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

TL;DR

This Letter demonstrates the existence of multi-level control-Z (CZ) gates for qudits encoded in multiple spatial modes in LOQC, and presents 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.

Abstract

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.

Figures (5)

• Figure 1: A schematic description of a qudit compression scheme. Here the blue wires represent the encoding of qubits onto qudits, with the brackets to the left indicating the qubits encoded onto a qudit and the purple and red wires represent the corresponding optical circuit implementation using existing and our proposed circuit compression schemes respectively. (a) Represents $g$ qubits mapped to the same qudit comprising of $2^{g}$ spatial modes. (b) A non-local CZ gate between two qubits mapped to the same qudit is transformed into a local gate. (c) Shows the optical implementation of $4$ qubit CCCZ gate where $2$ qubits using circuit compression. (d) Shows the optical implementation of a compressed circuit where two control qubits are removed from the multi-level CZ gate (one from each qudit).
• Figure 2: Schematic of the state dependent multi-level CZ gate. Two input-ancilla pairs in (input in blue, ancilla in red) serve as inputs to the two SMRs. The SMRs execute a partial swap operation on the defined trigger sets. Only the case in which one photon exits the port of the two SMRs is post-selected. The ancilla qudits $3$ and $4$ are then fed into unitaries $O_1$ and $O_2$ which, based on the knowledge about input qudits, map them to a two-dimensional subspace while the remaining $k_i-1$ modes are unoccupied. Finally, a Hadamard gate is applied to photon $4$ before a BSM is implemented on the qudits $3$ and $4$. Depending on the outcomes, the local unitaries $U_1$ and $U_2$ are then implemented, finally realizing the desired operation.
• Figure 3: Schematic of the state independent multi-level CZ gate. For $k_1$ ($k_2$) number of trigger modes in qubit $1$ ($2$), a sequence of $k_1$ ($k_2$) two-level CZ gates between the input qubit $1$ and the ancilla qubit $3$ (along with two Hadamard gates), selectively flips the ancilla qubit whenever input qudit is in the trigger state. The green boxes show two-level CZ gates, each triggered by $\ket{c_i}_1$ and $\ket{1}_3$ ($\ket{t_j}_2$ and $\ket{1}_4$). The grey boxes constitute the sequence of operations $\Tilde{O}_1$ and $\Tilde{O}_2$ respectively. A BSM is then performed on the ancilla qubits followed by the required unitary operations.
• Figure 4: Linear optical realization of SMR. The two input ports $A$ and $B$ are fed the input qudit (represented by solid blue lines) and ancilla qudit (represented by dashed red lines) modes respectively. Only the trigger modes undergo the swap operation, which is facilitated by Mach-Zehnder interferometers (represented in blue boxes) parameterized by a phase shift $\theta_i$, $i \in \lbrace 0, 1, \ldots, d - 1 \rbrace$. For a $k$th mode which is chosen as a trigger mode, the parameter $\theta_k = \pi$, while for modes which are not triggers, $\theta_j = 0$.
• Figure 5: (a) A schematic of the QFA circuit using (b) currently available non-local CZ gates, (c) our proposed state-independent multi-level CZ gate and, (d) state-dependent multi-level CZ gate. The input qubits $q_0, q_1$, and $q_2$ are encoded onto a single qudit. (b) In qudit circuit compression schemes, the operation corresponding to C is a non-local CNOT gate that must be decomposed into two non-local entangling gates for a linear optical realization. (c) For qudit compression scheme, the same circuit would require three non-local entangling gates using our state-independent scheme, followed by the Bell state measurement on the ancilla qubits, and the feedforward corrections $U_1$ and $U_2$. (d) The same operation corresponding to C can be implemented using a single multi-level entangling gate via our state-dependent scheme. In order to prepare the ancilla qudits for the multi-level entangling gate they are first initialized in the same state as the input qudits and then subsequently subject to the local operations A and B.