Resolving topological order crossovers on NISQ hardware

Technical Summary: Resolving Topological Order Crossovers on NISQ Hardware

Problem Statement
While topological phases of matter offer a promising route for robust quantum information processing, their experimental realization on present-day Noisy Intermediate-Scale Quantum (NISQ) devices faces a critical challenge: determining whether signatures of topological crossovers remain resolvable under realistic imperfections. Theoretical models often assume ideal conditions, but the transition from topological to trivial regimes in finite-size systems is sensitive to static disorder, accumulated circuit noise, and non-Hermitian effects (loss). This work addresses the question of whether finite-size diagnostic signatures of the Wen–plaquette model can be distinguished from trivial behavior on actual quantum hardware amidst these imperfections.

Methodology
The authors employ a two-stage strategy using IBM Quantum hardware to characterize and scale the simulation of the Wen–plaquette model:

1. Variational Characterization on a Small System:

   • Model: A $3 \times 3$ open lattice of nine qubits implementing the Wen–plaquette Hamiltonian ($H_{wen}$), extended with a transverse field ($g$), on-site disorder ($h$), and an effective non-Hermitian imaginary field ($\gamma$).
   • State Preparation: Target states (equilibrium ground states and time-evolved states under unitary and non-unitary quenches) are approximated using a hardware-efficient variational quantum circuit ($V_n(\theta)$). The parameters are optimized classically to minimize the infidelity between the circuit output and the exact target state.
   • Diagnostics: Two observables are measured:
     • Local Plaquette Stabilizer ($F_1$): Probes the local stabilizer sector structure.
     • Nonlocal Wilson Loop ($W$): Probes the persistence of finite-size loop response and topological order.
   • Noise Amplification: To quantify robustness, "identity" layers (gate sequences that act as the identity in the noiseless limit) are inserted into the circuit to systematically amplify accumulated gate and decoherence errors without altering the ideal unitary evolution.

2. Scalable Projection-Based Implementation:

   • Model: A larger $5 \times 5$ square lattice on a 2D IBM processor (IBM Miami).
   • State Preparation: Instead of variational optimization, a "representative-qubit" construction is used. This method coherently enforces local plaquette constraints ($\Pi_{i,j} = (I + F_{i,j})/2$) to project a product state directly into the desired $Z_2^A$ stabilizer sector. This avoids mid-circuit measurement and variational training.
   • Robustness Test: Controlled, amplified local coherent perturbations (random single-qubit $R_x$ rotations) are applied to the prepared state to test the stability of the lattice-averaged plaquette response.

Key Results

• Noise Resilience of Diagnostics:

  • The finite-size crossover between the stabilizer-dominated regime and the trivial polarized regime remains clearly resolved even when circuit depth is significantly increased via identity-layer insertion.
  • The nonlocal Wilson loop signal ($\langle W \rangle$) is preserved under systematic noise amplification, demonstrating that the hardware implementation retains the relevant topological signal despite accumulated errors.

• Equilibrium Diagnostics:

  • Local vs. Nonlocal Robustness: The local plaquette stabilizer ($\langle F_1 \rangle$) remains a robust diagnostic even in the presence of strong disorder ($h=5$), clearly distinguishing the $Z_2^A$ and $Z_2^B$ sectors. In contrast, the nonlocal Wilson loop response is strongly suppressed by disorder, indicating that nonlocal signatures are more fragile to static inhomogeneity than local ones.
  • Crossover Resolution: The transition from a strong loop response to a trivial regime is observable in clean systems but washed out in strongly disordered systems.

• Dynamical Stability (Quenches):

  • Unitary Quenches: The dynamical survival of the plaquette signature depends on the competition between disorder strength and the stabilizer energy scale. Near the finite-size crossover (weak coupling), the plaquette response decays rapidly under disorder. Deep in the strong-coupling regime, the response is substantially more robust, decaying slowly and remaining nonzero over long times.
  • Non-Hermitian Quenches: Similar trends are observed under non-unitary evolution (imaginary fields). States deep in the stabilizer regime retain a large plaquette response under weak non-Hermitian effects, whereas states near the crossover are rapidly destabilized. The loss of the plaquette response correlates with the emergence of complex eigenvalues in the spectrum.

• Scalable Implementation:

  • The projection-based preparation on the $5 \times 5$ lattice successfully generates a state in the target stabilizer sector.
  • The lattice-averaged plaquette response ($\bar{F}$) exhibits only weak degradation under intentionally amplified local coherent perturbations, confirming the local stability of the prepared topological structure.
  • The layered local structure of the circuit allows for parallel execution, suggesting a favorable path for scaling to larger systems without requiring fully nonlocal circuits.

Significance and Claims
The paper claims to provide a practical route for preparing and probing topological signatures on near-term quantum processors. Its primary contribution is not the demonstration of thermodynamic topological order (which is beyond current device capabilities) but a hardware-resolved characterization of the parameter and noise regimes where finite-size topological features remain distinguishable from trivial behavior.

Key claims include:

1. Feasibility: Finite-size topological diagnostics (both local and nonlocal) can be resolved on NISQ hardware despite realistic imperfections, provided the system is sufficiently deep in the stabilizer-dominated regime.
2. Robustness Hierarchy: Local stabilizer signatures are significantly more robust against disorder and noise than nonlocal Wilson loop signatures.
3. Scalability: A projection-inspired, non-variational preparation protocol offers a viable alternative to deep variational training for scaling topological simulations to larger 2D lattices on hardware with suitable connectivity.
4. Dynamical Memory: Non-equilibrium dynamics retain a clear memory of the underlying equilibrium crossover, with dynamical stability serving as a complementary probe for phase transitions.

The work connects controlled finite-size characterization with a scalable hardware implementation, offering a unified framework to study topological order under static disorder, accumulated noise, and effective non-Hermitian fields.