Adapting Quantum Approximation OptimizationAlgorithm (QAOA) for Unit Commitment

Samantha Koretsky∗, Pranav Gokhale†, Jonathan M. Baker∗, Joshua Viszlai∗, Honghao Zheng‡,Niroj Gurung‡, Ryan Burg‡, Esa Aleksi Paaso‡, Amin Khodaei§, Rozhin Eskandarpour¶, Frederic T. Chong∗†

∗University of Chicago†Super.tech

‡Commonwealth Edison§University of Denver¶Resilient Entanglement

Abstract—In the present Noisy Intermediate-ScaleQuantum (NISQ), hybrid algorithms that leverage clas-sical resources to reduce quantum costs are particu-larly appealing. We formulate and apply such a hybridquantum-classical algorithm to a power system opti-mization problem called Unit Commitment, which aimsto satisfy a target power load at minimal cost. Ouralgorithm extends the Quantum Approximation Opti-mization Algorithm (QAOA) with a classical minimizerin order to support mixed binary optimization. UsingQiskit, we simulate results for sample systems to validatethe effectiveness of our approach. We also compareto purely classical methods. Our results indicate thatclassical solvers are effective for our simulated UnitCommitment instances with fewer than 400 power gen-eration units. However, for larger problem instances, theclassical solvers either scale exponentially in runtime ormust resort to coarse approximations. Potential quantumadvantage would require problem instances at this scale,with several hundred units.

Index Terms—QAOA, hybrid algorithm, unit commit-ment, smart grid

I. BACKGROUND

Quantum computing has the potential to revolu-tionize computation in specific domains. Google’srecent quantum supremacy experiment demonstratedthat quantum computers have the potential to performcomputations that outclass the world’s fastest super-computers [1]. Though still in early stages, quantumcapabilities are advancing rapidly by providing novelways to solve previously intractable problems. In thispaper, we aim to apply quantum computing to achievea practical advantage over current methods in a powersystems application.

Current quantum machines are limited in that theyare noisy and only accommodate a small number ofqubits, generally under 100. Additionally, gates areprone to error. For this reason, current approachesfavor hybrid algorithms that combine classical andquantum methods to minimize required qubit and gatecounts. We apply this hybrid architecture to smartgrid optimization, specifically for the unit commitmentproblem.

A. Unit Commitment

Unit Commitment (UC) is an important optimiza-tion problem in the electrical power industry. It aimsto minimize operational cost while meeting a targetpower load using a number of power-generating unitsthat are subject to constraints [2]. As the size and com-plexity of current energy systems increase, improvingefficiency of solving UC becomes more important toindustry.

In UC, the main goal is to meet the power loadL while minimizing the total cost. Each unit’s costfunction is defined by three coefficients: A is the fixedcost coefficient that a unit necessarily incurs when itis turned on, regardless of the power it contributes.B and C are the linear and quadratic coefficients,respectively, and contribute to the unit’s cost basedon its power level. Each unit is further subject toa set of operational constraints, including but notlimited to minimum and maximum generation limits,ramping up and down limits, minimum on and off timeconstraints, and reserve. In this paper, we only focus onminimum and maximum power generation limits, pminand pmax. Additionally, individual power units may beturned on or off, adding another layer of complexityto the problem. The total cost is given by the sum ofthe costs only of the units that are on.

UC aims to determine the optimal combination ofunits to use and the power levels at which they shouldoperate, all while minimizing the total cost, meetingthe load L, and following the constraints defined bythe coefficients pmin and pmax for each unit. Fig. 1demonstrates two ways that the load of a 4-unitsystem could be met while keeping within each unit’sconstraints, and gives the cost of each configuration.

B. Quantum Approximation Optimization Algorithm

The Quantum Approximation Optimization Algo-rithm (QAOA) [3] is a quantum algorithm that can beused to approximate solutions to optimization prob-lems. It is a hybrid algorithm that uses both quantumand classical resources with the aim of reducing the re-source requirements on the quantum computer relative

1

arX

iv:2

110.

1262

4v1

[qu

ant-

ph]

25

Oct

202

1

4-unitsystem,L=1000MW

A=30$/hB=25$/MWhC=100$/MW Pmin=100MWPmax=300MW

h2

A=50$/hB=10$/MWhC=80$/MW Pmin=100MWPmax=300MW

h2

A=20$/hB=30$/MWhC=120$/MW Pmin=500MWPmax=900MW

h2

A=20$/hB=20$/MWhC=70$/MW Pmin=0MWPmax=900MW

h2

1 1 01

p=250MW

p=500MW

p=250MW

0 0 11p=600MW p=400

MW

C=30+25(250)+100(250)2

C=50+10(250)+80(250)2

C=20+30(500)+120(500)2

C=20+30(600)+120(600)2

C=20+20(400)+70(400)2

Totalcost=$4.1*107 Totalcost=$5.4*107

Figure 1. A diagram of two ways that units in a 4-unit system could meet a given power load, subject to the constraints of the units, inone time step. The total cost of each configuration is given by the sum of the cost of each unit that is turned on. Out of these two systems,the left is more cost efficient. More details on how the cost of a unit is calculated will be given in Sec. III.

to a quantum-only algorithm. QAOA is designed forquadratic unconstrained binary optimization (QUBO)problems, which are a specific kind of combinatorialoptimization problem that are somewhat similar toUC. Both problems feature binary choices, althoughQUBO is defined by only discrete variables, while UCcontains both continuous and discrete variables. Thesimilarities between QUBO and UC make QAOA acompelling algorithm to use in solving UC.

The job of the classical computer in QAOA isto optimize a set of variational parameters: γ =(γ1, γ2, ..., γP ) and β = (β1, β2, ..., βP ) whereγi, βi ∈ [0, 2π). A quantum computer will execute acircuit which is a function of these vectors of anglesγ and β. The length of parameter vectors, P , is pro-portional to the depth (runtime) of the QAOA circuitand does not depend on the total number of qubits.Initially, the quantum computer puts the N qubits intoa uniform superposition over all 2N bitstrings. Next,the quantum computer executes a two-step sequence.First a cost Hamiltonian is applied that phases1 eachbitstring by a quantity related to that bitstring’s costand to γ1. Second, a mixing Hamiltonian is appliedso that probability amplitude can transfer between the2N bitstrings. This two-step sequence—cost Hamil-tonian parametrized by γi and mixing Hamiltonianparametrized by βi—is repeated for i = 2, 3, ..., P .

By sampling from the output of the quantum circuitwith different γ,β vectors, a score can be recordedfor each choice of variational parameters. The classicalcomputer then performs an outer loop (with feedback)to optimize over the 2P -dimensional space, with theaim of producing the optimal parameters: γ∗,β∗.

1This is a uniquely quantum phenomenon that enables destructiveinterference when summing bitstrings.

When the quantum computer is evaluated with theseoptimal (or near-optimal parameters), it will outputbitstrings that approximately extremize the QUBOobjective function.

At P approaches ∞, QAOA can recover the Quan-tum Adiabatic Algorithm, which would exactly solvethe target optimization problem [4]. However, thiswould also take infinite circuit depth. Instead, we areinterested in the performance of QAOA at small, finiteP . Choices such as P = 1 or P = 2 are particularlyfavorable because they require low gate count andcircuit depth. This is suitable for near-term quantumhardware since (a) gates are significant error ratesand (b) qubit lifetimes are short, which prevents deepcircuits from running effectively.

II. PRIOR WORK

There are a variety of classical approaches to solv-ing or approximating Unit Commitment instances.For example, Generic Algebraic Modeling System(GAMS) can solve mixed-binary optimization pro-gramming problems like UC through tools like DI-COPT a DIscrete and Continuous OPTimizer [5]. Asin our quantum approach presented in Section III,DICOPT relaxes equality constraints and penalizingviolations of constraints using slack variables [5].Thus, the program ultimately approximates a solutionto a given Unit Commitment instance. Other classicalsolvers include IBM’s CPLEX which is a popularchoice for both exactly and approximately solvingMBO problems in addition to Mixed-Integer Program-ming (MIP) problems and other more general linearprogramming problems [6]. CPLEX makes use of abranch-and-bound approach to solving these problemsclassically.

2

To the best of our knowledge, [2] is the only priorwork on solving Unit Commitment with a quantumapproach. The authors propose an algorithm basedon quantum annealing. To cope with the mixture ofcontinuous and binary variables, [2] opts to discretizethe continuous variables with a one-hot encoding. As aresult, N(`+2) qubits are required to solve an N -unitsystem discretized to have ` partitions between pmin,iand pmax,i. While the results are promising at small-scale, the one-hot encoding of continuous variables isexpensive in qubit count. Our approach aims to avoidthis discretization cost. In addition, we use QAOA(instead of the quantum adiabatic algorithm), whichenables our approach to be executed on gate-modelquantum computers.

Finally, three recent papers have proposed quantumalgorithms for mixed binary optimization problemsmore broadly. [7] uses QAOA as a lever for translatingthe mixed binary optimization problem into a contin-uous optimization problem. The approach is demon-strated for transaction settlement, a financial applica-tion. Though developed independently, our approach inIII is similar to [7], but applied to Unit Commitment.[8] takes another approach, by using the AlternatingDirection Method of Multipliers (ADMM) to convertan input mixed binary optimization problem into acollection of QUBOs. Finally, [9] applies Bender’sdecomposition [10] to divide an input mixed-integerlinear program into a collection of quantumly-solvableQUBOs and classically-solvable linear programs.

III. OUR QAOA-BASED APPROACH

The critical insight behind our work is that QAOAconverts discrete optimization problems (e.g. over bit-strings) into a continuous optimization problem (overthe variational parameters γ,β). Our approach tosolving UC combines quantum and classical methods.We use QAOA to handle the binary variables of theproblem (whether a unit is on or off) and a classicaloptimizer to handle the continuous variables (howmuch power each unit should provide). The classicaloptimizer comprises the outer loop of our algorithm,and QAOA the inner. Thus, QAOA never needs todiscretize continuous variables into qubits. Meanwhile,in the frame of reference of the classical optimizer, themixed integer problem becomes a continuous one. Thiscombination of classical and quantum algorithms ap-proximates a solution while both limiting the numberof gates used in the quantum computation and allowingfor a plausible quantum advantage.

A. Formulation

In the Unit Commitment problem we have N units.Each unit may be turned on or off, given by a binaryvariable yi ∈ { 0, 1 }. When yi = 1, the power ofthe unit pi is a continuous real value constrained aspmin,i ≤ pi ≤ pmax,i. When yi = 0, pi = 0.Together these restrictions gives a quadratic constraint:

pmin,iyi ≤ pi ≤ pmax,iyi. UC aims to turn on certainunits (decide which yi should be 1) and set theircorresponding power values (decide the value of pi)so that the sum pi is exactly L.

Each power unit has a corresponding functionH(yi, pi) which specifies the cost of turning on unit iand for generating a certain amount of power pi. Thiscost function is quadratic for unit commitment:

H(yi, pi) = Aiyi +Bipi + Cip2i (1)

where Ai, Bi, Ci ∈ R are constant. This overarchingobjective of a UC problem then is to minimize thesum of these cost functions. The following equationssummarize a generic UC problem.

minN∑i=1

H(yi, pi) (2)

s.t.N∑i=1

pi = L (3)

piyi ≤ pmax,i, piyi ≥ pmin,i (4)pi ∈ R yi ∈ { 0, 1 } (5)

We convert this formulation into a QUBO problem.The goal is to convert the constrained variables yi, pito become unconstrained by modifying the objectivefunction to include penalty terms so that when thevalues yi, pi are outside of their desired ranges theobjective suffers. Now we let pi ≥ 0. For each contin-uous variable pi we introduce two slack variables si,1,si,2 ∈ R≥0, the first to penalize if pi is less than theminimum and the second to penalize if pi is larger thanits maximum. Consider the following penalty term

λ(pi − si,1 − pmin,iyi)2 (6)

where λ is some fixed constant which must be setempirically. Suppose yi = 1 and pi is greater thanpmin,i then this equation can be minimized by settingsi,1 = pi − pmin,i > 0 resulting in 0 penalty. How-ever, if pi < pmin,i then this equation is minimizedwhen si,1 = 0 resulting in a positive penalty term.This equation effective encodes the previous constraintbounding pi ≥ pmin,i by guiding the objective into asolution which minimizes this penalty term. We cando the same by adding in a penalty term using si,2to penalize values of pi > pmax,i. Finally, we need apenalty term to encode the constrain that the sum ofthe pi is the power load L. The following term

λ( N∑i=1

piyi − L)2

(7)

is minimized when the sum is exactly L. In practice,we may want to choose different λ values for eachpenalty term to enforce certain constraints more orless. Putting it all together we obtain the following

3

objective function which ideally encodes the same UCproblem

minN∑i=1

(Aiyi +Bipi + Cip2i )

+ λ1(

N∑i=1

piyi − L)2

+ λ2[

N∑i=1

(pi − si,1 − pi,minyi)2]

+ λ3[

N∑i=1

(pi + si,2 − pi,maxyi)2] (8)

Where si,j , pi ∈ R≥0, yi ∈ { 0, 1 } and empiricallydetermined λi. We have now rewritten the the UCproblem as a QUBO which will be useful for con-verting to a QAOA formulation so that all constraintsare now contained in the objective. Since y2i = yi,the above problem reduces to a quadratic program bynoting for fixed pi, si,j .

Once the problem is reformulated, we use a SciPyclassical minimizer to find the individual unit powervalues that minimize the cost function, as well as theoptimal γ,β variational parameters to use in QAOA.We use the minimize function from SciPy’s optimizepackage [11], specifically using the Nelder-Mead al-gorithm [12] with the γ,β,p, s as the arguments. Foreach iteration that the classical minimizer completes,we simulate a quantum processor with IBM Qiskitto execute QAOA [13]. This returns a probabilitydistribution of the optimal combinations of units tobe used, given the power values that the classicalminimizer has designated to each unit.

B. Quantum CircuitWhen the classical outer loop optimizer proposes

p and s, it induces a QUBO objective function, asdescribed by Equation 8. The classical optimizer isalso responsible for proposing the 2P continuous vari-ational parameters, γ,β. To comply with the standardform for QUBOs solved by QAOA, we convert fromyi ∈ {0, 1} binary variables to zi ∈ {+1,−1}variables via the transformation zi = 2yi − 1.

The actual QAOA circuit, given γ,β, is depictedin Fig. 2. Each Cost Hamiltonian step applies the op-eration with diagonal unitary matrix eiγiH(γ,β) whereH(γ,β) is a diagonal unitary matrix correspondingto Equation 8. This step is the most expensive oper-ation. Since the objective function is a QUBO, thesub-operations in the Cost Hamiltonian consist ofa sequence of eiγiZ⊗Z operations, where Z is thePauli-Z matrix. We used the sympy library [12] tomanage the construction of the QUBO and tracking ofits coefficients. After the Cost Hamiltonian step, themixing Hamiltonian is applied, which simply executesRx(βi) on each qubit. These Cost and Mixing Hamil-tonian steps are repeated P times before terminatingmeasurements on each qubit.

IV. CLASSICAL BASELINE

To benchmark how existing classical approachesperform on the Unit Commitment problem, we solvedUC instances of varying sizes both exactly and approx-imately using IBM’s CPLEX Optimizer [6]. Experi-ments were run using version 20.1 of IBM’s CPLEXOptimizer on a single core of an Intel Core i7-8750HCPU. We used CPLEX to solve the UC instances intheir MBO form. Fig. 3 shows results from solvingthese various UC problems. Approximate solving wasdone by searching for solutions with objective functionvalue within 8% of the optimal solution.

CPLEX uses a branch-and-bound approach to solv-ing the UC problems in MBO form, which althoughefficient in practice when compared to other classicalmethods, still has worst-case exponential scaling inruntime. The results show that, as expected, exact solv-ing scales exponentially in runtime, quickly becomingintractable for large problem sizes. We also observethat approximate methods perform better in runtimecompared to exact solving at the cost of solution qual-ity, however still eventually see exponential scaling forlarger problem sizes, as expected.

V. RESULTS

To confirm our algorithm does approximate solu-tions to UC adequately, we ran different example sys-tems and tracked the progress of numerous variables toensure our algorithm was converging on approximatesolutions each time it ran. One of our main exampleswas a real 10-unit system, specified with the parame-ters given in Table I.

Fig. 4 demonstrates the results of our quantumapproach on the 10-unit system in Table I. The left andright plots corresponding to P = 1 and P = 2 depthfor QAOA respectively. The top plots show the totalprobability of our algorithm returning a near-optimalsolution 2, i.e. an assignment of units to ON/OFF.Given a near-optimal bitstring, the actual power levelassignments can be obtained by solving the inducedquadratic program.

As the plots indicate, the probability of near-optimalsolution increases over the course of variational op-timization iterations in our hybrid approach. At iter-ation 0, the probability of achieving a near-optimalsolution is essentially the same as random guessing.After 1500 iterations, the probability of near-optimalsolution exceeds 4% for P = 1 and 6% P = 2. Thisboth validates our general approach and demonstratesthat increased performance is possible at higher P .

The bottom two plots show the average Hammingdistance between each of the top 50 bitstrings returnedby our algorithm and the near optimal solution that

2The set of near-optimal solutions were generated by classicalbrute force, with a the cutoff for “near-optimal” determined by bestjudgment. As a result, some systems may have many near optimalsolutions, while others have only a few. The cutoff selection wasdone as fairly as possible, but note that these selections will affectthe shape of the plot.

4

|0〉 H

CostHam(γ1,p, s)

Rx(β1) . . .

CostHam(γP ,p, s)

Rx(βP )

|0〉 H Rx(β1) . . . Rx(βP )

......

|0〉 H Rx(β1) . . . Rx(βP )

Figure 2. Circuit diagram

0 100 200 300 4000

200

400

600

800

Runt

ime

(s)

Exact Solving

2500 5000 7500 10000 12500 150000

25

50

75

100 Approximate Solving

Classical Solving of Unit Commitment Problem

Number of Units

Figure 3. Classical solving of UC problems with IBM’s CPLEX optimizer. For approximate solving, the algorithm finished with anobjective function value within 8% of the optimal solution.

i 1 2 3 4 5 6 7 8 9 10pmax,i (MW) 455 455 130 130 162 80 85 55 55 55pmin,i (MW) 150 150 20 20 25 20 25 10 10 10

Ai ($) 1000 970 700 680 450 370 480 660 665 670Bi ($/MW) 16.19 17.26 16.60 16.50 19.70 22.26 27.74 25.92 27.27 27.79Ci ($/MW 2) .00048 .00031 .002 .00211 .00398 .00712 .0079 .00413 .00222 .00173

Table ITHE PARAMETERS OF THE 10-UNIT SYSTEM

Iterations

Ham

min

g Di

stan

cePr

ob. o

f Nea

r-Opt

imal

Sol

utio

n P = 1 P = 2

Figure 4. For a 10-unit system, both using P = 1 (left) and P = 2 (right), the total probability of finding a near optimal bitstring increasesas our algorithm completes more iterations (top). Additionally, the average Hamming distance between each of the top 50 bitstrings andtheir closest near optimal solution decreases over time (bottom).

5

Iterations

Hamming DistanceProb. of Near-Optimal Solution

6-unit system

7-unit system

8-unit system

Figure 5. As in the case of the 10-unit system, when using P = 2 steps of QAOA for a 6-, 7- and 8-unit system, the total probability offinding a near optimal bitstring increases over time (left), and the average Hamming distance between each of the top 50 bitstrings andtheir closest near optimal solution decreases over time (right).

it is closest to. This metric decreases over the courseof the variational optimization iterations, corroboratingthat our algorithm is converging to better solutions (i.e.ON/OFF assignments).

We also produced similar plots for sample 6-, 7-,and 8-unit systems. For each system, we used P =2 QAOA, and we found the same trends as for the10-unit system, further supporting that our algorithmsucceeds in driving towards a near-optimal solution(assignment of units to ON/OFF). For brevity, we leaveout details of the parameters for these systems; the plotof results are in Fig. 5.

VI. CONCLUSION

We have introduced a hybrid quantum-classicalapproach to Unit Commitment. Our approach usesQAOA to turn a QUBO instance into a continuousoptimization problem over variational parameters γ,β.A classical optimizer is then able to simultaneouslyoptimize over these continuous parameters as well asthe power assignments for each unit.

Figures 4 and 5 validate the correctness and promiseour approach. As demonstrated for 6-, 7-, and 8-, and10- unit sytems (which entails quantum simulation ofa circuit with equal number of qubits), our variationalapproach boosts the probability of finding a near-

optimal bitstring (i.e. assignment of units to ON/OFF)far beyond naive brute force or random guessing.Moreover, increasing the QAOA depth parameter, P ,improves performance.

For smaller systems with only a few hundred units,existing classical solvers are able to perform well.However, as our classical simulation results indicate,these classical solvers are limited for larger systems,because their runtime scales exponentially. By con-trast, our hybrid quantum-classical algorithm couldmaintain effectiveness at larger system sizes, sinceeach QAOA circuit has logical depth of just O(N2P ).

In the future, it will be important to run our al-gorithm on a real machine rather than simulating thequantum circuits. We anticipate that real evaluationwill pose a variety of challenges not seen in our idealclassical simulation. For example, superconductingqubits have sparse connectivity, which could incur alarge SWAP overhead (though this could be mitigatedwith techniques like swap networks [14, 15]). In ad-dition, the recently discovered phenomenon of noise-induced barren plateaus [16] could hinder successfulexecution in near-term quantum computers. However,we expect quantum error correction to emerge inhardware over the next decade; in this medium-termera, noise would no longer be an obstacle.

6

ACKNOWLEDGEMENT

This work is funded in part by EPiQC, anNSF Expedition in Computing, under grantsCCF1730082/1730449; in part by STAQ undergrant NSF Phy-1818914; in part by DOE grantsDE-SC0020289 and DE-SC0020331; and in part byNSF OMA2016136 and the Q-NEXT DOE NQICenter. This material is based upon work supportedby the National Science Foundation under Grant No.2110860.

Disclosure: Fred Chong is Chief Scientist at Su-per.tech and an advisor to Quantum Circuits, Inc.

REFERENCES

[1] Frank Arute et al. “Quantum supremacy using aprogrammable superconducting processor”. In:Nature 574.7779 (2019), pp. 505–510.

[2] Akshay Ajagekar and Fengqi You. “Quantumcomputing for energy systems optimization:Challenges and opportunities”. In: Energy 179(2019), pp. 76–89.

[3] Edward Farhi, Jeffrey Goldstone, and SamGutmann. A Quantum Approximate Optimiza-tion Algorithm. 2014. arXiv: 1411 . 4028[quant-ph].

[4] Edward Farhi et al. “A quantum adiabatic evo-lution algorithm applied to random instances ofan NP-complete problem”. In: Science 292.5516(2001), pp. 472–475.

[5] Ignacio E Grossmann et al. “GAMS/DICOPT:A discrete continuous optimization package”.In: GAMS Corporation Inc 37 (2002), p. 55.

[6] IBM. IBM CPLEX Optimizer v20.1. URL: https://www.ibm.com/analytics/cplex-optimizer.

[7] Lee Braine et al. Quantum Algorithms forMixed Binary Optimization applied to Trans-action Settlement. 2019. arXiv: 1910 . 05788[quant-ph].

[8] Claudio Gambella and Andrea Simonetto.“Multiblock ADMM Heuristics for Mixed-Binary Optimization on Classical and QuantumComputers”. In: IEEE Transactions on Quan-tum Engineering 1 (2020), pp. 1–22.

[9] Chin-Yao Chang, Eric Jones, and Peter Graf.“On Quantum Computing for Mixed-Integer Programming”. In: arXiv preprintarXiv:2010.07852 (2020).

[10] Jacques F Benders. “Partitioning proceduresfor solving mixed-variables programming prob-lems ‘”. In: Numerische mathematik 4.1 (1962),pp. 238–252.

[11] The SciPy community. Optimization and rootfinding (scipy.optimize). URL: https://docs.scipy.org/doc/scipy/reference/optimize.html.

[12] The SciPy community.minimize(method=’Nelder-Mead’). URL:https : / / docs . scipy. org / doc / scipy / reference /optimize.minimize-neldermead.html.

[13] IBM. Qiskit. URL: https://qiskit.org.[14] Ian D Kivlichan et al. “Quantum simulation

of electronic structure with linear depth andconnectivity”. In: Physical review letters 120.11(2018), p. 110501.

[15] Teague Tomesh et al. “Coreset Clustering onSmall Quantum Computers”. In: arXiv preprintarXiv:2004.14970 (2020).

[16] Samson Wang et al. “Noise-induced barrenplateaus in variational quantum algorithms”. In:arXiv preprint arXiv:2007.14384 (2020).

7