Numeric Experimentson the CommercialQuantum ComputerRichard H. Warren

This paper describes the creativity that isneeded to solve an optimization prob-lem on an adiabatic quantum computer.Current features of adiabatic quantumcomputing are discussed. Research ques-

tions are posed. Quantum complexity is brieflyaddressed.

We will also discuss the mathematics at theforefront of this new era in computing, describesome of the initial work done at Lockheed Martin onan adiabatic quantum computer, and indicate thedistinct mindset that is used when programmingthis type of machine. The paper begins with adescription of the commercial, adiabatic quantumcomputer and concludes with a discussion ofcurrent limitations that point to future research.

Quantum computing is motivated by an expectedspeedup for solving large problems. In 2010experimenters estimated the time for adiabaticquantum optimization would be about 4 to 6 ordersof magnitude faster than for classical solvers oflarge problems [8]. New results show a variation ofspeedup. The largest reported is a speedup of about3600 times for two types of problems on specificsoftware packages [10]. The adiabatic quantumcomputing model is polynomially equivalent to thequantum circuit model [1], which is the standardquantum computation method. This means thatShor’s factoring algorithm can be implementedon an adiabatic quantum machine, but to ourknowledge no one has done this.

Richard H. Warren is retired after thirty years with Lock-heed Martin as an applied mathematician. He learned hisapplied skills at Wright-Patterson Air Force Base, Ohio, as anew Ph.D. His email address is rwarre01@villanova.edu.

DOI: http://dx.doi.org/10.1090/noti1059

Background InformationCommercial quantum computers are constructedonly by D-Wave Systems, a Canadian technologyfirm. Lockheed Martin Corporation purchased thefirst one in 2011 for a reported 10 million dollars.The machine was moved from Canada to the Univer-sity of Southern California’s School of Engineering[15]. A large percentage of the operating timeis devoted to academic computing, with the restreserved for Lockheed Martin applications.

This computer is an adiabatic quantum com-puter, not the popularized gate array type. Es-sentially, it does one thing: it solves discreteminimization problems extremely fast. In orderto do this, it needs an objective function andconstraints, both expressed in binary variables.This requires declaring the variables, their types,and their coefficients. There are no proceduralinstructions, such as “if X, then do Y ”.

In 2013 it was announced that a consortium ofGoogle, NASA AMES, and the nonprofit UniversitiesSpace Research Association had purchased anadiabatic quantum computer from D-Wave [7].

Adiabatic Quantum ComputingIn the commercial adiabatic quantum computer,the quantum bits (qubits) are loops of super-conducting wire, the coupling between qubits ismagnetic wiring, and the machine is supercooled.Reference [8] describes this superconducting adi-abatic quantum processor. Fabrication limits thenumber of pairwise-coupled qubits, which in turnlimits the number of variables for problems thatare implemented on the computer.

Quantum annealing is a process where thequbits achieve an optimal state of low energy whensupercooled. The Ising objective function for this

1434 Notices of the AMS Volume 60, Number 11

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On the left: Professor Daniel Lidar, scientificdirector of the Quantum Computing Center, withUSC engineering dean Yannis Yortsos in front ofan adiabatic quantum computer.

optimal state is

(1) min

∑(i,j)siJijsj +

∑ihisi

,where i and j are qubits, si is the input state of qubiti (either 0 or 1), hi is the energy bias for qubit i,and Jij is the coupling energy between qubits i andj . Quantum annealing can be thought of as a pathfrom an initial state to a final state according to theweights hi and Jij which minimizes energy. Thereare difficulties achieving the theoretical featuresof quantum annealing, but there are techniques tomitigate the obstacles [10].

Also, there are questions whether D-Wave ma-chines are true quantum systems. Boixo and hiscolleagues report some evidence that they are [3],[4]. The discussion is ongoing [16], [17].

Next we will describe how to transform optimiza-tion problems into the form of the Ising function (1)and solve them on an adiabatic quantum computer.This means the optimization problem needs to beexpressed in binary variables corresponding to sin the Ising function and is limited to linear andquadratic terms corresponding to hisi and siJijsjin the Ising function. This requires creativity toexploit a problem’s structure, often in new ways, inorder to adapt it to an adiabatic quantum computer.Thus, a unique mindset is used to program thismachine.

Reference [9] indicates that many NP -completeproblems have been transformed into quadraticbinary problems that the Ising model (1) requires.In addition, the work at Lockheed Martin hastransformed the traveling salesman problem [13],

the job shop problem for one machine, and alogistics problem into the form of the Ising model(1). The first two problems were implemented onthe Lockheed Martin adiabatic quantum machine.Numeric results showed that the probability offinding an optimal solution increases as the quan-tum annealing process is done more frequently.This correlates with the theory in [12].

Other problems that have been implemented onan adiabatic quantum computer include machinelearning and anomaly detection [11], quadraticunconstrained binary optimization, weighted maxi-mum 2-satisfiability, and the quadratic assignmentproblem [10]. Also, [10] cites finding Ramsey num-bers, binary classification in image matching, and3D protein folding on earlier D-Wave machineswith fewer qubits.

Adapting an Optimization Problem to anAdiabatic Quantum ComputerAs in 0-1 integer programming, an initial stepidentifies binary variables for the problem, i.e.,variables that are restricted to 0 or 1 and representthe problem. The goal is to use the binary variables(with real coefficients) to write an objective functionthat represents a solution to the optimizationproblem. It is important to describe the conditionthat the binary variables represent when they are1 and when they are 0. Next we suggest describingthe (real) coefficients of the variables and whatthey represent. Using the binary variables and theircoefficients, we write an objective function thathas the pattern of (1); i.e., the degree of each termis at most 2. Since x2 = x when x is binary, thisreduction may be useful for obtaining a quadraticobjective function.

For example, in the traveling salesman problem[13], the distance dij to proceed directly from cityi to city j is known for all cities i and j . Theobjective is to find a route through all the citiesthat returns to the starting city and is minimumin length. Our binary variables are xij for the pathdirectly from city i to city j . Our variable xij is 1 ifthe salesman proceeds directly from city i to cityj ; otherwise the value of xij is 0. The coefficientsof the variables are the dij . Assuming the numberof cities is n, our objective function is

(2)n∑

i,j=1dijxij

such that i 6= j . We want the adiabatic quantumcomputer to assign 0, 1 to the variables xij so thatthe objective function (2) is a minimum over alln-cycles representing routes for the salesman.

The next step is to identify constraints that needto be satisfied to prevent minimum solutions thatare not feasible. What are the boundaries for the

December 2013 Notices of the AMS 1435

optimization problem? What type of minimizationsmight occur that should be excluded? For example,in the traveling salesman problem, we need toensure that the route for a solution enters eachcity once and departs from each city once. Also,we need to prevent subtours, which are cyclesusing fewer than all of the cities. In the case for 6cities, this includes preventing a solution that hasa 3-cycle through cities 1, 2, and 3 and another3-cycle through cities 4, 5, and 6.

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Usually the constraints are equations or in-equalities with real coefficients and the previouslyidentified variables. The degree of a constraint isrestricted by (1) to be at most quadratic. For a6-city traveling salesman problem, the constraintx21+x31+· · ·+x61 = 1 ensures that the salesmanuses exactly one route into city 1. The inequalityx12 + x23 + x31 ≤ 2 will prevent a 3-cycle throughcities 1, 2, and 3 in a solution for a 6-city problem.

The final step in the binary characterizationis verification that the objective function andconstraints represent the optimization function.

Forming the HamiltonianThe Hamiltonian is a square, symmetric matrixwith a row and column for each variable. Thediagonal entries of the Hamiltonian are the valuesassigned to qubits. The off-diagonal entries inthe Hamiltonian are the values assigned to theconnections between qubits. The entries in theHamiltonian are the coefficients of the terms fromthe sum of the objective function and the penaltyfunctions (to be described in the next paragraphs).

Since equations and inequalities cannot becombined with a function, the usual technique is toreformulate the constraints as penalty functions.The constraints that are equations can be changedto penalty functions by reversing the algebraic signof all terms on one side of the equation and deletingthe equality sign. Most likely, the minimum forthe result is not the same as the solution for theconstraint equation. Thus we square the result,simplify it with the property x2 = x for binaryvariables, and delete the constant term.

In summary, we have shown a method to converta constraint equation to a penalty function. Thepurpose is to add the objective function and thepenalty functions to obtain an expression for anoptimization problem that corresponds to (1). The

coefficients in this sum are the entries in theHamiltonian.

Exercise 1. Let x and z be binary variables. Find apenalty function for the constraint z = ¬x.1

Exercise 2. Let x, y , and z be binary variables. Finda penalty function for the constraint z = x∨ y .2

Next we describe a technique to convert aconstraint inequality to a penalty function. Weinsert slack variables to change the inequality toan equation, as is done in the simplex algorithmfor linear programming. We require the slackvariables to be binary 0, 1. Then the above methodcan be used to change the equation to a penaltyfunction. For example, consider the constraintx+y +z ≤ 2 where x, y , and z are binary variables.We insert slack, binary variables s and t to obtainx + y + z + s + t = 2. Exercises 3 and 4 verifycorrectness.

Exercise 3. Let x, y , and z be binary variables suchthat x+y+z ≤ 2. Show that there are binary valuesfor s and t such that x+ y + z + s + t = 2.Exercise 4. Let x, y , z, s, and t be binary variablessuch that x + y + z + s + t = 2. Show that thevariables x, y , and z satisfy x+ y + z ≤ 2.

If the weight of the objective function greatlyexceeds the weight of the constraints, then thequantum solution is likely to be tilted towarda solution that satisfies the objective functionbut violates a constraint. On the other hand,if the weight favors the constraints, then thequantum solution is apt to find a suboptimalsolution that satisfies the constraints. As in thesimplex algorithm for linear programming, alambda factor is used to balance the coefficientsin the objective function and the coefficients inthe penalty functions. See the discussion in [9,pp. 239–40] about the scalar P in Transformation 1.Lockheed Martin experiments on the quantummachine showed that the lambda factor has anarrow range that is dependent on the coefficients.

Complexity ConsiderationsAfter creating the objective function and con-straints, the major steps to solve a minimizationproblem on an adiabatic quantum computer are:

• Form the Hamiltonian.• Input the Hamiltonian and computing parame-

ters to the machine.• Wait while the Hamiltonian transitions to a

final state on the machine.• Read the final Hamiltonian on the machine.• Interpret the final Hamiltonian to the variables.

1A solution for Exercise 1: 2xz − x− z.2A solution for Exercise 2: xy + (x+ y)(1− 2z)+ z.

1436 Notices of the AMS Volume 60, Number 11

• Verify that the result is optimal.A critical input parameter is the number of timesto iterate a problem on the quantum machine. Thisis due to errors [18] and the analog nature of themachine when it treats input parameters [5].

The Hamiltonian is usually formed by a prepro-cessor. This is not considered part of quantumcomplexity [5]. Input and output steps are treatedas instantaneous in [5]. A postprocessor may inter-pret the final Hamiltonian and prove the result isoptimal. Thus, in both theory and practice, adia-batic quantum complexity is viewed as a functionof the time to evolve the initial Hamiltonian tothe final Hamiltonian. The (scaled) Schrödingerequation [5, equation (2)] contains a term τ repre-senting the time for a quantum system to evolvefrom an initial state to a final state. The adiabatictheorem [2], [14] in quantum mechanics ensuresthat under certain conditions the evolution is closeto the ground state of the final Hamiltonian, i.e.,evolves to an optimal solution.

Cao and Elgart [5] cite the literature trail fortheoretic bounds onτ for variations of the quantumdatabase search algorithm of Grover. This workassumes conditions on a final Hamiltonian and askswhat initial Hamiltonians and input parametersminimize τ and what the optimal value of τ is.Partial answers are cited. Cao and Elgart [5] extendthe results.

Turning to the experimental side of quantumcomplexity, the current adiabatic quantum com-puter is too small to verify the theory. It has 512qubits. If all qubits are operational, the largestcomplete graph that can be embedded in it has 33vertices (qubits).3 This limits the traveling salesmanproblem (TSP) to 6 cities on the current adiabaticquantum machine, since the TSP requires a com-plete graph and an n-city problem has n(n − 1)variables xij for the objective function (2).

In summary, Cao and Elgart [6] point out thatquantitative characterization of the speedup ofadiabatic quantum computing is largely unknown.

Quantum Hardware Effects on ComputationThere is an accuracy difficulty, since for a singleprogram loaded into the adiabatic quantum com-puter and iterated 100 times, about 1 percent ofthe time none of the 100 solutions are optimal. Thetheoretic work in [5] acknowledges the accuracyproblem by indicating “the AQC algorithm is prob-abilistic in the sense that it gives a correct answerwith the probability γ2. The probability of failurecan be decreased to the desired value (namelyO(1/N)) by repeating the algorithm (lnN)/γ2times. We set γ = 1/5 throughout this paper.” Here

3Source is W. G. Macready, D-Wave Systems.

AQC = adiabatic quantum computation, N = 2n,and n is the dimension of the input space.

There is another hardware difficulty. The numberof fully connected qubits limits the number ofvariables. This is significant, because the numberof variables may grow rapidly as the size of theproblem increases. Variables in the Ising model(1) equate to qubits that need to be connected,physically or logically, in order to represent theirrelationships. Current development methods areimpractical for fabricating connectivity betweenqubits as the number of qubits increases [8]. Thisleads to the following open problems.

Find a decomposition algorithm that breaks alarge optimization problem into small problemsthat an adiabatic quantum computer can solve.Find a reassembly algorithm that combines thesolutions into a global optimal solution. Earlywork indicates that the algorithms may depend onproperties of the Hamiltonian.

Summary and Research AreasAn adiabatic quantum machine provides a re-markable setting for computing some discreteoptimization problems. The machine needs a qua-dratic objective function, binary variables, andquadratic penalty functions that are balanced withthe objective function. Then quantum annealingcan be expected to produce a near optimal solutionextremely fast.

Theory and experimental verification are neededto refine the quantum annealing process so thatan optimal solution is obtained in one iteration. Adual problem technique has been recommended sothere is a simple, uniform method to verify that theresult is optimal. Much work is needed to achievea quantified method to express complexity foradiabatic quantum computing. Lastly, compilersare needed to simplify the pre- and postprocesses.

AcknowledgmentsDaniel M. Davenport, Maxwell D. Henderson, andJoseph A. Angelozzi, who were my colleaguesat Lockheed Martin, are thanked for discussionsand ideas that improved the foundations for thisreport.

References[1] D. Aharonov et al., Adiabatic quantum computa-

tion is equivalent to standard quantum computation,Proceedings of the 45th Annual IEEE Symposium onFoundations of Computer Science (2004), pp. 42–51.

[2] J. E. Avron and A. Elgart, The adiabatic the-orem of quantum mechanics (1998), http://www.ma.utexas.edu/mp_arc/c/98/98-370.ps.gz

[3] S. Boixo, T. F. Rønnow et al., Quantum anneal-ing with more than one hundred qubits (2013),arXiv:1304.4595v1.

December 2013 Notices of the AMS 1437

http://www.ma.utexas.edu/mp_arc/c/98/http://www.ma.utexas.edu/mp_arc/c/98/

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Positions:

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The MSC invites applications for the above positions in the

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applications before February 28, 2014.

[4] S. Boixo, T. Albash et al., Experimental signatureof programmable quantum annealing (2013), Nat.Commun. 4.2067, DOI: 10.1038/ncomms3067.

[5] Z. Cao and A. Elgart, On efficiency of Hamiltonian-based quantum computation for low-ranked matrices,J. Math. Phys. 53 (2012), 032201.

[6] , Adiabatic Quantum Computation: Enthusiastand Skeptic’s Perspectives, Virginia Tech. University,Blacksburg, VA, 2010, arXiv:1004.5409.

[7] N. Jones, Google and NASA snap up quan-tum computer, Nature (2013), DOI: 10.1038nature.2013.12999, http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999.

[8] K. Karimi et al., Investigating the performance of anadiabatic quantum optimization processor, QuantumInf. Process. 11 (2012), 77–88, DOI:10.1007/s11128-011-0235-0.

[9] G. A. Kochenberger et al., A unified modeling andsolution framework for combinatorial optimizationproblems, OR Spectrum 26 (2004), 237–250.

[10] C. C. McGeoch and C. Wang, Experimental evaluationof an adiabatic quantum system for combinatorialoptimization, Proceedings of the ACM Interna-tional Conference on Computing Frontiers (2013),DOI:10.1145/2482767.2482797.

[11] K. L. Pudenz and D. A. Lidar, Quantum adiabaticmachine learning, Quantum Inf. Process. 12 (2013),2027–2070.

[12] A. T. Rezakhani et al., Accuracy versus run time inan adiabatic quantum search, Phys. Rev. A 82 (2010),052305.

[13] R. H. Warren, Adapting the traveling salesman prob-lem to an adiabatic quantum computer, Quantum Inf.Process. 12 (2013), 1781–1785, DOI:10.1007/s11128-012-0490-8.

[14] Wikipedia, Adiabatic theorem, available at http://en.wikipedia.org/wiki/Adiabatic_theorem.

[15] Operational Quantum Computing Center estab-lished at USC Viterbi School of Engineer-ing, news release, October 29, 2011, http://viterbi.usc.edu/news/news/2011/operational-quantum-computing334119.htm.

[16] J. Smolin and G. Smith, Classical signature of quantumannealing, IBM T. J. Watson Research Center, YorktownHeights, NY, 2013, arXiv:1305.4904v1.

[17] L. Wang et al., Comment on: “Classical signature ofquantum annealing”, 2013, arXiv:1305.5837v1.

[18] K. L. Pudenz et al., Error corrected quantum annealingwith hundreds of qubits, 2013, arXiv:1307.8190v1.

1438 Notices of the AMS Volume 60, Number 11

http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999http://en.wikipedia.org/wiki/Adiabatic_theoremhttp://en.wikipedia.org/wiki/Adiabatic_theoremhttp://viterbi.usc.edu/news/news/2011/operational-quantum-computing334119.htmhttp://viterbi.usc.edu/news/news/2011/operational-quantum-computing334119.htmhttp://viterbi.usc.edu/news/news/2011/operational-quantum-computing334119.htm