Efficient circuit compression by multi-qudit entangling gates in linear optical quantum computation

Linear optical quantum computation (LOQC) offers a promising platform for scalable quantum information processing, but its scalability is fundamentally constrained by the probabilistic nature of non-local 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 non-local entangling gate, leading to an exponential increase in the number of non-local gates. In this Letter, we address this bottleneck by demonstrating the existence of multi-level control-Z (CZ) gates for qudits encoded in multiple spatial modes in LOQC. Unlike conventional two-level CZ gates, which act only on a single pair of modes, multi-level 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 non-local entangling gate, at the cost of state dependence, which is significantly better than the current success probability of 1/9. Our second scheme provides a fully state independent realization reducing the number of non-local gates to mathcal{O}\(2^r₁+2^r₂\) as compared to the existing bound of mathcal{O}\(2^r₁+r₂\) where r₁ and r₂ are the number of qubits to be removed as control in the qudits. The success probability of the realization is frac{1}{2} left\(frac{1}{8}right\)^{2^r₁+2^r₂}. When combined with qudit circuit compression schemes, our results improve upon a key scalability limitation and significantly improve the efficiency of LOQC architectures.