Holonomic multi-controlled gates for single-photon states

TL;DR

This work addresses implementing quantum gates for single-photon qubits using non-Abelian holonomies in photonic waveguide networks. It introduces a two coupled M-pod architecture that yields a four-dimensional zero-energy degenerate subspace for encoding, and shows how adiabatic driving cycles realize a universal gate set, including CNOT, SWAP, and Toffoli, with extensions to OR and Deutsch-type query algorithms. The authors generalize to larger M-pods to enable multiple controlled operations and discuss Deutsch algorithm implementation via holonomic oracles. The results suggest a path toward robust holonomic linear-optics quantum computing and scalable holonomic quantum algorithms.

Abstract

Controlled and multi-controlled quantum gates, whose action on a target qubit depends on the state of multiple control qubits, represent a fundamental logical building block for complex quantum algorithms. We propose a scheme for realizing this class of gates based on non-Abelian holonomies in modulated photonic waveguide networks. Our approach relies on linear photonic cicuits formed by two star networks coupled via a two-path circuit. A star network with M peripheral waveguides coupled to a single central site, or M-pod, naturally generalizes the tripod structure used in non-Abelian Thouless pumping and stimulated Raman adiabatic passage (STIRAP). In the present work, we first analyze the minimal case of two connected tripods and design adiabatic driving loops that implement single-qubit, CNOT, and SWAP gates. We then show how extending the approach to larger M-pod structures enables the realization of multiply controlled operations, which we exemplify by designing Toffoli and the OR gate on two coupled pentapods. Finally, we demonstrate that networks of connected tripods can implement the Deutsch quantum query algorithm.

Figures (9)

• Figure 1: (a) Scheme describing the structure of the Hamiltonian of two-coupled tripods. (b) Schematic of the array composed of two tripods respectively colored in blue and green. The two tripods are connected via the hopping $J_{x_0}$ and $J_{x_1}$. (c) Two qubits encoding states $|s,p\rangle$ for the target qubit $p=0$ (upper row) and $p=1$ (lower row). At the initial parameters $J_{\ell_p} =0$ for $\ell = c,d,x$ and $J_{b_p}=J$ for $p=0,1$ of the driving cycle, the states correspond to the four zero-energy degenerate eigenstates.
• Figure 2: (a) Driving cycle in the parameter space (the circle indicates the initial point) and quantum state tomography for the single qubit Hadamard gate. (b) Same as (a) for the single qubit phase gate.
• Figure 3: (a) Illustration of the lattice pumped to implement a ${\rm CNOT}$ gate. The left tripod $p=0$ is colored in green while the right tripod $p=1$ is colored in blue. (b) Driving cycle in the parameter space. The circle indicates the initial point. (c,d) Propagation of $|1,0\rangle$ and $|1,1\rangle$ states respectively over one cycle period. The yellow lines separate the 0 and the 1 tripods. (e,f) Quantum state tomography for the ${\rm CNOT}$ gate and the ${\rm SWAP}$ gate respectively.
• Figure 4: (a) Schematic of the lattice composed of left (0) and right (1) pentapods respectively colored in blue and green. The two tripods are connected via the hopping $J_{x_L}$ and $J_{x_R}$. (b) Three qubits encoding states $|b(s_1,s_2,p)\rangle$ with two control qubits $s_1,s_2=\pm$ and a target $p=0,1$ qubits. At the initial parameters $J_{\ell_p} =0$ for $\ell = c,d,e,f,x$ and $J_{b_p}=J$ for $p=L,R$ of the driving cycle, the states correspond to the eight zero-energy degenerate eigenstates. (c) Quantum state tomography for the Toffoli gate. The driving cycle is shown in the top-right corner. (d) Same as (c) for a quantum version of the OR gate.
• Figure 5: (a) Illustration of the Deutsch's algorithm. (b,c) Superposition states $|+,-\rangle$ and $|-,-\rangle$. (d,e) Propagation of $|-,-\rangle$ over one cycle period for $\alpha_0=\pm 1$ and $\alpha_1=\pm 1$ which respectively implement the constant functions $f_1$ and $f_4$. (f,g) Same as (d,e) for $\alpha_0=\pm 1$ and $\alpha_1=\mp 1$ which respectively implement the balanced functions $f_2$ and $f_3$. The yellow lines separate the 0 and the 1 tripods.