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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 [7−24]

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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