Generalized Type II Fusion of Cluster States

TL;DR

The paper expands measurement-based quantum computation by generalizing the type-II fusion gate beyond Bell-projection limits, introducing a fusion matrix $U$ that acts on the two measured qubits before projection. It classifies all possible fusion outcomes into stabilizer states, weighted graph states, and cluster states (up to rotations), and develops a Schmidt-decomposition framework to analyze these outputs. Analytically, it proves a universal upper bound $P\le \tfrac{1}{2}$ on the success probability for producing maximally entangled fusion links in the ancilla-free setting, and shows that $P=1$ requires $S=0$ (i.e., product states). Numerically, it explores the trade-off between entanglement entropy and fusion probability, illustrates constructive examples for weighted-graph outputs, and demonstrates that including generalized outputs does not exceed the $50\%$ bound without ancilla, while offering richer resource-state possibilities for MBQC. The work also provides a concrete pathway to realize weighted graph states and discusses implications for scaling MBQC with or without ancilla resources and potential extensions to qudits and experimental implementations.

Abstract

Measurement based quantum computation is a quantum computing paradigm that employs single-qubit measurements performed on an entangled resource state in the form of a cluster state. A basic ingredient in the construction of the resource state is the type-II fusion procedure, which probabilistically merges two separate photonic cluster states by a quantum measurement. We generalize the type-II fusion procedure by generalizing the measurement setup, and classify the resulting final states, which also include cluster states up to single-qubit rotations. We prove that the probability for the success of the generalized type-II fusion is bounded by fifty percent, and classify all the possibilities to saturate the bound. We analyze the enhancement of the fusion success probability above the fifty percent bound, by the reduction of the entanglement entropy of the resulting state. We prove that the only states that can be obtained with a hundred percent probability of success, are product states.

Figures (13)

• Figure 1: Generalized type-II Fusion of cluster states: while the type-II fusion process merges two separate cluster states to a larger one by a quantum measurement, we implement a rather general measurement setup which generalizes the resulting final states. The lower the entanglement entropy between the merged clusters, the higher the probability for the success of the fusion process. The precise definitions of the entanglement entropy, the success probability and their relationship is presented in section \ref{['sec:prob_def']}.
• Figure 2: Graph states: (a) A graph state is generated by assigning to each node a single qubit in the state $\ket{+}$, and applying a $CZ$ gate to two neighboring (connected by an edge) vertices $i,j$. This construction can be described recursively as follows. Choose a vertex $a$ in the graph, set the qubits without $a$ and its edges in the graph state $\ket{\phi}_{V_a\setminus \{a\}}$, and apply $CZ$ gates to $a$ and its neighbors. The resulting graph state is as in Eq. (\ref{['Graph state recursive definition']}). We denote by $K_a$ the stabilizer of qubit $a$, and the graph wave function is an eigen-state of this stabilizer with eigenvalue $1$. (b) Examples of one-dimensional graph states of $n=2,3,4$ qubits. The corresponding wave functions are evaluated recursively using Eq. (\ref{['Graph state recursive definition']}).
• Figure 3: Type-II fusion optical setup, where $a$ and $b$ are the incoming photon modes that pass a diagonal polarization beam splitter and exit as $c$ and $d$ modes which are measured by the detectors $D_2$ and $D_1$.
• Figure 4: Type-II Fusion: (a) Two one-dimensional clusters. An $X$ measurement is performed on a qubit that belongs to one of the clusters, erases it and connects its two neighbors $a$ and $e$ to one logical qubit $L$. (b) Type-II fusion is performed on qubit $a$ in $L$ and qubit $b$ from the second cluster. (c) The result of the fusion: qubits $a,b$ are deleted, and qubit $e$ is connected to the previous neighbors $n(L)$ of the logical qubit $L$, and to the previous neighbors $n(b)$ of the qubit $b$.
• Figure 5: All the possible final states after performing a generalized type-II fusion. The black outer set is the set of all the final states. The yellow set is the set of all the final states that are stabilizer states. The red set is the set of all the final states that are weighted graph states. The green set is the set of all the final states that are cluster states up to one two-qubit rotation when $|n(b)|=2$, and single-qubit rotations when $|n(b)|=1$.

Theorems & Definitions (51)

• Remark 1
• Theorem 1: Stabilizer state
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 2
• Remark 3
• Remark 4