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A Hybrid Approach on Conditional GAN for Portfolio Analysis

Anonymous Authors1

††footnotetext:  1Anonymous Institution, Anonymous City, Anonymous Region,
Anonymous Country.

###### Abstract

Over the decades, the Markowitz framework has been used extensively in
portfolio analysis though it puts too much emphasis on the analysis of the
market uncertainty rather than on the trend prediction. While generative
adversarial network (GAN), conditional GAN (CGAN), and autoencoding CGAN
(ACGAN) have been explored to generate financial time series and extract
features that can help portfolio analysis. The limitation of the CGAN or ACGAN
framework stands in putting too much emphasis on generating series and finding
the internal trends of the series rather than predicting the future trends. In
this paper, we introduce a hybrid approach on conditional GAN based on deep
generative models that learns the internal trend of historical data while
modeling market uncertainty and future trends. We evaluate the model on
several real-world datasets from both the US and Europe markets, and show that
the proposed HybridCGAN and HybridACGAN models lead to better portfolio
allocation compared to the existing Markowitz, CGAN, and ACGAN approaches.

#### Keywords:

Synthetic series, Hybrid CGAN, Autoencoding conditional GAN (ACGAN),
Conditional GAN, Portfolio analysis and allocation, Time series, Sharpe ratio,
Markowitz framework.

##  1 Introduction

Financial portfolio management is largely based on linear models and the
Markowitz framework (Markowitz, 1968; 1976) though the underlying data and
information in today’s market has increased countless times over that of many
years ago. The framework, often known as the modern portfolio theory (MPT),
has become one of the cornerstones of quantitative finance. The fundamental
idea behind the MPT is to create portfolio diversification while reducing
specific risks and assessing the risk-return trade-offs for each asset. The
MPT, on the other hand, has been criticized for making ideal assumptions about
the financial system and data: the expected mean returns, and the covariance
matrix of the return series are estimated from historical observations and
assumed constant in the future. This is such a strong assumption, though, that
it will be impossible for the market to actually achieve this demand.

As the evaluation outcomes of cross-section risk, the traditional portfolio
assessment approach creates portfolio risk indicators based on asset price
series over the previous period, such as variance, value at risk, and expected
loss. The conventional method where the classical mean-variance optimization
approach employed in MPT, however, has two clear shortcomings. First,
historical data typically cannot be simply utilized to indicate the future due
to the capital market’s quick-change since financial returns are notoriously
stochastic with an extremely low signal-to-noise ratio. When a reliable long-
term prognosis is made available in a highly efficient market, traders
immediately act on this forecast, which directly affects the price at hand;
while future price variations are unpredictable again (Kallberg & Ziemba,
1981; 1984; Green & Hollifield, 1992; Best & Grauer, 1991; Timmermann &
Granger, 2004). Second, the linear components in the historical series are
typically all that are included in the risk measuring indicators evaluated
using conventional methods, leaving out the nonlinear information. This causes
a discrepancy between the evaluation results and the actual situation (Tsay,
2005).

The financial sector, on the other hand, has been significantly impacted by
advances in machine learning, deep learning, artificial intelligence. Machine
learning in general has been used in a variety of applications, including
forecasting, series generation, risk management, customer service, and
portfolio management (Huang et al., 2005; Kara et al., 2011; Takahashi et al.,
2019). Especially among these advancements, the generative adversarial
networks (GANs) are a sort of neural network architectures that have shown
promise in image generation and are now being used extensively to produce time
series and other financial data (Goodfellow et al., 2014; Esteban et al.,
2017; Eckerli & Osterrieder, 2021). While models of the ARCH and GARCH
families use traditional statistics to explain the change in variance over
time in a time series by describing the variance of the current error
component as a function of prior errors (Engle, 1982; Bollerslev, 1986; Lu &
Yi, 2022b). GANs are being used to address the problem of paucity of real
data, as well as to optimize portfolios and trading methods which achieve
better results (Takahashi et al., 2019; Mariani et al., 2019). However, due to
its highly stochastic, noisy, and chaotic nature, market price forecasting is
still a major topic in the time series literature. While previous work have
tried to generate financial data based on historical trend, the generation
still lacks guidance on the potential trend of the future series (Mariani et
al., 2019; Lu & Yi, 2022a).

In this light, we focus on GANs for better portfolio allocation that can both
capture historical trends and generate series based on past data. We present a
novel framework about portfolio analysis based on conditional GAN (CGAN) that
incorporates a proposer providing a potential mean value of future series for
data normalization to achieve stable strategies, hence the name HybridCGAN.
Similar to the CGAN and ACGAN models for portfolio analysis (Mariani et al.,
2019; Lu & Yi, 2022a), HybridCGAN can also directly model the market
uncertainty via its complex multidimensional form, which is the primary driver
of future price trends, such that the nonlinear interactions between different
portfolios can be embedded effectively. We evaluate the proposed HybridCGAN
method on two separate portfolios representing different markets (the US and
the European markets) and industrial segments (e.g., Technology, Healthcare,
Basic materials, and Industrials sectors). The empirical results show that the
proposed approach is capable of realizing the risk-return trade-off and
outperforms the classic MPT, CGAN- and ACGAN-based methodologies considerably.

##  2 Related Work

As aforementioned, there are several methods delving with portfolio
allocation, including the Markowitz framework, the CGAN and ACGAN
methodologies (Markowitz, 1968; Mariani et al., 2019; Lu & Yi, 2022a). The
Markowitz framework relies on the assumption that the past trend can be
applied in the future. While the CGAN and ACGAN methodologies partly solve the
drawback in the Markowitz framework by simulating future data based on
historical trends, it still lacks full ability to capture the information and
features behind the past data. The proposed HybridCGAN (and HybridACGAN) model
introduces an extra proposer that can help the constructed networks to capture
historical features and propose future trends.

###  2.1 Markowitz Framework

Portfolio allocation is a kind of investment portfolio where the market
portfolio has the highest Sharpe ratio (SR (Sharpe, 1966)) given the
composition of assets (Markowitz, 1968). For simplicity, we here only consider
long-only portfolio. Denote $\bm{r}$ as the return on assets vector,
$\bm{\Sigma}$ as the asset covariance matrix, $\bm{v}$ as the weight vector of
each asset, and $r_{f}$ as the risk-free interest rate. If we measure
portfolio risk by variance (or standard deviation), then the overall return
and risk of the portfolio are:

$\displaystyle
r=\bm{v}^{\top}\bm{r};\,\,\,\,\,\,\,\,\sigma^{2}=\bm{v}^{\top}\bm{\Sigma}\bm{v}.$
(1)

The Sharpe ratio is simply calculated by

$\text{SR}=\frac{r-r_{f}}{\sigma}=\frac{\bm{v}^{\top}\bm{r}-r_{f}}{\sqrt{\bm{v}^{\top}\bm{\Sigma}\bm{v}}}.$
(2)

According to the definition of asset allocation, the weight of each asset in
the market portfolio is the solution to the following optimization problem:

$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop{\arg\max}_{\bm{v}}\frac{\bm{v}^{\top}\bm{r}-r_{f}}{\sqrt{\bm{v}^{\top}\bm{\Sigma}\bm{v}}};$
(3) $\displaystyle\text{s.t. }\sum_{i=1}^{N}v_{i}=1;\,\,0\leq v_{i}\leq
1,\forall\,i\in\\{1,2,\ldots,N\\},$

where $N$ is the number of assets, and $v_{i}$ is the $i$-th element of the
weight vector $\bm{v}$.

Figure 1: A conceptual overview of the CGAN, ACGAN, and the proposed
HybridCGAN and HybridACGAn generators’ inputs and outputs.

###  2.2 Portfolio Analysis with GAN

Given the number of assets $N$, we consider the matrix $\bm{X}$ to span the
whole analysis length: $w=h+f$ and $\bm{X}\in{}^{N\times w}$. The matrix
$\bm{X}$ contains two components, the known historical series
$\bm{X}_{h}\in{}^{N\times h}$, and the unknown future series
$\bm{X}_{f}\in{}^{N\times f}$.

Considering the (known) historical series $\bm{X}_{h}\in{}^{N\times h}$ and a
prior distribution of a random latent vector $\bm{z}\in{}^{m}$, we use a
generative deep-neural network $G$ to learn the probability distribution of
future price trends $\bm{X}_{f}$ within the target future horizon $f$. Figure
1 provides a conceptual representation of the matrix $\bm{X}$, and the inputs
and outputs of the generator $G$. Formally the generative model simulates a
fake future matrix $\widetilde{\bm{X}}_{f}$ by

$\widetilde{\bm{X}}_{f}=G(\bm{z},\bm{X}_{h}),$ (4)

where $\bm{z}\in{}^{m}$ is the latent vector sampled from a prior distribution
(e.g., from a normal distribution). In practice, The unanticipated future
events and phenomena that will affect the market are represented by the latent
vector $\bm{z}$. Based on the most recent market conditions, the known
historical series $\bm{X}_{h}$ is used to extract features and condition the
probability distribution of the future $\bm{X}_{f}$. Given the historical
observation $\bm{X}_{h}$ and following the Wasserstein GAN-GP (WGAN-GP) by
Gulrajani et al. (2017), the generative $G$ is trained in adversarial mode
against a discriminator network $D$ with the goal of minimizing the
Wasserstein distance between the real future series $\bm{X}_{f}$ and the fake
series $\widetilde{\bm{X}}_{f}$. Formally, the procedure is described by the
following optimization problem:

$\displaystyle\mathop{\max}_{D}$ $\displaystyle\mathbb{E}_{\bm{x}\sim
p(\text{data})}\bigg{\\{}D(\bm{x})-\mathbb{E}_{\bm{z}\sim
p(\bm{z})}\big{[}D(G(\bm{z},\bm{x}_{h}))\big{]}\bigg{\\}}-$ (5)
$\displaystyle\,\,\,\lambda_{1}\cdot\mathbb{E}_{\overline{\bm{x}}\sim
p(\epsilon\,\text{data}+(1-\epsilon)G(\bm{z}))}\big{[}||\nabla_{\overline{\bm{x}}}D(\overline{\bm{x}})||_{2}-1\big{]}^{2};$
$\displaystyle\mathop{\max}_{G}$ $\displaystyle\mathbb{E}_{\bm{x}\sim
p(\text{data})}\bigg{\\{}\mathbb{E}_{\bm{z}\sim
p(\bm{z})}\big{[}D(G(\bm{z},\bm{x}_{h}))\big{]}\bigg{\\}},$

where $\bm{x}_{h}$ contains the historical parts of the data $\bm{x}$
($\bm{x}_{h}\in\bm{x}$), $G(\bm{z},\bm{x}_{h})$ indicates that the generator
depends on the (historical) data $\bm{x}_{h}$, and $\lambda_{1}$ controls the
gradient penalty. Theoretically, the optimization process finds the surrogate
posterior probability distribution $p(\widetilde{\bm{X}}_{f}|\bm{X}_{h})$ that
approximates the real posterior probability distribution
$p({\bm{X}}_{f}|\bm{X}_{h})$.

#### Discriminator

The discriminator shown in Figure 2(b) (for both CGAN and ACGAN) takes as
input either the real data matrix
$\bm{X}=[\bm{X}_{h},\bm{X}_{f}]\in{}^{N\times w}$ or the synthetic data matrix
$\widetilde{\bm{X}}=[\bm{X}_{h},\widetilde{\bm{X}}_{f}]\in{}^{N\times w}$.

The main drawback of the CGAN methodology is in that it puts too much emphasis
on the conditioner to extract features that can “deceive” the discriminator
(Figure 2(a)). When the discriminator is perfectly trained, this issue is not
a big problem. However, in most cases, especially due to the scarcity of
financial data, the discriminator works imperfectly such that the conditioner
may lose important information of the historical data.

(a) Generator

(b) Discriminator

Figure 2: Architectures of the CGAN generative and discriminative models for
portfolio analysis. Figure 3: Architecture of the ACGAN generative model for
portfolio analysis.

###  2.3 Autoencoding CGAN

The Autoencoding CGAN (ACGAN) model partly solves the problem in the CGAN
methodology, in which case it finds a balance between the information
extraction and generation for cheating the discriminator via an embedded
autoencoder providing the capability of keeping the intrinsic information of
historical data (Lu & Yi, 2022a). The ACGAN model has the same discriminator
structure as the CGAN. However, it contains an extra decoder in the generator
as shown in Figure 3. And therefore we call the conditioner an encoder in the
ACGAN context.

We use an encoding deep-neural network $E$ to learn the features that can help
the generator cheat the discriminator and can find the internal information
itself; and a decoding deep-neural network $F$ to reconstruct the historical
series so as to force the encoder to do so. Formally the encoding and decoding
models reconstruct the historical matrix by

$\displaystyle\bm{y}=E(\bm{X}_{h}),\,\,\,\,\,\,\,\,\widetilde{\bm{X}}_{h}=F(\bm{y}).$

This process is known as autoencoding, hence the name autoencoding conditional
GAN (ACGAN). In a non-GAN context, the autoencoder is typically done by
nonnegative matrix factorization or general matrix decomposition via
alternative least squares or Bayesian inference (Lee & Seung, 1999; Lu, 2021;
2022; Lu & Ye, 2022). Since we need to use the encoding part of the
autoencoder to help trick the discriminator as well, the autoencoder is then
constructed by deep-neural networks instead. Formally the process is described
by the following optimization problem:

$\displaystyle\mathop{\max}_{D}$ $\displaystyle\mathbb{E}_{\bm{x}\sim
p(\text{data})}\bigg{\\{}D(\bm{x})-\mathbb{E}_{\bm{z}\sim
p(\bm{z})}\big{[}D(G(\bm{z},\bm{x}_{h}))\big{]}\bigg{\\}}-$ (6)
$\displaystyle\,\,\,\lambda_{1}\cdot\mathbb{E}_{\overline{\bm{x}}\sim
p(\epsilon\,\text{data}+(1-\epsilon)G(\bm{z}))}\big{[}||\nabla_{\overline{\bm{x}}}D(\overline{\bm{x}})||_{2}-1\big{]}^{2};$
$\displaystyle\mathop{\max}_{G,E,F}$ $\displaystyle\mathbb{E}_{\bm{x}\sim
p(\text{data})}\bigg{\\{}\mathbb{E}_{\bm{z}\sim
p(\bm{z})}\big{[}D(G(\bm{z},\bm{x}_{h}))\big{]}-$
$\displaystyle\,\,\,{\color[rgb]{0,0,0}\lambda_{2}\cdot
g\big{(}\underbrace{F(E(\bm{x}_{h}))}_{\widetilde{\bm{x}}_{h}},\,\,\bm{x}_{h}\big{)}}\bigg{\\}},$

where $g(\cdot)$ denotes the error function; the authors apply the mean
squared error as the loss function in Lu & Yi (2022a). The parameter
$\lambda_{2}$ controls how large the penalization by the autoencoder, and the
term is thus known as the autoencoding penalty (AP). In the original CGAN
methodology, the conditioner is used to extract features that can help the
generator to cheat the discriminator; however, it may lose some important
information that captures the internal features of the market trend. The ACGAN
then finds a balance between cheating the discriminator and keeping its market
information.

###  2.4 Data Normalization

Following Mariani et al. (2019); Lu & Yi (2022a), Given the frame window of
$w=h+f$ days 111$h$ for the historical length, $f$ for the future length. The
historical series is denoted by $\bm{p}_{1:h}\in{}^{h}$, and the real future
series can be obtained by $\bm{p}_{h+1:w}\in{}^{f}$., we consider the adjusted
closing price $\bm{p}\in{}^{w}$ series for each asset. Then we unit-normalize
the price series $\bm{p}$ for each asset to fill in the range $[-1,1]$ for the
initial $h$ days. This normalization procedure can help us to expose the
values of neural networks limited within a suitable range that removes price-
variability over multiple assets within the specified window. In practice, the
unit-normalization can be done by 3-sigma normalization: given the mean $\mu$
and standard deviation $\sigma$ of $\bm{p}_{1:h}\in{}^{h}$, the normalization
is done by

$\widetilde{\bm{p}}=(\bm{p}-\mu)/(3\sigma).$ (7)

After generating the surrogate future series $\widetilde{\bm{p}}_{h+1:w}$, we
apply again a de-normalization procedure:

$\widehat{\bm{p}}_{h+1:w}=\widetilde{\bm{p}}_{h+1:w}\times 3\sigma+\mu.$ (8)

##  3 Hybrid Methods

###  3.1 CGAN with Eavesdropping

The proposed hybrid approaches highly rely on the data normalization
procedure. Suppose in Eq. (7), we obtain the mean $m$ of the whole asset
$\bm{p}\in{}^{w}$ in the window $w$ rather than the historical one
$\bm{p}_{1:h}\in{}^{h}$, and we apply the data normalization by this value:

$\widetilde{\bm{p}}=(\bm{p}-{\color[rgb]{0,0,1}m})/(3{\sigma}).$ (9)

Since the $\bm{p}_{h+1:w}$ in $\bm{p}$ cannot be obtained in practice and we
thus call this method eavesdropping. A simple experiment on this approach,
comparing the CGAN and CGAN with Eavesdropping, the return-SR (Sharpe ratio)
plots in Figure 4 shows that this eavesdropping procedure can increase the
performance to a large extent, enforcing the mean Sharpe ratio from about 1.0
to 2.5. And the random draws are more clustered with small deviations such
that the end strategy is more stable. This is reasonable since the GAN finds
the future means of the assets from this normalization and generates the
series whose means are closer to these values.

(a) US, rebalance every 15 days

(b) US, rebalance every 20 days

Figure 4: Eavesdroping: (Annual) return-SR measured on the test period in US
region by randomly sampling 1000 series for CGAN and CGAN with Eavesdropping.
Similar results can be observed for ACGAN and ACGAN with Eavesdroppoing
models.

###  3.2 Proposed Methodology

Though the eavesdropping methods are not practical since we use future data,
the idea can be applied to the proposed hybrid methods. This is known as the
forward bias issue in quantitative finance. We incorporate an extra proposal
network $P$ to find the surrogate of the mean values for each asset in
corresponding windows as shown in Figure 5. The proposer $P$ is trained by
minimizing the mean squared error between the real mean value $m$ of the whole
period and the prediction $\widetilde{\mu}$. Note that we incorporate the
historical mean into the input of the proposal network to make the
optimization easier:

$\widetilde{\mu}=P(\bm{X}_{h},\mu),$ (10)

i.e., we concatenate $\bm{X}_{h}$ and $\mu$ to predict $\widetilde{\mu}$.
After finding the surrogate mean value $\widetilde{\mu}$, the normalization
follows:

$\widetilde{\bm{p}}=(\bm{p}-{\color[rgb]{0,0,1}\widetilde{\mu}})/(3{\sigma}).$
(11)

And the de-normalization:

$\widehat{\bm{p}}_{h+1:w}=\widetilde{\bm{p}}_{h+1:w}\times
3\sigma+{\color[rgb]{0,0,1}\widetilde{\mu}}.$ (12)

Since this approach combines the CGAN and deep neural network regression, we
call it the HybridCGAN model. This hybrid approach can be easily extended into
the ACGAN methodology in a similar way, termed the HybridACGAN model.

(a) Proposer

(b) Generator

(c) Discriminator

Figure 5: Architectures of the HybridCGAN generative and discriminative models for portfolio analysis. | Ticker | Type | Sector | Company | Curr.
---|---|---|---|---|---
US Region | MSFT | Share | IT | Microsoft | USD
GOOG | Share | IT | Alphabet | USD
XOM | Share | Energy | Exxon Mobil | USD
HES | Share | Energy | Hess | USD
PFE | Share | Healthcare | Pfizer | USD
WBA | Share | Consumer staples | Walgreens Alliance | USD
KR | Share | Consumer staples | The Kroger | USD
IYR | ETF | Real estate | iShares US Real Estate | USD
IYY | ETF | Dow Jones | iShares Dow Jones | USD
SHY | ETF | US treasury bond | iShares Treasury Bond | USD
EU Region | VOW3.DE | Share | Automotive | Volkswagen | EUR
BMW.DE | Share | Automotive | BMW | EUR
VK.PA | Share | Industrials | Vallourec S.A. | EUR
SOI.PA | Share | Industrials | Soitec S.A. | EUR
DTE.DE | Share | Technology | Deutsche Telekom AG | EUR
SAP.DE | Share | Technology | SAP SE | EUR
BAS.DE | Share | Basic materials | BASF SE | EUR
BAYN.DE | Share | Healthcare | Bayer AG | EUR
$~{}^{\wedge}$FCHI | Index | French market | CAC 40 | EUR
$~{}^{\wedge}$GDAXI | Index | German market | DAX | EUR
Table 1: Summary of the underlying portfolios in the US and EU markets, 10
assets for each market respectively. In each region, we include assets from
various sectors to favor a somehow sector-neutral strategy.

Figure 6: Different portfolios for the US (upper) and EU (lower) markets with
a unit initial value.

##  4 Experiments

Algorithm 1 Training and testing process for the HybridCGAN, HybridACGAN,
CGAN, and ACGAN models.

1: General Input: Choose parameters $w=h+f$; number of assets $N$; number of
epoches $T$; latent dimension $m$;

2: Training Input:  Training data matrix $\bm{M}\in{}^{N\times D}$;

3: Decide index set $\mathcal{S}_{1}=\\{1,2,\ldots,D-w+1\\}$ and draw without
replacement;

4: for $t=1$ to $T$ do

5: for $i$ $\in$ $\text{{\color[rgb]{0,0,1}random}}(\mathcal{S}_{1})$ do

6: $\bm{X}=[\bm{X}_{h},\bm{X}_{f}]=\bm{M}[:,i:i+w-1]\in{}^{N\times w}$;

7: Randomly sample latent vector $\bm{z}\in{}^{m}$;

8: Backpropatation for generator in Eq. (6) or (5);

9: Generate surrogate
$\widetilde{\bm{X}}_{f}=G(\bm{z},\bm{X}_{h})\in{}^{N\times f}$;

10: Backpropatation for discriminator in Eq. (6) or (5);

11: end for

12: end for

13: Inference Input:  Testing data matrix $\bm{A}\in{}^{N\times K}$;

14: Inference Output:  Testing data matrix $\bm{B}\in{}^{N\times K}$;

15: Decide index set
$\mathcal{S}_{2}=\\{{\color[rgb]{0,0,1}h+1},h+f+1,\ldots\\}$;

16: Copy the first $h$ days data $\bm{B}[:,1:h]=\bm{A}[:,1:h]$;

17: for $i$ $\in$ $\text{{\color[rgb]{0,0,1}ordered}}(\mathcal{S}_{2})$ do

18: $\bm{X}=[\bm{X}_{h},\bm{X}_{f}]=\bm{A}[:,i:i+w-1]\in{}^{N\times w}$;

19: Randomly sample latent vector $\bm{z}\in{}^{m}$;

20: Generate $\bm{B}[:,i:i+f-1]=G(\bm{z},\bm{X}_{h})\in{}^{N\times f}$ with
de-normalization in Eq. (8);

21: end for

22: Output the synthetic series $\bm{B}$;

Figure 7: Actual price trend (black solid line) of the US assets, five
representative simulations (colored dashed lines) generated by HybridCGAN
(upper two rows), and five representative simulations generated by CGAN (lower
two rows) for the first 100 trading days in the test set.

(a) US, rebalance every 10 days

(b) US, rebalance every 15 days

(c) US, rebalance every 20 days

(d) EU, rebalance every 10 days

(e) EU, rebalance every 15 days

(f) EU, rebalance every 20 days

Figure 8: (Annual) return-SR measured on the test period by randomly sampling
1000 series.

To evaluate the strategy and demonstrate the main advantages of the proposed
HybridCGAN and HybridACGAN methods, we conduct experiments with different
analysis tasks; datasets from different geopolitical markets including the US
and the European (EU) markets, and various industrial segments including
Healthcare, Automotive, Energy and so on. We obtain publicly available data
from Yahoo Finance 222https://finance.yahoo.com/.. For the US market, we
obtain data for a 17-year period, i.e., from 2005-05-24 to 2022-05-27, where
the data between 2005-05-24 and 2019-03-28 is considered training data; while
data between 2019-03-28 and 2022-05-27 is taken as the test set (800 trading
days). For the EU market, we obtain data for a 16-year period, i.e., from
2006-07-18 to 2022-06-07, where the data between 2006-07-18 and 2019-04-09 is
considered training data; while data between 2019-04-10 and 2022-06-07 is
taken as the test set (800 trading days). The underlying portfolios are
summarized in Table 1:

  * •

US market: 10 assets of US companies from different industrial segments, i.e.,
GOOG and MSFT (from IT sector), PFE (from Healthcare sector), XOM and HES
(from Energy sector), WBA and KR (from Consumer staples sector), and three
ETFs (IYY, IYR, SHY).

  * •

EU market: 10 portfolios of EU companies from different industrial segments,
i.e., VOW3.DE and BMW.DE (from Automotive sector), VK.PA and SOI.PA (from
Industrials sector), DTE.DE and SAP.DE (from Technology sector), BAS.DE (from
Basic materials sector), BAYN.DE (from Healthcare sector), and two indices,
$~{}^{\wedge}$FCHI and $~{}^{\wedge}$GDAXI, that track the German and French
stock markets respectively.

The specific time periods and assets are chosen by following the four
criteria. 1). Data diversity: in each market, we include companies from
different sectors so that the final strategies are somewhat sector-neutral
with fewer risks; 2). Data availability: we cover as a longer period as
possible to make a decent prediction; the periods are selected to make all the
assets have same frame length; 3). Data correctness: given the Yahoo Finance
data source, we only include the data that do not have NaN values; 4).
Currency homogeneity: in each region, the traded currencies are the same.
Figure 6 shows the series of different assets where we initialize each
portfolio with a unitary value for clarity. In all experiments, the same
parameter initialization is adopted when conducting different tasks. We
compare the results in terms of performance of portfolio allocation and
diversification of the assets. In a wide range of scenarios across various
tasks, HybridCGAN and HybridACGAN improve portfolio evaluations, and lead to
return-risks performances that are as good or better than the existing
Markowitz framework, CGAN, and ACGAN methodologies.

(a) US, rebalance every 10 days, Sharpe ratios of HybridCGAN, HybridACGAN,
CGAN, ACGAN, and Markowitz are 1.32, 0.95, 1.22, 1.72, and 0.82 respectively.

(b) US, rebalance every 15 days. Sharpe ratios of HybridCGAN, HybridACGAN,
CGAN, ACGAN, and Markowitz are 1.47, 1.18, 0.49, 0.82, and 0.56 respectively.

(c) US, rebalance every 20 days. Sharpe ratios of HybridCGAN, HybridACGAN,
CGAN, ACGAN, and Markowitz are 0.74, 1.20, 0.71, 1.17, and 0.45 respectively.

(d) EU, rebalance every 10 days, Sharpe ratios of HybridCGAN, HybridACGAN,
CGAN, ACGAN, and Markowitz are 1.33, 1.22, 0.95, 1.02, and 0.74 respectively.

(e) EU, rebalance every 15 days. Sharpe ratios of HybridCGAN, HybridACGAN,
CGAN, ACGAN, and Markowitz are 0.96, 1.01, 0.76, 0.96, and 0.64 respectively.

(f) EU, rebalance every 20 days. Sharpe ratios of HybridCGAN, HybridACGAN,
CGAN, ACGAN, and Markowitz are 1.09, 1.06, 1.05, 1.27, and 1.06 respectively.

Figure 9: Portfolio values for different diversification risk settings.
Reference benchmarks are shown with dashed lines (Index or ETF assets).
HybridCGA, HybridACGAN, CGAN, ACGAN, and Markowitz with solid lines.

(a) US, $\eta=10$

(b) US, $\eta=15$

(c) US, $\eta=20$

(d) EU, $\eta=10$

(e) EU, $\eta=15$

(f) EU, $\eta=20$

Figure 10: The hybrid approaches surpass the non-hybrid alternatives reference
approach on financial performance where the shaded bars are the standard
deviation over means. Red horizontal lines are the relative Markowitz results.

Network structures for the conditioner (in HybridCGAN and CGAN), encoder,
decoder (in HybridACGAN and ACGAN), generator, discriminator (in HybridCGAN,
HybridACGAN, CGAN, and ACGAN), and proposer (in HybridCGAN and HybridACGAN)
are provided in Appendix A. In all experiments, we train the network with
1,000 epochs. For simplicity, we set the risk-free interest $r_{f}=0$ to
assess the Sharpe ratio evaluations.

Figure 11: Portfolio values of 10 assets, and weights distribution over time
on the test period for HybridCGAN, HybridACGAN, CGAN, ACGAN, and Markowitz
models for the US region with rebalancing every 20 days. Portfolios from the
HybridCGAN and HybridACGAN are more diversified than those from CGAN, ACGAN,
and Markowitz.

###  4.1 Generating Analysis

We follow the training and testing procedures in Algorithm 1. Given the
training matrix $\bm{M}$ of size $N\times D$ (where $N$ is the number of
assets and $D$ is the number of days in the daily analysis context) and the
window size $w$ ($w=h+f$ where $h$ is the length of the historical window and
$f$ is the length of the future window), we define the index set
$\mathcal{S}_{1}=\\{1,2,\ldots,D-w+1\\}$ so that $D-w$ samples can be
extracted for each training epoch. While at the testing stage, given the
testing matrix $\bm{A}\in{}^{N\times K}$, the index set is obtained by
$\mathcal{S}_{2}=\\{{\color[rgb]{0,0,0}h+1},h+f+1,\ldots\\}$ so that $(K-h)/f$
samples can be obtained (supposed here $(K-h)$ can be divided by $f$). The
output $\bm{B}\in{}^{N\times K}$ of Algorithm 1 is the financial market
simulation of the $N$ assets in $K$ days (here $N=10$ and $K=800$ in our
datasets for both US and EU regions). To be more concrete, the first $h$ days
of $\bm{B}$ are just copies of $\bm{A}$, while the next $f$ days are the
synthetic series based on the data of the first $h$ days; the next $f$ days
are the synthetic series based on the data between the $f$-th and $(f+h)$-th
days; and so on.

We set window size $h=40,f=20$ and $w=60$ in all experiments. Figure 7 shows
the actual price trend (black solid line) of the US assets for the first 100
trading days in the test set, and five representative simulations generated by
HybridCGAN and CGAN models (colored dashed lines). The proposed hybrid methods
are not seeking simulations that are closer to the real series, but find the
typical trends of the series, e.g., there is a big drawdown for GOOG, MSFT,
PFE, HES, XOM, WBA, and IYY around 80-th day; and an increase for SHY around
80-th day.

###  4.2 Portfolio Analysis

After generating the synthetic series for each asset, we optimize over the
fake series to generate minimal Sharpe ratio weight allocations (for
HybridCGAN, HybridACGAN, CGAN, and ACGAN). For Markowitz framework, the
optimization is done over the past data (here we use $h$ days). We consider
three rebalance settings: a defensive setting with rebalancing every $\eta=10$
days; a balanced setting with $\eta=15$; and an aggressive setting with
$\eta=20$. Figure 8 presents the distribution of return-SR (Sharpe ratio)
scatters with 1,000 draws from HybridCGAN, HybridACGAN, CGAN, and ACGAN
models, and the one from Markowitz framework. The points in the upper-right
corner are the better ones.

Figure 10 shows the distribution of Sharpe ratios over 1,000 draws where the
shaded bars are the standard deviation over means. We observe that all the
HybridCGAN models surpass the alternative CGAN approaches; and in most cases,
the HybridACGAN models perform better than the ACGAN model except when
$\eta=10$ in the US region and $\eta=20$ in the EU region. The hybrid approach
has a smaller variance such that the final strategies are more stable.

The end strategies from HybridCGAN, HybridACGAN, CGAN, and ACGAN are the ones
by taking average weight from these 1,000 draws on each rebalancing date (we
call it mean strategy). Figure 9 shows the portfolio value series of mean
strategies for HybridCGAN, HybridACGAN, CGAN, ACGAN, and the one from
Markowitz framework along the test period where we initialize each portfolio
with a unitary value. The hybrid approaches dominate the other approaches in
terms of the final portfolio values and Sharpe ratios.

When we apply the mean strategy in the US region with $\eta=15,20$, the hybrid
versions of CGAN and ACGAN achieve both better return and Sharpe ratio
evaluations compared to the mean strategies of non-hybrid approaches. Similar
results are observed in the EU region with $\eta=10,15$. The Sharpe ratio of
ACGAN model for US region with $\eta=10$ obtains best performance among other
results (SR=1.72); however, its final portfolio value is not the best where
the HybridCGAN obtains the best final portfolio value with a decent Sharpe
ratio of 1.32.

###  4.3 Weight Distribution

In Figure 11, we present the distributions of weights over time on the test
period for the US region with rebalancing every 20 days and the corresponding
portfolio values of the 10 assets. Since we use the first 40 trading days as
the historical series, weights of only 760 days are shown in the figure. We
observe that the Markowitz model puts a large weight on the SHY asset for the
first 300 trading days; while in this period, companies in IT sector (GOOG and
MSFT) receive large positive returns making the Markowitz result less
competitive.

Moreover, we find that there is a big drawdown for the IT companies since the
600-th day and 700-th day. The HybridCGAN and HybridACGAN perform well
compared to their non-hybrid versions in that they put less weight on these
companies during the drawdown periods. Further, one can easily observe that
the portfolios from the HybridCGAN and HybridACGAN are more diversified than
those from CGAN, ACGAN, and Markowitz. Hybrid methods are able to
systematically improve the returns achievable.

For other rebalancing days and weights distribution for the EU region, results
are provided in Figure 12, 13, 14, 15, and 16; and we shall not repeat the
details.

##  5 Conclusion

The paper aims to solve the issue of poor prediction ability in the CGAN and
ACGAN methodology for portfolio analysis. We propose a simple and
computationally efficient algorithm that incorporates a deep neural regression
model and requires little extra computation. A potential future work on the
HybridCGAN and HybridACGAN models is to further reduce the variance of
different draws such that the end strategy will be more stable and consistent.

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##  Appendix A Network Structures

We provide detailed structures for the neural network architectures we used in
our experiments in this section. Given the number of assets $N$, historical
length $h$, future length $f$ ($w=h+f$), and latent dimension $m$ for the
prior distribution vector $\bm{z}$, we consider multi-layer perceptron (MLP)
structures, the detailed architecture for each fully connected layer is
described by F$(\langle\textit{num inputs}\rangle:\langle\textit{num
outputs}\rangle:\langle\textit{activation function}\rangle)$; for an
activation function of LeakyRelu with parameter $p$ is described by
$\text{LR}(\langle p\rangle)$; and for a dropout layer is described by
DP$(\langle\textit{rate}\rangle)$. The conditioner in CGAN shares the same
structure as the encoder in the ACGAN model (see Figure 2 and Figure 3). Then
the network structures we use can be described as follows:

$\displaystyle\textbf{Conditioner}=\textbf{Encoder}=\text{F\big{(}$N\cdot h$:
512:
LR(0.2)\big{)}}\cdot\text{F\big{(}512:512:LR(0.2)\big{)}}\cdot\text{DP(0.4)}\cdot\text{F(512:16)}$
(13)
$\displaystyle\textbf{Decoder}=\text{F\big{(}16:512:LR(0.2)\big{)}}\cdot\text{F\big{(}512:512:LR(0.2)\big{)}}\cdot\text{DP(0.4)}\cdot\text{F(512:$N\cdot
h$)}$ (14) $\displaystyle\textbf{Simulator (in CGAN or
ACGAN)}=\text{F\big{(}$m$+16:128:LR(0.2)\big{)}}\cdot\text{F\big{(}128:256:LR(0.2)\big{)}}\cdot$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
F\big{(}256:512:LR(0.2)\big{)}}\cdot\text{F\big{(}512:1024:LR(0.2)\big{)}}\cdot\text{F\big{(}1024:$N\cdot
f$:TanH\big{)}}$ (15) $\displaystyle\textbf{Simulator (in HybridCGAN or
HybridACGAN)}=\text{F\big{(}$m$+16:128:LR(0.2)\big{)}}\cdot\text{F\big{(}128:256:LR(0.2)\big{)}}\cdot$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
F\big{(}256:512:LR(0.2)\big{)}}\cdot\text{F\big{(}512:1024:LR(0.2)\big{)}}\cdot\text{F\big{(}1024:$N\cdot
f$:TanH\big{)}}\cdot{\color[rgb]{0,0,1}100}$ (16)
$\displaystyle\textbf{Discriminator}=\text{F\big{(}$N\cdot(h$+$f)$:512:LR(0.2)\big{)}}\cdot\text{F\big{(}512:512:LR(0.2)\big{)}}\cdot\text{
DP(0.4)}\cdot\text{F\big{(}256:512:LR(0.2)\big{)}}\cdot\text{F(512:1)}$ (17)
$\displaystyle\textbf{Proposer}=\text{F\big{(}$N\cdot(h+{\color[rgb]{0,0,1}1})$:
512:
LR(0.2)\big{)}}\cdot\text{F\big{(}512:512:LR(0.2)\big{)}}\cdot\text{DP(0.4)}\cdot\text{F(512:16)},$
(18)

where the highlighted 1 in proposer network is the input of the historical
mean values (Eq. (10)). Due to effect of the proposal network, the outputs of
the generator (or simulator) are not constrained into the range of [-1,1], we
also multiply the result by 100 in the simulator of HybridCGAN or HybridACGAN.
We trained networks using Adam’s optimizer with learning rate $2\times
10^{-5}$, $\beta_{1}=0.5$, and $\beta_{2}=0.999$. We set the penalization
parameters $\lambda_{1}=10$ and $\lambda_{2}=3$. The latent dimension is
$m=100$. And we trained models for 1,000 epochs.

##  Appendix B Weight Distributions under Different Settings

As shown in the main paper, Figure 11 presents the weight distribution over
time on the test period for the US region with $\eta=20$. Figure 12 and 13
show the weight distributions for the US region with $\eta=10,15$
respectively. Again, we observe that the HybridCGAN and HybridACGAN still have
a more diverse portfolio allocation. Figure 14, 15, and 16 then present the
weight distributions for the EU region with $\eta=10,15,20$ respectively.

Figure 12: Weights distribution over time on the test period for HybridCGAN,
HybridACGAN, CGAN, ACGAN, and Markowitz models for the US region with
rebalancing every 10 days.

Figure 13: Weights distribution over time on the test period for HybridCGAN,
HybridACGAN, CGAN, ACGAN, and Markowitz models for the US region with
rebalancing every 15 days.

Figure 14: Weights distribution over time on the test period for HybridCGAN,
HybridACGAN, CGAN, ACGAN, and Markowitz models for the EU region with
rebalancing every 10 days.

Figure 15: Weights distribution over time on the test period for HybridCGAN,
HybridACGAN, CGAN, ACGAN, and Markowitz models for the EU region with
rebalancing every 15 days.

Figure 16: Weights distribution over time on the test period for HybridCGAN,
HybridACGAN, CGAN, ACGAN, and Markowitz models for the EU region with
rebalancing every 20 days.


# Universal Euler-Cartan Circuits for Quantum Field Theories

Ananda Roy<EMAIL_ADDRESS>Department of Physics and
Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA  Robert M. Konik
Division of Condensed Matter Physics and Material Science, Brookhaven National
Laboratory, Upton, NY 11973-5000, USA  David Rogerson  Department of Physics
and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

###### Abstract

Quantum computers can efficiently solve problems which are widely believed to
lie beyond the reach of classical computers. In the near-term, hybrid quantum-
classical algorithms, which efficiently embed quantum hardware in classical
frameworks, are crucial in bridging the vast divide in the performance of the
purely-quantum algorithms and their classical counterparts. Here, a hybrid
quantum-classical algorithm is presented for the computation of non-
perturbative characteristics of quantum field theories. The presented
algorithm relies on a universal parametrized quantum circuit ansatz based on
Euler and Cartan’s decompositions of single and two-qubit operators. It is
benchmarked by computing the energy spectra of lattice realizations of quantum
field theories with both short and long range interactions. Low depth circuits
are provided for false vacua as well as highly excited states corresponding to
mesonic and baryonic excitations occurring in the analyzed models. The
described algorithm opens a hitherto-unexplored avenue for the investigation
of mass-ratios, scattering amplitudes and false-vacuum decays in quantum field
theories.

## Introduction.

Quantum field theory (QFT), which reconciles quantum mechanics and the special
theory of relativity, is fundamental to the description of a vast array of
physical phenomena. The latter range from confinement and asymptotic freedom
in theories of elementary particles to high-temperature superconductivity and
topological order in condensed matter systems. The investigation of non-
perturbative characteristics of generic QFTs remains an outstanding challenge
with ab-initio computations likely to stay beyond the reach of the most
powerful classical computers. Quantum computation [1] provides an alternate
paradigm for the analysis of QFTs with quantum algorithms boasting exponential
speedup over classical ones for the computation of scattering amplitudes in
massive scalar [2, 3] and fermionic [4] theories.

Despite their potential, it is a daunting challenge to implement quantum
algorithms on state-of-the-art quantum simulators and solve QFT problems of
widespread interest. This is because the current quantum simulators contain
only a few hundred qubits with modest error rates. In the current noisy
quantum era, hybrid quantum-classical algorithms, which combine the power of a
quantum processor with that of a classical one, are crucial in bridging the
enormous divide between purely-quantum algorithms and their classical
counterparts. Notable examples include the variational quantum eigensolver [5,
6] and its modifications [7, 8] as well as the quantum approximate
optimization algorithm [9, 10], which have been successful in analyzing
problems in quantum chemistry [11] and combinatorial optimization [12].

There are two main elements of a hybrid algorithm that govern its efficacy: i)
a parametrized quantum circuit ansatz built with a predetermined pool of
unitary operators, ii) an efficient optimization procedure for the parameters
specifying the unitary operators. While the second element is typically based
on variations of gradient descent methods [13, 14, 15], there is no natural
prescription for the first. The circuit ansatz either relies on the
Hamiltonian whose eigenstate is being sought alongside a heuristically-chosen
mixer Hamiltonian [9, 16, 17] or variations of the coupled cluster theory
relevant mostly for chemistry problems [5]. Neither of the aforementioned are
suitable for generic QFT problems whose Hamiltonians, when mapped to registers
of qubits in a quantum computer, could involve long-range couplings [18, 19]
or might not even be Hermitian [20, 21]. Furthermore, many interesting
properties of QFTs such as mass-ratios and scattering amplitudes remain
encoded in highly-excited states [22, 23, 24], which have largely eluded
existing hybrid algorithms. Last but not least, restricting the circuit ansatz
to a given pool of unitary operators that does not encompass the entire
manifold of possible unitary operators necessarily implies that the result so
obtained may not be the globally optimum solution which could potentially be
found by considering a larger ansatz.

Figure 1:  (a) Schematic of the proposed algorithm initialized with a state
$|{\rm\psi_{i}}\rangle$. The quantum processor (maroon dotted line) is
involved in the application of $N$ layers of parametrized unitary operators
yielding the state $|{\rm\psi_{f}}\rangle$, followed by measurements of a
cost-function and relevant gradients. A classical processor (blue dotted line)
verifies the iteration convergence criterion. This checks if the results for
the cost-function are converged when compared to that obtained in the previous
iteration step for a given $N$. If the iteration convergence criterion is not
met, the parameters of the $N$ unitary operators are updated using an
optimization routine. Some typical examples are BFGS and the quantum natural
gradient (QNG). The updated set of parameters are used in the next iteration
and this process is repeated until the desired iteration convergence criterion
is reached. Subsequently, a second layer convergence criterion is checked by
comparing the result to that obtained with $N-1$ layers. If the convergence
criterion is satisfied, the entire process terminates, else the number of
layers is grown by 1 and the previous steps are repeated. (b) Decomposition of
each layer unitary operator in terms of general two-qubit unitary operators
(orange rectangles) and those after Euler and Cartan’s KAK decompositions and
compression (orange ellipses). (c) Cartan’s KAK decomposition of a general
SU(4) two-qubit unitary operator in terms of four single-qubit SU(2) operators
and the SU(4) entangler. (d) Euler’s decomposition of a general SU(2) operator
in terms of rotations by three angles about two non-parallel axes (chosen here
to be $\hat{y}$ and $\hat{z}$). The SU(4) entangler is decomposed into three
controlled-NOT gates and five single-qubit rotations. (e) The two-qubit
operator after Euler’s and Cartan’s decompositions used for the optimization
process. (f) Legends for the different symbols used. The symbol $R_{\alpha}$
stands for the SU(2) operator $e^{-i\theta\alpha}$, where $\alpha$ is one of
the three Pauli matrices and $\theta$ is an unspecified angle determined by
the optimization procedure. Two boxes of the same color perform different
amounts of rotations about the same axis.

In light of the above, it is natural to ask: is there a quantum-classical
algorithm based on a universal parameterized quantum circuit ansatz that is
tractable with established optimization strategies and applicable to a broad
set of QFT problems? The goal of this work is to answer this question. The
presented solution computes the optimal circuit over all possible two-qubit
unitary rotations required to realize an arbitrary $L$-qubit unitary rotation
[25]. The unitary operators are chosen to couple neighboring qubits, which is
imposed for potential implementation on majority of quantum computing
architectures and could be relaxed in a more general setting. The model under
investigation affects the parameters of the circuit, but not the ansatz
itself. Importantly, the proposed ansatz explores the largest possible
manifold of allowed unitary operators and is closer to the globally optimal
solution than what could be obtained by considering a more restrictive ansatz.
The potential of the algorithm is demonstrated by computing not only ground
states of strongly-interacting QFTs, but also highly excited states that
correspond to bound-states of topological excitations occurring in these
theories.

## Algorithm.

The algorithm is implemented on a quantum processor coupled with a classical
one (Fig. 1). The former is involved in applying the parametrized circuit and
conducting measurements required for the optimization process. The classical
processor receives the measurement outcomes, performs the optimization and
feeds the outcome back into its quantum counterpart.

The algorithm begins with the application of $N$ (see Figs. 2, 3 for
representative choices of $N$) layers of unitary operators on $L$ qubits of
the quantum processor initialized in the state $|\psi_{\rm i}\rangle$. Each of
the $N$ layers is built, in principle, out of two sublayers of two-qubit
operators [orange rectangles in Fig. 1(b)]. Each two-qubit operator is an
element of the group SU(4). It can be written, using Cartan’s KAK
decomposition [26, 27], as a product of four single-qubit SU(2) rotations and
an SU(4)-entangler [Fig. 1(c)]. The latter when involving qubits $i,i+1$
equals $e^{-i\left(\alpha X_{i}X_{i+1}+\beta Y_{i}Y_{i+1}+\gamma
Z_{i}Z_{i+1}\right)}$ where $X_{i},Y_{i}$ and $Z_{i}$ are the Pauli operators
of the $i^{\rm th}$ qubit. The SU(4) entangler can be realized by three
controlled-NOT gates and five single-qubit rotations [28, 29] [Fig. 1(d)].
Finally, each of the four single-qubit rotations depicted in panel Fig. 1(c)
can be further decomposed into three single qubit rotations by the well-known
Euler angles [25]. Using the above, it can be shown that the general circuit
ansatz is constructed with two-qubit operators [orange ellipses in Fig. 1(b,
e)] with ten undetermined angles.

After the application of the $N$ layers, measurements are performed to
determine the relevant gradients and the cost-function ${\cal
L}(\vec{\theta})$, where $\vec{\theta}$ contains all parameters of the $N$
layer quantum circuit. To obtain the ground state of a given Hamiltonian $H$,
the cost-function is chosen to be ${\cal L}(\vec{\theta})=\langle\psi_{\rm
f}(\vec{\theta})|H|\psi_{\rm f}(\vec{\theta})\rangle$. To obtain the $n^{\rm
th}$ excited state, the cost-function is augmented to ${\cal
L}(\vec{\theta})\rightarrow{\cal
L}(\vec{\theta})+\sum_{m<n}\lambda_{m}|\langle\psi_{m}|\psi_{\rm
f}\rangle|^{2}$, with $\lambda_{m}>0,m=0,\ldots,n-1$. This choice amounts to
simultaneously minimizing the energy and the overlap to the $n-1$ lower-energy
states that have been already obtained.

The measurement results are communicated to the classical processor which
checks for the convergence of the cost-function for the given choice of $N$
(iteration convergence in Fig. 1). If the condition is not satisfied, the
parameters of the circuit are updated using an optimization routine. In this
work, the quantum natural gradient optimizer [15] is used due to its superior
performance in obtaining eigenstates of many-body Hamiltonians [30, 31]. In
contrast to ordinary gradient descent, this ensures that optimization follows
the steepest descent taking into account the geometry of the space of
wavefunctions [15, 32] and can viewed as the quantum analog of the classical
information geometric approach [33]. The parameters at the $(t+1)^{\rm th}$
iteration are related to the $t^{\rm th}$ one by

$\vec{\theta}_{t+1}=\vec{\theta}_{t}-\eta g^{-1}\frac{\partial{\cal
L}}{\partial\vec{\theta}_{t}},$ (1)

where $\eta$ is the learning rate (typically a real number $<1$) and $g$ is
the Fubini-Study metric [15, 31]. The latter is given by the real part of the
quantum geometric tensor $G$:

$G=\bigg{\langle}\frac{\partial\psi_{\rm
f}}{\partial\theta_{i}}\bigg{|}\frac{\partial\psi_{\rm
f}}{\partial\theta_{j}}\bigg{\rangle}-\bigg{\langle}\frac{\partial\psi_{\rm
f}}{\partial\theta_{i}}\bigg{|}\psi_{\rm
f}\bigg{\rangle}\bigg{\langle}\psi_{\rm f}\bigg{|}\frac{\partial\psi_{\rm
f}}{\partial\theta_{j}}\bigg{\rangle}.$ (2)

Note that both the gradients need to be evaluated using measurements for both
the cost-function and the Fubini-Study metric [34, 35]. Subsequently, the
updated set of parameters are fed back to the quantum processor and the
process is repeated until the iteration convergence criterion is satisfied.
Once the latter is satisfied, a second layer convergence criterion is
verified. This ensures that the number of layers chosen is sufficient to
attain the desired degree of convergence. If this condition is not satisfied,
the number of layers is grown by 1 and the earlier steps are repeated. Else,
the algorithm terminates. When the number of layers is increased from $N$ to
$N+1$, the optimized parameters obtained for the $N$-layer circuit, augmented
by an initial guess for the added layer, were used as the choice for the
optimization process for $N+1$ layers. This improved the stability of the
algorithm.

For $N$ layers and a ring of $L$ qubits, the total number of parameters
determined by the above algorithm is $10LN$, where the factor 10 originates
from the number of free parameters in the two-qubit unitary operator of Fig.
1(e). Despite being tractable [see Fig. 3(c) for demonstration], conservation
of symmetries present in the target Hamiltonian can drastically reduce the
number of parameters to be determined. Symmetries can be conserved by
modifying the general circuit ansatz of Fig. 1 and allowing only operators
that respect this symmetry. Alternatively, the cost-function can be augmented
to ensure that the obtained state lies in the correct symmetry sector. For
example, states which are invariant under translation by $r$ lattice site(s)
can be obtained by starting with a state and circuit ansatz respecting this
symmetry. For $r=1(2)$, the number of undetermined parameters for $N$ layers
reduces to $10N~{}(20N)$. Alternatively, in order to obtain a $\pm 1$
eigenstate of the translation operator $T$, the cost-function can be augmented
as: ${\cal L}(\vec{\theta})\rightarrow{\cal
L}(\vec{\theta})\mp\mu\langle\psi_{\rm f}(\vec{\theta})|T|\psi_{\rm
f}(\vec{\theta})\rangle$, where $\mu$ is a positive real number. Similar
considerations apply to symmetry operators related to parity/particle number
conservation.

Figure 2:  (a) - (c) Results obtained for the ground state energies ($E_{0}$)
of the Ising, 3-state Potts and the massive Schwinger models using the
optimized Euler-Cartan circuits. A periodic chain of $L$ qubits is subject to
$N$ layers for all three models. A translation-invariant Euler-Cartan circuit
ansatz is optimized for panels (a, b), while the most general ansatz is
considered in panel (c). The different values of $N~{}(L)$ are distinguished
by the different colors (markers). For the lowest value of $N$, a uniform
choice of $\theta_{0}=0.1$ was used as a starting guess for all of the angles
being optimized. Once the iteration convergence criterion is met for a chosen
value of $N$, the number of layers is increased by one: $N\rightarrow N+1$.
While optimizing for $N+1$ layers, the previously obtained results for the $N$
layers was used as a starting guess, augmented by an initial guess of
$\theta_{0}/10$ for the last layer. This process was continued until the
layer-convergence criterion was reached. Both convergence criteria were chosen
to be $5\times 10^{-4}$. The learning rates were chosen to be $0.04$ for
panels (a), (b) and 0.1 for panel (c). (d) Energies for the smallest and
largest values of $N$ for $L=16$ are compared to the DMRG results. The
smallest choice of $N=2$ layers yielded results within $1\%$ error of the
target energies, while the additional layers caused changes in the third and
fourth significant digits.

## Results.

The proposed algorithm is benchmarked using three paradigmatic one-dimensional
QFTs. The results were obtained by modifying standard time-evolution routines
for matrix product states. In all three cases, the quantum processor
considered comprises a periodic ring with a fixed number of qubits with
nearest-neighbor connectivity. This is independent of the model under
consideration. The results obtained using the optimized quantum circuit are
compared with those from density matrix renormalization group (DMRG) and exact
computations.

First, the critical Ising model in a longitudinal magnetic field is analyzed.
In the continuum limit, the QFT contains eight mesonic excitations which
scatter with a diagonal scattering matrix [36]. These mesons can be viewed as
an extremal ($g\rightarrow 1$) limit of bound states formed by the Ising
domain walls in the ferromagnetic phase $g<1$. The magnetic field leads to a
linearly growing energy of separation of two domain walls [37, 38], akin to
what happens in quarks in quantum chromodynamics [39], see Fig. 3(c). The QFT
describing this model is integrable and the number of mesons and their mass
ratios are given by the rank and the exponents of the Lie algebra $E_{8}$
respectively [36]. The relevant lattice Hamiltonian, imposing periodic
boundary condition, is:

$H_{\rm
I}=-\sum_{j=1}^{L}X_{j}X_{j+1}-g\sum_{j=1}^{L}Z_{j}-h\sum_{j=1}^{L}X_{j},$ (3)

where $g$ and $h$ were chosen to be $1$ and $\simeq 0.156$ respectively. Figs.
2(a) and 3(a) together present the results for the lowest eight translation-
invariant states obtained using the proposed algorithm. The circuit ansatz was
chosen to be invariant under translation by 2 sites. Furthermore, the cost-
function was augmented to ensure that only $+1$ eigenstates of the operator
for single-site translation were obtained. Fig. 2(a) shows the results for the
ground state energy for system-sizes between $L=8$ and 20 in steps of 4. The
number of layers required to obtain a reasonable approximation of the ground
state do not scale with the system-size, which is compatible with the
observations of Ref. [31]. In fact, $N=2$ is sufficient to obtain the ground
state with error in the energy $<1\%$ [Fig. 2(d)]. Fig. 3(a) shows the results
for the first seven excited states for $L=8$. For the chosen convergence
criterion of $10^{-3}$, the maximum error in the obtained energies was
$0.58\%$. In contrast to the ground state, the number of layers required to
achieve this precision ranged between $L$ and $3L$. The different translation-
invariant excited states can be classified as lattice counterparts of the
single and multiple meson states existing in the QFT. The mass ratio for the
lowest two mesonic excitations [square and diamond markers in Fig. 3(a)]
obtained for L $=$ 8 obtained using the proposed algorithm is 1.39 which is
close to the expected value of 1.41 obtained using exact diagonalization of
the corresponding spin-chain. This should be compared with the QFT prediction
[36] given by $2\cos(\pi/5)\simeq 1.618$.

Figure 3:  (a) - (b) Results for the first seven excited states obtained using
optimized Euler-Cartan circuits for the Ising [(a)] and Potts [(b)] models
encoded in a periodic chain of $L=8$ qubits. The error percentages when
compared to exact diagonalization results (black dotted lines) are shown in
the top left insets. The variation of the obtained energies with number of
layers is shown in the bottom right insets. The energies were obtained with a
convergence criterion of $10^{-3}(10^{-4})$ for the Ising (Potts) models with
the same learning rate of 0.02. In both cases, a translation-invariant circuit
ansatz was used. The Euler-Cartan circuit for the $i^{\rm th}$ state was
obtained by minimizing the Hamiltonian cost-function while orthogonalizing
against the previously obtained states. Furthermore, in the Potts model,
another term was added to the cost-function to obtain states with $+1$
eigenvalue for the operator $C$ (see maintext). The seventh translation-
invariant excited state in the Ising (Potts) models corresponds to the
$48^{\rm th}(42^{\rm nd})$ excited state in the spectrum. (c) - (d) Schematic
of the confinement phenomena in the Ising and Potts models in the presence of
a magnetic field. The latter breaks the two (three)-fold degeneracy of the
ground state leading to one true ground state and one (two) false vacua in the
Ising (Potts) case. As a consequence, the domain-walls experience a confining
force (yellow lines with stripes), leading to the formation of mesonic and
baryonic excitations.

Next, the three-state Potts model in the presence of a magnetic field [40, 41,
42, 43] is analyzed. In contrast to its two-state counterpart described above,
this model exhibits a richer state of excitations. The three states of each
Potts spin are encoded in the triplet of states of two qubits:

$|0\rangle=|\uparrow\uparrow\rangle,|1\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle\right),|2\rangle=|\downarrow\downarrow\rangle.$
(4)

The relevant Hamiltonian is

$H_{\rm
P}=-\sum_{j=1}^{L^{\prime}}\left(\sigma_{j}\sigma_{j+1}^{\dagger}+g\tau+h\sigma+{\rm
h.c.}\right),$ (5)

where $\sigma$ and $\tau$ are the shift and clock matrices respectively [44].
Periodic boundary condition was imposed and $L^{\prime}=L/2$ is the number of
Potts spins encoded in $L$ qubits. For $g<1,h=0$, the model exhibits a
ferromagnetic phase with three degenerate vacua and six domain walls [the
three shown in Fig. 3(d) and those obtained by reflection about the domain
wall]. The presence of the magnetic field breaks the degeneracy of the ground
space, giving rise to one true vacuum and two false vacua. The domain-walls
experience a confining potential as in the Ising case. However, now there are
not only mesonic excitations formed by two domain walls, but also baryonic
excitations formed by three domain walls [Fig. 3(d)]. Notice that $H_{\rm P}$
commutes with the $\mathbb{Z}_{2}$ charge operator $C$ which acts trivially on
the state $|0\rangle$ and interchanges $|1\rangle$ and $|2\rangle$. Figs. 2,3
together show the results for $g=h=0.1$ for the lowest eight translation-
invariant states with eigenvalue $+1$ for the operator $C$. The 2-site
translation-invariant circuit ansatz was chosen with the $C$-eigenvalue
imposed using the cost-function. As in the Ising case, the ground state is
reached with depth independent of the system-size, with errors in energy less
than $0.1\%$ for two layers [Fig. 2(b, d)]. The excited states were obtained
using between $L$ and $3L$ layers with maximum error in the obtained energies
being $2.26\%$ [Fig. 3(b)]. The first excited state is one of the false vacua,
with the rest comprising mesonic and baryonic excitations. This is verified by
counting the average number of domain-walls in the corresponding state.

Finally, results are presented for the canonical model for quantum
electrodynamics in two space-time dimensions: the one-flavor massive Schwinger
model [45, 46]. After mapping to an array of qubits, the resulting Hamiltonian
with open boundary conditions is given by [18, 19]:

$\displaystyle H_{\rm MS}$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{L-1}\left(X_{j}X_{j+1}+Y_{j}Y_{j+1}\right)+m\sum_{j=1}^{L}P_{j}$
$\displaystyle\quad+\frac{g^{2}}{2}\sum_{j=1}^{L-1}\left[\sum_{k=1}^{j}(-1)^{k}P_{k}\right]^{2},$
(6)

where $P_{j}=[1+(-1)^{j}Z_{j}]/2$. In contrast to the Ising and Potts models,
this model involves an all-to-all coupling Hamiltonian. To obtain the ground
state of this model, the general circuit ansatz of Fig. 1 is used. Note the
existence of the conserved U(1) charge $Q$, $[H,Q]=0$, where
$Q=\sum_{j=1}^{L}Z_{j}$. To test the algorithm under the most challenging of
conditions, the ground states were obtained for system sizes between L = 8 and
16 without conserving this charge. For these simulations, $m=0.5,g=0.3$ were
chosen. Despite the long-range nature of the model Hamiltonian, reasonable
approximations of the ground-state were obtained with circuit-depths that are
independent of the system-size with energies within $1\%$ accuracy obtained
within application of 2 layers. Excited states were also obtained similar to
the Ising and Potts models and are not shown here for brevity.

## Conclusion.

To summarize, a hybrid quantum-classical algorithm is presented for the
investigation of strongly-interacting QFTs. In contrast to existing approaches
that rely on heuristic arguments for choosing the parametrized quantum circuit
ansatz, a universal ansatz is described relying on Euler’s and Cartan’s
decomposition of single and two-qubit operators. The algorithm is benchmarked
by computing eigenstates of lattice models for three paradigmatic one-
dimensional QFTs: the Ising, Potts and one-flavor massive Schwinger models.
Quantum circuits were obtained for the realization of ground states, false
vacua, mesonic and baryonic excitations occurring in these models.

The presented results demonstrate the power of geometric control theory [47]
for the analysis of QFTs. While geometric quantum control is known to be the
central concept in the search for an optimal circuit for quantum computation
[48, 49, 50], previous proposal are centered around minimization of the
distance in the Finsler space of unitary operators. In contrast, the current
work presents an alternative framework based on the Fubini-Study metric and a
universal circuit ansatz. The proposed framework is not only numerically-
tractable, but also readily-implementable on current quantum machines, opening
the door to hybrid quantum-classical analysis of a large class of QFTs.

## Acknowledgements.

This work was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, under Contract No. DE-SC0012704.

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# COVID-Net UV: An End-to-End Spatio-Temporal Deep Neural Network Architecture
for Automated Diagnosis of COVID-19 Infection from Ultrasound Videos

Hilda Azimi<EMAIL_ADDRESS>National Research Council Canada
Ashkan Ebadi<EMAIL_ADDRESS>National Research Council Canada
Jessy Song<EMAIL_ADDRESS>University of Waterloo  Pengcheng Xi
<EMAIL_ADDRESS>National Research Council Canada  Alexander Wong
<EMAIL_ADDRESS>University of Waterloo

###### Abstract

Besides vaccination, as an effective way to mitigate the further spread of
COVID-19, fast and accurate screening of individuals to test for the disease
is yet necessary to ensure public health safety. We propose COVID-Net UV, an
end-to-end hybrid spatio-temporal deep neural network architecture, to detect
COVID-19 infection from lung point-of-care ultrasound videos captured by
convex transducers. COVID-Net UV comprises a convolutional neural network that
extracts spatial features and a recurrent neural network that learns temporal
dependence. After careful hyperparameter tuning, the network achieves an
average accuracy of 94.44% with no false-negative cases for COVID-19 cases.
The goal with COVID-Net UV is to assist front-line clinicians in the fight
against COVID-19 via accelerating the screening of lung point-of-care
ultrasound videos and automatic detection of COVID-19 positive cases.

##  1 Introduction

The Coronavirus Disease 2019 (COVID-19) has resulted in a dramatic loss of
life worldwide and posed an unprecedented public health challenge. There is no
doubt that vaccination has been helping in mitigating the further spread of
COVID-19. However, fast screening individuals to test for the disease is still
necessary to ensure public health safety [1]. Chest x-ray (CXR) and computed
tomography (CT) are two modalities that are often used for screening patients
suspicious for COVID-19 infection. Another imaging modality for diagnosing
lung-related diseases is the lung point-of-care ultrasound (POCUS). This
modality has been suggested as the most helpful in contexts that are resource-
limited, such as emergency settings or low-resource countries [1]. Compared to
CXR and CT, POCUS is much cheaper to acquire and has higher portability and
accessibility, thus enhancing the ability for possible COVID-19 screening [2].
Deep learning (DL) networks have been applied to POCUS images for different
tasks and analyses such as segmentation, disease classification, and detection
[3]. However, the protocol for physicians to perform an ultrasound (US)
examination requires them to capture and analyze the US video, often from
various angles, views, and positions [2, 1]. This means that the sequences of
US video frames from one position or view to another can provide physicians
with more information to make an accurate diagnosis; and perhaps not all
frames of US videos contain signs and symptoms of a suspected disease.
Therefore, applying DL to frames of US videos only and without considering
their temporal information is not the ideal solution to adopt POCUS data for
screening and diagnostics purposes. Motivated by this challenge, we propose
COVID-Net UV, an end-to-end spatio-temporal deep neural network architecture
to detect COVID-19 positive cases from POCUS videos. Our contributions can be
summarized as follows: 1) COVID-Net UV is an effective tool for automatic
detection of COVID-19 positive cases from POCUS videos without requiring the
need for technician intervention and any further processing, 2) it also
bridges the gap in the current diagnostic procedure of POCUS data by
eliminating the need for the time-consuming and costly training of human
experts, as interpreting US data requires domain knowledge [4].

Figure 1: COVID-Net UV: a CNN-RNN architecture to classify POCUS videos in to
two class of positive, i.e., COVID-19 infection, and negative, i.e., pneumonia
or normal.

##  2 Related Work

A number of techniques have been proposed so far for classification of various
features in POCUS images and videos related explicitly to COVID-19 Disease,
where some are models learned by DL, mostly using frame-based data as input.
Roy et al. applied a DL model, derived from spatial transformer networks, to
predict the COVID-19 severity score associated with POCUS videos frames,
provide localization of pathological artifacts in a weakly-supervised way, and
adopted a uninorms-based method for frame score aggregation at the video-level
[5]. In [6], authors presented a technique for classifying POCUS videos, based
on a Two-Stream Inflated 3D ConvNet (I3D) to categorize the main imaging
features seen in POCUS scans, such as Alines, B-lines, consolidation, and
pleural effusion, which unveil the degree to which the lungs have been
affected by the infection. Both these works had different approaches to
analyzing PUCUS videos in the presence of COVID19 disease. We aim to bridge
the gap in the current diagnostic procedure of POCUS Videos to detect and
classify COVID-19 disease by learning spatio-temporal features, combining
Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN)
architectures.

##  3 Data and Methods

To train and evaluate the COVID-Net UV, we used the COVIDx-US dataset v1.4.
[7] that contains 242 US videos curated and integrated from 9 different data
sources. The videos comprised four different classes: COVID-19 infection, non-
COVID-19 infection, other lung diseases/conditions, and normal control cases.
We filtered out the other class, i.e., other lung diseases/conditions, due to
the heterogeneity of the cases. And, we only included lung US video captured
with a convex probe. We formulated the problem as a binary classification
problem, i.e., the COVID-19 cases were labeled as positive and the normal and
non-COVID-19 cases as negative. This resulted in 119 videos in total that were
split into a training set with 76 videos (38 positives and 38 negatives), a
validation set with 25 videos (12 positives and 13 negatives), and a test set
with 18 videos (10 positives and 8 negatives).

We employed a hybrid architecture that included convolutional and recurrent
layers to process spatial and temporal aspects of videos, respectively (Fig.
1). We adopted the InceptionV3 model pre-trained on the ImageNet data set [8],
as the spatial feature extraction backbone and added 2 GRU units to capture
temporal features. Since a video is an ordered sequence of frames, the frames
can be extracted and placed on a 3D tensor. However, the number of frames may
vary from video to video, making it impossible to stack them in batches. To
overcome this problem, we first captured the frames of a video. Next, we
extracted frames from the videos until a maximum frame count was reached. In
this case, if the frame count was lower than the maximum frame count, the
video was padded with zeros. We chose 60 as the maximum frame count,
considering the characteristics of the videos in the dataset. Our training
strategy involved two callbacks, i.e., learning rate scheduler (decay learning
rate by factor of 0.5 after three epochs with no performance improvement on
validation set), and early stopping (stop training after seven epochs with no
performance improvement on validation set), to optimize our network and avoid
over-fitting. The initial learning rate and the maximum number of epochs were
set at 0.001 and 30, respectively. The network was trained for 18 epochs
following the early stopping strategy.

##  4 Results

The learning curves through the process of training and optimizing the network
are illustrated in Fig. 2. The training of the network was stopped right
before the loss on the validation set started increasing and avoided
overfitting, Fig. 2-b. Following the learning rate scheduler strategy, during
the process of training, the learning rate was decayed two times through
epochs of 15 and 18. The network learned to classify two classes with an
overall accuracy of 94.44% among all the classes. We received a sensitivity of
100% and 87.50% for positive and negative classes, respectively, meaning no
false-negative detection for COVID-19 cases. The network achieved a precision
of 90.91% for the positive class and 100% for the negative class.

Figure 2: Learning curves through the process of training and optimizing the
network. (a) Accuracy, (b) Loss and (c) Learning rate

##  5 Discussion and Future Work

In this work, we design COVID-Net UV, a hybrid end-to-end network architecture
to classify lung POCUS videos for the diagnosis of COVID-19. Our network
comprises two modules: pre-trained InceptionV3 to extract spatial features
from video frames and RNN structure containing GRU units to learn the temporal
dependence between video frames. Our results with no false-negative cases for
COVID-19 indicate COVID-Net UV is a powerful tool that exceeds human experts
with a sensitivity of 86.4% [9] and models with purely spatial architecture
(with highest accuracy of 83.2%) [10].

Admittedly, we are aware that our proposed network architecture is evaluated
on a small dataset containing 119 POCUS videos. However, we believe our
proposed methodology and results are the desired baseline for our future work
in examining more complex models on the larger POCUS video dataset. The
network architecture can be easily retrained by adding new disease categories
or feeding new data and be employed for different classification purposes.

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