special issue on financial signal processing and machine ... · onsider an asset manager wishing to...

This is the author’s version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/JSTSP.2016.2574703

Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

SPECIAL ISSUE ON FINANCIAL SIGNAL PROCESSING AND MACHINE LEARNING FOR ELECTRONIC TRADING 1

Solving the Optimal Trading TrajectoryProblem Using a Quantum AnnealerGili Rosenberg, Poya Haghnegahdar, Phil Goddard*, Peter Carr, Kesheng Wu,

Marcos López de Prado

Abstract—We solve a multi-period portfolio opti-mization problem using D-Wave Systems’ quantumannealer. We derive a formulation of the problem,discuss several possible integer encoding schemes, andpresent numerical examples that show high successrates. The formulation incorporates transaction costs(including permanent and temporary market impact),and, significantly, the solution does not require theinversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve issignificantly harder than the continuous variableproblem. We present insight into how results may beimproved using suitable software enhancements andwhy current quantum annealing technology limits thesize of problem that can be successfully solved today.The formulation presented is specifically designed tobe scalable, with the expectation that as quantumannealing technology improves, larger problems willbe solvable using the same techniques.

Index Terms—Optimal trading trajectory, portfoliooptimization, quantum annealing.

Copyright (c) 2016 IEEE. Personal use of this material ispermitted. However, permission to use this material for any otherpurposes must be obtained from the IEEE by sending a requestto [email protected].

G. Rosenberg and P. Goddard* are with 1QBit, Suite 458–550 Burrard Street, Vancouver, BC V6C 2B5, Canada.

P. Haghnegahdar is with the University of BritishColumbia, 2329 West Mall, Vancouver, BC V6T 1Z4, Canada.

P. Carr is with New York University (NYU) – CourantInstitute of Mathematical Sciences, 251 Mercer Street, NewYork, NY 10012, USA.

K. Wu is with Lawrence Berkeley National Laboratory,One Cyclotron Road, Berkeley, CA 94720, USA.

M. López de Prado is with Guggenheim Partners LLC, 330Madison Avenue, New York, NY 10017, USA and is a ResearchFellow at Lawrence Berkeley National Laboratory’s Computa-tional Research Division, One Cyclotron Road, Berkeley, CA94720, USA.

* Corresponding author ([email protected])

I. THE PROBLEM

A. Introduction

CONSIDER an asset manager wishing to investK dollars in a set of N assets with an invest-

ment horizon divided into T time steps. Given aforecast of future returns and the risk of each assetat each time step, the asset manager must decidehow much to invest in each asset at each timestep, while taking into account transaction costs,including permanent and temporary market impactcosts.

One approach to this problem is to computethe portfolio that maximizes the expected returnsubject to a level of risk at each time step. Thisresults in a series of “statically optimal” portfolios.However, there is a cost to rebalancing from aportfolio that is locally optimal at t to a portfoliothat is locally optimal at t + 1. This means thatit is highly likely that there will be a differentseries (or, a “trajectory”) of portfolios that will be“globally optimal” in the sense that its risk-adjustedreturns will be jointly greater than the combinedrisk-adjusted returns from the series of “staticallyoptimal” portfolios.

Mean-variance portfolio optimization problemsare traditionally solved as continuous-variable prob-lems. However, for assets that can only be tradedin large lots, or for asset managers who are con-strained to trading large blocks of assets, solvingthe continuous problem yields an approximation,and a discrete solution is expected to give betterresults. For example, institutional investors are oftenlimited to trading “even” lots (due to a premiumon “odd” lots), that is, lots that are an integer

http://dx.doi.org/10.1109/JSTSP.2016.2574703




multiple of a standard lot size, in which case theproblem becomes inherently more discrete as thetrade size increases versus the lot size. This couldoccur, for example, due to the trading of illiquidassets. Two common examples of block trading areETF-creation and ETF-redemption baskets, whichcan only be traded in large multiples, such as fundunits of 100,000 shares each.

The discrete problem is non-convex due to thefragmented nature of the domain, and is thereforemuch harder to solve than a similar continuousproblem. Furthermore, our formulation allows thecovariance matrix to be ill-conditioned or degener-ate. This complicates the finding of a solution usingtraditional optimizers since a continuous relaxationwould still be non-convex, and therefore difficult tosolve.

B. Previous work

The single-period discrete portfolio optimizationproblem has been shown to be NP-complete, regard-less of the risk measure used [1], [2]. Jobst et al.[3] showed that the efficient frontier of the discreteproblem is discontinuous and investigated heuristicmethods of speeding up an exact branch-and-boundalgorithm for finding it. Vielma et al. presented abranch-and-bound algorithm and results for up to200 assets [4]. Heuristic approaches, including anevolutionary algorithm, were investigated by otherauthors [1], [5], [6].

Bonami and Lejeune [7] solved a single-periodproblem with integer trading constraints, minimiz-ing the risk given a probabilistic constraint onthe returns (with no transaction costs), and findingexact solutions via a branch-and-bound method forproblems with up to 200 assets. They consideredfour different methods, of which one was able tosolve the largest problems to optimality in 83% ofthe cases, but the other three failed for all problems(the average run time for the largest problems was4800 seconds). They found that solving the integerproblem was harder than solving a continuous prob-lem of the same size with cardinality constraints orminimum buy-in thresholds.

Gârleanu and Pedersen [8] solved a continuousmulti-period problem via dynamic programming,

deriving a closed-form solution when the covariancematrix is positive definite, thereby offering insighton the properties of the solutions to the multi-periodproblem. A multi-period trade execution problemwas treated analytically by Almgren and Chriss[9], motivating our inclusion of both temporary andpermanent price-impact terms.

The connection between spin glasses andMarkowitz portfolio optimization was shown byGalluccio et al. [10]. The discrete multi-periodproblem was suggested by López de Prado [11]as being amenable to solving using a quantumannealer. The contribution of this paper is to inves-tigate the implementation and solution of a similardiscrete multi-period problem on the D-Wave quan-tum annealer.

C. Integer formulation

The portfolio optimization problem describedabove may be written as a quadratic integer op-timization problem. We seek to maximize returns,taking into account the risk and transaction costs,including temporary and permanent market impact(the symbols are defined in Appendix A),

(1)w = argmaxw

T∑t=1

(µTt wt −

γ

2wT

t Σtwt

−∆wTt Λt∆wt + ∆wT

t Λ′

twt

),

subject to the constraints that the sum of holdingsat each time step be equal to K,

∀t :

N∑n=1

wnt = K, (2)

and that the maximum allowed holdings of eachasset be K ′,

∀t, ∀n : wnt ≤ K ′. (3)

The first term in Eq. 1 is the sum of the returnsat each time step, which is given by the forecastreturns µ times the holdings w. The second term isthe risk, in which the forecast covariance tensor isgiven by Σ, and γ is the risk aversion. The third and





fourth terms encapsulate transaction costs. Specif-ically, the third term includes any fixed or relativedirect transaction costs, as well as the temporarymarket impact, while the fourth term captures anypermanent market impact caused by trading activity.The transaction cost term is square in the change inthe holdings, so it penalizes changes in the holdingsif the corresponding entry in Λt is positive [8]. Thepermanent market impact term allows for the factthat increasing a large holding requires executinga large buy order, which increases the price, andhence the returns [9].

D. Extensions

A straightforward extension can be made to solveoptimal trade execution problems. For example, inorder to solve a problem in which the asset managerhas K units invested and would like to liquidatethem over T time steps, the constraint in Eq. 2would change to

∀t :

N∑n=1

wnt ≤ K (4)

and the sum in the transaction cost and permanentimpact terms would extend to time step T + 1 withwT+1 = 0 (the zero vector).

The risk term in Eq. 1 requires the estimationof a covariance matrix for each time step. Thereare cases in which this is problematic: for example,if not enough data exists for a good estimate orif some of the assets were not traded due to lowliquidity. An alternative and more direct way toquantify risk is via the variance of the returnsstream of the proposed trajectory. This avoids theissues with the estimation of covariance matrices.An additional advantage of this method is that itdoes not assume a normal distribution of returns.The disadvantage is that if the number of time stepsT is small, the estimate of the true variance of theproposed trajectory will be poor. A high variance ofreturns is penalized regardless of whether it occursdue to positive or negative returns.

The variance is quadratic in the returns, so it is asuitable term to include in a quadratic integer for-mulation. We use the identity Var(r) = 〈r2〉−〈r〉2,

and note that the returns stream is given by thevector r[w] = diag(µTw). We find the alternativerisk term

(5)

risk[w] =γ

T

T∑t=1

[(µTt wt

)2− 1

T

T∑t′=1

(µTt wt

) (µTt′wt′

)].

II. SOLUTION USING A QUANTUM ANNEALER

A. Quantum annealing

Quantum annealing is a process which can beused to find the optimal solution to optimizationproblems, if these problems can be encoded as aHamiltonian [12], [13]. To this end, the quantumsystem is first prepared such that it represents atrivial problem, and is in the ground state of thatproblem, which is an equally weighted superposi-tion of all possible states. The system is then trans-formed continuously to the point that it representsthe optimization problem that we want to solve. Ifthis process is done slowly enough, the adiabatictheorem guarantees that the system will remain inthe ground state, as long as external disturbancesare absent. The state of the system is then read,and in the ideal case it would correspond to theoptimal solution of the optimization problem wewish to solve [14]. This process is referred to as“adiabatic quantum computation”. In a real device,external interference always exists to some degree,so the result is probabilistic, and annealing the sameproblem multiple times increases the probability offinding the optimum. Therefore, quantum annealersare effectively heuristic solvers.

It has been argued that quantum annealing hasan advantage over classical optimizers due to quan-tum tunnelling. Quantum tunnelling allows an op-timizer to more easily search the solution space ofthe optimization problem, thereby having a higherprobability of finding the optimal solution. Thismight provide a speed improvement over classicaloptimizers, at least for certain problem classes [12],[13], [15]–[17].

D-Wave Systems has developed a scalable quan-tum annealer. Mathematically, this is a device





which minimizes unconstrained binary quadraticfunctions,

minxTQx (6)

s.t. x ∈ {0, 1}N ,

where Q ∈ RN×N [18]–[20]. In order to keepexternal disturbances to a minimum, D-Wave Sys-tems’ quantum annealer is cooled to 15 mK (about180 times colder than interstellar space), is shieldedfrom RF signals due to its being housed inside ametal enclosure, is shielded from external magneticfields larger than 1 nT (about 50,000 times lessthan Earth’s magnetic field), and operates in a high-vacuum environment in which the pressure is 10billion times lower than atmospheric pressure [21].

There is strong evidence that the D-Wave ma-chine is indeed quantum [22], [23]. Recently, therehas been significant interest in benchmarking theD-Wave machines using different metrics, and of-ten against classical solvers [24]–[29]. There isan ongoing debate on how to define quantumspeedup, and on which problems a noisy quantumannealer would be expected to show such a speedup[30]–[32]. Recently, Denchev et al. claimed a 108

speedup over simulated annealing when solving aspecially constructed class of problems on a single-core machine [33]. It is still an open questionwhether D-Wave Systems’ quantum annealer showsa quantum speedup. We expect new results to shedlight on this in the near future.

The connectivity of the qubits in D-Wave’s quan-tum annealer is currently described by a squareChimera graph [34]. This hardware graph is com-posed of a lattice of bipartite unit cells containingeight qubits. Qubits in adjacent unit cells are con-nected if they are in the same position in the unitcell (see Fig. 1 for an example).

If we label the number of unit cells along an edgeas s, then the total number of qubits is q = 8s2. Thehardware graph is sparse and in general does notmatch the problem graph, which is defined by theadjacency matrix of the problem matrix Q. In orderto solve problems that are denser than the hardwaregraph, we identify multiple physical qubits with asingle logical qubit (a problem known as “minorembedding” [35], [36]), at the cost of using many

more physical qubits. For square Chimera hardwaregraphs, the size V of the largest fully dense problemthat can be embedded on a chip with q qubits isV =

√2q+1 = 4s+1, assuming no faulty qubits or

couplers. For example, the latest chip is the D-Wave2X1, which has s = 12 unit cells along each side,giving q = 1152 qubits, for which we get V =49. Lower-density problems of significantly largersize can be embedded. For example, on one of theannealers used in this study, which has 1100 qubits,problems with Vb ' 140 and a density of ' 0.1were able to be embedded.

B. From integer to binary

To solve this problem using the D-Wave quantumannealer, the integer variables of Eq. 1 must berecast as binary variables, and the constraints mustbe incorporated into the objective function. We haveinvestigated four different encodings: binary, unary,sequential, and partitioning. The first three can bedescribed by writing the integer holdings as a linearfunction of the binary variables,

wnt[x] =

D∑d=1

f(d)xdnt, (7)

Fig. 1: An example hardware graph, showing theconnectivity of the qubits for a Chimera graph withs = 4 unit cells in each row/column, giving a totalof q = 128 qubits.

1as of April 2016





TABLE I: Encodings: f(d) is the encoding functionand D is the bit depth.

Encoding f(d) D

Binary 2d−1 log2(K ′ + 1)Unary 1 K ′

Sequential d(√

1 + 8K ′ − 1)/2

where xdnt ∈ {0, 1} and the encoding function f(d)and the bit depth D for each encoding are given inTable I.

The fourth encoding involves finding all parti-tions of K into N assets with K ′ or less units ineach asset, and assigning a binary variable to eachencoding at each time step.

We summarize the properties of each of thefour encodings described above in Table II. Whichencoding is preferred will depend on the problembeing solved and the quantum annealer being used.Table III presents the number of variables requiredfor some example multi-period portfolio optimiza-tion problems for each of these encodings.

For binary, unary, and sequential encodings,there is a trade-off between the efficiency of theencoding—the number of binary variables neededto represent a given problem—and the largest inte-ger that can be represented. The reason is that foran encoding to be efficient it will typically intro-duce large coefficients, limiting the largest integerrepresentable (due to the noise level in the quantumannealer). For example, the binary encoding is themost efficient of the three (that is, it requires thefewest binary variables); however, it is the most sen-sitive to noise, and hence can represent the smallestinteger of the three, given some number of qubits.Conversely, the unary encoding is the worst of thethree in efficiency, but can represent the largestinteger. An additional consideration is that someencodings introduce a redundancy that biases thequantum annealer towards certain solutions. Briefly,each integer can be encoded in multiple ways, thenumber of which is (in general) different for eachinteger. In this scenario, the quantum annealer isbiased towards integers that have a high redundancy.

The partition encoding is different in that it re-

quires an exponential number of variables; however,it allows a straightforward formulation of com-plicated constraints, like cardinality, by excludingpartitions that break the constraints, which alsolowers the number of variables required. For theother three encodings, constraints can be modelledthrough the encoding (for example, a minimum ormaximum holdings constraint), or through linear orquadratic penalty functions. We note that the actualnumber of physical qubits required could be muchlarger than the number of variables indicated inTable III due to the embedding (see Section II-A).

In many cases, the maximum holdings K ′, whichis also the largest integer to be represented, willnot be exactly encodable using the binary andsequential encodings. For example, using a binaryencoding one can encode the values 0 to 7 usingthree bits, and 0 to 15 using four bits, but integerswith a maximum value between 7 and 15 arenot exactly encodable. In order to avoid encodinginfeasible holdings, these can be penalized by anappropriate penalty function. However, this penaltyfunction will typically be a high-order polynomialand require many auxiliary binary variables in orderto reduce it to a quadratic form. Instead, we proposeto modify the encoding by adding bits with thespecific values needed. For example, {1, 1, 2, 2, 4}represents a modified binary encoding for the values0 to 10.

The constraints of Eq. 2 can be incorporated intothe objective function by rearranging the equations,squaring them, and summing the result, obtainingthe penalty term

penalty[w] = −MT∑

t=1

(K −

N∑n=1

wnt

)2

, (8)

where M > 0 is the strength of the penalty. Intheory, M can be chosen large enough such thatall feasible solutions have a higher value than allinfeasible solutions. In practice, having an overlylarge M leads to problems due to the noise inthe system (see Section II-D), so we choose itempirically by trial and error. In the future, itmight be possible to include equality constraints ofthis form via a change in the quantum annealing





TABLE II: Comparison of the four encodings described in Section II-B. The column “Variables” refers tothe number of binary variables required to represent a particular problem. The column “Largest integer”refers to a worst-case estimate of the largest integer that could be represented based on the limitationimposed by the noise level ε and the ratio between the largest and smallest problem coefficients δ andn ≡ 1/

√εδ.

Encoding Variables Largest integer Notes

Binary TN log2(K ′ + 1) b2nc − 1Most efficient in number of variables; allowsrepresenting of the second-lowest integer.

Unary TNK ′ No limit

Biases the quantum annealer due to differingredundancy of code words for each value;encoding coefficients are even, giving no de-pendence on noise, so it allows representingof the largest integer.

Sequential 12TN

(√1 + 8K ′ − 1

)12 bnc (bnc+ 1)

Biases the quantum annealer (but less thanunary encoding); second-most-efficient innumber of variables; allows representing ofthe second-largest integer.

Partition ≤ T(K+N−1N−1

)bnc

Can incorporate complicated constraints eas-ily; least efficient in number of variables;only applicable for problems in whichgroups of variables are required to sum toa constant; allows representing the lowestinteger.

process, allowing us to drop this term [37].We note that an alternative approach, involving

the tiling of the integer search space with binaryhypercubes, was investigated by [38]; however, itrequires an exponential number of calls to thequantum annealer. In addition, a

C. Numerical Results

The results for a range of portfolio optimizationproblems are presented in Table IV. For each prob-lem 200 random instances were generated. Eachinstance was solved using one query to the quan-tum annealer, with 1000 reads per query (whichinvolved either 1 call or 5 calls if averaging overgauges)2. All results were obtained from chips witha hardware graph with either 512 or 1152 qubits.(The number of active qubits was a little smaller.)

2We have observed that the success rate rises when increas-ing the number of reads, for a fixed problem size. If the numberof reads is fixed, the success rate is expected to decrease asproblem size increases, as seen in the results.

For validation purposes, each instance was alsosolved using an exhaustive integer solver to findthe optimal solution. For the larger problems, aheuristic solver was run a large number of timesin order to find the optimal solution with highconfidence.

As a solution metric, we used the percentage ofinstances for each problem for which the quantumannealer’s result fell within perturbation magnitudeα% of the optimal solution, denoted by S(α). Thismetric was evaluated by perturbing each instanceat least 100 times, by adding Gaussian noise withstandard deviation given by α% of each eigenvalueof the problem matrix Q. Each perturbed problemwas solved by an exhaustive solver, and the optimalsolutions were collected. If the quantum annealer’sresult for that instance fell within the range ofoptimal values collected, then it was deemed suc-cessful (within a margin of error). This procedurewas repeated for each random problem instance,giving a total success rate for that problem. For thecase α = 0, this reduces to defining success as the





TABLE III: Dependence of the number of binary variables required on the number of units K, numberof assets N , and the number of time steps T for some example values (here we assumed K ′ = K/3).The number of variables is given for the three linear encodings: binary Vb, unary Vu, and sequential Vs.

N T K K ′ Vb Vu Vs5 5 15 5 75 125 75

10 10 15 5 300 500 30010 15 15 5 450 750 45020 10 15 5 600 1000 60050 5 15 5 750 1250 75020 15 15 5 900 1500 90050 10 15 5 1500 2500 150050 15 15 5 2250 3750 2250

TABLE IV: Results using D-Wave’s 512-qubit quantum annealer (with 200 instances per problem): N isthe number of assets, T is the number of time steps, K is the number of units to be allocated at each timestep and the maximum allowed holding (with K ′ = K), “encoding” refers to the method of encodingthe integer problem into binary variables, “vars” is the number of binary variables required to encodethe given problem, “density” is the density of the quadratic couplers, “qubits” is the number of physicalqubits that were used, “chain” is the maximum number of physical qubits identified with a single binaryvariable, and S(α) refers to the success rate given a perturbation magnitude α% (explained in the text).

N T K encoding vars density qubits chain S(0) S(1) S(2)2 3 3 binary 12 0.52 31 3 100.00 100.00 100.002 2 3 unary 12 0.73 45 4 97.00 99.50 100.002 4 3 binary 16 0.40 52 4 96.00 100.00 100.002 3 3 unary 18 0.53 76 5 94.50 99.50 100.002 2 7 binary 12 0.73 38 4 90.50 100.00 100.002 5 3 binary 20 0.33 63 4 89.00 100.00 100.002 6 3 binary 24 0.28 74 4 50.00 97.50 99.503 2 3 unary 18 0.65 91 6 38.50 72.50 91.503 3 3 binary 18 0.45 84 5 35.50 66.50 82.503 4 3 binary 24 0.35 106 6 9.50 50.50 84.50

finding of the optimal solution.

The chosen approach relaxes the success metricin a problem-instance-specific way. An alternativewould be to define success as finding a solutionwithin ε of the optimum. However this alternativesuccess metric has the disadvantage that ε could besmall or large compared to the energy scale of aparticular problem instance, and so the metric canbe misleading for problems of the type being solvedhere.

Although the variance of the success rate wouldalso be interesting to observe, the number of exper-imental runs required for this metric to be statisti-cally significant were not able to be performed dueto a limited availability of machine time.

In order to investigate the quality of the solutionon software enhancements, we replaced the “em-bedding solver” supplied with the D-Wave quan-tum annealer with a proprietary embedding solverdeveloped by 1QBit, tuned the identification cou-





TABLE V: Results using D-Wave’s 512-qubit quantum annealer, with custom-tuned parameters andsoftware (with 200 instances per problem): an improved embedding solver, fine-tuned identificationcoupling strengths, and averaging over 5 random gauges (200 reads per gauge, giving a total of 1000reads per call). Columns are as in Table IV, and the hardware used was the same as in that table.

N T K encoding vars density qubits chain S(0) S(1) S(2)2 3 3 binary 12 0.52 31 3 100.00 100.00 100.002 4 3 binary 16 0.40 52 4 99.50 100.00 100.003 2 3 unary 18 0.65 91 6 99.00 100.00 100.002 3 3 unary 18 0.53 76 5 98.50 99.50 100.002 5 3 binary 20 0.33 63 4 96.00 100.00 100.002 6 3 binary 24 0.28 74 4 80.00 100.00 100.002 4 3 unary 24 0.41 104 6 70.50 96.50 99.503 4 3 binary 24 0.35 106 6 44.00 92.50 99.003 3 3 binary 18 0.45 84 5 65.50 98.00 100.003 6 3 binary 36 0.24 196 7 0.50 74.50 99.004 4 3 binary 32 0.32 214 8 1.50 13.50 60.504 5 3 binary 40 0.26 281 10 0.00 3.00 24.00

TABLE VI: Results using D-Wave’s 1152-qubit quantum annealer (with 200 instances per problem), withlower noise and higher yield, and with custom-tuned parameters and software (as in Table V). Columnsare as in Table IV.

N T K encoding vars density qubits chain S(0) S(1) S(2)3 2 3 unary 18 0.65 86 5 99.50 100.00 100.003 3 3 binary 18 0.45 61 4 83.50 99.00 100.003 3 3 unary 27 0.46 146 7 81.50 92.50 97.003 4 3 binary 24 0.35 87 5 75.50 100.00 100.003 4 3 unary 36 0.36 209 8 40.50 52.50 60.504 4 3 binary 32 0.32 157 6 27.00 46.50 64.503 6 3 binary 36 0.24 143 5 15.50 59.00 78.504 5 3 binary 40 0.26 210 7 4.00 34.50 64.005 5 3 binary 50 0.25 321 8 4.00 13.00 27.006 5 3 binary 60 0.24 492 10 2.00 20.50 49.506 6 3 binary 72 0.20 584 11 0.00 6.50 21.00

pling strength, and combined results from calls withmultiple random gauges. A gauge transformationis accomplished by assigning +1 or −1 to eachqubit, and flipping the sign of the coefficients inthe problem matrix accordingly, such that the op-timization problem remains unchanged. We founda large improvement for all problems. Results withthese improvements are presented in Table V.

To investigate the dependence of the successrate on the noise level and qubit yield of thequantum annealer, several problems were solved ontwo different quantum annealing chips. Results forthe second chip, the D-Wave 2X, which has 1152qubits, a lower noise level and fewer inactive qubitsand couplers, are presented in Table VI. We foundan increase in all success rates, as well as the ability





to solve larger problems.These investigations highlight the importance

of having an in-depth understanding of D-Wave’squantum annealer in order to be able to achieve thebest results possible.

D. DiscussionThe success rate and the ability to solve larger

problems are affected by certain hardware parame-ters of the quantum annealer. First, there is a level ofintrinsic noise which manifests as a misspecificationerror—the coefficients of the problem that the quan-tum annealer actually solves differ from the problemcoefficients by up to ε. 3 For future chip generationsthe expectation is that ε will decrease, and hence thesuccess rate will increase by virtue of the quantumannealer solving a problem that is closer to theproblem passed to it. In addition, the problem coef-ficients on the chip have a defined coefficient range,and if the specified problem has coefficients outsidethis range, the entire problem is scaled down. Thiscan result in coefficients becoming smaller thanε, affecting the success rate. These factors areespecially relevant for high-precision problems suchas the multi-period portfolio optimization problemsolved here.

The quantum annealer has a hardware graphthat is currently very sparse and in general doesnot match the problem graph. In order to solveproblems that are denser than the hardware graph,multiple physical qubits are identified with a singlebinary variable, at the cost of using more physicalqubits. In order to force the identified qubits to allhave the same value, a strong coupling is needed.If the required coupling is outside of the range ofthe couplings in the problem, the result will bean additional scaling, possibly reducing additionalcoefficients to less than ε, again impacting thesuccess rate. Generally, the denser the hardwaregraph is, the fewer identifications are needed, andthe weaker the couplings are required to be toidentify the qubits.

To solve larger problems, the number of qubitsmust be greater. More qubits would also allow the

3The intrinsic noise for the current generation of chip isestimated to be around 2%–4% of the full scale.

use of an integer encoding scheme that is lesssensitive to noise levels (such as unary encodingversus binary encoding). The fabrication process isnot perfect, resulting in inactive qubits and couplers.The more inactive qubits and couplers there areon a chip, the lower the effective density and thehigher the number of identifications required, whichtypically reduces the success rate.

In addition, custom tuning, through software, canbe used to enhance the results. In particular, whenthe problem to be solved has a graph that differsfrom the hardware graph, a mapping, referred to asan “embedding”, must be found from the problemgraph to the hardware graph. The development ofsophisticated ways to find better embeddings (forexample, with fewer identified qubits) would beexpected to increase the success rate—often thestructure of the problem can be exploited in orderto find better embeddings. In addition, when anembedding is used, there are different ways inwhich the couplings of the identified qubits shouldbe set, controlled by the “embedding solver”. Forexample, the strength of the couplings could be fine-tuned further to give higher success rates, or tunedseparately for each set of identified qubits [39].

The issue of which embedding properties aremost desirable is still under active research. Herethe pi-elite metric has been used to select the bestscaling of the problem versus the strength of thequbit identification chains, and to choose the bestembedding amongst the highest-ranked embeddings[39]. The pi-elite score is determined by comparingthe mean energy of the best (“elite”) states (forexample, the lowest 2%) found by using eachscale/embedding. The embeddings were ranked us-ing a scheme that combines equal weights based onthe shortest of the longest chains, the total numberof qubits, and the variance of the lengths of thechains, after which the highest-ranked embeddingswere compared using their pi-elite scores.

In addition it has been observed that there isa gauge transformation under which the problemand solution remain invariant, but the success ratesvary strongly (due to imperfections in the annealingchip). Combining results from multiple calls tothe solver with random gauges, as we did, could





provide a large improvement [39]. Software errorcorrection, such as majority voting (which we em-ployed) when the identified physical qubits do notagree, as well as calibration, could also lead toimproved solutions [40]–[46]. We also note that itmay be possible to use the quantum annealer to findgood local minima, which could then be used tospeed up deterministic or heuristic classical solvers[47], [48].

Although the core contributions of this paper arethe formulation of the general multi-period opti-mization strategy and discussion of the issues in-volved in solving that problem using available quan-tum annealing hardware, a brief comment regardingthe time taken to calculate a solution is warranted.In general, benchmarking of the time to solutionof a D-Wave quantum annealer against classicalhardware is an area of ongoing and active research.For the small-scale problems solved in this study,the time to solution, on both classical hardwareand using the quantum annealer, is comparable. Therecent results by Denchev et al. [33], which showthat for a specific class of problems the D-Wavemachine has the potential to provide speedup overcomputations on classical hardware, are encourag-ing. However, only after quantum speedup has beendemonstrated for general problems, and specificallythose requiring a high precision of couplings, is itexpected that a quantum speedup for the optimaltrading trajectory problem will be observed.

III. CONCLUSIONS

In this limited experiment we have demonstratedthe potential of D-Wave’s quantum annealer toachieve high success rates when solving an impor-tant and difficult multi-period portfolio optimizationproblem. We have also shown that it is possibleto achieve a considerable improvement in successrates by fine-tuning the operation of the quantumannealer.

Although the current size of problems that can besolved is small, technological improvements in fu-ture generations of quantum annealers are expectedto provide the ability to solve larger problems, andat higher success rates.

Since larger problems are expected to be in-tractable on classical computers, there is muchinterest in solving them efficiently using quantumhardware.

APPENDIX ADEFINITION OF SYMBOLS

The symbols used above are defined in Table VII.In addition, we use wt to denote the t-th column ofthe matrix w (and similarly for µ), and Σt to denotethe covariance matrix (N × N ) which is the t-thpage of the tensor Σ. For convenience of notation,the temporary transaction costs c are representedusing the tensor Λ, where Λtnn′ = cntδnn′ (andsimilarly for the permanent price impact c′ andΛ′). The difference in holdings between two timeperiods is defined as ∆wt ≡ wt − wt−1.

ACKNOWLEDGMENT

The authors would like to thank Marko Bucykfor editing a draft of this paper, and Robyn Foerster,Pooya Ronagh, Maxwell Rounds, and Arman Zarib-afiyan for their useful comments. Clemens Adolphsand Dominic Marchand provided invaluable helpin obtaining the numerical results, and MohammadVazifeh’s code was used for the Chimera-graph fig-ure. This work was supported by 1QB InformationTechnologies and Mitacs.

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Gili Rosenberg is a Senior Researcherat 1QBit. His research focuses on theapplication of quantum annealing tech-nology to solve problems in computa-tional finance that are traditionally for-mulated using machine learning tech-niques such as neural networks, andheuristic algorithms such as simulatedannealing.

Dr. Rosenberg holds a Ph.D. inPhysics from the University of British

Columbia (2012) and a B.Sc. in Physics from Ben-GurionUniversity (2004).

Poya Haghnegahdar is currently com-pleting a Ph.D. degree in Physics atthe University of British Columbiawhere his research focuses on themeasurement-based model of quantumcomputing, Monte Carlo simulationtechniques, and percolation theory.

He was formerly the Applicationsand QA Lead at 1QBit and has practi-cal experience in machine learning anddata analysis techniques. He holds an

M.Sc. in Physics from the University of British Columbia (2009)and a B.Sc. in Physics from the University of Waterloo (2005).


http://link.aps.org/doi/10.1103/PhysRevA.92.042325

http://www.sciencemag.org/content/345/6195/420.abstract

http://link.aps.org/doi/10.1103/PhysRevApplied.5.034007

http://link.aps.org/doi/10.1103/PhysRevApplied.5.034007



http://link.aps.org/doi/10.1103/PhysRevX.5.031026




Phil Goddard is Head of Researchat 1QBit. His research focuses on theapplication of heuristic and machinelearning algorithms to topics found incomputational finance and traditionalengineering. Working with clients andpartners spanning a range of businesssectors, he leads teams focused on ap-plying quantum annealing technologyto a variety of traditional optimizationproblems.

With 25 years of experience in designing and developing cus-tom software applications for modelling, simulation, data analy-sis, and visualization, Dr. Goddard is also President and PrincipalConsultant at Goddard Consulting. He was formerly a SeniorConsultant at The MathWorks, where he worked with clientsin the financial services, automotive, aerospace, telecommuni-cations, and process industries. He has also worked for BritishAerospace (Dynamics) and Woodside Offshore Petroleum. Since2006 he has held a position as Visiting Lecturer within theBeedie School of Business at Simon Fraser University, wherehe lectures on topics in numerical analysis and software devel-opment for computational finance.

Dr. Goddard holds a Ph.D. in Engineering from CambridgeUniversity (1995), and a B.Eng. from the University of WesternAustralia (1991).

Peter Carr currently serves as theExecutive Director of the Math Fi-nance program at NYU’s Courant In-stitute. Formerly a Managing Directorand Global Head of Market Model-ing at Morgan Stanley, Dr. Carr hasapproximately 20 years of experiencein the financial services industry. Heis also a trustee for the Museum ofMathematics in New York.

Prior to joining the financial indus-try, Dr. Carr was a finance professor for 8 years at CornellUniversity, after obtaining his Ph.D. from UCLA in 1989. Hehas over 75 publications in academic and industry-orientedjournals and serves as an associate editor for 8 journals relatedto mathematical finance. He was selected as Quant of the Yearby Risk Magazine in 2003 and Financial Engineer of the Yearby IAFE and Sungard in 2010. For the last 4 years, Dr. Carr wasincluded in Institutional Investor’s Tech 50, an annual listing ofthe 50 most influential people in financial technology.

Kesheng Wu is the Group Leader ofthe Scientific Data Management Groupat Lawrence Berkeley National Labo-ratory (U.S. Department of Energy’sOffice of Science). His current researchfocuses on indexing technology forsearching large data sets, including im-proving bitmap index technology withcompression, encoding, and binning.Dr. Wu is the key developer of Fast-Bit bitmap indexing software, which is

used in a number of applications including high-energy physics,combustion, network security, and query-driven visualization. Hehas also worked on various scientific computing projects includ-ing developing the Thick-Restart Lanczos (TRLan) algorithmfor solving eigenvalue problems and devising statistical tests fordeterministic effects in broad band time series.

Dr. Wu holds a Ph.D. in Computer Science from the Universityof Minnesota, an M.Sc. in Physics from the University ofWisconsin-Milwaukee, and a B.Sc. in Physics from NanjingUniversity.

Marcos López de Prado is Se-nior Managing Director at GuggenheimPartners. He is also a Research Fellowat Lawrence Berkeley National Lab-oratory’s Computational Research Di-vision (U.S. Department of Energy’sOffice of Science), where he conductsunclassified research in the mathemat-ics of large-scale financial problemsand supercomputing.

Previously, Dr. López de Prado wasHead of Quantitative Trading and Research at Hess EnergyTrading Company (the trading arm of Hess Corporation, aFortune 100 company) and Head of Global Quantitative Researchat Tudor Investment Corporation. In addition to his 17 yearsof trading and investment management experience at someof the largest corporations, he has received several academicappointments, including Postdoctoral Research Fellow of RCC atHarvard University and Visiting Scholar at Cornell University.

Dr. López de Prado holds a Ph.D. in Financial Economics(2003), and a second Ph.D. in Mathematical Finance (2011) fromComplutense University. He is a recipient of the National Awardfor Excellence in Academic Performance by the Government ofSpain (National Valedictorian, 1998) among other awards, andwas admitted into American Mensa with a perfect test score.

He serves on the Editorial Board of the Journal of PortfolioManagement (IIJ), the Journal of Investment Strategies (Risk)and the Big Data and Innovative Financial Technologies Re-search Series (SSRN). He has collaborated with many leadingacademics, resulting in some of the most read papers in finance(SSRN), four international patent applications on high frequencytrading, three textbooks, and numerous publications in top math-ematical finance journals.


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