Non-Clifford symmetry protected topological higher-order cluster states in multi-qubit measurement-based quantum computation

1. Problem Statement

Measurement-based quantum computation (MBQC) typically relies on cluster states generated by applying Controlled-Z (CZ) gates to a product state of $|+\rangle$ qubits. These standard cluster states are protected by a $\mathbb{Z}_2^{\text{even}} \times \mathbb{Z}_2^{\text{odd}}$ symmetry and exhibit a Symmetry-Protected Topological (SPT) phase. However, standard cluster states are Clifford states, meaning they can be efficiently simulated by classical computers and do not provide universal quantum computation on their own without additional non-Clifford resources (magic states).

The paper addresses the following challenges:

• How to generalize cluster states to include non-Clifford operations while preserving the SPT protection.
• How to generate higher-order entanglement (involving more than two qubits) to create "higher-order" cluster states.
• How to utilize these generalized states to provide multi-qubit input and output channels (specifically $N$-qubit edges) for MBQC, rather than the standard single-qubit edge.

2. Methodology

The author employs a systematic generalization of the standard cluster state construction using unitary mappings and higher-order controlled gates.

• Generalized Cluster State Construction:
  Instead of applying only CZ gates, the author defines a general cluster state $|\psi\rangle$ by applying a finite-depth local-unitary $N$-qubit gate $U\{j; N\}$ to the state $|+\rangle^{\otimes 2L}$:
  $$|\psi\rangle = \left( \prod_{j=1}^{2L-N} U\{j; N\} \right) \bigotimes_{j=1}^{2L} |+\rangle_j$$
  Here, $U\{j; N\}$ acts on qubits $\{j, j+1, \dots, j+N\}$.

• Hamiltonian Construction:
  A corresponding cluster Hamiltonian $H_U$ is constructed by mapping the trivial Hamiltonian $H_X = -\sum X_j$ via the unitary transformation $V$ (where $V$ is the product of the gates used to create the state). The ground state of $H_U$ is the generalized cluster state.

• Specific Gate Models Investigated:
  The paper analyzes several specific instances of $U\{j; N\}$:

  1. Standard & Modified Models: ZXZ (standard), XZX, and ZZZ-XXX models (involving Hadamard transformations).
  2. Non-Clifford Perturbations: Bit-flip ($e^{-i\phi X}$) and Phase-flip ($e^{i\phi Z}$) models to test robustness.
  3. Controlled Phase (CP) Gate: Generalizing CZ to arbitrary phase angles $\phi$.
  4. Higher-Order Gates:
     • CCZ (Controlled-Controlled-Z): Generates 5-body interactions ($N=2$).
     • CNZ (Controlled-N-Z): Generalizes to $(2N+1)$-body interactions.
     • CNP (Controlled-N-Phase): Generalization of CP to $N$-qubit controls.

• Symmetry Analysis:
  The author investigates the preservation of:

  • $\mathbb{Z}_2^{\text{even}} \times \mathbb{Z}_2^{\text{odd}}$ Symmetry: The generator is transformed via the unitary mapping.
  • Non-Invertible Symmetry: A generalized duality transformation (related to the Kennedy-Tasaki transformation) is constructed for these new models.
  • String Order Parameters: Defined to detect the SPT phase in the generalized models.

3. Key Contributions

A. Generalization to Non-Clifford Higher-Order Cluster States

The paper demonstrates that by replacing CZ gates with non-Clifford gates (like CP with arbitrary angles) or higher-order gates (CCZ, CNZ), one can generate a new class of cluster states.

• CCZ Model: Generates a state with 5-body entanglement. The Hamiltonian involves complex multi-spin interactions (e.g., $Z_j X_{j+1} Z_{j+2}$ terms).
• CNZ Model: Generates states with $(2N+1)$-body entanglement.
• Non-Clifford Nature: For $N \geq 3$, the symmetry generators become non-Clifford, distinguishing these states from standard stabilizer states.

B. Emergence of $N$ Free Spins at Edges

A central finding is the behavior of open chains.

• In the standard cluster state ($N=1$), an open chain has a 4-fold degenerate ground state, corresponding to 1 free spin at each edge (total 2 logical qubits).
• In the generalized CNZ model, an open chain exhibits $2^{2N}$-fold degenerate ground states.
• Physical Interpretation: This degeneracy indicates the emergence of $N$ free spins at each edge (left and right). These edge modes are protected by the $\mathbb{Z}_2^{\text{even}} \times \mathbb{Z}_2^{\text{odd}}$ symmetry.

C. Robustness and Symmetry Protection

• The paper proves that the topological edge states remain robust even when the system is perturbed by random bit-flip or phase-flip errors (non-Clifford gates), provided the symmetry is preserved.
• The non-invertible symmetry is shown to persist in these generalized models, mapping local operators to non-local string operators, confirming the self-dual nature of the models.

D. Kennedy-Tasaki (KT) Transformation

The author extends the KT transformation (which maps SPT phases to symmetry-broken phases) to these higher-order models.

• For the standard ZXZ model, the transition occurs at $\alpha = 1/2$ (duality point).
• For the CCZ model, the topological phase transition occurs at $\alpha = 0$, indicating a different topological structure compared to the standard cluster state.

4. Results

• Degeneracy Scaling: The ground state degeneracy for an open chain scales as $2^{2N}$. For $N=2$ (CCZ model), this is 16-fold degeneracy (4 edge spins). For general $N$, it is $2^{2N}$.
• Logical Qubits: The edge states form logical qubits that satisfy the Pauli algebra. Specifically, the left edge supports $N$ logical qubits ($X^L_k, Y^L_k, Z^L_k$) and the right edge supports $N$ logical qubits.
• Phase Transitions:
  • ZXZ Model: Topological phase transition at $\alpha = 1/2$ (gap closing).
  • CCZ Model: Topological phase transition at $\alpha = 0$. The gap does not close at $\alpha = 1/2$, but the string order parameter shows a jump at $\alpha = 0$.
• Experimental Feasibility: The paper discusses experimental realizations using:
  • Superconducting Qubits: Cross-resonance gates and tunable couplers for CCZ/CP.
  • Ion Traps: Mølmer-Sørensen gates.
  • Rydberg Atoms/Ions: Native generation of multi-qubit gates (CCZ, CNZ).
  • Photonic Systems: Generation of weighted graph states.

5. Significance

• Enhanced MBQC Capability: This work proposes a framework for MBQC where the input and output are not single qubits but $N$-qubit registers. This allows for more complex data processing at the boundaries of the quantum computer without needing to measure the entire bulk.
• Beyond Clifford Computation: By utilizing non-Clifford cluster states, the paper explores a regime where the system is not efficiently simulable by classical computers (unlike standard 1D cluster states which are Matrix Product States with bond dimension 1). While not universal on its own, it enhances computational power.
• New Topological Phases: The discovery of $2^{2N}$ degenerate edge states and the specific behavior of non-invertible symmetries in higher-order interactions expands the classification of SPT phases in 1D systems.
• Error Resilience: The demonstration that these complex, non-Clifford states remain robust against local noise (bit/phase flips) is crucial for practical quantum hardware implementation.

In summary, Ezawa generalizes the concept of the 1D cluster state to higher-order, non-Clifford variants, proving that they host robust, multi-qubit edge modes protected by symmetry, thereby offering a promising architecture for advanced measurement-based quantum computation.