Introduction

Solving the portfolio optimization is computationally costly. Finding the optimal solution requires solving problems of N (number of assets) x Nq (number of qubits to represent the maximum investment on individual asset) variables times the number of periods, considering the case where the transaction cost is not present.

Quantum Counselor for Portfolio Investment

The Quantum Counselor for portfolio investment is a tool with two main objectives: forecasting the trend of assets price and optimizing portfolio returns both using quantum computing techniques. For the case of the forecasting method, we use a hybrid method for a Quantum neural network (QNN) that combines a deep learning model of classical LSTM layers with quantum layers. For the case of portfolio optimization, we convert the optimization problem into a Quadratic unconstrained binary optimization (QUBO) problem and using the quantum algorithms of Quantum Approximate Optimization Algorithm (QAOA) and the variational quantum eigensolver (VQE) solve the problem. Additionally, we use the classical solver CPLEX for comparison with the other two methods. Both tools are deeply connected because the forecasted price of the different assets is used for the optimization protfolio cost function construction.

Requirements

A library was built with the name tfqml.py to store all the dependencies, classes and methods that are used in this work, these classes are

ClassicalPreprocessing	QuantumPreprocessing	CircuitLayer	QuantumModel
create_dataset()	convert2circuit()	add_layer()	quantum_circuit()
preprocessing()	data2qubits()
	print_circuit()

and the methods

• visualization()
• save_data()

Scripts:

• normalization.py script that generate a npy file with the original data after train the model.
• quantum _stock_price_simulation_mean.py script to generate 10 trianing of each stocks.

the dependencies are:

Procesing data	machine learning	quantum frame works
numpy: 1.20.1	sklearn: 0.24.1	cirq : 0.13.1
pandas: 1.2.3	sympy: 1.7.1	pennylane: 0.21.0
docplex version: 2.22.213	tensorflow_quantum: 0.6.1	qiskit version: 0.19.2
matplotlib: 3.4.1	tensorflow: 2.7.0	qiskit_optimization version: 0.3.1

Outline

1. Stocks forecasting using a QNN File: stock_price_classical_algorithm.ipynb stock_price_hibryd_algorithm.ipynb stock_price_hibryd_algorithm_with_noise.ipynb stock_price_hybrid_quantum_fair_value.ipynb pennylane_stock_price_hybrid_algorithm.ipynb Functions: tfqml.py quantum_stock_price_simulation_mean.py normalization.py

2. Portfolio Optimization. File: Portfolio-Optimization.ipynb Functions: portfolioFunctions.py

   2.1 Model XS (3 Stocks, 2 periods), QAOA and VQE with SPSA and COBYLA classical optimizers.

   2.2 Model S (5 Stocks, 3 periods), QAOA and VQE with SPSA and COBYLA classical optimizers.

   2.3 Model M (8 Stocks, 3 periods), QAOA and VQE with SPSA and COBYLA classical optimizers.

3. A novel approach for the Portfolio Optimization

4. Conclusion and Future Work

5. References

1. Stocks forecasting using a QNN

One of the principal requirements for portfolio optimization is the ability to predict the price trend P_{n,t} for N assets during some periods of time. In this work, our first objective is to show the capabilities of a quantum neural network (QNN) to predict the trend of a set of stocks. Specifically, we select 8 stocks from 8 conglomerates based on the dataset of Xu et al. [2]:

• Basic Materials: TOTAL S.A. "TOT"

• Consumer Goods: Appel Inc. "AAPL"

• Healthcare: AbbVie Inc. "ABBV"

• Services: Wall-Mart Stores Inc. "WMT"

• Utilites: Duke energy corporation "DUK"

• Financial: HSBS Holding pcl "HSBC"

• Industrial Goods: ABB Ltd. "ABB"

• Technology: China Mobile Limited "CHL"

The information comes from Sep 2012 to Sep 2017 with daily Technical information of Open, High, Low, Close, Adj Close, and Volume for the stocks price.

Fig.1 - Stocks trend for the 8 conglomerates. We used the closed price as the prediction parameter.

Creating our hibryd model this is use of an Parametric Quantum circuit (PQC) and classical model using LSTM. Dropout, and Dense functions.The first layer is our proposal which input is 30 parameters per instance and output is 10, followed by LSTM and Dropout layers, at the end a Dense layer to have an output value.

We use a noise Parameter Quantum circuit , this is posible use the NoisyPQC method, and is important add repetitions for the time is 10 and consider a the fag dample_based= True, this las one parameter is improtant to run our noise simulation. More information you can find here

From here, we use a QNN to predict the trend in the price of the stock. The results of the forecasting are stored in stocks_forecasting

We design a quantum circuit for encoding all the values in the data set, using 30 data per instance with this proposal we reduce to an output of 10. The ansatz for the QNN follows the structure of the figure below structure, and using for the model 2 layers of this ansatz.

we design four hybrid proposal of solutions with the first using a PQC:

• Hybrid Model using Tensorflow Quantum
• Hybrid Model using Tensorflow Quantum with noise
• Hybrid Model using Pennylane and Tensorflow
• Hybryd Model using Tensorflow Quantum and the minimal model.

and two classical propsal using tensorflow , this with a big model and minimal model to predict the same stock.

Considering the capacity of our PQC we have minimal models of the classical and quantum part with 82 and 80 parameters and we can identify their performance, their error per 1 measurement is shown in the figure below, where using PCA as preprocessing gives us better results in the quantum part giving us a MAE error in half of the test predictions.

For the following experiments, the same was done for a more robust model and equivalent values were obtained for both the classical and quantum parts. It should be noted that the classical model uses about 200 thousand parameters and the hybrid model uses only 126 thousand parameters to reach similar results.

In addition to the hybrid part for the case of tensorflow quantum a noise is performed, for this implementation it takes about 10 times longer due to the time limit, only 3 stocks were identified to identify the effect that our model could have when used in a real computer. Giving us a case of up to 20% error, increasing 10 times the error for our models.

We focus on the Tensorflow Quantum model, for most of our experiments, since this only takes 3s with respect to the model implemented between pennylane-Tensorflow takes on average 121 seconds, being approximately 40 times what tensorflow quantum takes, therefore in time-limited problems it is better to base the entire project on Tensorflow Quantum.

When reaching the conclusion that we can identify with our proposed model eight different stocks, with a MAE in the worst case close to 7%, we consider that our architecture and circuit design give us enough to be able to continue to the Portfolio optimization stage.

We must emphasize that the model was made in two frameworks: Tensorflow Quantum and Pennylane, the latter gave us from 3 experiments similar results to Tensorflow Quantum. We can confirm that our quantum circuit manages to support as a QNN so that the classical model based on LSTM converges.

Finally, we obtain the prediction for all the stocks with our proposal hybrid model.

2. Portfolio optimization

The results of three portfolio cases are presented in this section:

• XS model with 3 stocks and 2 periods of time
• S model with 5 stocks and 3 periods of time
• M model with 7 stocks and 3 periods of time.

For portfolio optimization, we use the modern portfolio theory where it is wanted to maximize the return of an investment while keeping the risk of losing money low. We based or cost function in the work of Mugel et al. [1] where the cost function is described by:

The equation shown above is encoded using the function Model from docplex a classical library from IBM for optimization problems. Next, using the qiskit_optimization function QuadraticProgramToQubo, the problem is translated to the QUBO representation. The QUBO is the input to the functions QAOA and VQE from qiskit. They translate the problem to the needed Hamiltonian and solve it using a hybrid model. In our case, we test two classical solvers the Simultaneous Perturbation Stochastic Approximation (SPSA) and Constrained Optimization BY Linear Approximation (COBYLA).

Fig. 2 shows the training of the S -model and the S -NEW model (explained in Section 3) using VQE and QAOA. Here, it is observed that the VQE has a smoother way to come to the optimal point for both cases COBYLA and SPSA while the QAOA for 1 repetition is not comming to the optimal point contrary to the 2 repetitions where the optimal is found.

Fig.2 - Cost function optimization using two quantum algorithms VQE and QAOA with classical optimizers SPSA and COBYLA. The red results are for the model S while the green for a novel approach explained in **Section 3** a

2.1 Model XS

The Table below shows the results of the XS model, with 3 stocks "AAPL","ABB", and "ABBV". We consider a transaction cost of 0.1% and holding period of 30 (this means that the stock should remain with us 30 days). From the perspective of the forecasting data, we take the last two months of prediction.

2.2 Model S

Model S is similar to model XS the only condition that changes are the stocks chosen in this case they are "AAPL","ABB", "ABBV","TOT", and "DUK". In this case, the only quantum algorithm that converges to the optimal is QAOA with SPSA. However, we only consider one repetition of the mixer and the Hamiltonian of this case, as we show un Fig.2 for 2 repetitions the QAOA converges. Similarly, the VQE in this case takes an ansatz from the function TwoLocal and it is truncated to a maximum number of 50 iterations, increasing the number of iterations will make the

2.3 Model M

Model M is similar to model S and XS the only variation is the stocks chosen in this case they are "AAPL","ABB", "ABBV","CHL", "DUK", "HSBC", "TOT", and "WMT". In this case, as the problem size includes 24 qubits and we found problems executing it on the IBM quantum simulator ibmq_quasm_simulator, we have to run it locally and we could only get results for QAOA COBYLA and SPSA, in the coming days the results for VQE will be presented. Eventhough QAOA does not converge for any of the two cases, it only uses one repetiton incresing this number and the maximum number of iterations will improve the result. However, this is still a good approximation of the optimal.

3. A novel approach to the portfolio cost function

We introduce two modifications to the objective function that we think will improve the profit reached by a single investment. We call it, the budget increment opportunity. In this case, we see some situations where the market offers advantages to invest in some specific periods of time. In such periods, we want the budget constraint to be weak, allowing surpassing the budget established in the problem. This move is possible thanks to two considerations, the bare return of all the stocks is high and the uncertainty in the forecasting price of the stocks is low. We call the forecasting uncertainty with the greek letter kappa and it is the mean relative error mre of the test cases in the QNN training. The second modification is the addition of a second risk term, this term is based on the uncertainty mentioned before. Therefore, if the forecasting prediction is low for some assets, the optimization will avoid them. The Equation below describes our new approach:

Here, we present results only for model S. Fig. 3 shows the different investment trajectories based on our new approach. VQE SPSA and CPLEX choose a trajectory that increases to 125% of the budget. This is a strategy that increases the profit with a small risk. Once, the two first periods of good prospects have passed, the budget investment goes back to the maximum budget of 100%.

Fig.3 - New method investment trayectory

Finally, Fig. 4 shows the profit made with the method described in section 2 and our new approach. Here, increasing the budget invested in the two first periods makes a higher profit compared with the first approach.

Fig.4 - Profit made with the two different approaches

Conclusions and future work

We have come up with a QNN model capable of forecasting the price trend for different assets. This model presents some advantages when compared with classical approaches.

We implement satisfactorily the problem of the optimal portfolio using qiskit with two quantum solvers QAOA and VQE, and we compare the results with a classical solver CPLEX. Even though we select a small number of maximal iterations, the quantum models come to the optimal solution.

We implement a new approach for the objective function called the budget increment opportunity, where if there is a great opportunity of investment (high bare return and low uncertainty in the forecasting) the budget constraint becomes weak. This approach allows us to get a considerable increment in profits.

For the next work, we want to implement these methods on real hardware. Unfortunately, we couldn’t make it because of some technical difficulties with the two backends where we tried it. Additionally, we want to add fundamental analysis as input to the QNN, to explore new ways of improving the forecasting ability.

References

[1] Mugel, S., Kuchkovsky, C., Sánchez, E., Fernández-Lorenzo, S., Luis-Hita, J., Lizaso, E., & Orús, R. (2022). Dynamic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor networks. Physical Review Research, 4(1), 1–13. https://doi.org/10.1103/PhysRevResearch.4.013006.

[2] Chen, Samuel & Yoo, Shinjae & Fang, Yao-Lung. (2020). Quantum Long Short-Term Memory.

[3] Bausch, Johannes. (2020). Recurrent Quantum Neural Networks.

[4] Verdon, Guillaume & Broughton, Michael & Mcclean, Jarrod & Sung, Kevin & Babbush, Ryan & Jiang, Zhang & Neven, Hartmut & Mohseni, M.. (2019). Learning to learn with quantum neural networks via classical neural networks.