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Compact quantum algorithms that can potentially maintain quantum advantage for solving time-dependent differential equations
Authors:
Sachin S. Bharadwaj,
Katepalli R. Sreenivasan
Abstract:
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 governing fluid flow problems. We build on an idea based on linear combination of unitaries to simulate non-unitary, non-Hermitian quantum systems, and generate hybrid quantum-classical a…
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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 governing fluid flow problems. 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 lead to low-depth quantum circuits that protect quantum advantage, with the best-case asymptotic complexities that are near-optimal. We demonstrate the performance of the algorithms by conducting: (a) ideal state-vector simulations using an in-house, high performance, quantum simulator called $\textit{QFlowS}$; (b) experiments on a real quantum device (IBM Cairo); 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.
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Submitted 15 May, 2024;
originally announced May 2024.
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Two quantum algorithms for solving the one-dimensional advection-diffusion equation
Authors:
Julia Ingelmann,
Sachin S. Bharadwaj,
Philipp Pfeffer,
Katepalli R. Sreenivasan,
Jörg Schumacher
Abstract:
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--Hassid…
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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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Submitted 30 December, 2023;
originally announced January 2024.
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Hybrid quantum algorithms for flow problems
Authors:
Sachin S. Bharadwaj,
Katepalli R. Sreenivasan
Abstract:
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…
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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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Submitted 21 November, 2023; v1 submitted 1 July, 2023;
originally announced July 2023.
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Quantum Computation of Fluid Dynamics
Authors:
Sachin S. Bharadwaj,
Katepalli R. Sreenivasan
Abstract:
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 ke…
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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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Submitted 17 July, 2020;
originally announced July 2020.
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The area rule for circulation in three-dimensional turbulence
Authors:
Kartik P. Iyer,
Sachin S. Bharadwaj,
Katepalli R. Sreenivasan
Abstract:
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 f…
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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.
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Submitted 19 October, 2021; v1 submitted 13 July, 2020;
originally announced July 2020.
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Synthesis and Characterization of Copper Doped Zinc Oxide Thin Films for CO Gas Sensing
Authors:
Sachin S Bharadwaj,
Shivaraj B W,
H N Narasimha Murthy,
M Krishna,
Manjush Ganiger,
Mohd Idris,
Pundaleek Anawal,
Vitthal Sangappa Angadi
Abstract:
Objective of this work was to synthesize Copper doped Zinc Oxide (CZO) films and optimization of process parameters by varying molarity of zinc acetate dehydrate from 0.5 M to 1.0 M, concentration of copper acetate monohydrate from 1% to 5 % and annealing temperature from 200 C to 300 C to measure the sensitivity of CZO films for CO (Carbon Monoxide) gas. The concentration of CO gas was maintained…
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Objective of this work was to synthesize Copper doped Zinc Oxide (CZO) films and optimization of process parameters by varying molarity of zinc acetate dehydrate from 0.5 M to 1.0 M, concentration of copper acetate monohydrate from 1% to 5 % and annealing temperature from 200 C to 300 C to measure the sensitivity of CZO films for CO (Carbon Monoxide) gas. The concentration of CO gas was maintained at 5 ppm and operating temperature of 250 oC was used for sensing. Analysis for sensitivity showed highest grading for parametric combination of 0.75 molarity, 3% copper concentration and 300 C annealing temperature with surface roughness of 3.90 nm and grain size of 256 nm. TEM image revealed the crystalline grain size was 5 nm. ANOVA showed that annealing temperature influenced the sensitivity by 69.06 % .
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Submitted 19 February, 2018;
originally announced March 2018.
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Effect of RF Sputtering Process Parameters on Silicon Nitride Thin Film Deposition
Authors:
Sachin S Bharadwaj,
GR Rajkumar,
B W Shivaraj,
M Krishna
Abstract:
The objective of this work was to study the RF sputtering process parameters optimisation for deposition of Silicon Nitride thin films. The process parameters chosen to be varied were deposition power, deposition duration, flow rate of argon and flow rate of nitrogen. The parameters were varied at three levels according to Taguchi L9 orthogonal array. Surface topology, film composition, coating th…
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The objective of this work was to study the RF sputtering process parameters optimisation for deposition of Silicon Nitride thin films. The process parameters chosen to be varied were deposition power, deposition duration, flow rate of argon and flow rate of nitrogen. The parameters were varied at three levels according to Taguchi L9 orthogonal array. Surface topology, film composition, coating thickness, coating resistivity and refractive index were determined using SEM, XRD, profilometer, Semiconductor device analyser and UV spectrometer respectively. The measured film thickness values ranged from 127.8nm to 908.3nm with deposition rate varying from 1.47nm/min to a maximum value of 10.1nm/min. The resistivity of the film varied between 1.53x1013ohm-m to 7.85 x1013ohm-m. Refractive index of the film was calculated to be between 1.84 to 2.08. From the results, it was seen that film properties tend to be poor when there is no nitrogen flow and tend to improve with small input of nitrogen. Also, SEM images indicated amorphous structure of silicon nitride which was confirmed by XRD pattern.
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Submitted 28 November, 2017; v1 submitted 27 November, 2017;
originally announced November 2017.
