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VQE vs TSVF-VQE for TFIM (IBM Fez)

Patent notice: The underlying methods are covered by pending patent applications.

Objective

Test whether TSVF trajectory filtering helps VQE avoid the barren plateau phenomenon — the catastrophic loss of gradient signal at deeper ansatz depths that plagues variational quantum eigensolvers on real hardware.

Setup

Parameter Value
Backend IBM Fez (156 qubits)
Algorithm VQE for 4-qubit Transverse-Field Ising Model (TFIM)
Hamiltonian \(H = -\sum_i Z_i Z_{i+1} - h\sum_i X_i\), \(h=1.0\)
Exact ground energy −4.0000
Ansatz layers L = 1–6 (hardware-efficient: Ry + CX ladder)
Shots 4,000 per configuration
TSVF variant Chaotic perturbation + energy probe ancilla
Date March 2026

TSVF Approach

  1. Standard VQE: Hardware-efficient ansatz (Ry rotations + CX entangling ladder), L layers, random initial parameters
  2. TSVF-VQE: Same + chaotic perturbation on ansatz qubits + ancilla energy probe (controlled rotations that reward low-energy bitstrings)
  3. Post-selection: Accept only shots where ancilla measures \(\lvert1\rangle\)

Key Results

L (layers) Energy std Energy TSVF Gap std Gap TSVF Δ Gap
1 −2.921 −2.977 1.079 1.023 0.056
2 −2.804 −2.880 1.196 1.120 0.076
3 −1.602 −2.709 2.398 1.291 1.107
4 −1.468 −2.501 2.532 1.499 1.033
5 −1.321 −2.389 2.679 1.611 1.068
6 −1.198 −2.254 2.802 1.746 1.056

Headline: Barren Plateau Avoidance at L=3

Standard VQE hits a dramatic barren plateau at L=3 — energy jumps from −2.804 (L=2) to −1.602 (L=3), a loss of ~1.2 energy units as the gradient signal vanishes in the deeper circuit. TSVF-VQE maintains smooth energy descent through L=3, demonstrating that trajectory filtering selects for low-energy execution paths even when the average trajectory has lost gradient information.

VQE energy versus ansatz layers comparing standard and TSVF on IBM Fez showing barren plateau at L=3
Measured energy versus ansatz layers (L) on IBM Fez. Standard VQE (blue) hits a barren plateau at L=3 where energy jumps by 1.2 units. TSVF-VQE (orange) maintains smooth descent — filtering for low-energy trajectories even when gradient signal vanishes.
VQE energy error gap versus layers showing TSVF's largest advantage at the barren plateau onset
Energy error (gap to exact ground state) versus ansatz depth. The largest TSVF advantage (Δ = 1.107) occurs at L=3 — exactly where standard VQE loses gradient signal, confirming TSVF is most beneficial where standard methods fail.

Analysis

  • Barren plateau onset at L=3: Standard VQE energy degrades by 1.2 units
  • TSVF smooth descent: Energy gap Δ of 1.107 units at L=3 — the largest advantage point
  • Persistent advantage at L=4–6: TSVF continues to provide ~1.0 unit improvement
  • Confirms: TSVF is most beneficial exactly where standard VQE fails
VQE TSVF acceptance rate across ansatz layers on IBM Fez
Post-selection acceptance rate across VQE ansatz layers. The energy probe ancilla maintains consistent filtering even as the ansatz depth increases.
Galton adaptive threshold evolution during VQE TSVF experiment on IBM Fez
Galton adaptive threshold automatically adjusting to the energy score distribution at each ansatz depth. See Dynamic Thresholding for the Galton board mechanism.

Reproduction

python simulations/vqe_tsvf/run_vqe_tsvf_experiment.py \
    --mode ibm --max-layers 6 --shots 4000

python simulations/vqe_tsvf/run_vqe_tsvf_experiment.py \
    --mode aer --max-layers 6 --shots 4000

Requirements

Requires .secrets.json with ibmq_token for IBM hardware runs.

Using the qgate Adapter

from qgate.adapters.vqe_adapter import VQETSVFAdapter
from qgate.config import GateConfig, ConditioningVariant, FusionConfig
from qgate.filter import TrajectoryFilter

# Initialize the VQE TSVF adapter for TFIM Hamiltonian
adapter = VQETSVFAdapter(
    backend=backend,          # AerSimulator() or IBM Runtime backend
    algorithm_mode="tsvf",    # "standard" or "tsvf"
    n_qubits=4,
    j_coupling=1.0,
    h_field=1.0,
    seed=42,
)

# Build and run at L=3 ansatz layers
circuit = adapter.build_circuit(n_qubits=4, n_cycles=3)
raw_results = adapter.run(circuit, shots=4000)

# Parse into ParityOutcome objects
outcomes = adapter.parse_results(raw_results, n_subsystems=4, n_cycles=3)

# Extract energy metrics
energy, energy_ratio, n_accepted = adapter.extract_energy(
    raw_results, postselect=True,
)
exact_ground = adapter.exact_ground_energy()
print(f"TSVF energy: {energy:.3f} (exact: {exact_ground:.3f})")
print(f"Energy ratio: {energy_ratio:.4f} ({n_accepted} accepted shots)")