Dynex technology presents a revolutionary tool for research and academia, offering robust computational capabilities that can dramatically accelerate scientific discovery and innovation. By facilitating complex simulations and data analysis at unprecedented speeds, Dynex enables researchers and academics to tackle large-scale problems that were previously too time-consuming or computationally demanding. This technology supports a broad range of disciplines, from physics and chemistry to social sciences and humanities, empowering scholars to explore new theories, test hypotheses, and achieve breakthroughs more efficiently than ever before. As such, Dynex is not just enhancing the pace of research but is also broadening the horizons of what can be explored and discovered in academic settings.

##### Grover Integer Factorisation on Dynex

Grover's algorithm is a quantum search algorithm that offers a quadratic speedup over classical search methods, making it one of the most significant algorithms in quantum computing. Traditionally, searching through an unsorted database of 𝑁 elements requires 𝑂(𝑁) time, as each element must be checked individually. However, Grover's algorithm can find the desired element in 𝑂(√𝑁) time, using the principles of quantum superposition and interference. The algorithm works by iteratively applying two main operations: the oracle, which marks the correct solution by flipping its phase, and the amplitude amplification, which increases the probability amplitude of the correct solution while decreasing that of the incorrect ones. These steps are repeated a number of times proportional to 𝑁, and a measurement at the end of the process will reveal the correct solution with high probability. This makes Grover's algorithm particularly powerful for applications like cryptography, where it can significantly reduce the time required to search through large keyspaces, posing a potential threat to classical encryption methods.

> Simple Grover Integer Factorisation Circuit

**Scientific background**: Grover, L.K. From Schrödinger’s equation to the quantum search algorithm. *Pramana - J Phys* **56**, 333–348 (2001). https://doi.org/10.1007/s12043-001-0128-3

##### Shor Integer Factorisation on Dynex

Shor's algorithm is a groundbreaking quantum algorithm that efficiently factors large integers, a problem that is classically hard to solve and forms the basis of many encryption schemes, such as RSA. The algorithm leverages quantum parallelism and the Quantum Fourier Transform (QFT) to find the period of a specific function related to the integer to be factored. This period is crucial for determining the factors of the integer. Shor's algorithm runs exponentially faster than the best-known classical algorithms, solving the factorization problem in polynomial time. This poses a significant threat to classical cryptographic systems, as it can break widely-used encryption methods by efficiently discovering prime factors of large numbers, something that would take classical computers an infeasible amount of time. Shor's algorithm consists of two main parts: classical pre- and post-processing, and a quantum phase estimation subroutine, which finds the order of a modular exponentiation function. Once the order is determined, the factors of the number can be computed using classical methods. Shor's algorithm is one of the most significant demonstrations of the potential power of quantum computing and has profound implications for the future of cryptography and information security.

> Simple Shor Integer Factorisation Circuit

**Scientific background**: Shor, P.W. (1994). "Algorithms for quantum computation: Discrete logarithms and factoring". *Proceedings 35th Annual Symposium on Foundations of Computer Science*. pp. 124–134.

##### Grover Quantum Search on Dynex

Grover’s search algorithm is a quantum algorithm designed to search an unsorted database or solve unstructured search problems with quadratic speedup compared to classical algorithms. In classical computing, finding a specific item in an unsorted list of N items requires O(N) time in the worst case. Grover’s algorithm, however, can accomplish this task in O(√N) time. The algorithm operates by initializing a superposition of all possible states and then iteratively amplifying the probability amplitude of the correct solution while diminishing the amplitudes of the incorrect ones. This process is achieved through the application of two main operations: the Oracle, which marks the correct solution, and the Diffusion operator, which amplifies the marked state’s probability. After approximately √N iterations, measuring the quantum state yields the correct solution with high probability. Grover’s algorithm exemplifies the power of quantum computing to solve specific problems more efficiently than classical counterparts, making it a cornerstone of quantum search techniques.

> Grover Quantum Search on the Dynex Platform

**Scientific background**: Grover, L.K. From Schrödinger’s equation to the quantum search algorithm. *Pramana - J Phys* **56**, 333–348 (2001). https://doi.org/10.1007/s12043-001-0128-3

##### Efficient Quantum State Tomography on Dynex

Quantum state tomography is a process used in quantum physics to characterize and reconstruct the quantum state of a system. In simple terms, it's like taking a snapshot of a quantum system to understand its properties fully. In quantum mechanics, a quantum state represents the complete description of a quantum system, including its position, momentum, energy, and other physical quantities. However, unlike classical systems where properties are well-defined, quantum systems often exist in superposition states, meaning they can simultaneously be in multiple states until measured. While traditional training methods perform rather poorly, Dynex computed training achieves near perfect fidelity.

> Quantum Mode-assisted unsupervised learning of Restricted Boltzmann Machines

**Scientific background**: Yuan-Hang Zhang. Efficient Quantum State Tomography with Mode-assisted Training. Physical Review A. 106. 10.1103/PhysRevA.106.042420.

##### Quantum Single Image Super-Resolution on the Dynex Platform

Implementation of a Quantum Single Image Super-Resolution algorithm to use on the Dynex platform. One of the well-known classical approaches for SISR relies on the well-established patch-wise sparse modeling of the problem. Yet, this field’s current state of affairs is that deep neural networks (DNNs) have demonstrated far superior results than traditional approaches. Nevertheless, quantum computing is expected to become increasingly prominent for machine learning problems soon. Among the two paradigms of quantum computing, namely universal gate quantum computing and adiabatic quantum computing (AQC), the latter has been successfully applied to practical computer vision problems, in which quantum parallelism has been exploited to solve combinatorial optimization efficiently. This algorithm demonstrates formulating quantum SISR as a sparse coding optimization problem, which is solved using the Dynex Neuromorphic Computing Platform via the Dynex SDK. This AQC-based algorithm is demonstrated to achieve improved SISR accuracy.

**Scientific background**: Choong HY, Kumar S, Van Gool L. Quantum Annealing for Single Image Super-Resolution. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition 2023 (pp. 1150-1159)

##### Quantum Job/Flow Shop Scheduling

Job shop scheduling (JSS) is an optimization challenge focused on scheduling jobs with varied processing orders on multiple machines, aiming to minimize the makespan or the completion time of the final task. Flow shop scheduling (FSS), a specialized form of JSS, requires every job to be processed in the same order across all machines. This demo showcases the application of Dynex's quantum computing technology to efficiently solve both JSS and FSS problems. By leveraging the power of quantum optimization, Dynex significantly reduces the makespan, demonstrating superior scheduling efficiency and operational performance. This capability is crucial for industries requiring precise and optimized scheduling of tasks across different machines and processes.

##### Quantum Workforce Scheduling

This advanced solution leverages the power of quantum algorithms to optimize the scheduling and allocation of workforce resources, ensuring maximum efficiency and productivity. By addressing complex scheduling challenges, such as shift assignments, task allocation, and resource management, Dynex's Workforce Scheduling application enhances operational performance, reduces costs, and improves employee satisfaction. This innovative use case is ideal for industries that require precise and dynamic workforce management, offering a robust and scalable solution to meet the demands of modern business operations.

##### Quantum Integer Factorization with n.Quantum Annealing

Identifying new methods for integer factorization plays an important role in modern information security. Shor’s algorithm is perhaps the most well-known method for integer factorization. An equally powerful model of quantum computing is the adiabatic quantum computing (AQC) model, which can also solve the integer factorization problem. In this example, we show how to convert an arbitrary integer factorization problem to an executable Ising model and tested it on the Dynex Neuromorphic Platform.

> Jupyter Notebook**Scientific background**: Jiang, S., Britt, K.A., McCaskey, A.J. *et al.* Quantum Annealing for Prime Factorization. *Sci Rep* **8**, 17667 (2018)

##### Quantum NP Problems

Explore our examples demonstrating how to solve NP problems using quantum algorithms, including number partitioning, vertex cover, and more. These examples showcase the power of our quantum computing platform in tackling complex computational challenges efficiently. Dive into these practical applications to understand how quantum algorithms can revolutionize problem-solving in various fields.

> Quantum n Queen Problem

> Quantum Binary Integer Linear Programming

> Quantum Graph Partitioning

> Quantum Job Sequencing

> Quantum Number Partitioning

> Quantum Set Cover

> Quantum Vertex Cover

> Quantum k-means Clustering

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