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Portfolio Optimization of 60 Stocks Using Classical and Quantum Algorithms

Abstract and Figures

We continue to investigate the use of quantum computers for building an optimal portfolio out of a universe of 60 U.S. listed, liquid equities. Starting from historical market data, we apply our unique problem formulation on the D-Wave Systems Inc. D-Wave 2000Q (TM) quantum annealing system (hereafter called D-Wave) to find the optimal risk vs return portfolio. We approach this first classically, then using the D-Wave, to select efficient buy and hold portfolios. Our results show that practitioners can use either classical or quantum annealing methods to select attractive portfolios. This builds upon our prior work on optimization of 40 stocks.
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Portfolio Optimization of 60 Stocks Using
Classical and Quantum Algorithms
Chicago Quantum
email the authors
August 21, 2020
Contents
1 Introduction 2
2 Market Context 2
3 Shape of the Energy Landscape 3
4 Classical and Hybrid Methods 3
4.1 Fat Tailed Monte Carlo Analysis . . . . . . . . . . . . . . . . . . 3
4.2 GeneticAlgorithm .......................... 4
4.3 SimulatedAnnealer.......................... 4
4.4 D-Wave Tabu Multistart MST2 Sampler . . . . . . . . . . . . . . 6
4.5 D-Wave Hybrid Sampler . . . . . . . . . . . . . . . . . . . . . . . 7
5 D-Wave Systems Quantum Annealer 7
6 Quality of Solutions Found 12
7 Comparative Analysis 15
8 Scale-Up Potential 15
9 Conclusion 16
Abstract
We continue to investigate the use of quantum computers for build-
ing an optimal portfolio out of a universe of 60 U.S. listed, liquid equities.
Starting from historical market data, we apply our unique problem formu-
lation on the D-Wave Systems Inc. D-Wave 2000QTM quantum annealing
Jeffrey Cohen, Alex Khan, Clark Alexander
1
arXiv:2008.08669v1 [q-fin.GN] 19 Aug 2020
system (hereafter called D-Wave) to find the optimal risk vs return port-
folio. We approach this first classically, then using the D-Wave, to select
efficient buy and hold portfolios. Our results show that practitioners can
use either classical or quantum annealing methods to select attractive
portfolios. This builds upon our prior work on optimization of 40 stocks
1 Introduction
Our work is inspired by the notion that we can find attractive investment port-
folios from a universe of US equities leveraging a quantum computer. As we
scale the problem with the number of equities analyzed (portfolio choices are
2n), we investigate whether quantum annealing can scale up vs. classical meth-
ods and select a reasonable sized grouping of attractive portfolios, as opposed
to just one ideal solution.
We advanced our classical methods. We now have five methods to find the
ideal portfolio from 60 stocks in under a minute. This raises the bar for quantum
annealing, which takes about 0.13 seconds (or 130,000µs) to sample 200 to
500 times within one run (to find the best Nout of 60 stocks). The D-Wave
time, 21 seconds, is an accumulation of all runs used in this research in our
primary account, and the results are an accumulation of all valid portfolios
found in this research.
We retain our prior formulations; The Chicago Quantum Net Score and the
Chicago Quantum Ratio defined by the following
CQNS(Rw, α) = V ar(Rw)E[Rw]2+α(1)
and
CQR(Rw) = C ovim(Rw)
σ(Rw)(2)
where Rwis a weighted portfolio, α > 0 is a real number, which we generally
set to 1. C ovim is the covariance of our portfolio against the entire market, which
we take as the S&P 500 for this article.
In this paper we will provide our progress, and setbacks, with classical meth-
ods, then focus attention on our quantum annealing on the D-Wave DW 2000Q 6
2,048 qubit, D-Wave 2000Q lower-noise system, with the [16,16,8] Chimera
topology. We will then provide a comparative analysis and discuss the scale-up
potential of the D-Wave.
2 Market Context
We performed our research during a time of market increases for the largest
companies, and a relatively low interest rate environment. Our analysis used
a risk-free rate of 1%. The markets have seen a rise of 18.60% over the past
year, when taken as the average of the increases of the four equity indices we
use: S&P 500, Nasdaq Composite, Russell 2000 and Wilshire 5000. The range
2
of βfor the 60 stocks was [0.417,2.12]. The variance of the S&P 500, used in
the Chicago Quantum Ratio, is 0.00045105. The 60-asset, all in portfolio of
equally weighted stocks, has an expected future return of 22.09% (using the
Capital Asset Pricing Model), and the standard deviation is expected to be
2.48%, yielding a Sharpe Ratio of 8.92. Our model does use prior year trading
history to pick its portfolios. The parameters of our model remain unchanged
since our last paper in July 2020.
3 Shape of the Energy Landscape
We start with a Monte Carlo random sampling analysis to understand the high-
level shape of the energy landscape before we look to the different methods to
solve it. We score in excess of 220 thousand portfolios across all portfolio sizes
and store the average, minimum and maximum CQNS values which we can plot
and review. As shown in 1 the blue bar shows the CQNS average, which is
stable across portfolio sizes, which implies there is no bias toward any portfolio
of a particular magnitude. Additionally, in green we see the CQNS minimum
values which have a few special cases, which are highly attractive portfolios
where the expected return greatly outweighs the expected risk. The shape may
be different for each set of stocks analyzed.
Figure 1: CQNS minima and averages
4 Classical and Hybrid Methods
4.1 Fat Tailed Monte Carlo Analysis
We keep track of our random samples in an array which holds the minimum,
maximum and average CQNS score for each size portfolio analyzed. We initial-
ize this with random samples in a discrete distribution (centered around N/2
assets), then run a random sample for each portfolio size. In our research, these
two sampling methods generated 221,660 samples. In our last run, we found
the “ideal” portfolio as the solution is in a very small portfolio.
This “fat tailed MC” method will do well if the solution is either very large
or very small. The random sampler which is centered on the N/2 size portfolios,
will generally not do well. In a 60 asset universe, there are 1.15 ×1018 possible
3
combinations. We cannot count on sampling all of them. However, we ran this
experiment twice. In one case we found the best answer and in another we found
the 2nd best answer; both in 24 seconds. Again, this only worked because the
best answer was in a tail of the portfolio sizes with very few possibilities, and
the random sampler can find them.
Like in the simulated annealing case, the Monte Carlo analysis has the great-
est range of CQNS values at the smallest portfolios (best and worst portfolios).
The range and relative attractiveness narrows as we analyze larger portfolio
sizes. In the graphics, the sequence moves from small to large portfolios.
4.2 Genetic Algorithm
A genetic algorithm looks to bring the best attributes (stocks) forward from two
portfolios (parents) by a process of breeding them and creating new portfolios
(children and mutations) that we then score. We keep the best X portfolios as
scored by the CQNS as parents in the next generation, then breed again.
Our genetic algorithm (GA) is custom coded to start with an initial popula-
tion of parents (either 456 random portfolios or seeded with D-Wave solutions).
We tune it to run for 40 generations and pass the 40 best solutions (equal or
better values only) to the next generation. We breed a ratio of 3:2 children to
mutations. Experimentally, both solve the problem of optimizing the 60 asset
portfolio quickly. In our last run, the GA (456 random) took 7 seconds and GA
(2,588 D-Wave) took 48 seconds to find the best solution. Typically, however,
the GA (D-Wave) runs 20% faster than GA (random) as we start with a smaller
and better scoring initial population.
4.3 Simulated Annealer
A simulated annealer models the temperature-based evolution process, where
the algorithm is looking for the lowest energy solution where the ability to jump
to a new interim solution outside of a local minima decreases as the system
cools. At higher temperatures, it is more free to tunnel or jump to neighboring
energy levels and look for deeper energy minima. As the temperature “cools”,
it becomes harder to jump far and we look for the best solution in that neigh-
borhood. In this model, the Chicago Quantum Net Score is a fixed multiple of
the energy level at each (Nof 60-asset) QUBO. By minimizing the energy level,
we find the best CQNS and investment portfolio.
Our simulated annealer is custom coded in Python 3 in about a page of code.
In our tuned version, it finds the optimal 60 asset solution some of the times,
and a good solution otherwise. We can tune it to run longer, which increases
the frequency of success. In our last run, it found the optimal portfolio from 60
stocks in 15 seconds.
We tune our simulated annealer with four parameters. (1) When to jump
to a neighboring solution. When the temperature is warm, sometimes our algo-
rithm jumps even when the score is slightly worse. (2) Initial temperature and
minimum temperature which determines the temperature range for annealing.
4
(3) Cooling rate determines how fast the temperature cools per cycle, and is
used to determine the number of annealing cycles Tmax Tmin
cooling rate , and (4) the num-
ber of annealing trials, or as D-Wave calls them “sweeps” per annealing cycle.
We tune this mix to maximize the frequency of finding the optimal solution in
the shortest time. This is an ad-hoc exercise.
We also use the D-Wave Simulated Annealer as an alternative sampler. In
our most recent runs, it finds the optimal portfolio that we allowed it to find in
11 seconds, but modifies the energy level of that portfolio. We run the simulated
annealer by specifying the range of portfolio sizes, and resulting QUBO, to run
through the annealer. Normally we run [2,59], but in the last case to save time
we ran [2,50]. Like the quantum annealer, it has to match the number of assets
to the desired portfolio size for us to accumulate those answers.
Our settings have been tuned to speed up the annealer. We run a βrange,
or inverse of the temperature, at [0.000001,9]. We run the simulated annealer
to sample 200 portfolios for each portfolio size. We also set the number of
“sweeps” or times to look at each energy cycle at 200. We use a βschedule
of type “geometric.” With these settings, we run for 11 seconds, find 168 out
of 9,600 trials, and see the 2nd best result (and the best we allow it to find),
repeatedly in the results. If we want to look at good portfolios at different sizes,
like we can with the quantum annealing answers, we see valid portfolios at many
portfolio sizes.
In the graphic below we see that the smaller portfolios (run in order of size),
have the widest spread, and the best possible results. As we get larger, we see
an almost asymptotic tightening of the CQNS scores in the middle. We see this
with most samplers we use.
(a) Simulated Annealing Solutions (b) Monte Carlo Simulation
Figure 2: Simulated Annealing vs Monte Carlo Sampling
5
4.4 D-Wave Tabu Multistart MST2 Sampler
The D-Wave Tabu Sampler was run against our QUBOs and we saw the least
attractive portfolios from this method. The sampler picked most of its solutions
from the 20 to 40 asset size portfolios despite being run against the different
QUBOs with the penalty functions. We also see that the Tabu sampler started
by finding very large (poor) CQNS values, and quickly reached a plateau of its
best answer. Unfortunately, the best Tabu scores found were worse by a factor
of 10 or more.
(a) Assets Chosen vs QUBO size (b) Portfolio Sequence
Figure 3: On the right, we see the Tabu sampler picked portfolio sizes between
20 and 40 regardless of the desired portfolio sizes, and penalty functions. On
the left we see CQNS values chosen by the Tabu sampler, and how it starts with
poor (high) values.
We set the sampler to run from [2,60] assets, with 200 reads per QUBO, a
scale factor of 1, a maximum of 20µs to run, and a tenure of 50. The tenure is
the number of answers to store in memory to save time in the run. Like with
the D-Wave quantum annealer, we keep and accumulate valid portfolios from
each size portfolio, and in 11,600 trials we had 190 valid portfolios found. The
final run took 267 seconds. We have an open question of whether increasing
the run time, reads per QUBO, and lowering the tenure would help us find
better answers. We had initially set the Tabu Sampler to run longer with these
settings, but did not see better results.
6
4.5 D-Wave Hybrid Sampler
At this point we do not see valid results from the hybrid sampler. It is more of
a “black box” solver for us where we can set a few parameters and feed it the
same 60 ×60 ×60 matrix used by the other methods. We set it to run across
portfolio sizes using the same QUBO as the other methods, and it found no
valid portfolios that match the size required. It does find “good” portfolios, but
the CQNS scores are incorrect due to the penalty we apply.
5 D-Wave Systems Quantum Annealer
(a) 10 assets (b) 20 assets
(c) 30 assets (d) 40 assets
(e) 50 assets
Figure 4: Charts showing resulting energy levels from QUBO (N=
10,20,30,40,50 assets) before (blue) and after (green) affine transformation
based on random sampling.
7
Similar to our simulated annealer, a quantum annealer modifies the quantum
fluctuations of a physical set of qubits to allow the spin of the qubits to jump
to a neighboring solution to find a lower energy level. The resulting energy
level, which we convert to a CQNS score, is the value the quantum annealer is
minimizing by making those jumps between solutions. At the end, the system
reads the energy and spin of the qubits, rounds up or down, does a majority
vote if there is a chain break, and outputs a string in {0,1}60 (one for each
stock), the energy of the system, and the value of the chain break, if any. We
re-factor the energy to a CQNS score, and append the values to an output file.
We build a 60 ×60 ×60 “bigmatrix” of values which load the negative of
the expected return content on the linear terms, and load the variance and
covariance content on the quadratic terms.
We then apply an affine transformation to the values (shift) so the non-
desired number of assets have a penalty. We set the affine transformation with
three points. The zero asset point must have an energy value of zero, the desired
asset count must have an accurate count, and we can increase the energy level
for all other points. We create a shape like a pulley in the energy values that
result from executing the QUBO in simulation or in D-Wave.
In figure 4 we see the 10, 20, 30, 40 and 50 asset QUBO energy landscape. We
bend the shape of the curve so the desired number of assets is centered around
the lowest energy levels, while still staying accurate at that number of assets,
during a small random sampling of CQNS values. We ground the 0,0 point for
zero assets, set the point for the number of assets desired to be accurate, and
vary the energy level of the 60,60 asset point to the positive, or a penalty.
Sometimes creating an affine transformation with the apparent shape we
need is not enough. In other words we might not get any matching results for
the desired assets. As we run our QUBO on D-Wave, we also visualize the
results of all the samples received back from D-Wave. We use that visualization
to further tune the shift of the matrix. In the chart below, we see the orange
points which indicate the assets desired by each QUBO run. The blue points
are the size of the portfolios returned by D-Wave. In case of overlap, we see the
orange dots. You can see that as we exceed 36 assets, the portfolios received
were significantly smaller than desired, which required us to further adjust our
shift values.
After we apply the affine transformation, we scale the matrix values to be-
tween [0.9,0.9] before we send it to D-Wave. Experimentally we find D-Wave
does better when we apply the initial scaling to the data. We believe this is
because D-Wave does additional scaling while it feeds the data to the quantum
annealer, and if the values are divergent enough may apply different scaling fac-
tors to the linear and quadratic terms. This would nullify our CQNS score and
the predictive ability of our model. The D-Wave annealer can accept H Range
values between [2,2] in the linear term and J Range values between [0.9,0.9]
in the quadratic terms for Ising formulations. We find that if we perform our
scaling, we receive valid D-Wave energy levels, and resulting CQNS scores when
compared to classical validation.
We find it difficult to have D-Wave find larger-sized portfolios. It typically
8
(a) Quadratic Affine Transformation of tri-
als size 2 to 50
(b) Graduated Affine Transformations
of trials size 2 to 50
Figure 5: On the left: two trials: 2-50: resulting portfolios always lower than
desired number of assets. 20-40: after shift shows results in the number of assets
desired. On the right: displaying how a small shift in the affine transformation
allowed portfolios detected at the higher end 40 assets to increase. The desired
effect is having as little spread as possible around a centered orange line.
finds smaller portfolios despite the penalty on them. In our research on 60
assets, the largest valid portfolio we received from the D-Wave was 47 assets.
We believe we could continue to adjust chain strength and annealing time to
increase our chances of a matching portfolio. However, we believe we may need
to further refine the QUBO for larger portfolio sizes (e.g., 45 or more assets)
to continue to scale to 100 or more assets. This assumes we believe the largest
portfolios can also be the most efficient.
The challenge is our quadratic values become too small relative to the linear
terms. At 59 assets, the linear terms still vary from [1,0.2], but the quadratic
terms are [0.0037,0.0045].
We understand why this happens. The covariance is a quadratic term and
it decreases as the number of assets increases. At just a few assets, the linear
and quadratic terms are in the same order of magnitude, and D-Wave seems to
find excellent, or in one case the ideal value, in those smaller assets.
This is an open question in our continuing research...how to rescale or re-
formulate our QUBO so that portfolio sizes from N/2 to Nhave increasing
covariance values on the quadratic terms instead of decreasing as they do now.
We then convert the 60×60×60 shifted matrix to QUBO formulations which
9
(a) bigmatrix for 59 assets
(b) heat map for big matrix with 59 assets
Figure 6: Heat map for a big matrix with 59 assets.
are individually sent to D-Wave. D-Wave applies its own scaling to achieve
acceptable H range and J range values. This same matrix is used for all of
our classical models which require a matrix as input (e.g., simulated annealer,
TABU sampler, and the hybrid solver).
We run the quantum annealer repeatedly against our QUBOs in this exper-
iment and accumulate valid portfolios (where number of assets chosen matches
the number of assets desired in that QUBO). Our initial runs, which accumu-
lated 116 values, were on the 30 asset QUBO and were intended to calibrate and
tune the D-Wave and parameters we use. We adjust settings for chain strength,
post processing, shift, scale, annealing time, reduce intersample correlation, and
to look at the resulting solutions, and information about the programming of
the QUBO on the qubits. We use the D-Wave Inspector to better understand
the solution space and the loading of the QUBO. Including the calibration, we
10
(a) bigmatrix for 2 assets
(b) heat map for big matrix
Figure 7: Heat map for a big matrix with 2 assets.
accumulate a total of 3725 valid portfolios with portfolio size ranges of [2,47].
We find an inverse relationship between the chain strength setting and the
maximum chain length (and resulting chain breaks). We find an inverse relation-
ship between chain strength and sensitivity to find valid portfolios. We settled
into a chain strength range of [0.2,2.0] against a default value of 1.0. Chain
strengths >2.0 did not provide valid results in our runs. In this chart you can
see the layout of our 60 ×60 QUBO shifted for 58 assets after embedding 8,
or programming, the QUBO onto the D-Wave qubits and before the annealing
process begins. The yellow line traces the path of one asset being embedded
on multiple qubits in a chain, with covariance connectivity to all other stocks.
This embedding is required to ensure our covariance relationships are retained
with the limited number of connections available between qubits in the actual
hardware.
We also found a relationship between annealing time and number of valid
portfolios found in each run. We vary our annealing time in [20,200]µs, against
11
Figure 8: The connectivity of D-Wave in the [16,16,8] topology for 58 of 60
assets
a default of 20µs. The best results came from larger portfolios [40,47] and
the smallest portfolios [2,4] at 100µs and chain strength of 2. We did not try
annealing times >200µs, although they can go as high as 2,000µs. Longer
annealing times, and higher chain strength values, might be the way to produce
valid solutions at the largest portfolios [48,59]. The mid-sized portfolios [724]
did best with a chain strength of 0.2 and a fast annealing time of 20µs.
6 Quality of Solutions Found
In this paper, we use the CQNS, and the related Chicago Quantum Ratio, as
a proxy for the classical method of portfolio optimization. We consistently find
that the D-Wave system, with just a few valid portfolios, picks portfolios that
12
are on or ahead of the efficient frontier as found through random sampling.
In this case, 414 valid D-Wave quantum annealing portfolios compare favorably
against 40,000 random portfolios, and outperform at the higher risk (or standard
deviation) levels seen below in 1011. When we build investment portfolios, we
find the D-Wave selected CQNS portfolios tend to be relatively more risky than
CQR portfolios while both are still highly efficient.
Figure 9: Above: The comparison of 414 quantum portfolios vs 40000 Monte
Carlo samples using the Sharpe ratio. Below: The comparison of 414 quantum
portfolios vs 40000 Monte Carlo samples using CQR
In the graphic10, you can see the initial 116 valid portfolios with 30 assets
(out of 60) that we used for calibration compared to our 221,660 Monte Carlo
results. The quantum portfolios, in red, have relatively low CQNS values (-
13
0.0146 vs. -0.0437 for top 10 portfolios). D-Wave appears to be picking efficient
portfolios, even out of a population of average results. The blue dots reflect the
genetic algorithm and the simulated annealing results. They reflect the best, or
ideal values, and a sample of the values from the previous generations.
Figure 10: Several Methods using CQR
After accumulating 2,588 valid portfolios, we can see that the red dots cover a
significant portion of the efficient frontier. The quantum points provide investors
with a broad coverage of the efficient frontier after selecting just 2,588 portfolios.
(a) CQR (b) CQNS
Figure 11: Several Comparative methods with different scores.
14
7 Comparative Analysis
We measure success in whether the different methods find the ideal portfolio
and how long each method takes. Six methods 12, including D-Wave, found the
ideal portfolio, which is the portfolio we believe has the lowest (best) CQNS
score. For four of the methods 13, we had Python time the run duration. For
the D-Wave quantum annealer we used the percentage of account resources
used (against a 60 second budget) for the primary account used. The classical
methods were run on a 2013 iMAC with a 3.4GHz Quad-Core Intel Core i5,
with 16GB DDR3 RAM at 1600 MHz, running macOS Catalina and Python
3.7.7.
Figure 12: Table of Comparative methods
Figure 13: Analysis of Comparative methods
8 Scale-Up Potential
We see a few challenges in scaling up this analysis from 60 stocks to 100, and
up to 500 stocks to be analyzed at once. First, our formulation puts significant
strain on the system to evaluate portfolios with larger numbers of stocks (47+).
The variance and covariance content between each stock (and loaded onto qubit
15
coupling terms) becomes too small to be effectively read by the D-Wave system.
Values currently approaching 0.003 as compared to linear terms of -0.9. Second,
room on the D-Wave QPU is filling up. When we program and embed 60
stocks, we use between 1,300 and 1,700 of the available 2048 qubits. Using a
linear extrapolation, there may only be room for 20% more, or 72 stocks, at the
higher end of the loading.
9 Conclusion
Our aim is to extend our research from a 40 to 60 asset universe, to find the ideal
portfolio out of a possible 260 options. Using the CQNS formulation we extend
to 60 assets on both the D-Wave quantum annealer and five classical methods.
We do this through engineering improvements and applying a graduated penalty
to the 60 ×60 ×60 matrix. We improve our classical methods and all run in
under one minute.
We believe we can scale the problem further with improved embedding,
management of the energy delta between portfolios, and continued improvement
in the penalty function.
Our results show that D-Wave can be leveraged to generate an efficient fron-
tier landscape quickly with a small number of samples (e.g., <1,000) and create
a set of attractive portfolios, while potentially finding the “ideal” portfolio. Our
classical methods continue to validate the quantum results.
Addendum
Below are two charts 14, 15 with more complete details of our analyses, stock
info, date and time, and portfolios selected by various methods. We provide
these details for the sake of repeatability by interested readers.
Figure 14: Full Table of Comparative Methods
16
Figure 15: Details of Analysis
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This study examines the total return of investment portfolios composed of mutual funds and analyzes the contributions of strategic asset allocation (investment policy), tactical timing (the periodic over- or underweighting of asset classes relative to the strategic weightings), and security selection (the selection of individual mutual funds to represent asset classes). The results of Brinson, Hood and Beebower (1986) and Brinson, Singer and Beebower (1991) are confirmed using a sample free of several data limitations in their samples. Utilizing data from five model mutual fund portfolios, covering a wide range of asset class combinations over a three year period, I demonstrate that strategic asset allocation policy explains more than 90 per cent of the variation in total portfolio return, and that tactical timing decisions and security selection may also contribute significantly to the variation in total return. I expand the existing literature by performing a pooled regression, which incorporates all portfolios together (in addition to examining each portfolio individually and averaging the results, as in Brinson et al.) I further demonstrate that the results are also valid when examining risk-adjusted return: I extend the analysis to include several types of risk measure, examining the data in terms of absolute and relative variance, and conclude the analysis with an evaluation of the portfolios in terms of riskadjusted return, demonstrating that the results obtained vis-a-vis total return also apply on a risk adjusted return basis.
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Determinants of Portfolio Performance
  • G Brinson
  • L R Hood
  • G Beebower
G. Brinson, L.R. Hood, G. Beebower, Determinants of Portfolio Performance, June 1986, Financial Analysts Journal 42(4):39-44, DOI: 10.2469/ faj.v42.n4.39
Financial Portfolio Management using D-Wave's Quantum Optimizer: The Case of Abu Dhabi Securities Exchange
  • Nada Elsokkary
  • Davide La Shah Khan
  • Travis S Torre
  • Joel Humble
  • Gottlieb
Nada Elsokkary, Faisal Shah Khan, Davide La Torre, Travis S. Humble, Joel Gottlieb. Financial Portfolio Management using D-Wave's Quantum Optimizer: The Case of Abu Dhabi Securities Exchange. IEEE High Performance Extreme Computing Conference,HPEC, Sept. 2017 (edited)