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International Journal of Physics

Qudits of Multi-Single Slits, Multi-Double Slits and Multi-Triple Slits --- Corresponding to Octal System, Decimal System, Duodecimal System, Hexadecimal System

Hui Peng, James L. Peng

Published Mar 24, 2024• 0 citations
Publisher

Abstract

One of tasks of quantum computing research is to reduce the components utilized for reaching certain computation power. To significantly reduce the components, we propose four kinds of Qudits of higher-dimension, multi-folds higher than the two-dimension of the qubits, which are respectively formed by: (1) multi-single-slits; (2) multi-double-slits (multi-Qubits); (3) multi-triple-slits (multi-Qutrits); and (4) combination of above. The further advantage is that the so formed Qudits may adopt Octal Number System, Decimal Number System, Duodecimal Number System, and Hexadecimal Number System, respectively.

DOI
10.12691/ijp-12-2-1

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One of tasks of quantum computing research is to reduce the components utilized for reaching certain computation power. To significantly reduce the components, we propose four kinds of Qudits of higher-dimension, multi-folds higher than the two-dimension of the qubits, which are respectively formed by: (1) multi-single-slits; (2) multi-double-slits (multi-Qubits); (3) multi-triple-slits (multi-Qutrits); and (4) combination of above. The further advantage is that the so formed Qudits may adopt Octal Number System, Decimal Number System, Duodecimal Number System, and Hexadecimal Number System, respectively.
Abstract Quantum computing leverages the power of superposition, entanglement, and interference to surpass classical systems in certain computational tasks. The first studies revolved around two-state quantum systems (qubits), which are a superposition of the |0⟩ and |1⟩ states. These qubits are physically realized using subatomic particles (e.g., electrons or photons) with the two states corresponding to different energy levels or polarization directions. However, some of these particles can exist in more than two discrete states. For example, an electron may occupy several energy levels, or a photon may have more than two polarization states. As a result, there has been an interest in qudits, quantum systems with d different states. A fascinating subcase of this is a qutrit, a three-level quantum system with states |0⟩, |1⟩, and |2⟩ as its basis. It is understood that qutrits with their larger state space can improve the storage and processing of information, and Qutrit-based quantum computing can be realized by defining suitable quantum gates to concatenate the three-level systems. Researchers have proposed qutrit gates, some of which are direct generalizations of qubit gates, such as the Pauli-X gate, and others are entirely new gate constructs. This paper summarizes recent advances in qutrit-based quantum computing, where the definitions of various qutrit gates are presented, and their significance in the development of qutrit-based quantum computing, particularly in implementing Deutsch and Shor’s algorithms, is explored.
As quantum computing progresses into the new Quantum Utility era—marked by the ca- pability to execute increasingly complex computations with improved fault tolerance—systems exceeding 1,000 qubits have already been demonstrated by Atom Computing and IBM. While the majority of academic and industrial efforts remain focused on qubit-based archi- tectures, built on decades of work in two-state computational logic and algorithms, growing attention is now turning to the untapped potential of qudits—quantum units that ex- tend beyond the binary nature of qubits. Research into qudits is gaining momentum, as increasing the number of computational states per unit can significantly enhance compu- tational power and reduce circuit complexity. This paradigm shift opens the door to more efficient quantum algorithms and hardware architectures. Moreover, hybrid systems that integrate qubits and qudits may provide a powerful bridge between existing technologies and next-generation performance—making qudit research a critical frontier in unlocking the full potential of quantum computing.This dissertation presents a comprehensive framework for implementing a hybrid quan- tum architecture using fluxonium-based qudits. Fluxonium circuits, constructed from a superconducting loop interrupted by a Josephson junction and a large inductance, exhibit strong anharmonicity and non-fixed selection rules. These features enable multiple direct transitions between quantum states, which can be precisely tuned through a set of circuit parameters known as fluxonium characteristic values. A detailed study of these parameters was conducted to identify optimal qudit systems within a ten-dimensional Hilbert space (D = 10), with estimated coherence times comparable to those of conventional qubit sys- tems. Theoretical and numerical tools were developed to extend the approach to higher dimensions. A set of universal qudit logic gates was designed for arbitrary dimensions D = 2^n, as well as some gates and gate combinations in high dimension Hilbert space. The study demonstrates a significant reduction in gate complexity—particularly in the number of driving pulses required for multiply-controlled gates. Additionally, hybrid logic circuits using D = 8 and D = 4 qudits were analyzed, revealing reduced circuit complexity com- pared to qubit-only implementations. In summary, this work shows that fluxonium superconducting circuits are well-suited systems extending the traditional two-level qubit into multi-level qudit operation, where accessing higher energy levels can enhance computational capacity, reduce gate overhead, and enable more efficient quantum algorithms compared to standard qubit architectures.
Quantum computing holds great promise for surpassing the limits of classical devices in many fields. Despite impressive developments, however, current research is primarily focused on qubits. At the same time, quantum hardware based on multi-level, qudit, systems offers a range of advantages, including expanded gate sets, higher information density, and improved computational efficiency, which might play a key role in overcoming not only the limitations of classical machines but also of current qubit-based quantum devices. However, working with qudits faces challenges not only in experimental control but particularly in algorithm development and quantum software. In this work, we introduce MQT Qudits, an open-source tool, which, as part of the Munich Quantum Toolkit (MQT), is built to assist in designing and implementing applications for mixed-dimensional qudit devices. We specify a standardized language for mixed-dimension systems and discuss circuit specification, compilation to hardware gate sets, efficient circuit simulation, and open challenges. MQT Qudits is available at github.com/cda-tum/mqt-qudits and on pypi at pypi.org/project/mqt.qudits.