Generating redundantly encoded resource states for photonic quantum computing

TL;DR

This work addresses the challenge of building large photonic cluster states for MBQC by boosting fusion success through redundantly encoded vertices, and presents a deterministic protocol to generate such redundantly encoded photonic resource states using a single quantum emitter. The method encodes each vertex on GHZ states and builds a 1D chain of entangled, time-bin qubits, with detailed analysis of how realistic error sources and photon loss affect the generated state and subsequent type-II fusion. Key contributions include a concrete generation protocol, explicit spin-photon entangled-state forms, and quantitative fidelity and loss models for various error channels in two quantum-dot platforms. The results offer a practical route to efficiently construct complex entangled photonic states for MBQC and quantum repeaters, while also outlining the necessary considerations for translating state fidelity into system-wide fault tolerance. The work emphasizes that time-bin ordering and compatibility with multi-emitter schemes can reduce hardware demands and routing losses, advancing scalable photonic quantum computing.

Abstract

Measurement-based quantum computing relies on the generation of large entangled cluster states that act as a universal resource on which logical circuits can be imprinted and executed through local measurements. A number of strategies for constructing sufficiently large photonic cluster states propose fusing many smaller resource states generated by a series of quantum emitters. However, the fusion process is inherently probabilistic with a 50% success probability in standard guise. A recent proposal has shown that, in the limit of low loss, the probability of achieving successful fusion may be boosted to near unity by redundantly encoding the vertices of linear graph states using Greenberger-Horne-Zeilinger states [Quantum 7, 992 (2023)]. Here we present a protocol for deterministically generating redundantly encoded photonic resource states using single quantum emitters, and study the impact of protocol errors and photonic losses on the generated resource states and type-II photonic fusion. Our work provides a route for efficiently constructing complex entangled photonic qubit states for photonic quantum computing and quantum repeaters.

Figures (10)

• Figure 1: A diagram of the boosted fusion process proposed in hilaire2023near. Local Hadamard gates pull single photonic qubits (dark purple circles) out of the GHZ states (light purple hexagons) redundantly encoding logical qubits. The two photonic qubits pulled out by the Hadamard gates are consumed in a fusion measurement which, if the result indicates success, will generate bipartite entanglement between the logical qubits. If the fusion measurement fails the process is repeated.
• Figure 2: The energy level structures of (a) a single charged QD situated in an in-plane magnetic field and (b) a charged QDM. Hole and electron states are denoted by $\{\Downarrow,\Uparrow\}$ and $\{\downarrow,\uparrow\}$ respectively. In (b) the notation $e_Be_Th_Bh_T$ is used to denote the spatial position of electrons $e$ and holes $h$ in the top $T$ or bottom $B$ quantum dot of the QDM as in vezvaee2022deterministic.
• Figure 3: Depictions of the steps of the resource state generation protocol using the example of a charged single QD as the resource state generator. (a) Step 1a prepares the resource state generator (RSG) in a known state via known initialisation protocols sheldon2024optical. (b) In Step 1b control pulses prepare the RSG in a 50:50 superposition of the two ground states ($n\in\{1,3\}$). (c) In step 2 $M_i$ sequential excitation pulses incident on the RSG cause $M_i$ photons to be generated if the RSG initially occupies the ground state selected to be the bright state. (d) Step 3 inverts the populations of the ground states. (e) In step 4 a second set of $M_i$ sequential optical pulses cause $M_i$ photons to be sequentially generated if the RSG occupies the ground state selected to be the bright state after the prior control pulse. (f) A further control pulse prepares the RSG to generate the next photonic qubit(s) and determines the nature of the entanglement between photonic qubits in step 5 ($n\in\{1,2,3\}$).
• Figure 4: (a) A representative example of a redundantly encoded resource state entangled with a quantum emitter (QE). Each vertex of the photonic component of the state is redundantly encoded on a GHZ state (large hexagon) and entangled with its nearest neighbours. Each of the GHZ states is formed by a number of sub-vertices (small hexagons) which in turn are formed by individual photonic qubits (circles) entangled in a GHZ state (where appropriate). (b) The temporal ordering of the time-bins of the photonic qubits in $\mathcal{V}_1$ in (a).
• Figure 5: The calculated fidelity of the spin-photon resource state when the spin preparation process in step (1) of the protocol is subject to (a) spin initialisation and either a $y$-rotation or $z-$rotation error, and (b) step (1a) is performed with unity fidelity but step (1b) is subject to both a $y$-rotation and $z$-rotation error.