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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Hybrid Quantum-Classical Eigensolver without Variation or Parametric Gates

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H2 molecule.

Jouzdani, Pejman (ORCID:0000000256470381)↗
DOE OSTI2024

AXI4MLIR: User-Driven Automatic Host Code Generation for Custom AXI-Based Accelerators

Tensor algebra operations represent an important class of algorithms used across many applications, including machine learning, scientific computing, and data analytics. As a result, the efficient generation of custom accelerators for tensor operations has received increased attention. Previous efforts have produced automated tools enabling users to prototype and explore optimized accelerators. However, little effort has been focused on the host-accelerator interaction in these tools. Efficient use of hardware accelerators requires knowledge about the accelerator's capabilities (operations, data formats, and opcode support), the host CPU microarchitecture (e.g., memory hierarchy), the host-accelerator interface, and the application's features (which code regions should be mapped onto an accelerator). Manually rewriting the original applications to facilitate improved custom accelerator mapping is an error-prone and time-consuming endeavor. To cope with this, we propose AXI4MLIR, a new framework to automatically generate and optimize the communication between the host CPU and arbitrary accelerators that implement linear algebra algorithms. AXI4MLIR extends the MLIR compiler framework to automatically generate efficient host-accelerator driver code for accelerators with AXI-based interfaces. Our compiler extensions enable automatic driver code generation while carefully considering the host's memory hierarchy and target accelerator features. To demonstrate the flexibility and utility of AXI4MLIR, we test it with diverse use cases that include different types of accelerators, tiling scenarios, and dataflow schemes. We compare our experimental results to manual implementations of host-accelerator driver code and find that our approach can reduce CPU cache references by 56% and deliver up to a 1.65x speedup.

Bohm Agostini, Nicolas↗
DOE OSTI2025

Employing MACS/ViBRANT as a Surrogate MARVEL Reactor for Startup Reactivity Tuning and Supervisory Control Processes

Advanced nuclear reactors are a key part of the future of nuclear energy both in the United States and globally. They offer unique benefits for various energy-demanding applications, including use in remote locations, compact size, modular manufacturing, remote monitoring, low and/or variable power rating operation, and reliance on novel technologies to enhance operational safety. To achieve economic feasibility, advanced reactors must significantly reduce their workforces in comparison with the current fleet. Achieving this reduction will occur through reducing staff workloads using technology to achieve autonomous or semi-autonomous operations, demonstrated by comprehensive testing and validation activities. These operations will require both software and hardware platforms during the design and testing phases. While simulations are useful during the design phase, their performance can significantly deviate during actual deployment on hardware. This report presents the outcomes of a collaborative technical initiative between the U.S. Department of Energy (DOE) Microreactor Program (MRP) and Advanced Sensors and Instrumentation (ASI) Program. The collaboration utilized the Microreactor Automated Control System (MACS) hardware platform to bridge the gap between theoretical reactor design and actual startup and control operations. Two key use cases were investigated: facilitating the startup testing period and demonstrating supervisory control. The first use case details the key Microreactor Applications Research Validation and Evaluation (MARVEL) reactor startup physics testing activities conducted using the MACS platform. These activities included drum worth measurements, shutdown margin assessment, temperature feedback analysis, and scram time evaluation, as well as unique testing that would apply to the MARVEL reactor to demonstrate the testing methodologies in a low-risk environment. The MACS platform, serving as a surrogate representation of the MARVEL reactor, proved instrumental in performing these tests. The exercise revealed aspects that led to optimized processes, refined hardware design, and enhanced base software capabilities. By maturing methods and technologies in this manner, the initiative promises to reduce wasted time in the actual on-site reactor deployment effort, thereby saving significant time and resources. The second use case focuses on the development and implementation of supervisory control methods aimed at managing core tilt, which can result from asymmetrical operations or manufacturing imperfections in fuel rods or reactivity control devices. A key objective was to assess and compare the use of artificial intelligence (AI) for supervisory control. The effort aimed to define the role of supervisory control to enhance performance without risking control instability. This effort explored three distinct approaches: rules-based (RB) methods, optimization techniques, and reinforcement learning (RL) algorithms. Each approach was evaluated for its ease of implementation, its usability, and its effectiveness in responding to asymmetries in neutron flux. Comparative analysis of these approaches provided valuable insights into their applicability and effectiveness, offering a robust framework for advanced reactor operations. Together, these two use cases highlight the potential of hardware test beds to help streamline the design, operation, and control of advanced nuclear reactors. This collaborative effort underscores the importance of continued innovation and experimentation in achieving the next generation of safe, reliable, and economically viable nuclear energy solutions.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

Record acceleration of the two-dimensional Ising model using a high-performance wafer-scale engine

The versatility and wide-ranging applicability of the Ising model, originally introduced to study phase transitions in magnetic materials, have made it a cornerstone in statistical physics and a valuable tool for evaluating the performance of emerging computer hardware. Here, we present a novel implementation of the two-dimensional Ising model on Cerebras Wafer-Scale Engine (WSE) – a revolutionary processor that is opening new frontiers in computing. In our deployment of the checkerboard algorithm, we optimized the Ising model to take advantage of the unique WSE architecture. Specifically, we employed a compressed bit representation storing 16 spins on each int16 word, and efficiently distributed the spins over the processing units enabling seamless weak scaling and limiting communications to only immediate neighboring units. Our implementation can handle up to 754 simulations in parallel, achieving an aggregate of over 61.8 trillion flip attempts per second for Ising models with up to 200 million spins. This represents a gain of up to 148 times over previously reported single-devices with a highly optimized implementation on NVIDIA V100 and up to 88 times in productivity compared to NVIDIA H100. Our findings highlight the significant potential of the WSE in scientific computing, particularly in the field of materials modeling.

Ising model↗
DOE OSTI2022

Long-time simulations for fixed input states on quantum hardware

Publicly accessible quantum computers open the exciting possibility of experimental dynamical quantum simulations. While rapidly improving, current devices have short coherence times, restricting the viable circuit depth. Despite these limitations, we demonstrate long-time, high fidelity simulations on current hardware. Specifically, we simulate an XY-model spin chain on Rigetti and IBM quantum computers, maintaining a fidelity over 0.9 for 150 times longer than is possible using the iterated Trotter method. Our simulations use an algorithm we call fixed state Variational Fast Forwarding (fsVFF). Recent work has shown an approximate diagonalization of a short time evolution unitary allows a fixed-depth simulation. fsVFF substantially reduces the required resources by only diagonalizing the energy subspace spanned by the initial state, rather than over the total Hilbert space. We further demonstrate the viability of fsVFF through large numerical simulations, and provide an analysis of the noise resilience and scaling of simulation errors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Entity—Hardware-agnostic Particle-in-cell Code for Plasma Astrophysics. I. Curvilinear Special Relativistic Module

Entity is a new-generation, fully open-source particle-in-cell (PIC) code developed to overcome key limitations in astrophysical plasma modeling, particularly the extreme separation of scales and the performance challenges associated with evolving, GPU-centric computing infrastructures. It achieves hardware-agnostic performance portability across various GPU and CPU architectures using the Kokkos library. Crucially, Entity maintains a high standard for usability, clarity, and customizability, offering a robust and easy-to-use framework for developing new algorithms and grid geometries, which allows extensive control without requiring edits to the core source code. This paper details the core general-coordinate special relativistic module. Entity is the first PIC code designed to solve the Vlasov–Maxwell system in general coordinates, enabling a coordinate-agnostic framework that provides the foundational structure for straightforward extension to arbitrary coordinate geometries. The core methodology achieves numerical stability by solving particle equations of motion in the global orthonormal Cartesian basis, despite using generalized coordinates like Cartesian, axisymmetric spherical, and quasi-spherical grids. Charge conservation is ensured via a specialized current deposition technique using conformal currents. The code exhibits robust scalability and performance portability on major GPU platforms (AMD MI250X, NVIDIA A100, and Intel Max Series), with the 3D particle pusher and the current deposition operating efficiently at about 2 ns per particle per time step. Functionality is validated through a comprehensive suite of standard Cartesian plasma tests and the accurate modeling of relativistic magnetospheres in curvilinear axisymmetric geometries.

Hakobyan, Hayk [Flatiron Institute, New York, NY (↗
DOE OSTI2022

Calibrating the Classical Hardness of the Quantum Approximate Optimization Algorithm

The trading of fidelity for scale enables approximate classical simulators such as matrix product states (MPSs) to run quantum circuits beyond exact methods. A control parameter, the so-called bond dimension $\mathcal{χ}$ for MPSs, governs the allocated computational resources and the output fidelity. Here, we characterize the fidelity for the quantum approximate optimization algorithm by the expectation value of the cost function that it seeks to minimize and find that it follows a scaling law $\mathscr{F}$(ln $\mathcal{χ}$/N), where N is the number of qubits. With ln $\mathcal{χ}$ amounting to the entanglement that a MPS can encode, we show that the relevant variable for investigating the fidelity is the entanglement per qubit. Importantly, our results calibrate the classical computational power required to achieve the desired fidelity and benchmark the performance of quantum hardware in a realistic setup. For instance, we quantify the hardness of performing better classically than a noisy superconducting quantum processor by readily matching its output to the scaling function. Moreover, we relate the global fidelity to that of individual operations and establish its relationship with $\mathcal{χ}$ and N. We sharpen the requirements for noisy quantum computers to outperform classical techniques at running a quantum optimization algorithm in speed, size, and fidelity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Cyber-Physical Smart Light Control System Integration with Smart Grid using Zigbee

This paper presents a hardware-in-the-loop cyber-physical system architecture design to monitor and control smart lights connected to the active distribution grid. The architecture uses Zigbee-based (IEEE 802.15.4) wireless sensor networks and publish-subscribe architecture to exchange monitoring and control signals between smart-light actuators (SLAs) and a smart-light central controller (SLCC). Each SLA integrated into a smart light consists of a Zigbee-based endpoint module to send and receive signals to and from the SLCC. The SLCC consists of a Zigbee-based coordinator module, which further exchanges the monitoring and control signals with the active distribution management system over the TCP/IP communication network. The monitoring signals from the SLAs include light status, brightness level, voltage, current, and power data, whereas, the control signals to the SLAs include light intensity, turn ON, turn OFF, standby, and default settings. We have used our existing hardware-in-the-loop (HIL) cyber-physical system (CPS) security SCADA testbed to process signals received from the SLCC and respond suitable control signals based on the smart light schedule requirements, system operation, and active distribution grid dynamic characteristics. We have integrated the proposed cyber-physical smart light control system (CPSLCS) testbed to our existing HIL CPS SCADA testbed. We use the integrated testbed to demonstrate the efficacy of the proposed algorithm by real-time performance and latency between the SLCC and SLAs. The experiments demonstrated significant results by 100% realtime performance and low latency while exchanging data between the SLCC and SLAs.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

Classification of dynamical Lie algebras generated by spin interactions on undirected graphs

Dynamical Lie algebras (DLAs) are a versatile tool for various topics that span from the expressibility-trainability of variational quantum algorithms (VQAs), to simulation of many body Hamiltonians. Quantum gates and most of the Hamiltonians of interest consist of local interactions; therefore, the analysis of all possible DLAs generated by 1- and 2-local operators is crucial for quantum simulation and VQAs on current hardware. Previously in [R. Wiersema et al ., npj Quantum Inf. 10 , 110 (2024)], we analyzed the DLAs on linear, circular and all-to-all topologies, and obtained results about their dimensions and algebraic structure. Here, in this work, we extend our analysis into any possible hardware topology and provide a classification of all DLAs generated by Pauli strings on any undirected interaction graph. Our results indicate that the DLAs depend solely on whether the connectivity or interaction graph is bipartite or not. In addition, we find that the non-trivial polynomially scaling DLAs appear only on 1D line or circle topologies, and all other DLAs have dimensions scaling exponentially with the system size. Together with the current VQA literature, our results imply that either the majority of VQAs are non-trainable, or we are yet to understand the role of DLAs on the trainability of VQAs.

Algebraic structures↗
DOE OSTI2025

Autonomous organic synthesis for redox flow batteries via flexible batch Bayesian optimization

Traditional trial-and-error methods for materials discovery are inefficient to meet the urgent demands posed by the rapid progression of climate change. This urgency has driven the increasing interest in integrating robotics and machine learning into materials research to accelerate experimental learning. However, idealized decision-making frameworks to achieve maximum sampling efficiency are not always compatible with high-throughput experimental workflows inside a laboratory. For multi-step chemical processes, differences in hardware capacities can complicate the digital framework by introducing constraints on the maximum number of samples in each step of the experiment, hence causing varying batch sizes in variable selection within the same batch. Therefore, designing flexible sampling algorithms is necessary to accommodate the multi-step synthesis with practical constraints unique to each high-throughput workflow. In this work, we designed and employed three strategies on a high-throughput robotic platform to optimize the sulfonation reaction of redox-active molecules used in flow batteries. Our strategies adapt to the multi-step experimental workflow, where their formulation and heating steps are separate, causing varying batch size requirements. By strategically sampling using clustering and mixed-variable batch Bayesian optimization, we were able to iteratively identify optimal conditions that maximize the yields. Our work presents a flexible approach that allows tailoring the machine learning decision-making to suit the practical constraints in individual high-throughput experimental platforms, followed by performing resource-efficient yield optimization using available open-source Python libraries.

Tamura, Clara [Univ. of Washington, Seattle, WA (U↗
DOE OSTI2020

NISQ+: Boosting quantum computing power by approximating quantum error correction

Quantum computers are growing in size, and design decisions are being made now that attempt to squeeze more computation out of these machines. In this spirit, we design a method to boost the computational power of near-term quantum computers by adapting protocols used in quantum error correction to implement "Approximate Quantum Error Correction (AQEC)." By approximating fully-fledged error correction mechanisms, we can increase the compute volume (qubits × gates, or "Simple Quantum Volume (SQV)") of near-term machines. The crux of our design is a fast hardware decoder that can approximately decode detected error syndromes rapidly. Specifically, we demonstrate a proof-of-concept that approximate error decoding can be accomplished online in near-term quantum systems by designing and implementing a novel algorithm in Single-Flux Quantum (SFQ) superconducting logic technology. This avoids a critical decoding backlog, hidden in all offline decoding schemes, that leads to idle time exponential in the number of T gates in a program. Our design utilizes one SFQ processing module per physical qubit. Employing state-of-the-art SFQ synthesis tools, we show that the circuit area, power, and latency are within the constraints of contemporary quantum system designs. Under pure dephasing error models, the proposed accelerator and AQEC solution is able to expand SQV by factors between 3,402 and 11,163 on expected near-term machines. The decoder achieves a 5% accuracy-threshold and pseudo-thresholds of ~ 5%,4.75%,4.5%, and 3.5% physical error-rates for code distances 3,5,7, and 9. Decoding solutions are achieved in a maximum of ~20 nanoseconds on the largest code distances studied. By avoiding the exponential idle time in offline decoders, we achieve a 10x reduction in required code distances to achieve the same logical performance as alternative designs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Quantum Approximate Optimization Algorithm on Different Qubit Systems

Solving optimization problems is critical across many research domains, but the high dimensionality of parameter spaces often poses significant challenges. The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for accelerating optimization in the Noisy Intermediate-Scale Quantum (NISQ) era by leveraging both classical and quantum computational resources. However, its performance can vary depending on the underlying quantum hardware architecture. In this work, we evaluate the performance of QAOA on different quantum hardware platforms, specifically, superconducting transmon qubits and trapped-ion qubits, targetting real-world optimization problems formulated as fully connected Quadratic Unconstrained Binary Optimization (QUBO) instances. We evaluate both the solution quality and time-to-solution using dense QUBO matrices. Furthermore, we show that large-scale problems, such as a 100-bit QUBO instance, can be effectively tackled by integrating quantum computing with high-performance computing (HPC) resources. This study provides practical insights into the strengths and limitations of different qubit technologies and advances the application of quantum computing in solving real-world optimization problems.

Kim, Seongmin [ORNL] (ORCID:0000000159063004)↗
DOE OSTI2021

Port and optimize the CEED software stack to Aurora/Frontier EA (ECP Milestone Report)

The goal of this milestone was to port the CEED software stack, including Nek, MFEM and libCEED to the Frontier and Aurora early access hardware, work on optimizing the performance on AMD and Intel GPUs, and demonstrate impact in CEED-enabled ECP applications. As part of this milestone, we also performed a number of other activities, including continued optimizations for CEED applications at scale on Summit, performance evaluation on non-ECP hardware, including the Fugaku’s A64FX chip and the NVIDIA A100 GPU architecture, exploring the use of just-in-time compilations in applications at scale and potential of mixed precision optimizations in the CEED discretization and solver algorithms, and more. During the milestone period, the CEED team released new versions of 6 of its packages, including major releases of MFEM, NekRS and OCCA. We also organized the fifth CEED Annual meeting (CEED5AM) which included nearly 100 researchers from national labs, universities and industry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Promise of Graph Sparsification and Decomposition for Noise Reduction in QAOA: Analysis for Trapped-Ion Compilations

We develop new approximate compilation schemes that significantly reduce the expense of compiling the Quantum Approximate Optimization Algorithm (QAOA) for solving the Max-Cut problem. Our main focus is on compilation with trapped-ion simulators using Pauli-X operations and all-to-all Ising Hamiltonian HIsing evolution generated by Molmer-Sorensen or optical dipole force interactions, though some of our results also apply to standard gate-based compilations. Our results are based on principles of graph sparsification and decomposition; the former reduces the number of edges in a graph while maintaining its cut structure, while the latter breaks a weighted graph into a small number of unweighted graphs. Though these techniques have been used as heuristics in various hybrid quantum algorithms, there have been no guarantees on their performance, to the best of our knowledge. This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity. For quantum hardware that uses edge-by-edge QAOA compilations, sparsification leads to a direct reduction in circuit complexity. For trapped-ion quantum simulators implementing all-to-all HIsing pulses, we show that for a (1−ϵ) factor loss in the Max-Cut approximation (ϵ>0), our compilations improve the (worst-case) number of HIsing pulses from O(n2) to O(nlog(n/ϵ)) and the (worst-case) number of Pauli-X bit flips from O(n2) to O(nlog(n/ϵ)ϵ2) for n-node graphs. This is an asymptotic improvement for any constant ϵ>0. We demonstrate that significant improvements to the approximation ratio are obtained using decomposition in simulated trapped-ion experiments with dephasing noise. We further present a generic argument showing that sparsification results in an exponentially improved circuit fidelity lower bound in digital computing schemes based on one- and two-qubit gates, which are relevant to a wide variety of hardwares such as superconducting qubits and certain neutral atom or trapped ion setups, and more sophisticated noise models. We anticipate these approximate compilation techniques will be useful tools in a variety of future quantum computing experiments.

Moondra, Jai [Georgia Institute of Technology]↗
DOE OSTI2024

Graph Neural Networks for Parameterized Quantum Circuits Expressibility Estimation (Rev.1)

Parameterized quantum circuits (PQCs) are fundamental to quantum machine learning (QML), quantum optimization, and variational quantum algorithms (VQAs). The expressibility of PQCs is a measure that determines their capability to harness the full potential of the quantum state space. It is thus a crucial guidepost to know when selecting a particular PQC ansatz. However, the existing technique for expressibility computation through statistical estimation requires a large number of samples, which poses significant challenges due to time and computational resource constraints. This paper introduces a novel approach for expressibility estimation of PQCs using Graph Neural Networks (GNNs). We demonstrate the predictive power of our GNN model with a dataset consisting of 25,000 samples from the noiseless IBM QASM Simulator and 12,000 samples from three distinct noisy quantum backends. The model accurately estimates expressibility, with root mean square errors (RMSE) of 0.05 and 0.06 for the noiseless and noisy backends, respectively. We compare our model’s predictions with reference circuits from Sim et al. and IBM Qiskit’s hardwareefficient ansatz sets to further evaluate our model’s performance. Our experimental evaluation in noiseless and noisy scenarios reveals a close alignment with ground truth expressibility values, highlighting the model’s efficacy. Moreover, our model exhibits promising extrapolation capabilities, predicting expressibility values with low RMSE for out-of-range qubit circuits trained solely on only up to 5-qubit circuit sets. This work thus provides a reliable means of efficiently evaluating the expressibility of diverse PQCs on noiseless simulators and hardware.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Lossy compression of statistical data using quantum annealer

Abstract We present a new lossy compression algorithm for statistical floating-point data through a representation learning with binary variables. The algorithm finds a set of basis vectors and their binary coefficients that precisely reconstruct the original data. The optimization for the basis vectors is performed classically, while binary coefficients are retrieved through both simulated and quantum annealing for comparison. A bias correction procedure is also presented to estimate and eliminate the error and bias introduced from the inexact reconstruction of the lossy compression for statistical data analyses. The compression algorithm is demonstrated on two different datasets of lattice quantum chromodynamics simulations. The results obtained using simulated annealing show 3–3.5 times better compression performance than the algorithm based on neural-network autoencoder. Calculations using quantum annealing also show promising results, but performance is limited by the integrated control error of the quantum processing unit, which yields large uncertainties in the biases and coupling parameters. Hardware comparison is further studied between the previous generation D-Wave 2000Q and the current D-Wave Advantage system. Our study shows that the Advantage system is more likely to obtain low-energy solutions for the problems than the 2000Q.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Qu8its for quantum simulations of lattice quantum chromodynamics

We explore the utility of d = 8 qudits, qu8its, for quantum simulations of the dynamics of 1+1⁢D SU(3) lattice quantum chromodynamics, including a mapping for arbitrary number of flavors and lattice size and a reorganization of the Hamiltonian for efficient time evolution. Recent advances in parallel gate applications, along with the shorter application times of single-qudit operations compared with two-qudit operations, lead to significant projected advantages in quantum simulation fidelities and circuit depths using qu8its rather than qubits. The number of two-qudit entangling gates required for time evolution using qu8its is found to be more than a factor of 5 fewer than for qubits. Here, we anticipate that the developments presented in this work will enable improved quantum simulations to be performed using emerging quantum hardware.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Report for the ASCR Workshop on Basic Research Needs in Quantum Computing and Networking - 2023

Employing quantum mechanical resources in computing and networking opens the door to new computation and communication models and potential disruptive advantages over classical counterparts. However, quantifying and realizing such advantages face extensive scientific and engineering challenges. Investments by the Department of Energy (DOE) have driven progress toward addressing such challenges. Quantum algorithms have been recently developed, in some cases offering asymptotic exponential advantages in speed or accuracy, for fundamental scientific problems such as simulating physical systems, solving systems of linear equations, or solving differential equations. Empirical demonstrations on nascent quantum hardware suggest better performance than classical analogs on specialized computational tasks favorable to the quantum computing systems. However, demonstration of an end-to-end, substantial and rigorously quantifiable quantum performance advantage over classical analogs remains a grand challenge, especially for problems of practical value. The definition of requirements for quantum technologies to exhibit scalable, rigorous, and transformative performance advantages for practical applications also remains an outstanding open question, namely, what will be required to ultimately demonstrate practical quantum advantage?

97 MATHEMATICS AND COMPUTING↗