Comparing Schemes for Creating Qudit Graph States from 16- & 128-dimensional Hilbert Space using Donors in Silicon

TL;DR

This work analyzes two hardware approaches for generating qudit graph states in silicon using Sb donors: (i) a single Sb donor with time-bin multiplexing and fusion to scale to complex graphs, and (ii) two Sb donors sharing a single electron, coupled to separate cavities to deterministically realize arbitrary graphs via CZ operations. The first approach leverages a $d$-dimensional Fourier gate and permutation operations to emit multi-mode photons that form linear graphs and then uses type-II fusion to create higher-dimensional resource states, with fusion success scaling as $2/[d(d+1)]$ (odd $d$) or $2/d^2$ (even $d$). The second approach combines two emitters in a Sb$_2^+$ molecule (or paired donors) into a $128$-state system ($2\times 8\times 8$) that supports deterministic graph-state generation through CZ gates, enabling 6-ring and 2D ladder graphs while highlighting photon distinguishability and architectural constraints. The paper discusses coherence, timing, and loss considerations, compares the two paths across hardware metrics, and outlines challenges and future directions for scalable high-dimensional FBQC with Sb donors. Collectively, the work advances high-dimensional photonic graph-state generation in silicon and clarifies when fusion-based versus direct-coupling architectures are advantageous for scalable quantum computing with qudits.

Abstract

In this work, we compare two schemes for generating arbitrary qudit graph states using spin qudits in silicon. The first scheme proposes the creation of qudit linear graph states from a single emitter - a silicon spin qudit. By employing fusion - a destructive and non-deterministic measurement technique - these linear graphs can then be combined to form more complex resource states (multi-photon entangled states), such as ring or ladder structures, which are used to carry out the computation. The second scheme employs two spin qudits. Instead of relying on fusion, the two emitters are directly coupled via CZ to generate the same resource states, thereby eliminating the need for fusion. We compare the two schemes in terms of their ability to produce equivalent resource states and discuss their respective advantages and limitations for building scalable architectures.

Figures (10)

• Figure 1: a) represents a three-photon graph state where the purple spheres correspond to photons prepared in the $\ket{+}$ state. The photons are connected by edges, which correspond to CZ gates. b) shows the circuit representation of the graph state defined in (a). A Hadamard gate is applied to each photon, followed by CZ gates between the first and second photons, as well as between the second and third photons, forming a linear cluster state. Here, photons are treated as qubits; they are not yet qudits.
• Figure 2: Energy spectrum of the single neutral antimony donor. Antimony possesses a high nuclear spin, giving rise to eight distinct nuclear spin states. In its neutral charge configuration, the electron spin becomes hyperfine-coupled to the antimony nuclear spin, resulting in a Hilbert space of dimension 16. NMR transitions with $\Delta m \pm 1$ are represented by curved arrows and are labeled with a subscript 0 to denote the neutral charge state. ESR transitions are shown as vertical solid arrows, while EDSR transitions are indicated with dashed arrows.
• Figure 3: The antimony donor is assumed positioned at an antinode of the cavity electric field, to achieve capacitive coupling to the donor’s electric dipole and stimulate the EDSR transitions to emit photons coherently. The EDSR frequency between the states $\ket{7/2}\ket{\downarrow} \leftrightarrow \ket{5/2}\ket{\uparrow}$ is chosen as fixed frequency for emitting photons
• Figure 4: The circuit illustrates the steps of Protocol \ref{['p:photon_emission']}, in which we emit a 3-dimensional single photon from a single antimony donor. We begin by applying a generalized Hadamard gate ($F_3$) on the nucleus. Next, we apply an EDSR pulse at the frequency at which the cavity operates (between the states $\ket{7/2}$ and $\ket{5/2}$) on the electron to excite the population of state $\ket{7/2}\ket{\downarrow}$ to $\ket{5/2}\ket{\uparrow}$, allowing the electron to emit a photon coherently. We then apply a permutation operation between the $\ket{7/2}$ and $\ket{5/2}$ states to exchange the populations between states. In this case, the permutation is achieved via a single nuclear magnetic resonance (NMR) pulse, followed by another EDSR pulse. This process of applying a permutation and an EDSR pulse is repeated sequentially until the third state involved in the superposition, $\ket{1/2}$, is used. After these operations, a single photon is emitted in 3 modes but remains entangled with the antimony donor. To decouple the antimony from the photon, we apply a second $F_3$ gate and then measure the antimony. As a result, a single photon is emitted into many modes.
• Figure 5: Type-II Fusion of a Linear Eight-Qudit (Four-Dimensional) graph State. This figure illustrates the generation of high-dimensional resource states for photonic quantum computing. Initially, an eight-qudit linear graph state is prepared, with local dimension, qudit dimension, is being four. The antimony donor is then measured to decouple the nuclear spin from the photons, yielding a purely photonic graph state. Subsequently, the first and eighth photons are fused using a type-II fusion gate, which requires two single-photon ancilla states. This operation results in a six-ring qudit graph state, where each node (qudit) retains its four-mode encoding.