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Towards Scalable Quantum Unit Commitment in Power Systems: Encoding and Complexity Reduction Techniques

Marc Carrillo-MuÑoz, Sergio MartÍnez-Hermida, Sara Barja-MartÍnez, Antonio E. SaldaÑa GonzÁlez, MÒnica AragÜÉs-PeÑalba

arXiv:2609.36999v1quant-phcs.SYeess.SY

TL;DR

Restricted qubit availability and the cost of simulating or executing large Hamiltonians limit quantum optimisation as power-system problems grow. This paper combines PCE compression with SQD refinement for UC, and the combined workflow reaches fully concentrated feasible and optimal probability across the analysed system sizes.

  • Problem

    Quantum optimisation is constrained by limited available qubits and the cost of simulating or executing large Hamiltonians, while power-system binary-variable counts can grow with assets, time periods, and constraints.

    Source: §I · p. 2

  • Method

    The workflow combines PCE as a qubit-reduction preprocessing stage with SQD as a hybrid post-processing refinement for quantum UC solutions.

    Source: §I · p. 3

  • Results

    Popt = 1.0 and Pfeas = 1.0 across all analysed system sizes for PCE+SQD; PCE also permits original UC systems up to N = 20 in the present simulation environment.

    Source: §IV · p. 13, §V · p. 14

  • Takeaways & Limitations

    PCE reduces the qubits required during evolution, while SQD recovers a fully concentrated feasible and optimal solution, making the techniques complementary in the evaluated workflow.

    Source: §V · p. 14

    Reported computational-time benefits reflect classical state-space simulation; hardware evaluation is needed to determine how those reductions translate to practical resource and execution-time advantages.

    Source: §IV · p. 13

Abstract

from arXiv · show

This paper develops a Quantum optimisation Unit Commitment (UC) framework that benefits from Pauli Correlation Encoding (PCE) as a qubit reduction preprocessing technique and Sample-based Quantum Diagonalisation (SQD) as a hybrid post-processing refinement method. The UC problem is formulated in binary form, then transformed into a QUBO problem and mapped onto an Ising representation suitable for adiabatic quantum evolution within the Qibo framework. This study analyses how PCE and SQD techniques affect qubit scalability, computational effort, feasibility, and probability of recovering the optimal solution as the number of generators increases. The results show that PCE reduces the effective size of the encoded problem, SQD improves the concentration of probability on feasible and optimal solutions, and their combined application provides the best overall behaviour in several scalable case studies, achieving more reliable solutions with lower effective computational cost than the unreduced formulation. The repository is available open source in Github (link is not yet available until work is published. However, if requested by the reviewers, we can make it open for a scific time window, if requested).

I. INTRODUCTION

The paper addresses the absence of an integrated UC workflow for examining whether qubit reduction and post-processing preserve useful decoded schedules. It studies PCE and SQD in a scalable quantum optimisation pipeline.

I. INTRODUCTION

  • Existing studies leave unclear whether PCE-compressed UC Hamiltonians preserve feasible, near-optimal schedules and whether SQD improves decoded solutions.The paper studies these questions together in scalable UC instances as generator count increases.Source: §I · p. 3

II. METHODOLOGY

  • The workflow maps UC through binary, QUBO, and Ising forms, optionally compresses before evolution, optionally refines sampled states, then evaluates decoded schedules in the original problem space.The four configurations are direct adiabatic evolution, PCE-only, SQD-only, and combined PCE+SQD.Source: §II · pp. 3–4

MAPPING

  • For the commitment-based model, each committed unit contributes its available power, and omitting ramp costs makes each time step independent.Demand balance is incorporated as a QUBO penalty, with normalization used to reduce sensitivity to input cost and power-demand scales.Source: A. UNIT COMMITMENT FORMULATION AND… · p. 5
  • The QUBO is converted to an Ising Hamiltonian whose linear fields and pairwise couplings define the problem Hamiltonian for adiabatic evolution.The optional PCE stage can be applied before that evolution.Source: A. UNIT COMMITMENT FORMULATION AND… · pp. 5–6

B. PCE-BASED QUBIT REDUCTION

PCE reduces the register by encoding original logical spins as correlations among fewer reduced spins. This restricts which original UC schedules can be represented, while the encoding search prioritizes low-cost feasible regions.

B. PCE-BASED QUBIT REDUCTION

  • PCE maps an N-spin Ising Hamiltonian to a reduced Hamiltonian on m < N qubits, restricting evolution to logical configurations representable by the selected correlations.The encoded search space has at most 2^m states rather than 2^N, but the representation is not lossless.Source: B. PCE-BASED QUBIT REDUCTION · p. 6
  • Smaller m strengthens qubit reduction but limits the original UC schedules that can be recovered after decoding.Only schedules representable by the selected encoding can be recovered.Source: B. PCE-BASED QUBIT REDUCTION · p. 6
  • Replacing logical spins with Pauli-Z strings can turn quadratic Ising terms into higher-order interactions, while decoded candidates remain assignments in the original UC space.The correlation-order parameter controls encoding expressiveness but does not directly impose final Hamiltonian locality.Source: B. PCE-BASED QUBIT REDUCTION · pp. 6–7
  • The demand-shell-guided search scores candidate encodings by weighted coverage of feasible target-shell configurations, prioritizing low-cost regions.The weighting selects the encoding; the subsequent adiabatic evolution uses the resulting reduced Hamiltonian.Source: B. PCE-BASED QUBIT REDUCTION · p. 7

C. ADIABATIC QUANTUM EVOLUTION AND SAMPLING

The quantum solver evolves an initial state using either the original or PCE-compressed problem Hamiltonian, then measures the final state to obtain candidate bitstrings. Evolution settings are held constant when comparing the original and compressed Hamiltonians.

C. ADIABATIC QUANTUM EVOLUTION AND SAMPLING

  • The solver uses the original N-qubit Ising Hamiltonian without PCE and the reduced m-qubit Hamiltonian with PCE.The selected Hamiltonian becomes the problem Hamiltonian used for quantum evolution.Source: C. ADIABATIC QUANTUM EVOLUTION AND… · p. 7
  • The lowest-energy state of the problem Hamiltonian is targeted by interpolating it with an easy initial driver Hamiltonian.The driver Hamiltonian has a known lowest-energy state, and boundary conditions define the evolution path.Source: C. ADIABATIC QUANTUM EVOLUTION AND… · p. 7
  • Discrete evolution steps approximate continuous evolution, with final time T and time step Δt affecting approximation quality and simulation cost.Qibo implements the evolution, using the same settings for original and PCE-compressed Hamiltonians to isolate encoding effects.Source: C. ADIABATIC QUANTUM EVOLUTION AND… · p. 7
  • Repeated computational-basis measurements produce bitstring frequencies, which can be decoded directly or first refined by optional SQD post-processing.The most probable measured bitstrings are candidate solutions from the quantum evolution.Source: C. ADIABATIC QUANTUM EVOLUTION AND… · p. 7

D. SQD POST-PROCESSING

SQD refines measured configurations by projecting an auxiliary Hamiltonian into a subspace built from observed bitstrings and diagonalising it classically. It operates in the same effective space as the preceding evolution and cannot add configurations outside the sampled subspace.

D. SQD POST-PROCESSING

  • SQD uses measured bitstrings to build a reduced subspace, projects an auxiliary Hamiltonian into it, and diagonalises the projection classically without a new quantum simulation.SQD acts in the reduced m-qubit space with PCE, or the original N-qubit space otherwise.Source: D. SQD POST-PROCESSING · p. 7
  • The subspace retains the most frequently measured bitstrings according to a target fraction, but cannot exceed the distinct observed states.When fewer distinct states are observed than the target dimension, the sampled set caps the subspace size.Source: D. SQD POST-PROCESSING · p. 8
  • The auxiliary Hamiltonian balances the problem Hamiltonian and transverse-field contribution through β, with values near one favouring low-energy states under the original problem Hamiltonian.Smaller β increases the influence of one-bit-flip couplings between sampled configurations.Source: D. SQD POST-PROCESSING · p. 8
  • The projected matrix preserves sampled-state energies and couples two sampled configurations only when they differ by one bit, with coupling weight 1−β.Diagonal entries come from problem energies; off-diagonal entries come from the transverse-field contribution.Source: D. SQD POST-PROCESSING · p. 8
  • The ground-state eigenvector of the projected matrix supplies a refined amplitude distribution over the retained sampled states.SQD cannot introduce configurations outside the sampled subspace, so its quality depends on whether measurements include feasible or near-optimal candidates.Source: D. SQD POST-PROCESSING · p. 8

E. DECODE AND RECOVER UC SOLUTIONS

Decoding maps measured or SQD-refined states back into binary UC schedules, whose probabilities are aggregated when multiple effective states map to the same schedule. Evaluation uses the original UC objective and constraints and reports feasibility, optimality, qubit reduction, and runtime.

E. DECODE AND RECOVER UC SOLUTIONS

  • Decoding converts direct or SQD-refined outputs into binary UC variables, using the PCE correlation map when states are in the reduced space.Different reduced states may decode to the same schedule, so their probabilities are aggregated in the original UC decision space.Source: E. DECODE AND RECOVER UC SOLUTIONS · pp. 8–9
  • Decoded schedules are evaluated against the original UC formulation, regardless of whether candidates came from direct evolution, PCE, SQD, or their combination.Feasibility and optimality are not assessed solely through the Hamiltonian explored during quantum evolution.Source: E. DECODE AND RECOVER UC SOLUTIONS · p. 9
  • Feasibility requires demand balance and the remaining operational constraints; in the simplified commitment-based instances, feasibility is mainly determined by power balance at the natural discrete resolution.The demand-balance residual is computed for each decoded schedule.Source: E. DECODE AND RECOVER UC SOLUTIONS · p. 9
  • The evaluation reports probability mass on feasible and optimal or best-known decoded schedules, alongside the best feasible schedule and its UC cost.Costs use the original UC cost function or the corresponding generation-cost expression when dispatch or discretised production levels are included.Source: E. DECODE AND RECOVER UC SOLUTIONS · p. 9
  • Relative qubit reduction compares the original N logical qubits with effective qubits, while runtime includes evolution and applicable PCE and SQD processing times.The same criteria compare baseline, PCE-only, SQD-only, and combined workflows.Source: E. DECODE AND RECOVER UC SOLUTIONS · p. 9

III. CASE STUDIES AND RESULTS

The case studies compare PCE and SQD individually and together across progressively larger UC instances, tracking scalability, computational cost, feasibility, and optimal-solution probability. Direct evolution configurations are evaluated only up to N = 12, while PCE-based approaches extend to N = 20.

III. CASE STUDIES AND RESULTS

  • The evaluation compares PCE and SQD separately and in combination across system sizes, measuring scalability, computational cost, feasible-state probability, and optimal-solution probability.The number of generators and consequently the number of qubits increases across the analysis.Source: §III · p. 9
  • The analyzed instances range from N = 4 to N = 20 generators and progressively add units while preserving previously defined generators.This construction increases the number of binary variables and original Hilbert-space dimension with system size.Source: §III · pp. 9–10
  • The reported setup fixes η = 0.5, uses PCE correlation orders k = (1, 2, 3, 4), and applies 3000 trials per reduced dimension with random seed 0.The demand shell uses λcost = 0.20; the reduced dimension is calibrated and then fixed for each system size.Source: §III · p. 10
  • Direct AE and AE+SQD are evaluated up to N = 12, whereas PCE-based approaches are evaluated up to N = 20.The original-Hamiltonian state-vector simulation becomes increasingly demanding as 2^N grows; PCE compresses the problem before evolution.Source: §III · p. 10

A. PCE BASED UC

PCE compresses the UC Hamiltonian to fewer qubits while aiming to preserve useful feasible and high-quality schedules. Results show that the amount of compression alone does not determine solution quality.

A. PCE BASED UC

  • PCE compresses the original N-qubit Hamiltonian to m < N qubits, reducing the Hilbert-space dimension evolved while seeking to preserve relevant feasible and high-quality schedules.The study evaluates several reduced dimensions for each original system size.Source: A. PCE BASED UC · p. 10
  • The selected reduced dimension maximizes Popt, and the reported configuration metrics also include feasible probability, feasible-state coverage, preprocessing time, and evolution time.Cfeas is shown as the fraction of original feasible configurations represented by the selected encoding.Source: A. PCE BASED UC · p. 10, Table 1 · p. 11
  • The selected PCE reduction ranges from 40% to 60% across instances, but compression level does not predict solution quality linearly.Figure 3 summarizes feasible-state coverage, feasible and optimal probabilities, and reduced qubit count for the selected configurations.Source: A. PCE BASED UC · p. 10, Figure 3 · p. 11
  • For N = 18 and N = 20, PCE retains only 2/24 and 2/28 feasible configurations yet achieves Popt = 0.890 and 0.906; N = 8 performs worse despite retaining 4/9.The results indicate that preserving relevant feasible configurations and the resulting reduced Hamiltonian matter more than maximizing feasible-state coverage.Source: A. PCE BASED UC · p. 11

B. SQD UC

SQD post-processes measured distributions by constructing and diagonalising a reduced sampled subspace rather than changing the Hamiltonian before evolution. Across the analysed direct UC cases, it concentrates probability on feasible and optimal schedules with a small additional computational cost.

B. SQD UC

  • SQD refines the measured distribution by diagonalising a reduced sampled subspace; the selected operating point uses β = 1.0 and target retained fraction ρSQD = 1/8.Realized subspace size can differ from the nominal target depending on the number of distinct sampled configurations.Source: B. SQD UC · p. 11
  • SQD yields P_SQD_feas = 1.0 in every analysed direct case, while direct AE optimal probabilities range from 0.123 to 0.368.The reported SQD result concentrates the sampled distribution on the optimal schedule.Source: B. SQD UC · p. 11, Figure 4 · p. 11
  • For N = 12, SQD achieves a 98.9% dimensional reduction while retaining the optimal state.The SQD results cover direct UC cases from N = 4 to N = 12.Source: B. SQD UC · p. 11
  • SQD adds less than 0.02 s of computational cost in every analysed case, negligible relative to AE.Source: B. SQD UC · p. 11

C. PCE+SQD UC

The combined workflow first compresses the UC Hamiltonian with PCE, performs adiabatic evolution in the reduced space, and then applies SQD to the sampled distribution. Across analysed instances from N = 4 to N = 20, SQD recovers optimal and feasible probabilities of 1.0.

C. PCE+SQD UC

  • PCE compresses the N-qubit Hamiltonian to m qubits for adiabatic evolution, after which SQD refines the sampled reduced distribution.The PCE dimension is fixed to the configuration selected earlier; SQD uses β = 1.0 and ρSQD = 1/8.Source: C. PCE+SQD UC · p. 12
  • For every analysed instance from N = 4 to N = 20, combined PCE+SQD recovers Popt = 1.0 and Pfeas = 1.0.This holds even when PCE probabilities are moderate or low, and as retained feasible-state coverage decreases with problem size.Source: C. PCE+SQD UC · p. 12, Figure 5 · p. 12
  • The combined results show that full feasible-state coverage is not required for successful recovery of the optimum when the optimal configuration remains in the reduced sampled space.PCE supplies dimensional reduction for larger UC instances, while SQD is described as a low-cost refinement stage.Source: C. PCE+SQD UC · p. 12

IV. DISCUSSION

PCE reduces the evolved problem dimension, while SQD refines sampled distributions; together they provide the strongest balance of scalability and solution quality across the evaluated workflows. The reported timing advantages are specific to classical emulation, and direct AE is limited by emulator memory beyond N = 12.

PCE

  • PCE reduces the evolved dimension by approximately 40%–60%, while Popt reaches 0.890 at N = 18 and 0.906 at N = 20.The compressed evolution remains instance-dependent; the N = 8 case illustrates that solution quality is not uniform.Source: §IV · p. 12
  • At N = 18 and N = 20, PCE represents only 2/24 and 2/28 feasible schedules, respectively, despite optimal probabilities near 0.9.The results associate useful reduced representations with retaining relevant feasible configurations rather than maximizing feasible-state coverage.Source: §IV · p. 12

SQD

  • After SQD, feasible and optimal probabilities reach 1.0 for every analysed instance, with sampled-space reductions up to 98.9% at N = 12.SQD post-processing remains below 0.02 s in direct cases, compared with AE times above 260 s at N = 12.Source: §IV · pp. 12–13

Combined workflow

  • Across analysed sizes, PCE+SQD reaches Popt = 1.0 and Pfeas = 1.0 while PCE enables evaluation through N = 20.The comparison treats dimensional reduction and solution concentration as complementary contributions.Source: §IV · p. 13, Figure 7 · p. 14

Runtime and scalability

  • Direct AE and AE+SQD are limited to N ≤ 12 in the emulator, whereas PCE enables evaluation at N = 14, 16, 18, and 20.PCE runtime is not strictly monotonic in N; it depends on the selected reduced dimension and encoded-Hamiltonian complexity, while SQD adds almost no runtime.Source: §IV · p. 13, Figure 6 · p. 13

Evaluation boundary

  • Classical-emulator timings reflect state-vector memory limits and reduced simulation state space, not physical-processor performance; hardware evaluation is needed to assess practical savings.The paper identifies quantum hardware implementation as necessary to determine whether the observed reductions translate into resource and execution-time advantages.Source: §IV · p. 13

V. CONCLUSIONS AND FUTURE WORK

The conclusions frame PCE and SQD as complementary stages: PCE reduces the qubits evolved and SQD concentrates solutions from compressed samples. The authors identify feasible-space sensitivity, more realistic UC conditions, and hardware evaluation as future work.

PCE

  • PCE reduces the evolved Hamiltonian by approximately 40%–60%, with Popt = 0.890 at N = 18 and 0.906 at N = 20.The conclusions note that high feasible-state coverage is not required when relevant configurations remain encoded.Source: §V · p. 14

SQD

  • SQD raises Pfeas and Popt to 1.0 for every analysed instance using strongly reduced sampled subspaces and negligible computational overhead relative to evolution.This is presented as a refinement of direct AE distributions, which have moderate feasible and optimal probabilities.Source: §V · p. 14

Combined workflow

  • The combined PCE+SQD workflow treats original UC systems up to N = 20 in the simulation environment and recovers fully concentrated feasible and optimal solutions from compressed samples.The authors characterize the techniques as complementary: PCE addresses dimensional scalability, while SQD improves solution quality.Source: §V · p. 14

Future work

  • Future analysis will test how feasible-schedule number and distribution affect PCE representation and SQD refinement, and study reduced-dimension selection and encoded-Hamiltonian spectra.These are proposed investigations, not evaluated findings in the reported conclusions.Source: §V · p. 14
  • The authors propose extending UC to multi-period constraints and additional generator characteristics, then testing on quantum hardware to assess practical resource and execution-time reductions.The current reductions are observed under classical emulation.Source: §V · p. 14

VI. APPENDIX

The appendix describes a penalty construction for the normalized UC QUBO: a minimum penalty preserves the feasible optimum, and an automatic scaling is used across instances.

BASELINE

  • The minimum penalty is chosen to remove lower-cost infeasible competitors so the minimum-cost feasible schedule remains the QUBO ground state.The construction considers the limiting penalty associated with an infeasible competitor.Source: A. DERIVATION OF THE AUTOMATIC PENALTY… · p. 14
  • The automatic baseline includes a dimension-dependent safety margin that is larger for small problems and approaches one as logical-qubit count increases.This scaling is used to obtain a common automatic baseline across the analysed instances.Source: A. DERIVATION OF THE AUTOMATIC PENALTY… · pp. 14–15
  • With η = 0.5 in the case studies, the effective penalty is set above the limiting value while avoiding an excessive increase with problem dimension.This is the penalty construction used in the main-text numerical experiments.Source: A. DERIVATION OF THE AUTOMATIC PENALTY… · p. 15
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