Filtering VQE (F-VQE)

Track: exact-K binary selection. Method ID: f-vqe.

run_fvqe implements the normalized parameter-shift procedure of Amaro et al., Filtering variational quantum algorithms for combinatorial optimization, Eqs. (6)–(7). Every iteration takes one gradient-descent step toward a freshly filtered version of the current state. It does not run Adam on a transformed energy expectation or reuse the CVaR objective.

For the current parameters \(\theta\), the frozen target is \(F|\psi(\theta)\rangle/\sqrt{\langle F^2\rangle_\theta}\). The distance gradient and update are

\[ g_j=-\frac{\langle F\rangle_{\theta+\pi e_j/2} -\langle F\rangle_{\theta-\pi e_j/2}} {4\sqrt{\langle F^2\rangle_\theta}}, \qquad \theta\leftarrow\theta-\text{stepsize}\,g. \]

Both implemented filters are positive and decrease with energy:

  • filter_type="inverse": \(f(\widetilde E;\tau)=\widetilde E^{-\tau}\).

  • filter_type="exponential": \(f(\widetilde E;\tau)=\exp(-\tau\widetilde E)\).

The original penalized cost is \(E(x)=\lambda x^\top\Sigma x-\mu^\top x+\alpha(\sum_i x_i-K)^2\). Coefficient-wise QUBO bounds \(L,U\) give the affine scale \(\widetilde E=\epsilon+(1-\epsilon)(E-L)/(U-L)\), with spectrum_floor \(\epsilon=0.001\) by default. This makes inverse filtering valid even for negative costs, without finding an optimum or enumerating the spectrum to obtain bounds. A constant Hamiltonian produces an identity filter. Moments are evaluated in log space. The energy trace retains original penalized units.

Filter strength

By default adapt_tau=True selects strength using the gradient norm. Starting at tau, it doubles trial strengths up to tau_max; a threshold crossing is refined by tau_search_steps bisections. It retains the tried strength with the closest norm at or below gradient_target. If the threshold is unreachable, selection remains bounded by tau_max. This is a bounded version of the paper’s heuristic; nonmonotone gradient norms need not yield a global best strength. All trial strengths reuse the same measured distributions. Fixed-strength execution uses adapt_tau=False and tau directly.

Defaults are tau=1, tau_max=64, gradient_target=0.1, and tau_search_steps=20. The selected strength and actual gradient norm are recorded at every update. Filter choice and strength affect the update direction; the code does not simply rescale a gradient to the target norm.

Execution and returned candidates

Core dependencies suffice. The default device is default.qubit.

  • shots_train=None: evaluate computational-basis probabilities. This path enumerates costs and retains shifted distributions, requiring exponential storage in the number of assets. It is intended for small simulator studies.

  • Positive shots_train: execute computational-basis sampling circuits and evaluate filter moments only on measured bitstrings. No spectrum enumeration, statevector access, Hadamard test, or dense filter matrix is required by the algorithm. A simulator backend still has its own state-storage costs.

  • All three binary ansätze are supported. Each update uses \(2m+1\) training circuit evaluations for \(m\) rotation parameters, with one additional final training evaluation: steps * (2*m + 1) + 1. Adaptive strength selection adds classical processing, but no circuit evaluations.

energy_trace includes the initial state (step 0), logging checkpoints, and the final post-update state. It records mean penalized energy, rather than the changing state-distance objective. Finite-shot energy estimates use the same unshifted samples as the filter normalization.

Final candidate sampling uses shots_sample. The best feasible sampled portfolio is exposed as x_best_feasible; callers use x_topk as an explicit fallback when no feasible sample exists. Analytic inclusion probabilities use the cached final training distribution; finite-shot inclusion probabilities use the final candidate batch. Candidate callbacks publish at step 0, logging checkpoints, and completion, with separate sample batches and a source label. Their cost is included in budgeted execution. Training and candidate sampling use separate seeded device streams, so callbacks do not change training. Devices must support the selected measurements and a seed constructor option.

Neither exact filtering nor a variational projection guarantees that a finite run returns an optimum. Finite-shot estimates and large learning rates can increase energy. Compare feasible candidates using the shared independent evaluator and matched end-to-end budgets.

Examples

Use the shared inputs.

from vqe_portfolio import FVQEConfig, PortfolioProblem, run_fvqe

cfg = FVQEConfig(k=2, lam=4, steps=25, filter_type="inverse")
result = run_fvqe(mu, Sigma, cfg)
candidate = result.x_best_feasible
if candidate is None:
    candidate = result.x_topk
problem = PortfolioProblem(mu, Sigma, cfg.lam, "binary_qubo", cfg.k)
print(problem.evaluate(candidate))
print(result.tau_trace, result.gradient_norm_trace)

vqe-portfolio f-vqe --mu "0.1,0.2,0.15" \
  --sigma "0.05,0,0;0,0.06,0;0,0,0.04" --k 2 --lam 4 \
  --filter exponential --fixed-tau --tau 1 --shots-train 1024 --steps 25

f-vqe-data accepts the same solver options with the market-data arguments. Both commands support --budget-seconds. The generic budget API also accepts run_budgeted(problem, "f-vqe", 10, config=cfg).

For synthetic CSV comparisons use python scripts/generate_comparison_results.py --methods classical,f-vqe. The small-instance validation and fixed-deadline benchmark runners accept the f-vqe method ID. These runners use adaptive inverse filtering; use the Python API or solver CLI to compare other filtering settings. Historical saved studies are unchanged and do not contain F-VQE results.