Shot-based variational simulation of the toric code phase transition: Energy, global entanglement, and noise resilience

AI summary

Overview

Research area: Quantum simulation of topological phases of matter — specifically variational quantum eigensolver (VQE) simulation of the toric code in a uniform magnetic field, including entanglement-based detection of the phase transition and the effect of realistic gate and readout noise.

Technical level: Advanced. The paper assumes familiarity with stabilizer codes, VQE, entanglement measures, finite-size scaling, and quantum noise channels.

Scope (one sentence): The authors use shot-based (finite-measurement) simulation on Qiskit Aer's QASM Simulator to show that small toric-code lattices can reveal the topological phase transition through ground-state energy derivatives and global entanglement, and that the transition point moves linearly with noise strength.

What This Paper Is About

Topologically ordered phases cannot be described by any local order parameter, so detecting the transition out of such a phase requires non-local quantities or careful scaling analysis. The authors ask whether a simple, hardware-feasible variational quantum circuit — evaluated only through a finite number of measurement shots, as on a real device — is enough to locate the phase transition of the toric code when a uniform magnetic field is applied. A secondary goal is to test how much that detection survives realistic depolarizing and readout noise.

Key Contributions

1. A constructive, scalable circuit-building recipe for the toric code on a torus. Starting from the 2×2 lattice circuit of Chen and co-workers, the authors introduce "vertex inserting" and "face adding" lattice-expansion moves (adapted from triangulation transformations and from Vidal's renormalization framework) to grow the circuit to arbitrary L×L′ lattices with periodic boundary conditions.

2. A shot-based VQE study of the magnetic-field transition at three lattice sizes. The pure toric code circuit is converted into a variational ansatz by replacing Hadamard gates with single-qubit R_y(θ) rotations, and the ground-state energy is computed from finite measurement budgets (1024 shots for 2×2, 512 for 3×3, 32 for 4×4) over 0 ≤ h ≤ 2 sampled at 200 points.

3. Finite-size scaling of the critical field from noisy energy data. The first derivative dE/dh is fitted with a sigmoid, and the extracted critical fields for L = 2, 3, 4 are extrapolated linearly in 1/L to obtain h_c^(∞) ≈ 0.333, consistent with the established value h_c = 0.328.

4. A second variational circuit for global entanglement, plus a noise study. Because the state vector is unavailable in a shot-based setting, a secondary variational circuit minimizes the Shannon entropy of rotated single-qubit measurements to estimate the von Neumann entropy, giving global entanglement; one- and two-qubit depolarizing channels and readout errors are then added to assess hardware viability.

Main Findings

• Minimal lattices already show the transition: Even on the 2×2 lattice with 1024 shots, the energy derivative changes from values close to zero in the topological phase to large negative values in the polarized phase; a sigmoid fit gives h_c ≈ 0.404, slightly above the thermodynamic-limit value h_c ≈ 0.33–0.34, as expected for finite size.

• Derivatives are noisier than energies: The paper notes that sampling fluctuations are small in the energy itself but become more pronounced when numerical derivatives are evaluated, motivating the smooth sigmoid fit and the use of the analytic second derivative −∂²E/∂h² (which shows a peak near the transition) rather than raw numerical second derivatives.

• Sharpening with system size: Comparing 2×2, 3×3 and 4×4, the transition region becomes sharper and the peak in −∂²E/∂h² grows with lattice size, consistent with a developing singularity. The 3×3 data show noticeably stronger statistical fluctuations because the shot budget was reduced, and the 4×4 case used only 32 shots.

• Extrapolated critical field: Plotting h_c(L) against 1/L and fitting linearly yields h_c^(∞) ≈ 0.333 at 1/L → 0, indicating that deviations at small L are predominantly finite-size effects. The authors flag the limited number of accessible system sizes as a caveat.

• Global entanglement behaves as expected: On the 2×2 lattice, global entanglement is close to 1 at h ≈ 0 (its maximum for a single qubit with base-2 logarithm) and decreases continuously toward zero at larger field. The abstract reports that both global entanglement and its conditional counterpart show the expected qualitative behavior at the transition, with conditional global entanglement showing a peak at the transition point, in agreement with prior work on a nonlinear perturbation.

• Noise resilience: With one- and two-qubit depolarizing noise applied after the corresponding gates plus readout errors, the topological phase shows notable robustness, manifesting as a linear decrease of the transition point with increasing noise strength. The precise numerical noise probabilities and the fitted slope are not reported in the provided text.

Methodology in Plain English

The toric code is built on a square lattice wrapped onto a torus, with qubits on the edges and commuting stabilizer checks on vertices and plaquettes. First, the authors write down an explicit gate sequence that prepares the exact toric code ground state — the equal superposition of closed loops — on a 2×2 lattice. They then grow that circuit to larger lattices by two local maneuvers: inserting a new vertex by adding an ancilla qubit in |0⟩ and applying CNOTs from its neighbours into it, and adding a face by adding an ancilla in |+⟩ and applying CNOTs out of it to its neighbours. This makes the circuit pattern for any size systematic.

To handle the magnetic field, they keep the same circuit but swap every Hadamard for a rotation R_y(θ). At θ = π/2 the rotation is exactly a Hadamard and the pure toric code is recovered; at θ = 0 it is the identity and the output is the fully polarized state that the infinite-field ground state approaches. Intermediate θ values reweight the loop configurations, which is precisely what the field does physically.

They then run VQE: samples from the QASM Simulator estimate the expectation values of the Hamiltonian's Pauli terms (with basis rotations and CNOT fan-in so that a plaquette operator is read out from a single qubit), the energy is minimized over θ, and this is repeated for each field value. To get entanglement without access to the state vector, they append a second variational layer — a single-qubit rotation by angle α on each qubit — and use the fact that the von Neumann entropy equals the minimum over measurement bases of the Shannon entropy of the outcome distribution. Finally, they re-run the energy analysis with depolarizing and readout noise inserted into the circuit.

Why This Matters

The work bridges ideal-circuit proposals and what a real device can deliver: it shows that a phase transition in a topologically ordered system can be located with small lattices, modest circuits, and finite sampling, and that the signature degrades gracefully rather than catastrophically under noise. For topological quantum computing, this is evidence that the order being protected is not fragile to the very noise the code is meant to combat.

Real-world applications:

• Quantum error correction: The toric code underlies surface-code architectures; understanding its phase boundary under field-like perturbations informs how much biased noise or stray fields a encoded logical qubit can tolerate.
• Hardware benchmarking: The shot-based energy and entanglement protocol offers a concrete, scalable task for validating near-term superconducting and neutral-atom processors (the paper cites superconducting qubits and Rydberg atom arrays as relevant platforms).
• Condensed-matter simulation: The same VQE-plus-finite-size-scaling template transfers to other perturbed stabilizer Hamiltonians where classical methods are expensive.
• Entanglement metrology: The second variational circuit gives a measurement-only route to von Neumann entropy and multipartite entanglement measures, useful wherever state tomography is infeasible.

Industry relevance: The methods use standard tooling (Qiskit Aer's QASM Simulator, Clifford plus rotation gate sets, Pauli-term measurement groupings), so the protocol maps directly onto current cloud quantum backends and error-mitigation pipelines. The linear-in-noise shift of the critical point gives hardware teams a simple calibration-style metric for how much noise budget a topological-phase experiment can absorb.

Future Directions

• Larger lattices with better shot budgets. The 4×4 case used only 32 shots because of computational constraints, leaving the finite-size scaling fit based on three sizes (L = 2, 3, 4). More sizes and restored shot counts would test whether the linear 1/L ansatz holds and tighten the h_c^(∞) ≈ 0.333 estimate.
• Richer ansätze. The ansatz is a minimal parametric extension (a single rotation angle replacing each Hadamard), so it cannot represent every ground state; adding parameters or adaptive layers could improve energies away from the two limits.
• Beyond depolarizing noise. Only depolarizing channels and readout errors are modeled; biased noise, coherent errors, crosstalk, and device-specific calibration drift remain to be examined, as does whether the linear shift of the transition point persists.
• Entanglement measures beyond GE. The paper focuses on global entanglement and conditional global entanglement rather than topological entanglement entropy, which is known to jump at this transition; computing it under shot noise would be a natural, harder next test.

Target Audience

Quantum-information and condensed-matter theorists working on variational simulation, topological order, and entanglement measures; quantum-computing experimentalists and hardware teams looking for concrete benchmark protocols that run on today's noisy devices; and graduate students or advanced undergraduates with a background in stabilizer codes and VQE who want a worked example of extracting a phase transition from finite-shot data.

Authors’ abstract

We investigate the capability of variational quantum circuits in detecting a topological phase transition within the toric code model subjected to a uniform magnetic field. The detection is carried out via shot-based computations of both the ground state energy and the global entanglement. We begin by employing well-established renormalization techniques to construct a quantum circuit that accurately represents the pure toric code under periodic boundary condition. This circuit is subsequently parameterized to enable a shot-based variational simulation of the ground state in the presence of the magnetic field. Our results demonstrate that even for the minimal lattice sizes and a finite number of measurement shots, the ground state energy exhibits a distinct singularity at the transition point which can be reliably approximated through finite-size scaling analysis. Furthermore, we introduce a secondary variational circuit designed to compute the global entanglement of the generated ground state. We observe that both global entanglement and its conditional counterpart display the expected qualitative behavior at the transition point, consistent with the topological nature of the phase transition. To evaluate the practical viability of our variational approach on real quantum hardware, we systematically examine the detrimental effects of gate noise on the analysis. We find that the topological phase exhibits a notable robustness against noise, manifesting as a linear decrease in the transition point with increasing noise strength.