Measurement-based quantum computing with qudit stabilizer states

TL;DR

This work extends measurement-based quantum computing to qudits by introducing alternative stabilizer-resource states generated by diagonal or block-diagonal Clifford entangling gates. It establishes that each resource carries an intrinsic single-qudit gate $G_I$ that governs measurements, and it derives linear overhead bounds for implementing arbitrary single-qudit unitaries in prime-power dimensions, with potential self-inverse gates in even dimensions. The study shows when qudit graph states can mimic these resources by adjusting measurement bases and identifies optimal resources (e.g., light-shift and certain finite-field constructions) that can outperform the standard qudit cluster state in some dimensions. It also details two-dimensional resource geometries that enable universal MBQC and discusses which resources are graph-state-like versus non-graph-state-like, offering practical pathways for more efficient quantum information processing with qudit stabilizer states.

Abstract

We show how to perform measurement-based quantum computing on qudits (high-dimensional quantum systems) using alternative resource states beyond the cluster state. Estimating overheads for gate decomposition, we find that generalizing standard qubit measurement patterns to the qudit cluster state is suboptimal in most dimensions, so that alternative qudit resource states could enable enhanced computational efficiency. In these resources, the entangling interaction is a block-diagonal Clifford operation rather than the usual controlled-phase gate for cluster states. This simple change has remarkable consequences: the applied entangling operation determines an intrinsic single-qudit gate associated with the resource that drives the quantum computation when performing single-qudit measurements on the resource state. We prove a condition for the intrinsic gate allowing for the measurement-based implementation of arbitrary single-qudit unitaries. Furthermore, we demonstrate for prime-power-dimensional qudits that the complexity of the realization depends linearly both on the dimension and the Pauli order of the intrinsic gate. These insights also allow us to optimize the efficiency of the standard qudit cluster state by effectively mimicking more favorable intrinsic-gate structures, thereby reducing the required measurement depth. Finally, we discuss the required two-dimensional geometry of the resource state for universal measurement-based quantum computing. As concrete examples, we present multiple alternative resource states. In certain dimensions, we show the existence of resource states achieving optimal intrinsic gates, enabling more efficient measurement-based quantum information processing than the qudit cluster state and highlighting the potential of qudit stabilizer state resources for future quantum computing architectures.

Figures (8)

• Figure 1: Elementary building blocks for measurement-based quantum computing. $(a)$ We first consider the action of a single measurement on a single entangled state. $(b)$ Then, we concatenate entanglement and measurement to realize single-qudit gates on horizontal one-dimensional chains of arbitrary length. In addition, we require the measurement-based realization of an entangling gate between two qudits, which we achieve via transport through vertical edges using the configuration in $(c)$. Finally, we describe how to disassemble a two-dimensional resource state lattice into the previous building blocks via vertex deletion or the creation of new edges. Here, each horizontal line corresponds to the processing of a single qudit. For instance, via vertex deletion, one-dimensional chains of different lengths are cut out and used to implement single-qudit gates, while vertical edge creation allows the realization of entangling gates between two qudits in $(d)$.
• Figure 2: Concatenation of the two-qudit protocol for the measurement-based implementation of a single-qudit unitary. The first qudit $\ket{+}_1$ acts as an input state to be processed and is connected with a controlled-phase gate $CZ_{12}$ to $\ket{+}_2$ and subsequently measured in the basis $\{D_{\Vec{\phi}} \ket{k_X} \}_k$. Repeating this procedure, so entangling the second qudit, which is in the state $\ket{\psi} \coloneq H_d D_{\Vec{\phi}} Z^{-k} \ket{+}$ after the first measurement with outcome $k$, via $CZ_{23}$ with the third qudit $\ket{+}_3$ and measuring the second qudit subsequently is equivalent to first applying both entangling $CZ$ operations for preparing the three-qudit linear cluster state resource and afterwards performing both single-qudit measurements due to the commutation of all involved operations.
• Figure 3: Realization of a measurement-based $CZ_d$ gate acting on the two-qudit input $\ket{\psi}$ via a measurement pattern on a six-qudit cluster state, using an existing $CZ_d$ connection Zhou_2003.
• Figure 4: Implementation of a single-qubit unitary for generalized qubit resources, created by applying a diagonal Clifford $G_E$. If $G_I \propto G_I^{\dagger}$, so if the Pauli order of $G_I$ is two, four single-qubit measurements suffice. Below each qubit (green) in the figure, the gate executed with the respective single-qubit measurement is specified. For instance, measuring in the $X$ basis, rotated by $S^\dagger D_{\alpha}$ results in $G_I S^\dagger D_{\alpha}$ being realized. The fifth qubit, the output, then contains the state $G_I D_{\gamma} G_I (S G_I D_{\beta} G_I S^\dagger) D_{\alpha} \ket{\psi}$, assuming zero outcome for all four measurements.
• Figure 5: Two-dimensional resource state for diagonal entangling interaction on resource qudits, initialized in $\ket{+}$. The horizontal lines allow for the implementation of single-qudit unitaries and the vertical lines allow for two-qudit entangling gates. The directions on the vertical lines can, in principle, be chosen freely since that only affects the local Clifford operations required for vertex deletion or local complementation. Here, we chose a symmetric configuration that allows the propagation of quantum information along vertical lines.