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Advanced Quantum Technologies

A Computation‐Strengthened High‐Dimensional Three‐Qudit Toffoli Gate

Fang‐Fang Du, Xue‐Mei Ren, Qiulin Tan

Published Aug 20, 2024• 4 citations
Publisher

Abstract

Abstract A high‐dimensional quantum gate not only enables the processing of more information through parallel quantum channels but also enhances fault tolerance in a higher Hilbert space. In this paper, a protocol is presented for implementing a three‐qudit ‐Dimensional (D) Toffoli gate for a hybrid system, where the first control qudit, the second control qudit, and the target qudit of four dimension are encoded in the spatial‐polarization state of a flying photon, the electron‐spin state of the first two quantum dots (QDs), and the one of the remaining two QDs, respectively. Besides, the high‐dimensional Toffoli gate does not require any assistance. Moreover, the gate operates deterministically in principle, as the photon is easy to manipulate feasibly using simple optical elements, and four QDs have a long electron‐spin coherent time used for storage and manipulation. Furthermore, the success probability and fidelity of the high‐dimensional Toffoli gate, in alignment with current technological capabilities, demonstrate satisfactory results. This indicates that it is feasible in experimental settings and promises a quantum computing paradigm that excels in speed, error resilience, and scalability for intricate quantum operations.

DOI
10.1002/qute.202400313

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Abstract A high‐dimensional quantum gate not only enables the processing of more information through parallel quantum channels but also enhances fault tolerance in a higher Hilbert space. In this paper, a protocol is presented for implementing a three‐qudit ‐Dimensional (D) Toffoli gate for a hybrid system, where the first control qudit, the second control qudit, and the target qudit of four dimension are encoded in the spatial‐polarization state of a flying photon, the electron‐spin state of the first two quantum dots (QDs), and the one of the remaining two QDs, respectively. Besides, the high‐dimensional Toffoli gate does not require any assistance. Moreover, the gate operates deterministically in principle, as the photon is easy to manipulate feasibly using simple optical elements, and four QDs have a long electron‐spin coherent time used for storage and manipulation. Furthermore, the success probability and fidelity of the high‐dimensional Toffoli gate, in alignment with current technological capabilities, demonstrate satisfactory results. This indicates that it is feasible in experimental settings and promises a quantum computing paradigm that excels in speed, error resilience, and scalability for intricate quantum operations.
Abstract The high‐dimensional Toffoli gate functions in a broader Hilbert space, which empowers it to convey and handle greater amounts of information through parallel quantum pathways. In this study, a scheme of 3‐qudit ‐D Toffoli gate in the QD cavity‐coupling system with high fidelity is proposed. By modulating Purcell factor, the balanced condition is obtained so that the error induced by whether there is interaction between the QDs and cavities or not can be neglected. The scheme features a module circuit design, which not only ensures higher construction feasibility but also helps save resources. Meanwhile, it possesses technical scalability, enabling effective promotion to the application scenarios of n ‐qudit systems, which reduces the steps required to manipulate quantum states and makes it simpler to construct quantum computing circuits.
Abstract Qudit‐based quantum gates offer several advantages over qubit‐based counterparts, such as higher information density, the ability to address more complex problems, and richer quantum operations. In this study, three realistic protocols are presented for implementing a 44‐dimensional (16D) two‐qudit controlled‐SUM (CSUM) gate, where the 4D control qudit and 4D target qudit are encoded in the polarization degree of freedom (DoF) and spatial DoF of two photons, respectively. The first protocol is implemented exclusively using linear optical elements without auxiliary resources, making it feasible with current optical technologies and achieving an efficiency of 1/9. The second protocol utilizes photon scattering by a microcavity‐quantum‐dot system, enabling the 16D CSUM gate to operate deterministically without postselection. The third protocol introduces an error‐heralded mechanism based on the second protocol, theoretically achieving unity fidelity. Moreover, all protocols operate without ancillary photons, offering the advantages of compact circuits and low cost while further promoting the development of high‐dimensional quantum computation.
Abstract Qudit‐based quantum gates in high‐dimensional (HD) Hilbert space offer a viable route toward effectively accelerating the speed of quantum computing and performing complex quantum logic operations. In this study, an innovative 2‐qudit HD controlled‐SUM (CSUM) gate designed for four silicon‐vacancy spins is introduced, in which the first two electron‐spin states in silicon‐vacancy centers are encoded as the control qudit, and the other ones as the target qudit. The proposed protocol is implemented with assistance of an ancillary photon that serves as a common‐data bus linking four motionless silicon‐vacancy spins placed in four independent single‐sided optical nanocavities. Moreover, the CSUM gate works deterministically with the corresponding single‐qubit feed‐forward operation based on diverse outcomes of the single‐photon detectors to be directed against the ancillary photon. Further, it can be potentially generalized to other solid‐state quantum systems. Under current technological conditions, the 2‐qudit CSUM gate exhibits both high efficiency and fidelity.
High-dimensional quantum systems expand quantum channel capacity and information storage space. By implementing high-dimensional quantum logic gates, the speed of quantum computing can be practically enhanced. We propose a deterministic 4 × 4-dimensional controlled-not (CNOT) gate for a hybrid system without ancillary qudits required, where the spatial and polarization states of a single photon serve as a control qudit of four dimensions, whereas two electron-spin states in nitrogen-vacancy (NV) centers act as a four-dimensional target qudit. As the control qudits are easily operated employing simple optical elements and the target qudits are available for storage, the CNOT gate works in a deterministic way, and it can be flexibly extended to n × n-dimensional (n > 4) quantum gates for other hybrid systems or different photonic degrees of freedoms. The efficiency and fidelity of the CNOT gate are analyzed aligning with current technological capabilities, finding that they have satisfactory performances.
Abstract Enhancing the capabilities of quantum computing relies heavily on harnessing the power of qudit‐based high‐dimensional quantum gates. In the study, single‐qudit 4D , , and gates tailored for a two‐photon system in polarization states are presented. Furthermore, a two‐qudit ‐dimensional controlled‐not (CNOT) gate designed for a four‐photon system is introduced. These high‐dimensional gates can offer versatile and straightforward optical implementations, ensuring them to fulfill in a deterministic way. To facilitate these processes, an auxiliary system in the form of a ‐type atom residing in a cavity is employed. Remarkably, the auxiliary system retains its original state after the operation process ends, so it is not required to measure and plays a pivotal role in promoting effective interactions among distinct photons in its extended coherence time. Importantly, the in‐depth analysis of the fidelities and efficiencies of these quantum gates showcase remarkable outcomes, affirming the superiority of the proposed protocols. Therefore, these high‐dimensional gates not only amplify quantum parallelism, but also bolster the speed of quantum computations, fortify resilience against errors, and foster scalability for executing intricate quantum operations.
Abstract Quantum computation in high‐dimensional (HD) spaces holds significant promise for advancing a broad range of applications within quantum information technology (QIT), leading to processing more complex information and executing a wider array of quantum operations. In this study, a measurement‐free auxiliary cavity‐atom system is exploited to construct a three‐qudit photonic Toffoli gate in a ‐dimensional space. In addition, the auxiliary system maintains its initial state after a series of operations, eliminating the need for measurement and facilitating effective interactions between different photons by leveraging its extended coherence time. A comprehensive analysis of its fidelity and efficiency is performed, yielding impressive results that demonstrate the superiority and flexibility of this proposed protocol. Furthermore, the HD Toffoli gate lays the groundwork for realizing HD quantum computation for photon systems and significantly improves the efficiency of QIT.