Linear optical quantum computation (LOQC) offers a promising platform for quantum information processing, but its scalability is fundamentally constrained by the probabilistic nature of nonlocal entangling gates. Qudit circuit compression schemes mitigate this issue by encoding multiple qubits onto qudits. However, these schemes become inefficient when only a subset of the encoded qubits is required to participate in the nonlocal entangling gate, leading to an exponential increase in the number of nonlocal gates. In this paper, we address this bottleneck by demonstrating the existence of multilevel control-$Z$ (CZ) operations for qudits encoded in multiple spatial modes in LOQC. Unlike conventional two-level CZ gates, which act only on a single pair of modes, multilevel CZ gates impart a conditional phase shift for an arbitrarily chosen subset of the spatial modes. We present two explicit linear optical schemes that realize such operations, illustrating a fundamental trade-off between prior information about the input quantum state and the physical resources required. The first scheme is realized with a constant success probability of $1/8$ independent of the qudit dimension using a single nonlocal entangling operation, at the cost of input state dependence. Our second scheme provides a fully state-independent realization, akin to a standard quantum gate, reducing the number of nonlocal gates to $\mathcal{O}(2^{r_1}+2^{r_2})$ as compared to the existing bound of $\mathcal{O}\left(2^{r_1+r_2}\right)$ where $r_1$ and $r_2$ are the number of qubits to be removed as controls in the qudits. The success probability of the realization is $\frac{1}{2}{\left(\frac{1}{8}\right)}^{2^{r_1}+2^{r_2}}$. When combined with qudit circuit compression schemes, our results improve upon a key scalability limitation and significantly improve the efficiency of LOQC architectures.

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• Optical quantum information processing
• Quantum circuits
• Quantum computation