This paper studies the Graph Isomorphism Problem from a variational algorithmic perspective, specifically studying the Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Graph Isomorphism Problem and subsequent execution using the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE). This study presents the results of these algorithms and the variations that occur therein for graphs of four and five nodes. The main findings of this paper include the clustering in the energy landscape for the QAOA in isomorphic graphs having an equal number of nodes and edges. This trend found in the QAOA study was further reinforced by studying the ground state energy reduction using VQEs. Furthermore, this paper examines the trend under which isomorphic pairs of graphs vary in the ground state energies, with varying edges and nodes.
In recent years, numerous research advancements have extended the limit of classical simulation of quantum algorithms. Although, most of the state‐of‐the‐art classical simulators are only limited to binary quantum systems, which restrict the classical simulation of higher‐dimensional quantum computing systems. Through recent developments in higher‐dimensional quantum computing systems, it is realised that implementing qudits improves the overall performance of a quantum algorithm by increasing memory space and reducing the asymptotic complexity of a quantum circuit. Hence, in this article, QuDiet, a state‐of‐the‐art user‐friendly python‐based higher‐dimensional quantum computing simulator is introduced. QuDiet offers multi‐valued logic operations by utilising generalised quantum gates with an abstraction so that any naive user can simulate qudit systems with ease as compared to the existing ones. Various benchmark quantum circuits is simulated in QuDiet and show the considerable speedup in simulation time as compared to the other simulators without loss in precision. Finally, QuDiet provides a full qubit‐qudit hybrid quantum simulator package with quantum circuit templates of well‐known quantum algorithms for fast prototyping and simulation. Comprehensive simulation up to 20 qutrits circuit on depth 80 on QuDiet was successfully achieved. The complete code and packages of QuDiet is available at https://github.com/LegacYFTw/QuDiet.
In some quantum algorithms, arithmetic operations are of utmost importance for resource estimation. In binary quantum systems, some efficient implementation of arithmetic operations like, addition/subtraction, multiplication/division, square root, exponential and arcsine etc. have been realized, where resources are reported as a number of Toffoli gates or T gates with ancilla.Recently it has been demonstrated that intermediate qutrits can be used in place of ancilla, allowing us to operate efficiently in the ancilla-free frontier zone. In this article, we have incorporated intermediate qutrit approach to realize efficient implementation of all the quantum arithmetic operations mentioned above with respect to gate count and circuit-depth without T gate and ancilla. Our resource estimates with intermediate qutrits could guide future research aimed at lowering costs considering arithmetic operations for computational problems. As an application of computational problems, related to finance, are poised to reap the benefit of quantum computers, in which quantum arithmetic circuits are going to play an important role. In particular, quantum arithmetic circuits of arcsine and square root are necessary for path loading using the re-parameterization method, as well as the payoff calculation for derivative pricing. Hence, the improvements are studied in the context of the core arithmetic circuits as well as the complete application of derivative pricing. Since our intermediate qutrit approach requires to access higher energy levels, making the design prone to errors, nevertheless, we show that the percentage decrease in the probability of error is significant owing to the fact that we achieve circuit robustness compared to qubit-only works.
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