On-chip path encoded photonic quantum Toffoli gate

Fig.1 The circuit and experimental results of the path encoded three-qubit photonic quantum Toffoli gate fabricated by the FLDW technique. (a) The optimized 3D quantum circuit of the Toffoli gate with an overpass waveguide. (b) Micrograph of the coupling region of the directional coupler, whose interaction distance is 8 μm. (c) Micrograph of the overpass waveguide. (d) The experimental reconstructed truth table with the fidelity of 85.5%.


Fig.2 The circuit and experimental results of the path encoded four-qubit photonic quantum CCCNOT gate fabricated by the FLDW technique. (a) The optimized 3D quantum circuit of the CCCNOT gate with two overpass waveguides. (b) Theoretical output power distribution for each single port input. (c) Experimental output power distribution. The similarity of the experimental and theoretical values is 99.2%.

Femtosecond laser direct writing (FLDW) provides an important technique for the fabrication of integrated photonic quantum chips, but the produced universal photonic quantum computation chips are still limited to two-qubit logic gates, such as the most commonly used controlled NOT (CNOT) gates encoded by path or polarization. Multiqubit logic gates can be decomposed into the combination of a series of single-qubit and two-qubit logic gates in principle, but the circuits become much more complicated to write.

As for the most important three-qubit Toffoli gate, its decomposition according to the traditional quantum circuit generally requires six CNOT gates plus ten various single-qubit gates. Not only its circuit is very complex, but also its success probability is extremely low, even the heralded scheme using six entangled photon pairs just reaches (1/4)⁶=1/4096.

To address this problem, the research group led by Prof. Yan Li's from the School of Physics at Peking University, China, cooperating with Prof. Xi-Feng Ren's Group from the CAS Key Laboratory of Quantum Information at University of Science and Technology of China, using 3D configuration to optimize the quantum circuit, have designed and demonstrated a path encoded photonic quantum Toffoli gate chip based on FLDW technique for the first time.

The research results are published in Photonics Research, Volume. 10, Issue 7, 2022 (Meng Li, Chu Li, Yang Chen, Lan-Tian Feng, Linyu Yan, Qian Zhang, Jueming Bao, Bi-Heng Liu, Xi-Feng Ren, Jianwei Wang, Shufeng Wang, Yunan Gao, Xiaoyong Hu, Qihuang Gong, and Yan Li. On-chip path encoded photonic quantum Toffoli gate. Photonics Research, 2022, 10 (7): 1533-1542).

To reduce the circuit complexity, they utilize the probabilistic post-selected three-qubit controlled-controlled phase (CCZ) gate as the core to construct the Toffoli gate. By selecting path encoding and changing the target from the traditional two-level "qubit" into the three-level "qutrit", the circuit requires only two cascaded controlled phase (CZ) gates and three signal photons without auxiliary photon, and the success probability increases to 1/72.

Compared with the polarization encoded logic gate, it can also significantly shorten the chip length. Above all, it can fully take the advantages of the true 3D direct writing ability of FLDW to further upgrade the 2D circuit to a 3D one with more succinct configuration with only 10 directional couplers (Fig.1).

The key is to introduce a depth-varying 3D overpass waveguide to eliminate all undesired crossing and coupling with other waveguides, which are hard to avoid, so as to greatly improve the performance of the chip. What's more, the quantum interference in the chip happens no more than one time, instead of twice in the polarization encoded quantum Toffoli gate using bulk optical elements, so that it can achieve a relatively higher true-table fidelity, whose measured value is 85.5%.

In addition, they also demonstrate the potential in the fabrication of the path encoded four-qubit controlled-controlled-controlled NOT (CCCNOT) gate with two overpass waveguides, which further highlights the advantage of 3D quantum circuits (Fig.2).

Prof. Yan Li believes that by using the true 3D capability of FLDW and combining the 3D path with multiple degrees of freedom of photons such as polarization and orbital angular momentum, it is expected to further simplify the quantum circuit structures, reduce the photon resource requirements, and realize more complex and powerful 3D photonic quantum computation chips.

飞秒激光直写三维多比特光量子计算芯片

图1 飞秒激光直写制备的路径编码三比特光量子Toffoli门线路图和实验表征结果。(a)优化后的Toffoli门三维量子线路,含有三维立交桥状波导；（b）定向耦合器的耦合区显微图，耦合间距为8μm；（c）立交桥状波导的显微图；（d）实验重构真值表，保真度为85.5%


图2 飞秒激光直写制备的路径编码四比特光量子CCCNOT门线路图和实验表征结果。(a)优化后的CCCNOT门三维量子线路，含有2处三维立交桥状波导；（b）单端口注入激光情况下，芯片各端口理论输出功率分布；（c）实验测量的输出功率分布，实验值与理论值的相似度为99.2%

飞秒激光直写技术为制备集成光量子芯片提供了重要手段，而加工的通用光量子计算芯片还仅限于两量子比特门，比如路径或偏振编码的最常用的受控非（CNOT）门。 多量子比特逻辑门原则上可以分解为一系列单量子比特和两量子比特逻辑门的组合，但复杂度大大增加，制备极其困难。

最重要的三比特Toffoli门按照传统量子线路分解，一般需要6个两比特CNOT门和10个不同类型的单比特门，不仅线路复杂，而且操作成功率很低。即使利用纠缠光子预报方案提高成功率，使用6对纠缠光子的操作，成功率也仅为 (1/4)⁶=1/4096。

为解决以上问题，北京大学物理学院李焱教授和中国科学技术大学任希锋教授团队合作，利用三维构型优化量子线路，设计并首次展示了基于飞秒激光直写技术制备的路径编码光量子Toffoli门芯片。相关研究成果发表于Photonics Research 2022年第7期（Meng Li, Chu Li, Yang Chen, Lan-Tian Feng, Linyu Yan, Qian Zhang, Jueming Bao, Bi-Heng Liu, Xi-Feng Ren, Jianwei Wang, Shufeng Wang, Yunan Gao, Xiaoyong Hu, Qihuang Gong, and Yan Li. On-chip path encoded photonic quantum Toffoli gate. Photonics Research, 2022, 10 (7): 1533-1542）。

该团队以概率性的后选择三比特受控-受控相位（CCZ）门为核心构成Toffoli门，采用路径编码并将目标比特由传统的二能级的“qubit”更换为三能级的“qutrit”，仅需要2个级联的受控相位（CZ）门和3个信号光子，无需附属光子，并且操作成功率也提高至1/72。

与偏振编码的逻辑门相比，路径编码方式还可以显著缩短芯片长度。尤其重要的是，充分发挥飞秒激光制备的真三维直写能力，将二维量子线路进一步优化为更简洁的三维构型，所需要的定向耦合器（DCs）的数量减少到仅为10个（图1）。

最关键的是，该团队通过引入深度变化的三维立交桥状波导，消除了与其他波导间很难避免的交叉和耦合，大大提升了芯片的性能。同时，所制备的Toffoli门芯片中量子干涉最多仅需发生1次，而不是采用体光学元件实现的偏振编码的逻辑门中的2次，因此能够获得更高的真值表保真度，实测值为85.5%。

此外，他们还展示了利用两个立交桥状波导方案制备路径编码四量子比特受控-受控-受控非 (CCCNOT)门的潜力，进一步凸显了三维量子线路的优越性（图2）。

李焱教授表示：“利用飞秒激光制备的真三维直写能力，将三维路径和偏振、轨道角动量等多个自由度结合，有望进一步简化量子线路，降低光子资源需求，实现线路更复杂、功能更强大的三维光量子计算芯片。”