The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of $\mathcal{O}((\log N)^c)$, compared to $\mathcal{O}(N^c)$ in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated $\mathcal{O} \left ((\log N)^c \right)$ complexity of the proposed solver.
The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of O((logN)c), compared to O(Nc) in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated O((logN)c) complexity of the proposed solver.
The advent of quantum computers , operating on entirely different physical principles and abstractions from those of classical digital computers , sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of 𝒪 ( ( l o g 𝑁 ) 𝑐 ) , compared to 𝒪 ( 𝑁 𝑐 ) in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated 𝒪 ( ( l o g 𝑁 ) 𝑐 ) complexity of the proposed solver.
QAFE2: Quantum accelerated multiscale finite element analysis
The computational cost of concurrent multiscale finite element methods is dominated by the repeated solution of microscopic representative volume element (RVE) problems at macroscopic quadrature points. In this work, we introduce a quantum-classical framework for multiscale finite element analysis (QAFE2) that leverages quantum parallelism to fundamentally alter the scaling of RVE-based homogenisation. At the single-RVE level, the proposed quantum solver attains polylogarithmic complexity with respect to the microscopic discretisation size, yielding an exponential asymptotic speedup over the best available classical solvers. More importantly, QAFE2 exploits quantum superposition and entanglement to evaluate, in a single quantum execution, the entire ensemble of RVE problems associated with all macroscopic quadrature points. This capability is a form of intrinsic quantum concurrency with no classical analogue. Numerical experiments on one- and two-dimensional model problems with known analytical solutions confirm the accuracy of the proposed formulation and verify the theoretical computational scaling and parallel performance.
The advent of quantum computing has ushered in a new era of computational capabilities, promising to revolutionize the resolution of complex problems. Unlike classical computers, which rely on binary digits (bits) to process information, quantum computers utilize quantum bits (qubits) that can exist in multiple states simultaneously due to the principles of superposition and entanglement inherent in quantum mechanics. This fundamental difference allows quantum computers to perform certain calculations exponentially faster than their classical counterparts, offering profound implications for fields ranging from cryptography to material science. At the heart of this transformative potential lie quantum algorithms, specifically designed to harness the unique properties of quantum systems to tackle problems currently intractable for classical computers. Notable examples include Shor's algorithm for integer factorization, which threatens the security of widely used cryptographic schemes, and Grover's algorithm for unstructured search, which provides a quadratic speedup over classical approaches. Beyond these pioneering works, ongoing research continues to uncover new quantum algorithms capable of addressing a broad spectrum of applications. The proposed method leverages the principles of quantum mechanics to develop efficient algorithms that outperform classical approaches in solving complex computational problems. By harnessing the unique properties of quantum bits (qubits), such as superposition and entanglement, the method aims to achieve superior accuracy and efficiency in various applications. The performance of the proposed method was evaluated using several key metrics: an accuracy of 94.8%, a Root Mean Squared Error (RMSE) of 0.208, and a Mean Absolute Error (MAE) of 0.406. These metrics collectively underscore the robustness and reliability of the proposed quantum algorithms, highlighting their potential to significantly enhance computational efficiency and accuracy across various domains. This study delves into the development of quantum algorithms, exploring their theoretical foundations, practical implementations, and the challenges faced in their realization. By leveraging the principles of quantum mechanics, these algorithms hold the promise of not only solving existing problems more efficiently but also unlocking new computational paradigms previously deemed unattainable. Through a comprehensive analysis, this research aims to elucidate the current state of quantum algorithm research, highlight the advancements made, and identify the pathways forward in this rapidly evolving field.
The manuscript "A Comparison of Quantum and Traditional Fourier Transform Computations" discusses a very important and often overlooked aspect of quantum computing, namely a fair and detailed comparison of a quantum algorithm, its classical simulation, and its classical counterpart taking into account the complexity of I/O. Such an article is valuable and worth publishing. The current manuscript, however, still contains some inaccuracies that should be fixed and I will make suggestion on how the presentation can be improved.Let me first summarize the main result in my own words: a quantum Fourier transform (QFT) can calculate the Fourier transform of a vector with time complexity O(log2N), compared to the complexity O(NlogN) of a classical FFT. However, if one needs to read out the full vector instead, the complexity becomes O(N polylog(N)) again, without an advantage over the classical algorithm. It is actually worse than the author discusses, if one also takes into account the complexity of loading the initial state. Loading an initial classical data vector has complexity O(N), and this has to be repeated at every repetition, giving a complexity of O(N2polylogN), worse than classical. As the author correctly mentions, the QFT is thus a useful algorithm if the input data is prepared algorithmically, and limited sampling of the result vector is sufficient, as is the case for Shor's algorithm.This is a valuable observation that deserves a paper in CiSE, since these important considerations are not all known to people outside the quantum computing community, and are sometimes ignored even by quantum computing specialists. I thus want to encourage the author to improve the presentation to make it more accessible to a broader audience and fix a couple of technical flaws.Before discussing presentation issues, I want to address one technical flaw and a few points where more clarity is needed: This statement towards the end is too simplistic and needs to be clarified: "... if we want to measure each coefficient, we must redo our operations for each coefficient (since our wave function will collapse for every measurement)." The manuscript does not explain how to read out a specific coefficient. If one samples the wave function, one measures the result and gets a certain value (s1,⋯,sN) with a probability depending on the wave function. More complex amplitude-estimation algorithms are needed to read out specific coefficients, which then takes time O(N/ϵ). This needs to be better explained. This also raises another issue of normalization and precision. The quantum wave function has to be normalized to have an L2 norm of 1. We thus need to measure to a precision of epsilon divided by the L2 norm of the classical data, and if that increases with N the scaling is even worse. For example, if all entries are of order unity, the L2 norm is N. The one technical flaw in the paper is in preparing the input state to the QFT, and the complexity of the classical simulation of the quantum algorithm. There, the numbers are wrong. The classical complexity can easily be estimated by realizing that any sparse quantum gate (such as a 1-qubit or 1-qubit gate) can be simulated in time O(N)=O(2n), where N=2n. Thus, the overhead is just O(N), not O(N2), and the simulation complexity is just O(N) times the complexity of the quantum algorithm. This will be more apparent if, as I propose below, the author shows the quantum algorithm also through a sequence of quantum gates. However, note that in the simulation we then have the full vector, and thus do not incur the overhead of quantum state tomography. We need to simulate the algorithm only once and not O(N/ϵ) times. This means the simulation remains at O(Nlog2N) even if we read out all entries, which is not bad compared to the classical O(NlogN) of the FFT. The reason why the author seems to end with a different complexity is that his Matlab simulation does not simulate the quantum algorithm but the computation of the effect of the QFT applied to just one basis state. That is a suboptimal implementation. I thus propose to replace the Matlab code by a discussion of how a quantum gate (Hadamard or controlled phase rotation) can be implemented, and that will then nicely give the scaling discussed above.As mentioned in my summary at the beginning, the value of the paper can be increased if the author could also discuss the complexity of quantum state preparation from classical input data, e.g., following Shende-Bullock-Markov \citep{markov2006}, which has a complexity of O(Nlogϵ1). That means, that if the data is read from a classical vector (and not computed as, e.g., in Shor's algorithm), then the complexity of state preparation of O(Nlogϵ1) completely dominates the QFT itself, and if one furthermore wants to read out the full vector instead of sampling it, the complexity becomes O(N2log2(ϵ1)).In the conclusion the author writes that "QEC research is still in very early development and it is currently difficult to determine how the required resources for these corrections could scale with qubit usage." That is incorrect, as the overhead is pretty well known by now for certain QEC codes, such as the surface code. The asymptotic scaling of the overhead has long been known, and also detailed resource costs have been worked out for various algorithms. I suggest to focus the QEC section on the need for fault tolerance, and the large overhead associated with it. This is definitely not something that can be done on NISQ devices at an interesting scale. I would also use another reference for QEC than the Gil Kalai paper.Now, to suggestions for improved presentation:In the abstract, I suggest to present fewer technical details. "radix-2 DIT case of the Cooley-Tukey Algorithm" can just be called Fast Fourier transform, the QUBIT4MATLAB package does not need to be mentioned in the abstract, and neither does the "Master Theorem" have to be mentioned.In the abstract and in the rest of the paper, it is also better to talk about quantum speedup and not quantum supremacy. The paper is about asymptotic scaling (the author uses the big-O notation throughout), and not about supremacy, which is a specific size problem that is solved better on quantum hardware rather than classical hardware. A supremacy claim needs all constants to be worked out, and assumptions for the specific classical and quantum hardware. Replacing quantum supremacy by quantum speedup can fix this. The observations are still valuable and correct.The first section is called "The shortcomings of classical computing and the early developments of quantum computing". That title sounds strange to me. What are the shortcomings? The author may rather want to stress the additional capabilities of quantum computers. Given that most readers will not be physicists and have no experience with quantum mechanics, it would be valuable to expand this first section. Discuss equation (3) before equation (1), and then introduce equation (1), to show that this means that the state of a qubit is actually a two-dimensional complex vector, After that only go to the many-quit state. After equation 4 it may be good to talk about amplitudes first, and then probabilities. Saying that the state is determined by "the qubit simply has a probability...", ignores the all important phases that are the crucial difference between quantum computing and probabilistic classical computing. "Quantum gates, unlike classical gates, do not delete information and are fully reversible." Needs further explanation. This comes from the reversibility of the microscopic laws of physics. Here it may be worthwhile to mention that, as quantum gates are time evolution of a quantum system, they are unitary operators on the Hilbert space that was introduced above. The discussion in the next sentence about them not using energy is misleading as well. In fact, the control and operation of the quantum gate always needs energy. It is only the free evolution of a quantum system that conserves energy. The part on " subatomic systems such as a superposition of quantum states. " would also need more explanation. Does the author here want to describe specific implementations of qubits? If so, more detail is needed. If not, I propose to drop this part."The definition and use of the classical discrete Fourier transform": here is an abrupt jump from Shor's algorithm to the DFT. Since Shor's algorithm again gets mentioned later on as a case where the QFT is better than a classical DFT, it make sense to discuss more details of Shor's algorithm, and that in essence it solvers the factor problem by using a QFT to find the period of a function. In light of the above complexity discussion this is important: we don't deal with classical data that one has to load but with a state that is computed (the function f(x) = a^x mod N ), and one is not interested in the full vector but just in finding the period. This can be done just by measuring the result of the QFT, which in this particular case will return a random multiple of the base frequency, and with a few repeated measurements one can then extract the period).The Fourier transform should be well known to the reader, and it is thus not obvious to me that equation 6 is needed. I would instead introduce \omega in equation 5. For the discussion of the implementation I suggest to mention that the rest of the paper limits itself to the case of powers of 2, i.e. N=2^n, for which the radix-2 version applies. Since the implementation is well known for most computational scientists, it could be moved from to supplementary material, as can be the derivation of the O(N log N) complexity, which is also well known.In the discussion of the QFT, unfortunately the quantum algorithm is not presented. What is presented is the mathematical unitary operation. It would be very valuable for the reader to see an actual implementation of the quantum algorithm, i.e. the sequence of O(n^2) Hadamard gates and controlled rotations – either as a circuit drawing, or more useful as quantum code, either using a quantum programming language that allows loops (such as Q#), or as some quantum pseudo code with loops. Then the reader will see the sequence of O(n^2) operations and how it creates the unitary operation of the QFT."The implementation of the QFT in MATLAB...". This section can be improved as well., As discussed above, the Matlab code does not implement the quantum algorithm but rather implements the application of the unitary operation to one of the basis states. It would be more useful for the reader to instead get implementations of the Hadamard and controlled phase gates, which could then be combined with the above quantum algorithm (maybe all in Matlab even?) for a full simulation of the algorithm – and in time O(N n^2), not O(N^2 n^2)."Evaluating the complexity of the QFT Implementation" – I discussed the issues with the complexity estimates already above. If one replaces it by an estimate of the complexity of simulating the H and controlled phase gate each in O(N) operations then it both becomes simpler and the complexity becomes just O(N) times that of the quantum version"Evaluating the complexity of the theoretical form of the QFT Implementation”. I assume that by "theoretical" the author means the actual quantum operation? The comparison section needs to be fixed, since that's where there is the mentioned flaw based on the suboptimal Matlab code, and that will also change table 1. I suggest that table 1 could have three cases: A) ignoring state preparation and one only wants to sample the output a constant number of times; B) ignoring state preparation but one wants to read the full vector; C) including state preparation from classical data and one only wants to sample the output a constant number of times; D) including state preparation from classical data and one wants to read the full vectorThe reader will then see that only case A has a speedup, and that in the other cases even the simulation of the quantum algorithm is faster than the quantum algorithm on a quantum computer.The conclusions should then be updated, taking these slightly modified comparisons into account. The case of Shor's algorithm, which falls into category A) above, is indeed interesting but deserves a deeper explanation so that the reader understands that Shor's algorithm can compute the input with complexity O(N3) (up to log factors), and why only a few samples of the output are needed. That way, the exponential speedup over the FFT remains, and the super polynomial speedup over the name sieve. However, if classical data has to be loaded or should be returned then there is no speedup. I call this the I/O problem of quantum algorithms, which indeed—as the author writes—means that in these cases quantum computing is no better than classical computing.This is an important observation that the paper beautifully works out. Fixing the flaw in the simulation section and improving the presentation will make this a very nice paper.
Introduction: Quantum computing has the potential to transform scientific computing by providing efficient algorithms for solving large-scale numerical problems that arise in engineering and materials science. However, integrating quantum algorithms into multiscale mechanics remains largely unexplored due to the complexity of coupling atomistic and continuum models and the nonlinear nature of molecular dynamics. In this work, we develop a unified hybrid classical–quantum framework for molecular dynamics and multiscale materials modeling based on the Variational Quantum Linear Solver (VQLS). Materials and methods: The proposed framework reformulates three representative mechanics problems as quantum-compatible linear systems: static atomistic–continuum coupling through reduced stiffness equations, dynamic atomistic–continuum coupling using a boundary-reduced Berry–Childs–Ostrander–Wang formulation, and nonlinear molecular dynamics through Carleman linearization. These systems are solved using shot-free quantum-circuit simulations within a common VQLS framework. Results: For the static problem with 1024 free displacement degrees of freedom, the VQLS solution achieves R 2 = 0.99784 and a relative L 2 error of 2.70 % . For the dynamic problem, the five-qubit restarted systems produce maximum final-time displacement and velocity differences of 1.39 × 10 − 4 nm and 6.74 × 10 − 5 nm/ps, respectively. For the nonlinear problem, comparison of the six-qubit VQLS trajectory with a high-accuracy integration of the full Lennard–Jones equations over 150 time steps gives maximum displacement and velocity differences of 4.09 × 10 − 4 nm and 2.87 × 10 − 3 nm/ps, respectively. The reported velocities are reconstructed from the displacement histories using a common backward-difference rule with the prescribed zero initial velocity. The numerical comparisons quantify agreement with classical reference trajectories and distinguish selected modeling, temporal-discretization, and variational-solution errors. Conclusions: By providing a unified quantum linear-system formulation for representative multiscale mechanics problems, this deterministic, shot-free, simulator-based study establishes a methodological proof of concept for integrating VQLS formulations with computational materials models, rather than demonstrating near-term hardware performance or scalable quantum computational advantage.
Quantum computing (QC) represents an emerging frontier with significant potential for transformative advancements in Computational Fluid Dynamics (CFD). This potential has been explicitly recognized by the CFD Vision 2030 report, emphasizing the strategic importance of exploring QC for future breakthroughs. Among quantum algorithms relevant to CFD, the Variational Quantum Linear Solver (VQLS) is particularly promising due to its suitability for solving linear and Hamiltonian-based problems, characteristic of fundamental CFD equations such as the pressure-Poisson equation. However, the practical implementation of VQLS for large-scale CFD problems faces critical scalability challenges, most notably the exponential increase in the number of quantum circuit evaluations and the barren plateau phenomenon, which severely hampers trainability and algorithm convergence. In this study, two novel strategies are introduced to address these challenges. First, we utilize a block encoding technique based on efficient oracles, thereby significantly reducing quantum circuit complexity and qubit overhead while accessing system matrix. Second, we incorporate a quantum-inspired evolutionary optimization (QIEO) approach, demonstrating enhanced training efficiency and effectively mitigating the barren plateau issue. These improvements are integrated within a Hybrid Quantum-Classical Method (HQCM) linear solver framework and demonstrated through solving a 2D steady-state heat equation, representative of the pressure-Poisson equations common in incompressible CFD simulations. The proposed methodologies yield promising results, showcasing improved circuit efficiency, enhanced stability during optimization, and accurate numerical solutions. This work lays a foundational step toward realizing practical, scalable quantum-assisted CFD solvers and sets clear pathways for future research aimed at further algorithmic and architectural enhancements.