Methods from the literature

Every method on this page comes from a published paper with a power system application, ships as a first-class QuGrid API, and carries a measured effect on a bundled problem. The numbers are produced and asserted by example 12 and example 13 — run them to reproduce everything below (measured 2026-08-13).

Unbalanced penalization for inequalities

What it is. The textbook route for an inequality constraint on an annealer adds binary slack variables until the constraint becomes an equality, then squares it. The slack register costs qubits — PMU placement on the PJM 5-bus system needs 10 slack bits on top of 5 placement bits. Unbalanced penalization replaces the slack construction with the asymmetric penalty -l1*g + l2*g^2 on the constraint function g >= 0: zero extra qubits, at the price of an approximation whose penalty minimum does not exactly coincide with the constraint boundary.

API.

builder.add_inequality(terms, constant, weight=w)                          # slack (default)
builder.add_inequality(terms, constant, method="unbalanced", lam=(1, 2))   # slack-free

Papers. A. Lucas, "Ising formulations of many NP problems", Frontiers in Physics 2, 5 (2014), arXiv:1302.5843 (the slack construction). J. A. Montanez-Barrera, D. Willsch, A. Maldonado-Romo, K. Michielsen, "Unbalanced penalization: a new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms", Quantum Science and Technology 9, 025022 (2024), arXiv:2211.13914.

Measured effect (example 12). PMU placement, PJM 5-bus, depth-2 QAOA, seed 0:

encoding	qubits	dynamic range	P(optimum)	gap
slack	15	4.0	0.0014	0%
unbalanced	5	3.5	0.5364	0%

Both encodings agree with exhaustive enumeration (2 PMUs). The 383x success-probability gain is the removed slack register. Caveat: the lam pair needs per-problem-family tuning — the knapsack values from the paper (0.96, 0.0371) do not transfer to this coverage problem (they return 5 PMUs); the QuGrid default (1, 1) works here. Always check is_feasible on the returned answer.

Augmented Lagrangian outer loop for equalities

What it is. A fixed quadratic penalty forces a bad choice: too small returns infeasible answers, large enough for exactness stretches the QUBO coefficient range until sampling solvers cannot resolve the cost differences underneath. The augmented Lagrangian keeps the quadratic weight small and moves constraint pressure into linear multiplier terms, re-solving the QUBO between multiplier updates (u += 2*lam*g(x*), lam *= alpha).

API.

from qugrid.methods import AugmentedLagrangianLoop
loop = AugmentedLagrangianLoop(objective_qubo, constraints, solver="sa")
result = loop.run(seed=0)      # standard Result; loop.history has every iterate

Papers. J. Hong, Y. Xu, F. Teng, "Quantum annealing-aided aggregated unit commitment", arXiv:2502.15917. W. Feng, Y. Zhang, M. A. Bragin, Y. Zhou, "Scalability and performance of quantum computing for unit commitment", IEEE Transactions on Power Systems 38(3), 2023.

Measured effect (example 12). Unit commitment, 2 units x 2 periods, demand [60, 130] MW:

method	dynamic range	gap	feasible
fixed penalty (QAOA p=2)	3.9e3	383%	no
augmented Lagrangian (SA inner)	180	0%	yes

The loop reaches the feasible optimum (cost $2908) in 3 outer iterations; its objective QUBO stays 21x flatter than the fixed-penalty encoding. The qugrid.solve front door warns (RuntimeWarning) whenever a sampling solver receives a QUBO with dynamic_range() > 1e3, and QUBO.dynamic_range() is the diagnostic behind that warning.

Greedy repair and the random+repair baseline

What it is. Sampling solvers put probability mass on infeasible bitstrings whenever penalties compete with costs. Greedy repair post-processes each sample: flip the bit (or bit pair, when single flips stall) that most reduces the problem's constraint_residual, until feasible. The honest control is uniform random sampling pushed through the same repair — a hybrid pipeline that cannot beat random+repair has no quantum content.

API.

qg.solve(prob, solver="qaoa", repair="greedy")   # repairs answer + distribution
qg.solve(prob, solver="random+repair")           # the baseline to beat
result.extras["p_optimum_raw"], result.extras["p_optimum_repaired"]

Papers. Z. Jiang et al., "REGRID-QAOA" (grid islanding on IBM hardware, 9-57 buses, repair as the core contribution), arXiv:2606.15083. I. Gaidai, A. Mukherjee (negative result: hybrid pipelines at 5-13 qubits failed to beat random sampling plus classical evaluation), arXiv:2607.15543.

Measured effect (examples 07 and 12). Repair reaches feasibility from every uniform random state on all three bundled formulations (tested over seeded batches). On unit commitment it turns depth-2 QAOA's P(optimum) = 0.000 into P(optimum | repaired) = 0.011, and turns plain random sampling (gap 77%) into an exact solver (gap 0) at 256 samples. On islanding and PMU placement random+repair equals plain random sampling — almost every uniform sample is already feasible there, and repair restores feasibility, it does not optimize.

Tabu search and parallel tempering

What it is. Stronger classical opponents than exhaustive enumeration and plain simulated annealing. Tabu search is deterministic best-move local search with a recency memory; parallel tempering runs a ladder of Metropolis replicas with exchange moves, the strongest widely reported classical heuristic on sparse QUBOs. Any quantum claim built on QuGrid should survive both.

API.

qg.solve(prob, solver="tabu", seed=0)
qg.solve(prob, solver="pt", seed=0)

Papers. The comparison practice follows P. Kaseb et al., "Quantum annealing versus classical solvers for grid partitioning", arXiv:2505.15978.

Measured effect (example 07). Both reach gap 0 on all three bundled problems, every seed, in under 0.04 s — the honest bar that QAOA (255% mean gap on unit commitment) does not clear.

Warm-start QAOA

What it is. Vanilla QAOA starts from the uniform superposition and must find good regions from scratch at every depth. Warm-start QAOA starts from a product state biased toward a classical solution c in [0,1]^n (qubit i measures 1 with probability c_i) and replaces the mixer with one that keeps that state as its ground state. An epsilon clip keeps every bit movable.

API.

qg.solve(prob, solver="qaoa", warm_start=classical_x)     # any [0,1]^n vector
qg.solve(prob, solver="qaoa", warm_start="relaxation")    # built-in box relaxation

Papers. D. J. Egger, J. Marecek, S. Woerner, "Warm-starting quantum optimization", Quantum 5, 479 (2021). A. Salgado, A. Sequeira, J. Santos (unit commitment at p=1 within 5.1% of reference), arXiv:2412.11312.

Measured effect (example 13). Islanding, WSCC 9-bus, depth p=1: P(optimum) rises from 0.009 (vanilla) to 0.128 warm-started at the simulated-annealing answer — one order of magnitude from the same circuit depth. With epsilon=0 the start state freezes and P(optimum) = 1 trivially, which is why the clip defaults to 0.25.

XY mixer for one-hot structure

What it is. When variables form one-hot groups (a unit picks exactly one output level), the ring-XY mixer conserves Hamming weight inside each group: initialize each group in its W state and every reachable state satisfies the one-hot constraint with probability 1 — no penalty, no slack, no infeasible samples. The honest cost is qubit count: one-hot spends levels x units qubits where binary spends ceil(log2 levels) x units; the built-in 22-qubit statevector ceiling holds about 7 units at 3 levels.

API.

from qugrid.solvers import XYMixer
qg.solve(prob, solver="qaoa", mixer=XYMixer(groups))   # groups: list of qubit lists

Papers. Z. Wang, S. Hadfield, Z. Jiang, E. G. Rieffel, "XY-mixers: analytical and numerical results for the quantum alternating operator ansatz", Physical Review A 101, 012320 (2020). N. Mohseni et al. (biased variant, single-period unit commitment on IBM hardware), arXiv:2603.00260.

Measured effect (example 13). 2-unit, 3-level one-hot dispatch, p=2: the XY mixer holds 100.0% of the output distribution on one-hot-feasible states and hits the optimum exactly (gap 0); the penalty encoding of the same problem puts 22% of its mass on feasible states. Probability mass outside the feasible subspace is 0 to numerical precision (asserted at 1e-9 in the test suite).

Fixed transferred angles

What it is. The classical angle optimization is the expensive part of QAOA. Angle transfer reuses angles optimized on one instance for another instance of the same problem family, skipping the optimizer entirely.

API.

res = qg.solve(small_instance, solver="qaoa", p=1)
qg.solve(big_instance, solver="qaoa", p=1,
         angles=(res.resources["gammas"], res.resources["betas"]))

Every optimized run reports its best angles in resources["gammas"]/["betas"], so transfer experiments can harvest them.

Papers. H. Jing, Y. Wang, Y. Li, "Data-driven quantum approximate optimization algorithm for power systems", Communications Engineering 2, 12 (2023) — transfer keyed by normalized graph density on IEEE 24-bus maximum power sections.

Measured effect (example 13). Depth-1 angles optimized on 9-bus islanding, transferred to 14-bus islanding: expectation 211.9 against 202.9 for the full optimization (within 5%), at 1 circuit evaluation against 929. Arbitrary angles miss by ~90%. Caveat: the published transfer evidence is MaxCut-like; on penalty-stretched formulations it is untested — measure before trusting.

What QuGrid deliberately does not ship

Recursive QAOA, multi-angle QAOA, CVaR aggregation, reverse annealing, Grover mixers, and simulated bifurcation have no power or energy application paper in the survey behind this page; they stay out until one exists. ADMM and Benders-style decomposition are deferred to a later release — they deserve their own design round, not a corner of this one.