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arXiv:2602.20612

Non-Clifford symmetry protected topological hyper-cluster states and multi-qubit universal measurement-based quantum computation

Motohiko Ezawa

quant-phcond-mat.str-elhep-thSubmitted Feb 24, 2026• Announced Feb 25, 2026• 0 citations• Revised Oct 6, 2026 (v2)
v1 · Feb 24, 2026v2 · Oct 6, 2026
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Abstract

A cluster state is a highly entangled quantum state that serves as a universal resource for measurement-based quantum computation. It is generated by applying controlled-Z (CZ) gates to the product state $\left\vert ++\cdots +\right\rangle $, and its parent Hamiltonian is the ZXZ model. This model exhibits a topological phase protected by the $\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}$\ symmetry. By applying general quantum gates to the state $\left\vert ++\cdots +\right\rangle $, we systematically obtain a general short-range entangled cluster state. If we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain hyper-cluster states featuring five-body interaction terms. These states enable the implementation of the CZ gate within measurement-based quantum computation. Motivated by this, we propose a generalization to the C$^{N}$Z gate, leading to a hyper-cluster model, in which $(2N+1)$-body entangled states are generated. This model retains the $\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}$\ symmetry, which lies outside the Clifford group for $N\geq 3$. We demonstrate that an open chain in this model exhibits a $2^{2N}$ fold degenerate ground states protected by the symmetry, corresponding to the emergence of $N$ effective free spins at each edge. These boundary degrees of freedom can serve as $N$ qubits for measurement-based quantum computation. In particular, this framework makes it possible to implement the C$^{N-1}$Z gate in measurement-based quantum computation.

Comments
22 pages, 11 figures

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A cluster state is a highly entangled quantum state that serves as a universal resource for measurement-based quantum computation. It is generated by applying controlled-Z (CZ) gates to the product state ∣++⋯+⟩\left\vert ++\cdots +\right\rangle ∣++⋯+⟩, and its parent Hamiltonian is the ZXZ model. This model exhibits a topological phase protected by the Z2even×Z2odd\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}Z2even​×Z2odd​\ symmetry. By applying general quantum gates to the state ∣++⋯+⟩\left\vert ++\cdots +\right\rangle ∣++⋯+⟩, we systematically obtain a general short-range entangled cluster state. If we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain hyper-cluster states featuring five-body interaction terms. These states enable the implementation of the CZ gate within measurement-based quantum computation. Motivated by this, we propose a generalization to the CN^{N}NZ gate, leading to a hyper-cluster model, in which (2N+1)(2N+1)(2N+1)-body entangled states are generated. This model retains the Z2even×Z2odd\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}Z2even​×Z2odd​\ symmetry, which lies outside the Clifford group for N≥3N\geq 3N≥3. We demonstrate that an open chain in this model exhibits a 22N2^{2N}22N fold degenerate ground states protected by the symmetry, corresponding to the emergence of NNN effective free spins at each edge. These boundary degrees of freedom can serve as NNN qubits for measurement-based quantum computation. In particular, this framework makes it possible to implement the CN−1^{N-1}N−1Z gate in measurement-based quantum computation.
Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically nontrivial, short-ranged entangled states are promising candidates for such a resource. Miller and Miyake [npj Quantum Inf. 2, 16036 (2016)] recently constructed a particular ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-protected topological state on the Union Jack lattice and established its quantum-computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d\ensuremath{-}1)$ qudit nontrivial ${\mathbb{Z}}_{d}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{d}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{d}$ symmetry-protected topological states are universal on the triangular lattice, for ddd being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union Jack lattice.
The interplay of unitary evolution and projective measurements is a modern interest in the study of many-body entanglement. On the one hand, the competition between these two processes leads to the recently discovered measurement-induced phase transition (MIPT). On the other hand, measurement-based quantum computation (MBQC) is a well-known computational paradigm where measurements simulate unitary evolution by utilizing the entanglement of special resources such as the two-dimensional (2D) cluster state. The entanglement properties enabling MBQC may be attributed to symmetry-protected topological (SPT) orders, particularly subsystem-symmetry-protected topological (SSPT) orders. It was recently found that the one-dimensional cluster state may be associated with an SPT phase in random circuits respecting a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, and furthermore that all phase transitions in this scenario belong to the same universality class. As resources with greater computational power feature greater symmetry, it is fruitful to investigate further any relationship between levels of symmetry in MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels of symmetry-respecting unitary evolution interspersed by measurements. Although we find two area-law phases and one volume-law phase with distinct entanglement structures for each ensemble, the phase transition from the volume-law phase to the area-law phase associated with the 2D SSPT cluster state has variable correlation length exponent $\ensuremath{\nu}$. Whereas $\ensuremath{\nu}\ensuremath{\approx}0.90$ for unconstrained Clifford unitaries and $\ensuremath{\nu}\ensuremath{\approx}0.83$ for globally symmetric Cliffords, subsystem-symmetric Cliffords feature a much smaller value $\ensuremath{\nu}\ensuremath{\approx}0.38$. We discuss how these distinct $\ensuremath{\nu}$'s quantify spacetime response scales where quantum information is manipulated by single-qubit measurements as in MBQC.
Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.
Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2)({Z}^{1/2})(Z1/2), and TTT (Z1/4)({Z}^{1/4})(Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the TTT gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-ZZZ errors are calculated to be 2.7%--2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.
Despite growing interest in beyond-group symmetries in quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the microscopic level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is a G × Rep ( G ) -symmetric state: the G -cluster state introduced by Brell []. We show that this state lies in a symmetry-protected topological (SPT) phase protected by G × Rep ( G ) symmetry, distinct from the symmetric product state by a duality argument. We identify several signatures of SPT order, namely, protected edge modes, string order parameters, and topological response. We discuss how G -cluster states may be used as a universal resource for measurement-based quantum computation, explicitly working out the case where G is a semidirect product of Abelian groups.
Abstract Measurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. Although MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here we utilise recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behaviour differs fundamentally from that of the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X , Y , and Z operators. This novel computational feature is related to the ‘genuine’ 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity.