by Amara Graps
A CME (Coronal Mass Ejection) was headed our way to impact the Earth within 48 hours after emerging from Sunspot AR3842 on October 3, 2024. Solar physicists stated it was the strongest solar flare of this solar cycle, the strongest since 2017. A popular aurora chaser showed plasma simulations of shock waves in the interplanetary plasma that would soon be impacting our Earth’s magnetic field. Many seekers of Earth’s most colorful magneto-chemical displays headed to their dark country-sides for what they hoped would be a good aurora show. As the night sky was not yet cooperating, frustration was building. A popular aurora-chaser lectured on Twitter/X about NOAA’s unreliable tools to predict auroras. At about 3 AM, 48 hours later, the webcams in North America began to pick up the solar storm. It wasn’t a repeat of memorable May 10th auroras, however. Disappointment.
From my latitude at 57 degrees North in Europe, I periodically checked the sky outside my windows and replayed the plasma simulations of the CME from Sunspot AR3842.
Plasma Simulations on a Quantum Computer. Where to Start?
Simulations of plasma are governed by differential equations from fundamental physics with conservation laws. The numerical toolkits that computational physicists have implemented for their physics simulations for the last 80 years, include a list like the following of numerical methods for solving Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs), and Boundary Value Problems (BVPs).
My Master’s Physics thesis simulated plasma ions dynamics, and my PhD Physics thesis simulated cosmic dust moving in plasmas inside and outside of solar system magnetospheres. If you come from a classical computing world, how would you know where to begin to assess quantum computers’ usefulness in previous classical computer simulations?
I would begin with the quantum algorithm primitives. For differential equations, I would start small, with understanding the HHL= Harrow-Hassidim-Lloyd algorithm, as seen in the GQI Quantum Algorithms Framework.
Quantum Algorithm Primitives for Classical Scientific Computing
The Harrow-Hassidim-Lloyd algorithm manipulates the quantum state by accessing the several phases of eigenvalues in superposition in order to reach the desired result. In Dalzell et al., 2023, “Quantum Algorithms”, they classify the HHL as the original Quantum Linear System Solver (QLSS). This large 337-page reference article has a useful six-page section with a bibliography to describe Quantum Linear Solvers, starting on pg. 247.
The general idea of the QLSS is to take an N × N complex matrix A and a complex vector b of size N as inputs and give back a pure quantum state, that is close to the normalized solution vector of the linear system of equations Ax = b. In their most basic forms, QLSSs work by putting the normalized entries of the matrix A and the vector b into a unitary quantum circuit.
Let’s next go to Arnault et al., 2024’s paper to understand the class of solvers: QLSS. In QCR’s article: Quantum Technology Algorithm Trends – Tidying Up, we introduced Arnault et al., 2024 in A typology of quantum algorithms. Their figures and codes are at GitHub (updated since our QCR article). From Arnault et al.’s paper, on page 52-60 is a Classification Table of Algorithms (spreadsheet also at Github) from which we can search on “differential equations”, and “HHL”. In Arnault et al.’s paper, QLSS is called: QLSA = Family of Quantum Linear-System Algorithms, so we can search on “QLSA” in Arnault’s Table, as well.
Arnault et al.’s Classification Table shows the HHL as a highly-cited, primary quantum algorithm for solving linear sets of equations. It has inside a more quantum primitive: Quantum Phase Estimation (QPE) and Quantum Fourier Transform (QFT). However, the HHL algorithm is an LSQ algorithm, not a NISQ algorithm, according to the authors. Their Large-Scale Quantum (LSQ) computer definition follows GQI’s Fault-Tolerant Quantum Computer (FTQC), a theoretical, fully-error-corrected quantum computer with millions of qubits and high-fidelity gate operations. However, today’s NISQ devices of ~100 qubits are necessary for the journey to LSQ computers.
In Section 3, where Arnault et al., analyzes correlations between algorithmic primitives, mathematical classes, and application domains, is one not to miss. Their Figure 4: Correlations, show the same data as Figure 5’s Sankey Diagram, which is a more fluid representation. I’ve identified the Dynamical Systems and the Classical Scientific Computing Application Domains with a Red Rectangle. See the next Figure. There already may be quantum usefulness. In some instances, quantum algorithms in this application domain may outperform conventional algorithms for classical-mechanics simulations, empirical/semi-empirical numerical model prediction and optimization and experimental data processing/mining.
As the authors write in their abstract, the primary objective of their research article is to reveal trends of algorithms, identify promising fields for implementations in the NISQ era, and identify the key algorithmic primitives that would power quantum advantage. For classically trained physicists like those of us reading this article, it’s worth to look at our simulation codes and think about what parts could benefit from a quantum computer.
The LSQ characterization for the HHL algorithm hasn’t stopped researchers from working inside of NISQ device limitations, we’ll continue this Algorithm discussion about Differential Equations in Lapworth, 2022 A Hybrid Quantum-Classical CFD Methodology with Benchmark HHL Solutions in the next article.
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October 6, 2024