachin S. Bharadwaj.

10 recordsLinked to original sources

arXiv2026

Accelerated search for global extrema in structured quantum signals

Extreme values of a physical system are rare by definition, yet they can govern some of its most consequential and anomalous behavior. Their sparsity makes them particularly costly to identify in large data sets. Quantum computation offers a potential way to reduce this cost. The challenge is that quantum amplitudes cannot simply be read, copied, compared, or marked using conventional search oracles. Here we introduce an algorithm for searching amplitude-encoded quantum signals that exploits their underlying structure. Through a nonlinear amplitude transformation, the algorithm identifies and certifies a global extremum while exponentially reducing the effective search space for structured signals. We establish a polylogarithmic query complexity in the data size $N$, together with global completeness, efficient classical verification, and a measurable criterion for quantifying exploitable structure. We demonstrate the framework using computational turbulence data. Applied to line cuts of velocity gradients from a $1024^3$ simulation containing more than one billion grid points, the algorithm consistently isolates the extreme tail and reduces the median search space to $O(\log N)$. The clustered structure characteristic of turbulence further accelerates the search. These results establish a direct connection between physical structure and quantum advantage in extreme-value search. For accessibility, the main text emphasizes the central ideas and results, while the theoretical derivations are provided in the Appendices.

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arXiv2026

Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems

Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, nonlinear PDEs. We embed the PDEs in a truncated, high-dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods with a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of the time-marching system size, simulation time, accuracy parameters, and post-selection parameters. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts, and query complexity. A physically motivated measure of nonlinearity is connected to a parameter similar to the flow Reynolds number $Re_{\textrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of Burgers, Fisher--KPP, damped Kuramoto--Sivashinsky, and real Ginzburg--Landau/Allen--Cahn equations. Together, these examples encompass cubic nonlinearity and saturation, quadratic transport, and fourth-order dissipative stiffness. This work shows the potential of hybrid quantum algorithms for simulating nonlinear problems on near-term and fault-tolerant devices.

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arXiv2025

Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm

The time-dependent one-dimensional nonlinear Schrödinger equation (NLSE) is solved numerically by a hybrid pseudospectral-variational quantum algorithm that connects a pseudospectral step for the Hamiltonian term with a variational step for the nonlinear term. The Hamiltonian term is treated as an integrating factor by forward and backward Fourier transformations, which are here carried out classically. This split allows us to avoid higher-order time integration schemes, to apply a first-order explicit time stepping for the remaining nonlinear NLSE term in a variational algorithm block, and thus to avoid numerical instabilities. We demonstrate that the analytical solution is reproduced with a small root mean square error for a long time interval over which a nonlinear soliton propagates significantly forward in space while keeping its shape. We analyze the accuracy of the quantum algorithm and compare it with classical approaches. Furthermore, we investigate the influence of algorithm parameters on the accuracy of the results, including the temporal step width and the depth of the quantum circuit.

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arXiv2025

Compact quantum algorithms for time-dependent differential equations

Many claims of computational advantages have been made for quantum computing over classical, but they have not been demonstrated for practical problems. Here, we present algorithms for solving time-dependent PDEs, with particular reference to fluid equations. We build on an idea based on linear combination of unitaries to simulate non-unitary, non-Hermitian quantum systems, and generate hybrid quantum-classical algorithms that efficiently perform iterative matrix-vector multiplication and matrix inversion operations. These algorithms are end-to-end, with relatively low-depth quantum circuits that demonstrate quantum advantage, with the best-case asymptotic complexities, which we show are near-optimal. We demonstrate the performance of the algorithms by conducting: (a) fully gate level, state-vector simulations using an in-house, high performance, quantum simulator called QFlowS; (b) experiments on a real quantum device; and (c) noisy simulations using Qiskit Aer. We also provide device specifications such as error-rates (noise) and state sampling (measurement) to accurately perform convergent flow simulations on noisy devices. The results offer evidence that the proposed algorithm is amenable for use on near-term quantum devices.

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arXiv2024

Quantum Carleman linearisation efficiency in nonlinear fluid dynamics

Computational fluid dynamics (CFD) is a specialised branch of fluid mechanics that utilises numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing, which has the potential to resolve nonlinear differential equations more efficiently than classical computers. Here, we try to answer the question of which regimes of nonlinear partial differential equations (PDEs) for fluid dynamics can have an efficient quantum algorithm. We propose a connection between the numerical parameter, $R$, that guarantees efficiency in the truncation of the Carleman linearisation, and the physical parameters that describe the fluid flow. This link can be made thanks to the Kolmogorov scale, which determines the minimum size of the grid needed to properly resolve the energy cascade induced by the nonlinear term. Additionally, we introduce the formalism for vector field simulation in different spatial dimensions, providing the discretisation of the operators and the boundary conditions.

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arXiv2024

Simulating fluid flows with quantum computing

The applications and impact of high fidelity simulation of fluid flows are far-reaching. They include settling some long-standing and fundamental questions in turbulence. However, the computational resources required for such efforts are extensive. Here, we explore the possibility of employing the recent computing paradigm of quantum computing to simulate fluid flows. The lure of this new paradigm is the potentially exponential advantage in memory and speed, in comparison with classical computing. This field has recently witnessed a considerable uptick in excitement and contributions. In this work, we give a succinct discussion of the progress made so far, with focus on fluid flows, accompanied by an enumeration of challenges that require sustained efforts for progress. Quantum computing of fluid flows has a promising future, but the inherently nonlinear nature of flows requires serious efforts on resolving various bottlenecks, and on synthesising progress on theoretical, numerical and experimental fronts. We present certain critical details that have not yet attracted adequate attention.

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arXiv2023

Two quantum algorithms for solving the one-dimensional advection-diffusion equation

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection-diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithms (QLSA) based on the Harrow--Hassidim--Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to $N=64$ grid points on the unit interval. Both algorithms are of hybrid nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the sizes of an additional quantum register for the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement and decoherence circuit noise. We reflect the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

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arXiv2023

Hybrid quantum algorithms for flow problems

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with an end-to-end practical applications -- in this instance, fluid dynamics. We debut here a high performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates $T_{0}$ (a parameter crucial to Hamiltonian simulations) to the condition number $κ$ of the simulation matrix, and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for (a) the functional form or sparse quantum state preparation, and (b) \textit{in-situ} quantum post-processing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be $\mathcal{O}(\textrm{polylog} (N/ε)κ/ε_{QPP})$, where $N$ is the size of the linear system of equations, $ε$ is the the solution error and $ε_{QPP}$ is the error in post processing. This work suggests a path towards quantum simulation of fluid flows, and highlights the special considerations needed at the gate level implementation of QC.

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arXiv2021

The area rule for circulation in three-dimensional turbulence

An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called Area Rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the Area Rule, for both planar and non-planar loops, using high-resolution data from Direct Numerical Simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is further from unity, and when the moment-order increases. The differences do not exceed about $5\%$ for any condition examined here. The aspect-ratio dependence observed for the second-order moment are indistinguishable from results for the Gaussian Random Field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller $(< 2\%)$ but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence.

physics.flu-dyn↗

arXiv2020

Quantum Computation of Fluid Dynamics

Studies of strongly nonlinear dynamical systems such as turbulent flows call for superior computational prowess. With the advent of quantum computing, a plethora of quantum algorithms have demonstrated, both theoretically and experimentally, more powerful computational possibilities than their classical counterparts. Starting with a brief introduction to quantum computing, we will distill a few key tools and algorithms from the huge spectrum of methods available, and evaluate possible approaches of quantum computing in fluid dynamics.

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