Measurement-based quantum computing with qudit stabilizer states

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.

💡 Research Summary

The paper investigates measurement‑based quantum computing (MBQC) on qudits—quantum systems of dimension d > 2—by introducing a broad class of stabilizer‑state resources that go beyond the conventional qudit cluster (graph) state. In the standard MBQC model, a universal resource is created by applying a controlled‑phase (CZ₍d₎) gate on every edge of a two‑dimensional lattice and initializing each qudit in the equal superposition |+⟩. The intrinsic single‑qudit gate that automatically appears when a qudit is measured in this setting is the d‑dimensional Hadamard H₍d₎, which limits the efficiency of implementing arbitrary single‑qudit unitaries.

The authors propose to replace the CZ₍d₎ entangling gate with a block‑diagonal Clifford operation U_int. Different choices of U_int generate different stabilizer states, each characterized by its own “intrinsic gate” G_int. When a single‑qudit measurement is performed on such a resource, G_int is applied (up to Pauli by‑products) regardless of the measurement outcome. The central technical contribution is a rigorous condition for universality: if G_int raised to some integer power p equals a Pauli operator (G_int^p = τ·I, with τ a Pauli), then by concatenating measurements and adapting bases one can synthesize any element of SU(d). The smallest such p is called the Pauli order of G_int; a lower order directly translates into fewer measurement steps and less classical feed‑forward.

For prime‑power dimensions d = pⁿ the authors prove an upper bound on the number of measurements needed to implement an arbitrary single‑qudit unitary. The bound scales linearly with both the dimension d and the Pauli order of G_int, i.e. O(d·p). In the usual cluster state the Pauli order equals d, which is optimal only for even prime‑power dimensions (where H₍d₎ is self‑inverse). In odd dimensions the order is maximal, making the cluster state sub‑optimal. By selecting appropriate block‑diagonal Clifford entanglers, one can construct resources whose G_int has Pauli order 1 or 2 even in odd dimensions, thereby achieving a linear‑in‑d measurement depth that is strictly better than the cluster state.

The paper also clarifies which alternative resources can be mapped to conventional graph states by merely changing measurement bases. It shows that many optimal resources—particularly those with self‑inverse intrinsic gates in even dimensions—cannot be reproduced by any graph‑state transformation, highlighting the genuine novelty of the proposed stabilizer constructions. Physical relevance is underscored by noting that some platforms (e.g., trapped‑ion light‑shift gates) naturally implement non‑standard two‑qudit interactions that fit the block‑diagonal Clifford form.

Concrete examples are provided: a qutrit (d = 3) resource that mimics the cluster state but with a more favorable intrinsic gate; a four‑level (ququart) resource built from a diagonal phase gate combined with CZ, yielding Pauli order = 2; and a family of resources for any d = p^{2k} with even exponent, where the intrinsic gate is its own inverse. In each case the authors detail how to perform vertex deletion (to obtain one‑dimensional chains for single‑qudit gates) and edge creation (to enact two‑qudit entangling gates) using only single‑qudit measurements, preserving the two‑dimensional lattice geometry required for universality.

Finally, the authors discuss the overall architecture: a 2‑D lattice supplies both horizontal chains (for sequences of single‑qudit gates) and vertical connections (for entangling gates). By adapting measurement bases according to previous outcomes, the randomness of measurement results can be corrected deterministically, allowing the entire quantum circuit to be realized with a depth proportional to the number of measurement layers rather than the number of logical gates.

In summary, the work establishes a new design paradigm for qudit MBQC—choose an entangling gate → obtain its intrinsic gate → tailor measurement patterns—showing that, for many dimensions, alternative stabilizer resources dramatically reduce measurement overhead compared with the traditional cluster state. This advances the theoretical foundation for high‑dimensional quantum computing architectures and points to concrete experimental pathways in ion‑trap, superconducting, and photonic platforms.