Measurement-based quantum computation with cluster states

TL;DR

The paper establishes the universality of measurement-based quantum computation using cluster states (QC_C), showing how any quantum circuit can be simulated by adaptive single-qubit measurements on a fixed entangled resource. It introduces a graph-state perspective (QCmodel) that replaces the network view with a measurement-pattern framework, and demonstrates gate-by-gate constructions for a universal set, including CNOT and arbitrary rotations, while detailing byproduct propagation and adaptive basis selection. It provides concrete, scalable examples (QFT, addition, Hamiltonian simulation) and discusses resource scaling, the Clifford group's one-step realization, and robustness to decoherence via sub-cluster computations. The work connects quantum algorithms to graph theory, offering a practical pathway toward scalable, measurement-based quantum computation with finite resources and clarified composition principles.

Abstract

We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size.

Figures (27)

• Figure 1: Simulation of a quantum logic network by measuring two-state particles on a lattice. Before the measurements the qubits are in the cluster state $|\phi\rangle_{\cal{C}}$ of (\ref{['EVeqn']}). Circles $\odot$ symbolize measurements of $\sigma_z$, vertical arrows are measurements of $\sigma_x$, while tilted arrows refer to measurements in the x-y-plane.
• Figure 2: Realization of elementary quantum gates on the $\hbox{QC}_{\cal{C}}$. Each square represents a lattice qubit. The squares in the extreme left column marked with white circles denote the input qubits, those in the right-most column denote the output qubits.
• Figure 3: Here the exchange of the order of the measurements and the entanglement operations is shown. The crosses "$\times$" denote the one-qubit measurements and the horizontal lines between adjacent cluster qubits denote the unitary transformations $S^{a,a+1}$.
• Figure 4: Vertical cuts. The vertical cuts intersect each qubit line exactly once but do not intersect gates. Thus, ${\cal{O}}_i$, ${\cal{O}}_j$ and $\Omega$ are vertical cuts, but $\not\!\!{\cal{O}}\,\,$ is not. The cut ${\cal{O}}_i$ intersects the rotation $U_x$ just before its input. For two of the rotations in the displayed network, the sub-clusters on which these gates are realized are symbolically displayed in gray underlay. Via the measurement of the cluster qubits $a$ and $b$ (displayed as black dots with white border), the rotation angles of the respective rotations $U_x$ and $U_z$ are set.
• Figure 5: Useful identity for the realization of the rotation $U_z[\alpha]$ as the sequence $H\,U_x[\alpha]\,H$.

Theorems & Definitions (2)

• Definition 1
• Theorem 1