A Distributed Quantum Approximate Optimization Algorithm For Unit Commitment

arXiv:2608.04159v1 [cs.DC] 4 Aug 2026

A Distributed Quantum Approximate Optimization Algorithm For Unit Commitment Ali Rajabi

Milad Hasanzadeh

Amin Kargarian

Electrical and Computer Engineering Louisiana State University Baton Rouge, USA arajab2@lsu.edu

Electrical and Computer Engineering Louisiana State University Baton Rouge, USA mhasa42@lsu.edu

Electrical and Computer Engineering Louisiana State University Baton Rouge, USA kargarian@lsu.edu

Abstract—This paper presents a distributed quantum approximate optimization algorithm (DQAOA)-enabled three-block alternating direction method of multipliers (ADMM) framework for unit commitment (UC). The relaxed commitment and dispatch variables are solved in a continuous quadratic programming block, while the binary commitment block is formulated as a quadratic unconstrained binary optimization (QUBO) problem. The DQAOA interface allows this QUBO to be solved using brute-force enumeration, monolithic QAOA, or distributed QAOA, while the remaining ADMM updates are kept unchanged. In the distributed mode, the logical commitment qubits are allocated across multiple capacity-constrained quantum processing units (QPU), avoiding the requirement that the complete binary problem fits on a single device. The framework is evaluated on a five-unit UC instance containing 15 binary variables. All three solver modes reduce the ADMM primal residual below a certain tolerance and recover the same commitment schedule, dispatch, and operating cost. The results demonstrate solution consistency across the three solver modes and the multi-QPU capacity accommodation provided by the distributed QAOA method. Index Terms—Unit Commitment, Distributed Quantum Computing, Quantum Approximate Optimization Algorithm, Three-Block Alternating Direction Method of Multipliers, Quantum Processing Unit.

I. I NTRODUCTION Unit commitment (UC) is a fundamental scheduling problem in power system operation and electricity markets. It determines which generating units should be committed and how their outputs should be scheduled to meet system demand while satisfying operational requirements at minimum cost [1]. Because UC combines binary on/off decision variables with continuous dispatch variables, it is commonly formulated as a mixed-integer optimization problem [2]. The computational complexity of UC increases as the number of generating units, scheduling periods, and operating constraints grows. Consequently, the development of efficient UC solution methods remains an active research topic [3]. UC has traditionally been addressed using classical optimization techniques. Its binary decision component is This work was supported by the National Science Foundation under Grant ECCS-1944752 and Grant ECCS-2312086.

commonly handled through methods such as branch-and-bound, branch-and-cut, and decomposition algorithms, as well as heuristic and metaheuristic search procedures [3]. Although these approaches have been successfully applied to many UC formulations, their computational burden can increase considerably as the problem size and operational complexity grow. Therefore, decomposition methods have become particularly useful because they separate binary commitment decisions from continuous dispatch calculations and coordinate them via iterative updates. This structure also creates an opportunity to replace the binary optimization procedure while retaining efficient classical solvers for the continuous components. Among the available decomposition frameworks, three-block alternating direction method of multipliers (three-block ADMM) is particularly suitable for this purpose because it coordinates continuous and binary variable blocks within a unified iterative procedure [4]. In a three-block UC formulation, the relaxed commitment and dispatch variables are handled in the first block through a quadratic program (QP) with continuous variables. The binary commitment decisions are isolated in the second block, while the third block updates a slack variable to enforce consensus between the relaxed and binary commitment representations. This structure allows the solution method used for the binary block to be changed without modifying the remaining UC and ADMM calculations. A hybrid quantum–classical UC method based on this decomposition was previously presented in [5]. The second block can be expressed as a quadratic unconstrained binary optimization (QUBO) problem, which provides a standard representation for many combinatorial optimization models [6]. A QUBO can be transformed into an Ising cost Hamiltonian that is compatible with gate-based quantum optimization algorithms [7]. Among these algorithms, the quantum approximate optimization algorithm (QAOA) is designed specifically for discrete optimization and constructs a parameterized circuit by alternating problem-dependent cost operations with mixer operations [8]. A classical optimizer adjusts the circuit parameters to reduce the expected cost and increase the likelihood of observing low-cost binary solutions. QAOA and its variants have consequently received significant

attention for near-term combinatorial optimization [9]. Quantum approaches to UC have included annealing-based, variational circuit-based, and hybrid decomposition methods, as reviewed in [3]. The hybrid method in [5] demonstrated that a quantum routine can be used for the binary UC block while the continuous calculations remain classical. However, a monolithic quantum implementation requires the complete circuit and all logical data qubits to be accommodated by a single quantum processing unit (QPU). This requirement can become restrictive because near-term devices have limited qubit capacity, connectivity, and circuit fidelity [10]. Distributed quantum computing provides an alternative in which a quantum circuit is implemented across multiple connected QPUs [11]. Circuit operations involving qubits located on different QPUs can be implemented through distributed gate protocols and classical communication [12]. Distributed QAOA extends this idea to combinatorial optimization by assigning logical variables and circuit operations across multiple quantum devices [13]. Its main practical motivation is to use the combined qubit capacities of multiple QPUs rather than requiring the complete problem to fit on a single device. A recent study integrated a distributed variational quantum eigensolver (DVQE) into the binary block of a three-block ADMM formulation for UC [14]. Furthermore, the DQAOA framework introduced in [15] provides a common interface for solving QUBO problems using brute-force enumeration, monolithic QAOA, or distributed QAOA. The framework assigns logical data qubits based on the available QPU capacities and supports the distributed realization of interactions that span QPU boundaries. However, its integration as the binary solution routine within a complete three-block ADMM workflow for UC has not yet been evaluated in detail. This paper develops a DQAOA-enabled three-block ADMM framework for UC. At each ADMM iteration, the first block updates the relaxed commitment and dispatch variables. These updated values, together with the current slack and dual multiplier variables, are then used to construct the QUBO objective associated with the binary commitment block. The resulting QUBO is passed to the common DQAOA interface and solved using brute-force enumeration, monolithic QAOA, or distributed QAOA. The selected binary commitment solution is returned to the ADMM loop, after which the slack variable and dual multiplier updates are performed in the same manner for all three solver modes. Accordingly, the UC formulation, ADMM parameters, QUBO construction procedure, and convergence criterion remain unchanged across the three modes, while only the solution method for the binary block varies. The main contributions of this work are summarized as follows: •

The DQAOA package is integrated as a selectable solver for the iteration-dependent binary block within a three-block ADMM formulation for UC.

Brute force, monolithic QAOA, and distributed QAOA are evaluated using identical UC data, QUBO objective coefficients, ADMM settings, and convergence criteria. • The distributed implementation allocates the logical commitment qubits across multiple capacity-constrained QPUs while recovering the same commitment schedule, dispatch, and operating cost as the monolithic and brute-force modes for the tested case. •

II. P ROBLEM F ORMULATION This section introduces the unit commitment model considered in this work. It describes its decomposition into three coordinated subproblems: (i) a convex quadratic (QP) problem for relaxed binary commitment variables and continuous generation dispatch, (ii) a QUBO problem to determine the status of binary variables, and (iii) a linear optimization problem based on slack variable update with closed-form solution to enforce UC feasibility and satisfy consensus constraint. The adopted three-block ADMM formulation follows the general construction in [14], whereas the binary subproblem in the present work is solved using different modes including brute force, monolithic QAOA, or distributed QAOA. The remaining ADMM blocks and their associated constraints are kept unchanged across all solver modes. A. Unit Commitment Model Consider a scheduling horizon containing T time periods and a system with N generating units. The binary variable yi,t indicates whether unit i is committed during period t, and pi,t denotes its power output. The simplified UC problem is formulated as follows: min

yi,t , pi,t

s.t.

T X N X

Ai yi,t + Bi pi,t + Ci p2i,t ,

t=1 i=1 N X

pi,t = Lt , ∀t, i=1 pmin yi,t ≤ pi,t ≤ pmax yi,t , i i yi,t ∈ {0, 1},

∀i, t.

(1a)

(1b) ∀i, t,

(1c) (1d)

Here, Lt denotes the demand during period t. The coefficients Ai , Bi , and Ci represent the fixed, linear, and quadratic operating cost parameters of unit i, respectively. The quantities pmin and pmax specify the lower and upper generation i i limits of generating unit i. The objective function in (1a) minimizes the total operating cost over the scheduling horizon. Constraint (1b) maintains the supply–demand balance, while constraint (1c) couples the continuous dispatch variable to the binary commitment variable. When yi,t = 0, the corresponding generation output is forced to zero, while yi,t = 1 indicates that the unit is allowed to operate inside its admissible output range.

B. Three-Block ADMM Decomposition To separate the continuous generation variables from the binary commitment decisions, the binary variable yi,t in (1d) is first relaxed over the interval [0, 1]. An auxiliary binary commitment variable zi,t is then introduced to preserve the binary requirement. The relaxed and binary representations are coordinated through a slack variable ri,t and the associated consensus equality constraint. Following the three-block construction in [14], the binary constraint in (1d) is replaced by 0 ≤ yi,t ≤ 1,

∀i, t,

(2a)

zi,t ∈ {0, 1},

∀i, t,

(2b)

ri,t ∈ R,

∀i, t,

(2c)

yi,t − zi,t + ri,t = 0,

∀i, t.

(2d)

In this decomposition, yi,t remains as the continuous relaxed commitment, whereas zi,t carries the binary commitment requirement. The variable ri,t supports coordination between these two representations through (2d). Let λi,t denote the Lagrange multiplier associated with the consensus constraint, ρ > 0 denote the augmented Lagrangian penalty coefficient, and β > 0 denote the regularization parameter applied to r. Defining the continuous variables as ∆ = {y, p}, the augmented Lagrangian is expressed as Lρ (∆, z, r, λ) =

T X N X

where Q(k) , q (k) , and c(k) are obtained from the current ADMM variables and parameters. Equation (6) is the binary subproblem supplied to the selected solver. In this work, the QUBO problem is solved using brute-force enumeration, monolithic QAOA, or distributed QAOA. Therefore, only the solution method used in Block 2 changes among the three modes. c) Block 3: Slack variable update: Once the continuous and binary variables have been updated, the third block computes   r(k) = arg min Lρ ∆(k) , z (k) , r, λ(k−1) . (7) r

Since this subproblem is unconstrained and quadratic in r, its solution is obtained analytically as   (k) (k) (k−1) λi,t + ρ yi,t − zi,t (k) ri,t = − . (8) β+ρ d) Dual variable update: The Lagrange multipliers are subsequently adjusted using the consensus violation remaining after the three primal updates:  ρ  (k) (k) (k−1) (k) (k) λi,t = λi,t + yi,t − zi,t + ri,t . (9) 2 The ADMM iterations continue until the primal consensus residual (k) εpri = y (k) − z (k) + r(k) (10) 2

Ai yi,t + Bi pi,t + Ci p2i,t

t=1 i=1 T X N X

+

falls below a certain tolerance or the maximum number of ADMM iterations is reached. III. H YBRID DQAOA F RAMEWORK

λi,t (yi,t − zi,t + ri,t )

t=1 i=1

ρ β 2 2 + ∥yi,t − zi,t + ri,t ∥2 + ∥ri,t ∥2 (3) 2 2 The optimization remains subject to the power balance and generation limit constraints in (1b)–(1c), together with the domains specified in (2). The three-block ADMM procedure updates the continuous variables, binary variables, and slack variables sequentially, followed by a correction of the Lagrange multipliers. a) Block 1: Continuous variables update: With z, r, and λ fixed at their values from the preceding iteration, the relaxed commitment and dispatch variables are updated according to   ∆(k) = arg min Lρ ∆, z (k−1) , r(k−1) , λ(k−1) , (4)

At each ADMM iteration, the binary commitment block generates a QUBO objective whose coefficients depend on the current relaxed commitment, slack, and dual multiplier variables. The developed DQAOA package solves this QUBO problem using brute force, monolithic QAOA, or distributed QAOA. All three modes receive the same QUBO coefficients, while the continuous, slack, and dual updates remain unchanged. A. QUBO Encoding and Monolithic QAOA At iteration k, Block 2 is expressed as  T (k) F (k) (z) = z T H (k) z + f (k) z + c0 ,

subject to (1b)–(1c) and 0 ≤ yi,t ≤ 1. This block is a convex QP in the relaxed commitment and dispatch variables. b) Block 2: Binary commitment update: After obtaining ∆(k) , the binary commitment vector is determined from   z (k) = arg min Lρ ∆(k) , z, r(k−1) , λ(k−1) . (5) z∈{0,1}N T

The terms of (5) that depend on z can be collected in the standard QUBO form  T min z T Q(k) z + q (k) z + c(k) , (6) z∈{0,1}N T

(11)

z ∈ {0, 1}n .

∆

where n = N T is the number of binary commitment variables. Using the standard binary-to-Ising mapping zbj = (I − Zj )/2, the QUBO objective is transformed into a cost Hamiltonian expressed in terms of Pauli-Z operators. (k)

HC = α(k) I +

n X j=1

(k)

hj Zj +

X

(k)

Jjm Zj Zm ,

(12)

j<m

where I is the identity operator, Zj is the Pauli-Z operator (k) (k) acting on qubit j, and α(k) , hj , and Jjm are obtained from the QUBO coefficients.

QAOA alternates the unitaries generated by the cost and the mixer Hamiltonians. The mixer Hamiltonian is n X HM = Xj , (13) j=1

where Xj is the Pauli-X operator acting on qubit j. At depth p, the variational state is |ψp(k) (γ, β)⟩ =

p Y

(k)

e−iβℓ HM e−iγℓ HC |+⟩⊗n .

(14)

ℓ=1

A classical optimizer updates the variational parameters of the quantum circuit (γ and β) to minimize the expectation value of the cost Hamiltonian (k)

E (k) (γ, β) = ⟨ψp(k) (γ, β)|HC |ψp(k) (γ, β)⟩.

(a) Monolithic QAOA.

(15)

The optimized circuit is then sampled to obtain candidate commitment bitstrings. In monolithic QAOA, all n logical data qubits are placed on one QPU; therefore, the capacity of that QPU must satisfy cmono ≥ n. B. Capacity-Constrained Distributed QAOA DQAOA preserves the complete QUBO objective and the circuit depth of QAOA structure introduced in Section III-A, but distributes the logical data qubits across multiple QPUs. Thus, Block 2 is not separated into independent QUBO subproblems; instead, the circuit representing the original cost Hamiltonian is implemented over several capacity-constrained QPUs. Let V = {1, . . . , n} denote the binary variable indices and let cm be the available data qubit capacity of QPU m. The assigned variable subsets Vm satisfy M [

Vm = V,

Vm ∩ Vm′ = ∅,

m=1

|Vm | ≤ cm ,

∀m,

M X m=1

m ̸= m′ , (16)

cm ≥ n.

(b) Distributed QAOA.

Fig. 1: Monolithic and distributed QAOA circuits for a six-variable QUBO.

QAOA. The main benefit is therefore multi-QPU qubit capacity accommodation, rather than runtime acceleration, because cross-QPU operations introduce additional communication and circuit overhead. Fig. 1 compares monolithic and distributed QAOA circuits for a six-variable QUBO. The monolithic circuit uses a single QPU, whereas the distributed circuit distributes the logical data qubits across two QPUs and uses communication qubits for cross-QPU interactions. C. Integration Within the Three-Block ADMM

The developed DQAOA package is integrated into Block 2 Each binary variable is mapped to one logical data qubit. Consequently, monolithic QAOA requires a single QPU with as the binary solver of the three-block ADMM framework. at least n data qubits, whereas DQAOA accommodates the At iteration k, Block 1 first updates the continuous relaxed same number of data qubits using the combined capacities of commitment and dispatch variables. These values, together with the current slack and dual multiplier variables, determine the several QPUs. (k) (k) After allocation, each quadratic cost term Jjm Zj Zm is QUBO coefficients H (k) , f (k) , and c0 in (11). As illustrated classified according to the locations of its two qubits. If in Fig. 2, the coefficients and selected solver mode are passed both qubits are assigned to the same QPU, the interaction is to the DQAOA interface, which returns the binary commitment implemented locally. If they are assigned to different QPUs, the vector z (k) to the ADMM loop. For Distributed QAOA mode, corresponding operation is realized through the TeleGate-based QPU capacities are also provided to determine the structure of mechanism of the DQAOA package [15]. TeleGate uses variable-to-QPU allocation. communication qubits, intermediate measurements, and In brute-force mode, the binary search space is enumerated classically conditioned corrections to reproduce the required to obtain an exact Block 2 solution for sufficiently small cross-QPU interaction while preserving the logical data instances. In monolithic QAOA and Distributed QAOA qubits. The mixer operations remain local because they act modes, the variational parameters γ and β are trained by independently on each data qubit. All QPUs use the same repeatedly executing the corresponding parametrized quantum global QAOA parameters γ and β, and the classical optimizer circuit, estimating the expected QUBO cost and updating the minimizes the same expected QUBO cost as in monolithic parameters using a classical Adam optimizer. Once the training

Algorithm 1 DQAOA-Enabled Three-Block ADMM

H (k) f (k) (k)

c0

DQAOA Package BF / QAOA / DQAOA

z (k)

Solver mode QPU capacities

Fig. 2: Integration of the DQAOA package as the selectable Block 2 binary solver within the three-block ADMM framework.

stage is complete, the optimized circuit is executed using a certain number of final shots to produce sampled bitstrings and their frequencies. The measured outcomes are decoded into the logical ordering of the commitment variables. In Distributed QAOA mode, measurement bits associated with communication bits and intermediate TeleGate operations are excluded during this decoding step. Let S (k) denote the set of sampled distinct logical bitstrings obtained from the final circuit sampling. Each sampled bitstring is evaluated using the current Block 2 QUBO objective, and the best observed solution is selected as zopt (k) = arg min F (k) (z). z∈S (k)

(17)

The resulting commitment vector zopt (k) is returned to the ADMM loop and used in the subsequent slack variable and dual multiplier updates. Therefore, the developed package replaces only the solution procedure for Block 2, while the UC formulation and the remaining ADMM operations are identical across brute-force, monolithic QAOA, and Distributed QAOA modes. IV. C ASE S TUDY A. Test System and Settings The proposed framework is evaluated on a five-unit, three-period UC instance with the demand profile L = [60, 130, 280] MW.

(18)

The resulting problem contains n = N T = 15 binary variables. The same UC data, ADMM parameters, initialization, stopping tolerance, and Block 2 QUBO coefficients are used across the brute-force, monolithic QAOA, and distributed QAOA modes. Brute-force provides the exact reference for small problems. Monolithic QAOA represents all 15 commitment variables on one QPU, whereas distributed QAOA distributes the corresponding logical data qubits across multiple capacity-constrained QPUs. The primal residual tolerance is set to 10−3 .

1: Initialize: ∆(0) , z (0) , r(0) , λ(0) , ADMM parameters, solver mode, and QPU capacities. 2: Set ADMM iteration: k ← 1. 3: while stopping criterion is not satisfied do 4: Block 1: Solve (4) for (y (k) , p(k) ). (k) 5: Block 2: Construct (H (k) , f (k) , c0 ) for the QUBO. 6: if mode is brute-force then 7: Enumerate the binary space and obtain the bitstring z (k) with minimum QUBO cost. 8: else 9: if mode is Monolithic QAOA then 10: Allocate the logical data qubits on a single QPU according to (16) with M = 1. 11: Construct the monolithic QAOA circuit. 12: end if 13: if mode is Distributed QAOA then 14: Allocate the logical data qubits according to (16). 15: Construct the distributed QAOA circuit. 16: end if 17: Optimize (γ, β) to minimize E (k) (γ, β) using (15). 18: Sample the trained circuit and decode the logical bitstrings. (k) 19: Select zopt using (17). 20: end if 21: Block 3: Update r(k) using (8). 22: Dual: Update λ(k) using (9). 23: Compute the primal residual in (10). 24: Set k ← k + 1. 25: end while 26: Return: (y (k) , p(k) , z (k) , r(k) , λ(k) ).

TABLE I: Five-Unit System Data Unit Ai ($) Bi ($/MW) Ci ($/MW2 ) Pimin (MW) Pimax (MW) 1 185 25.0 0.0580 10 80 2 170 21.0 0.0160 5 50 3 275 22.0 0.0165 5 60 4 220 24.5 0.0110 10 85 5 150 24.7 0.0070 20 110

below a tolerance of 10−3 , indicating satisfaction of the consensus constraint. Although the quantum-based modes use finite-shot circuit evaluations, both monolithic QAOA and Distributed QAOA modes converge to the same commitment schedule as the brute-force reference. All three modes recover the commitment matrix   0 0 1 0 0 z ⋆ = 0 1 0 0 1 , (19) 0 1 1 1 1 where each row corresponds to one scheduling period and each column corresponds to one generating unit. Reading the commitment matrix row by row, with each row corresponding to one scheduling period, gives the bitstring z ⋆opt = 001000100101111,

(20)

B. Solution Consistency and ADMM Convergence

and the associated dispatch is  0 0 60 0 0 p⋆ = 0 50 0 0 50 60 71.6667

Fig. 3 indicates the ADMM primal residual for the solver quantum modes of Block 2, where the primal residual decreases

The dispatch satisfies the demand in every period, since the row sums are 60, 130, and 280 MW, respectively. The resulting

 0 80  MW. 98.3333

(21)

TABLE II: Comparison of Block 2 Solver Modes Solver Mode Brute force Monolithic QAOA Distributed QAOA

Bitstring 001000100101111 001000100101111 001000100101111

Cost ($) 12678.45 12678.45 12678.45

V. C ONCLUSION This paper developed a DQAOA-enabled three-block ADMM framework for unit commitment by integrating the DQAOA package into the binary QUBO block while keeping the remaining ADMM updates unchanged. The DQAOA interface enables solving the QUBO using brute-force enumeration, monolithic QAOA, or distributed QAOA with identical UC data, ADMM parameters, and convergence criteria. For the five-unit, three-period instance, brute-force enumeration, monolithic QAOA, and distributed QAOA all converged to a primal residual tolerance and recovered the same commitment schedule, dispatch, and operating cost of $12,678. Distributed QAOA achieved this result by allocating the logical commitment qubits across multiple capacity-constrained QPUs, demonstrating solution consistency and the ability to accommodate multi-QPU capacity constraints. Future work will consider larger UC formulations and noisy or hardware-based implementations. R EFERENCES

(a) Monolithic QAOA

(b) Distributed QAOA

Fig. 3: ADMM primal residual convergence for the five-unit, three-period UC case using (a) monolithic QAOA, and (b) distributed QAOA as the Block 2 solver. The dashed line denotes the convergence tolerance of 10−3 .

operating cost is $12,678 for all three modes, corresponding to a zero optimality gap relative to brute force. These results demonstrate two aspects of the proposed framework. First, replacing brute-force enumeration with monolithic QAOA or Distributed QAOA in Block 2 does not change the recovered commitment bitstring, dispatch, operating cost, or ADMM convergence for the tested case. Second, Distributed QAOA obtains the same solution by distributing the 15 logical commitment qubits across multiple QPUs, avoiding the requirement that a single QPU contain the complete binary problem. The result therefore demonstrates solution consistency and multi-QPU capacity accommodation.

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