YZ-plane measurement-based quantum computation: Universality and Parity Architecture implementation

Abstract

We define the class of register-logic graphs and prove that any uniformly deterministic measurement-based quantum computation (MBQC) where the inputs coincide with the outputs must be driven on such graphs by measurements in the $YZ$ plane of the Bloch sphere. This observation is revisited in the context that goes beyond uniform determinism, where we present a universal $YZ$-plane-only measurement pattern and establish a connection between $YZ$-plane-only and $XZ$-plane-only patterns. These results conclude the line of research on universal patterns with measurements restricted to one of the principal planes of the Bloch sphere. We further demonstrate, within the framework of the Parity Architecture, that $YZ$-plane patterns with the register-logic graph can be embedded into another graph with purely local interactions, and we extend this case to the scenario of universal quantum computation.

Figures (8)

• Figure 1: Open graph $(G, I, O)$ of the register-logic form. (a) General RL graph, where $O$ is formed by the nodes in the right box. If there were no edges between nodes in the right box, the open graph would be of the bipartite-register-logic form. (b) bRL graph that allows to implement an arbitrary unitary on four qubits that is diagonal in the $Z$ basis. All orange (dark gray) nodes are measured in the $YZ$-plane and correspond to parity qubits, where the numbers refer to the given parity for each qubit, see Sec. \ref{['sec:parity_architecture']}.
• Figure 2: Diagram explaining the mutual relations between Theorems \ref{['thm:io_g_to_rl_yz']}, \ref{['thm:rl_yz_impl']}, and \ref{['thm:mod_isaac_thm_1']}, denoted in the figure by T1, T2, and T3, respectively, and Lemma \ref{['thm:inputs_gflow']} denoted by L1. The asterisk marks that the theorem is valid only for $I = O$. Note that T3 contains two implications, either of which corresponds to one arrow.
• Figure 3: Parity Architecture in MBQC picture. (a) MBQC formulation of the LHZ triangle, where the base qubits are indexed by single-digit numbers. The two-digit numbers indexing each parity qubit represent the two base qubits, whose parity is affected by the local action on that parity qubit. The base qubits are simultaneously both inputs and outputs. Adapted from smith_parity_2024. (b) Triangular segment of a cluster state, where every other vertical edge in the last row is intentionally missing. The qubits in the $X$ sublattice, rendered in green (light gray), are measured in Pauli $X$ basis to arrive at the graph in (a). (c) Beveled cluster with four inputs. (d) The post-measurement state for (c) after the Pauli $X$ measurements are made. (e) Series of alternating beveled clusters that supports universal quantum computing. Note the missing vertical edges at the interface of two beveled clusters. Every other beveled cluster implements the $\exp(i \alpha X_i \otimes X_j)$ operations as marked by the parity labels of the form $\langle ij \rangle$, together with the $R_X$ rotations marked by $\langle i \rangle$. The graphical convention used throughout is as follows: the input qubits have an extra thick red (dark gray) frame, the outputs are blue with a dashed outline. The qubits that are to be measured in the course of computation are orange (with thin dark gray outline), while those measured in a Pauli basis prior to the computation are green (with thin light gray outline).
• Figure 4: Universal $YZ$-only measurement patterns. (a) The unit cell for complex MBQC with measurement bases and indices depicted. (b) Sheet of unit cells for complex MBQC. (c) Hexagonal grid. One of its two sublattices is subjected to the $Y$ basis measurements. (d) The post-measurement state when all such measurements reduce to $\ket{+i}$ projections. In green (light gray) are marked those qubits that are measured in the $Z$ basis in the second step of the construction of the real-valued universal $YZ$-only pattern. The letters denote the extra gates applied on top of the graph state.
• Figure 5: Examples and counterexamples of labeled open graphs. The graphics follows the convention of Fig. \ref{['fig:parity_arch']} except that the number in each qubit is merely an index, not a parity label. (a) Graph with $I \subsetneq O$ and gflow given by $g(1) = \{ 1 \}$ and $1 \prec 3$. (b) Graph with $I \subsetneq O$ that is not of the RL form and has no gflow. One can also remove qubit 3 and consider a similar graph with $I = O$, which also has no gflow. (c) Graph of the RL form with $I \subseteq O$ and no gflow. A valid gflow would have to satisfy $3 \in g(1)$ which is not allowed as qubit 3 is an input. The same conclusion holds when considering the same graph but with qubit 2 removed and so $I = O$. (d) Graph with $I \subsetneq O$ and gflow given by $g(2) = \{4\}$, $g(1) = \{1\}$ with $1 \prec 2 \prec 3,4$. (e) Graph with $I \subsetneq O$ and gflow that is not of the RL form. The gflow reads $g(1) = \{ 1 \}$, $g(2) = \{ 2,3 \}$ with $1 \prec 2 \prec 3, 4$. (f) Graph of the RL form with gflow, where $I \cap O = \emptyset$. The gflow reads $g(1) = \{ 3 \}$, $g(2) = \{ 4 \}$ with $1,2 \prec 3,4$. (g) Pattern with $I \subsetneq O$ and gflow, where some maximal elements are not measured in the $YZ$-plane. The gflow reads $g(1) = \{3\}$, $g(2) = \{2\}$ with $1,2 \prec 3,4$. (h) Graph with $I = O$ and gflow that is not bipartite with $I$ forming one part. The gflow is given by $g(1) = \{1\}$, $g(2) = \{2\}$ with $1, 2 \prec 3, 4$. (i) Bipartite graph with $|I| = |O|$ and $I$ forming one part, yet with no gflow as the inputs that are not also outputs must be measured in the $XY$-plane.

Theorems & Definitions (16)

• Lemma 1
• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3: Modification of Theorem 2 in smith_parity_2024
• Lemma 2: Generalization of Lemma \ref{['thm:inputs_gflow']}
• Definition 2: Gflow; adapted from backens_there_2021
• Definition 3: Pauli flow; adapted from simmons_relating_2021mcelvanney_complete_2023
• Definition 4: Maximally delayed gflow, mhalla_finding_2008
• Lemma 3: Lemma C.4 of Ref. backens_there_2021