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EPJ Quantum Technology

Robust-fidelity high-dimensional controlled-controlled-SUM gate with qudits

Fang‐Fang Du, Xin‐Shan Du, Zhuoya Bai, Qiang Zhu, Qiulin Tan

Published Aug 6, 2025• 0 citations
Publisher

Abstract

Abstract High-dimensional (HD) quantum systems are capable of processing more complex information and performing a wider array of quantum operations, in contrast to low-dimensional (LD) quantum systems, thereby improving the speed and fault tolerance of quantum computing. In the study, we propose a deterministic qudit-encoded $4\times 4\times 4$ 4 × 4 × 4 -dimensional (64D) controlled-controlled-SUM (CCSUM) gate based on weak Kerr effect. The 64D CCSUM gate leverages the polarization and spatial degrees of freedom (DoFs) of three photons to encode 4D control and target qudits, thereby optimizing quantum resources and reducing costs. Moreover, the HD CCSUM gate functionality is implemented in a deterministic way by employing the X-Homodyne detector to measure coherent states, combined with related classical feed-forward operations. Through detailed analyses, the proposed HD CCSUM gate, under current technological conditions, exhibits robust fidelity and feasibility, offering a promising path toward the realization of HD quantum computing.

DOI
10.1140/epjqt/s40507-025-00398-3

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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.
High-dimensional quantum gates not only extend the existing quantum computing framework but also serve as a vital component in a range of quantum technologies. In the study, a 4 × 4-dimensional controlled-NOT (CNOT) gate is presented based on the assistance of a weak cross-Kerr medium, utilizing only two degrees of freedom (DoFs) with two photons. Specifically, the control qudits are encoded in the photonic polarization DoF, while the target qudits are encoded in the path DoF of the two photons. Notably, the circuit is constructed using only two times of the Kerr medium, outperforming a previous design that required six times, which simplifies the entire circuits. Furthermore, the fidelity and success probability of the proposed protocol are near 1 when the influence factors take the appropriate value.
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 Remote implementation of high-dimensional quantum operation is pivotal for advancing quantum communication and distributed computing. Here, we propose a deterministic protocol capable of performing arbitrary four-dimensional unitary operations between two spatially separated nodes. The scheme utilizes hybrid encoding in both polarization and spatial degrees of freedom, which not only conserves quantum resources but also leverages the scalability of high-dimensional photonic systems. A key feature of the protocol is the use of weak cross-Kerr nonlinearities combined with X -quadrature homodyne detection. This design significantly reduces the reliance on strong nonlinear interactions and enhances robustness against photon loss. By performing classical feed-forward operations conditioned on coherent-state measurement outcomes, our approach ensures deterministic execution and high operational fidelity. Importantly, the protocol avoids the need for complex ancillary entanglement by relying solely on linear optics and simplified measurements. Due to the use of mature optical components and a loss-resilient architecture, numerical simulations under realistic experimental conditions indicate a high success probability (≈1) of this scheme. These advantages highlight the practicality and resource efficiency of our method, making it well-suited for the implementation of high-dimensional quantum communication protocols with current photonic technologies.
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.