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 $\mathcal{O}(2^{r_1}+2^{r_2})$ as compared to the existing bound of $\mathcal{O}(2^{r_1+r_2})$ where $r_1$ and $r_2$ are the number of qubits to be removed as control in the qudits. The success probability of the realization is $\frac{1}{2} \left(\frac{1}{8}\right)^{2^{r_1}+2^{r_2}}$. When combined with qudit circuit compression schemes, our results improve upon a key scalability limitation and significantly improve the efficiency of LOQC architectures.

💡 Research Summary

Linear‑optical quantum computation (LOQC) offers a compelling route to scalable quantum information processing, yet its progress is hampered by the inherently probabilistic nature of non‑local entangling gates. Existing qudit‑circuit‑compression techniques mitigate this issue by encoding several qubits into a single d‑dimensional qudit, thereby converting many non‑local two‑qubit gates into deterministic local operations. However, when only a subset of the encoded qubits must participate in a non‑local gate, the compression advantage collapses: removing each unwanted control qubit doubles the number of required non‑local gates, leading to an exponential overhead.

The authors address this bottleneck by introducing multi‑level controlled‑Z (CZ) gates for spatial‑mode encoded qudits. Unlike the conventional two‑level CZ, which flips the phase only when both qudits occupy their highest computational basis state (|d-1\rangle), a multi‑level CZ applies a (-1) phase whenever the two qudits occupy any pair of states drawn from pre‑selected trigger sets (C_{1}) and (C_{2}). Mathematically, \