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Optics Letters

Qudit-based high-dimensional controlled-not gate

Fang‐Fang Du, Xue-Mei Ren, Ming Ma, Gang Fan

Published Feb 7, 2024• 65 citations
Publisher

Abstract

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.

DOI
10.1364/ol.518336

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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.
Implementation of quantum logic gates with linear optical elements plays a prominent role in quantum computing due to their relatively easy manipulation and realization. We present efficient schemes to implement controlled-not (cnot) gates and controlled-swap (Fredkin) gates by using solely linear optics. We encode the control qubits and target qudits in photonic polarization (two-level) and spatial degrees of freedom (ddd level), respectively. On the basis of the hybrid encoding, cnot and Fredkin gates are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Remarkably, the number of linear optics required to implement a cnot gate has been reduced to one polarizing beam splitter, while only ddd polarizing beam splitters are necessary to implement a generalized Fredkin gate. The optical depths of all schemes are reduced to 1 and and are dimension independent. Besides, the fidelity of our three-qubit Fredkin gate is higher than 99.7% under realistic conditions, which is higher than in previous schemes.
Implementation of quantum logic gates with linear optical elements plays a prominent role in quantum computing due to the relatively easier manipulation and realization. We present efficient schemes to implement controlled-NOT (CNOT) gate and controlled-swap (Fredkin) gate by solely using linear optics. We encode the control qubits and target qudits in photonic polarization (two-level) and spatial degrees of freedom (ddd-level), respectively. Based on the hybrid encoding, CNOT and Fredkin gates are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Remarkably, the number of linear optics required to implement a CNOT gate has been reduced to one polarization beam splitter (PBS), while only ddd PBSs are necessary to implement a generalized Fredkin gate. The optical depths of all schemes are reduced to one and dimension-independent. Besides, the fidelity of our three-qubit Fredkin gate is higher than 99.7\% under realistic conditions, which is higher than the previous schemes.
Abstract In high-dimensional quantum systems, qudits offer a richer resource than traditional two-dimensional qubits, increasing the capacity of quantum channels and enhancing the efficiency of fault-tolerant quantum computation. These advantages can be utilized to solve complex problems across various fields. In the paper, we propose a 2-qudit controlled-NOT (CNOT) gate in a 4 × 4 -dimensional space and a 3-qudit controlled-controlled-NOT (Toffoli) gate in a 4 × 4 × 4 -dimensional space, both equipped with error-heralded units. Our designs do not require auxiliary photons or extra negatively charged nitrogen-vacancy (NV−) center, resulting in saving resources. Moreover, since the imperfect NV−-cavity interaction processes are predicted in real-time by sensitive single-photon detectors, both high-dimensional CNOT and Toffoli gates boast robust fidelities using existing technology. Furthermore, our protocols simplify circuits with error-heralded units, significantly contributing to the effectiveness of quantum information technology and paving the way for advanced high-dimensional quantum computing.
Implementing quantum logic gates with linear optical elements is a key approach in photon-based quantum information processing (QIP) due to its operational convenience and experimental flexibility. The adoption of high-dimensional (H-D) quantum computing with qudit encoding over conventional 2-D qubit encoding offers higher information encoding capability. In this paper, we present a linear-optics-based two-qudit 4×4-D (16-D) controlled-SUM (CSUM) gate for photon systems, where the polarization degrees of freedom (DoFs) of two photons encode the 4-D control qudit and the polarization and spatial DoFs of another single photon encode the 4-D target qudit. The 16-D CSUM gate demonstrates an efficiency of 1/243 and a theoretical average fidelity of 0.98395. Additionally, the H-D CSUM gate neither requires ancillary photons and pre-shared entanglement nor presents complex quantum circuits, thus significantly reducing the experimental overhead and enhancing its feasibility with current technology.
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.
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.