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Quantum Portfolio Optimizer: A Qiskit Function by Global Data Quantum

ملاحظة

Qiskit Functions are an experimental feature available only to IBM Quantum® Premium Plan, Flex Plan, and On-Prem (via IBM Quantum Platform API) Plan users. They are in preview release status and subject to change.

Overview

The Quantum Portfolio Optimizer is a Qiskit Function that tackles the dynamic portfolio optimization problem, a standard problem in finance that aims to rebalance periodic investments across a set of assets, to maximize returns and minimize risks. By deploying cutting-edge quantum optimization techniques, this function simplifies the process so that users, with no expertise in quantum computing, can benefit from its advantages in finding optimal investment trajectories. Ideal for portfolio managers, researchers in quantitative finance, and individual investors, this tool enables back-testing of trading strategies in portfolio optimization.

Function description

The Quantum Portfolio Optimizer function uses the Variational Quantum Eigensolver (VQE) algorithm to solve a Quadratic Unconstrained Binary Optimization (QUBO) problem, addressing dynamic portfolio optimization problems. Users simply need to provide the asset price data and define the investment constraint, then the function runs the quantum optimization process that returns a set of optimized investment trajectories.

The process consists of four main stages. First, the input data is mapped to a quantum-compatible problem, constructing the QUBO of the dynamic portfolio optimization problem, and transforming it into a quantum operator (Ising Hamiltonian). Next, the input problem and the VQE algorithm are adapted to be run in the quantum hardware. The VQE algorithm is then run on the quantum hardware, and finally, the results are post-processed to provide the optimal investment trajectories. The system also includes a noise-aware (SQD-based) post-processing to maximize the quality of the output.

This Qiskit Function is based on the published manuscript by Global Data Quantum.

Input

The input arguments of the function are described in the following table. The assets data and other problem specifications must be provided; additionally, the VQE settings can be included to customize the optimization process.

Name	Type	Description	Required	Default	Example
assets	json	Dictionary with the asset prices	Yes	-	-
qubo_settings	json	Settings of the QUBO	Yes	-	See the examples in the table below
ansatz_settings	json	Settings of the ansatz	No	None	See the examples in the table below.
optimizer_settings	json	Settings of the optimizer	No	None	See the examples in the table below.
backend	str	The QPU backend name	No	-	"ibm_torino"
previous_session_id	list of str	List of session IDs to retrieve data from previous runs(*)	No	Empty list	["session_id_1", "session_id_2"]
apply_postprocess	bool	Apply noise-aware SQD post-processing	No	True	True
tags	list of strings	List of tags to identify the experiment	No	Empty list	["optimization", "quantum_computing"]

*To resume an execution or retrieve jobs that were processed in one or more previous sessions, the list of session IDs must be passed in the previous_session_id parameter. This is particularly useful in cases where an optimization task failed to complete due to any error in the process, and execution needs to finish. To achieve this, you must provide the same arguments used in the initial execution, along with the previous_session_id list as described.

Warning

Loading data for previous sessions (for resuming an optimization) can take up to one hour.

assets

The data must be structured as a JSON object that stores information about the closing prices of financial assets on specific dates. The format is as follows:

• Primary key (string): The name or ticker symbol of the financial asset (for example, "8801.T").
• Secondary key (string): The date in YYYY-MM-DD format.
• Value (number): The closing price of the asset on the specified date. Prices can be entered either normalized or non-normalized.

Note that all dictionaries must have the same secondary key (dates). If a specific asset lacks a date that others have, the data must be filled to ensure consistency. For example, this can be done by using the last tracked closing price of that asset.

Example

{
    "8801.T": {
        "2023-01-01": 2374.0,
        "2023-01-02": 2374.0,
        "2023-01-03": 2374.0,
        "2023-01-04": 2356.5,
        ...
    },
    "AAPL": {
        "2023-01-01": 145.2,
        "2023-01-02": 146.5,
        "2023-01-03": 147.3,
        "2023-01-04": 148.1,
        ...
    },
    ...
}

{
    "asset_name": {
        "date": closing_value,
        ...
    },
    ...
}

Note

The asset data must contain, at least, the closing prices at (nt+1) * dt (see the qubo_settings input section) time stamps (for example, days).

qubo_settings

The next table describes the keys of the qubo_settings dictionary. Build the dictionary specifying the number of time steps nt, the number of resolution qubits nq, and the max_investment - or change other default values.

Name	Type	Description	Required	Default	Example
nt	int	Number of time steps	Yes	-	4
nq	int	Number of resolution qubits	Yes	-	4
max_investment	float	Maximum number of invested currency units across all assets	Yes	-	10
dt*	int	Time window considered in each time step. The unit matches the time intervals between the keys in the asset data	No	30	-
risk_aversion	float	Risk aversion coefficient	No	1000	-
transaction_fee	float	Transaction fee coefficient	No	0.01	-
restriction_coeff	float	Lagrange multiplier used to enforce the problem constraint within the QUBO formulation	No	1	-

ansatz_settings

To modify the default options, create a dictionary for the ansatz_settings parameter with the following keys. By default, the ansatz is set to "real_amplitudes", and both extra options (see the following table) are set to False.

Name	Type	Description	Required	Default
ansatz*	str	Ansatz to be used	No	"real_amplitudes"
multiple_passmanager**	bool	Enables multiple passmanager subroutine (not available for Tailored ansatz)	No	False
dd_enable	bool	Adds dynamical decoupling	No	False

* Available ansatzes

• real_amplitudes
• cyclic
• optimized_real_amplitudes
• tailored (Only for ibm_torino backend, 7 assets, 4 time steps, and 4 resolution qubits)

** If multiple_passmanager is set to False, the function uses the default Qiskit pass manager with optimization_level=3. If set to True, the multiple_passmanager subroutine compares three pass managers: the previous default Qiskit pass manager, a pass manager mapping qubits over the QPU first neighbors chain, and the AI transpiler services. Then, the pass manager with the estimated lower cumulative error is selected.

optimizer_settings

This parameter is a dictionary with some tunable options of the optimizing process.

Name	Type	Description	Required	Default
primitive_options	json	Settings of the primitive	No	-
optimizer	str	Selected classical optimizer	No	"differential_evolution"
optimizer_options	json	Configuration of the optimizer	No	-

Note

Currently, the only optimizer option available is "differential_evolution".

Under primitive_options and optimizer_options keys we set dictionaries with the following parameters:

primitive_options

Name	Type	Description	Required	Default	Example
sampler_shots	int	Number of shots of the Sampler.	No	100000	-
estimator_shots	int	Number of shots of the Estimator.	No	25000	-
estimator_precision	float	Desired precision of the expected value. If specified, the precision will be used instead of the estimator_shots.	No	None	0.015625 · (1 / sqrt(4096))
max_time	int or str	Maximum amount of time a runtime session can remain open before being forcibly closed. Can be given in seconds (int) or as a string, like "2h 30m 40s". Must be less than the system-imposed maximum.	No	None	"1h 15m"

optimizer_options

Name	Type	Description	Required	Default
num_generations	int	Number of generations	No	20
population_size	int	Size of the population	No	20
mutation_range	list	Maximum and minimum mutation factor	No	[0, 0.25]
recombination	float	Recombination factor	No	0.4
max_parallel_jobs	int	Maximum number of QPU jobs executed in parallel	No	3
max_batchsize	int	Maximum batch size	No	200

Note

• The number of generations evaluated by the differential evolution is num_generations + 1 since the initial population is included.

• The total number of circuits is calculated as (num_generations + 1) * population_size.

• Using a larger population size and more generations generally improves the quality of the optimization results. However, it is not recommended to exceed a population size of 120 and a number of generations greater than 20 (for example, 120 * 21 = 2520 total circuits), as this would generate an excessive number of circuits, which can be computationally expensive and time-consuming to process.

• The function allows you to resume previous optimization, and it is always possible to increase the number of generations (by providing the same input except for previous_session_id and an increased num_generations).

Note

Ensure compliance with Qiskit Runtime job limits.

• Sampler: sampler_shots <= 10_000_000.
• Estimator: max_batchsize * estimator_shots * observable_size <= 10_000_000 (for this function, all the terms of the observable commute, so observable_size=1).

See the Job limits guide for more information.

Output

The function returns two dictionaries: "result" dictionary, containing the best optimization results, including the optimal solution and its associated minimum objective cost; and "metadata", with data from all results obtained during the optimization process, along with their respective metrics.

The first dictionary focuses on the best-performing solution, while the second provides detailed information about all solutions, including objective costs and other relevant metrics.

Output dictionaries:

Name	Type	Description	Example
result	dict[str, dict[str, float]]	Contains the investment strategy over time, with each time-stamp mapping to asset-specific investment weights (each weight is the investment amount normalized by the total investment amount).	{'time_1': {'asset_1': 0.2, 'asset_2': 0.3, ...\}, ...\}
metadata	dict[str, Any]	Data generated during the analysis, including solutions, costs, and metrics.	See examples below

Description of the metadata dictionary

Name	Type	Description	Example
session_id	str	Unique identifier for the IBM Quantum session.	"d0h30qjvpqf00084fgw0"
all_samples_metrics	dict	Dictionary containing various metrics for each postprocessed sample, such as costs or constraints.	See description below
sampler_counts	dict[str, int]	Dictionary where keys are bitstring representations of sampled solutions and values are their counts.	{"101010": 3, "111000": 1\}
asset_order	list[str]	List with the corresponding investment order of assets at each time step within the investment strategies.	["Asset_0", "Asset_1", "Asset_3"]
QUBO	list[list[float]]	QUBO matrix of the problem.	[[-6.96e-01, 5.81e-01, -1.26e-02, 0.00e+00], ...]
resource_summary	dict[str, dict[str, float]]	Summary of CPU and QPU usage times (in seconds) across different stages of the process.	{'RUNNING: EXECUTING_QPU': {'CPU_TIME': 412.84, 'QPU_TIME': 87.22\}, ...\}

Description of the all_samples_metrics dictionary

Name	Type	Description	Example
investment_trajectories	list[list]	Investment strategies derived from decoded quantum states.	[[1, 2, 2], [1, 2, 1]]

| counts | list[int] | Number of times each investment trajectory was sampled. Index matches investment_trajectories. | [5, 3] | | objective_costs | list[float] | Value of the objective function for each investment trajectory, ordered from lowest to highest. | [0.98, 1.25] | | sharpe_ratios | list[float] | Risk-adjusted performance (Sharpe ratio) for each investment trajectory. Aligned by index. | [1.1, 0.7] | | returns | list[float] | Expected return for each investment trajectory. Aligned by index. | [0.15, 0.10] | | rest_breaches | list[float] | Maximum constraint deviation within each investment trajectory. Aligned by index. | [0.0, 0.25] | | transaction_costs | list[float] | Estimated transaction cost associated with each investment trajectory. Aligned by index. | [0.01, 0.02] |

Get started

Authenticate using your API key and select the Qiskit Function as follows. (This snippet assumes you've already saved your account to your local environment.)

from qiskit_ibm_catalog import QiskitFunctionsCatalog

catalog = QiskitFunctionsCatalog(channel="ibm_quantum_platform")

# Access function
dpo_solver = catalog.load("global-data-quantum/quantum-portfolio-optimizer")

Example: Dynamic portfolio optimization with seven assets

This example demonstrates how to execute the dynamic portfolio optimization (DPO) function and adjust its settings for optimal performance. It includes detailed steps for fine-tuning the parameters to achieve the desired outcomes.

This case involves seven assets, four time steps, and four resolution qubits, resulting in a total requirement of 112 qubits.

1. Read the assets included in the portfolio.

If all the assets in the portfolio are stored in a folder at a specific path, you can load them into a pandas.DataFrame and convert it to a dict format object using the following function.

import os
import glob
import pandas as pd

def read_and_join_csv(file_pattern):
    """
    Reads multiple CSV files matching the file pattern and combines them into a single DataFrame.

    Parameters:
    file_pattern (str): The pattern to match CSV files.

    Returns:
    pd.DataFrame: Combined DataFrame with data from all CSV files.
    """
    # Find all files matching the pattern
    csv_files = glob.glob(file_pattern)
    # Get the base file names without the .csv extension
    file_names = [os.path.basename(f).replace(".csv", "") for f in csv_files]
    # Read each CSV file into a DataFrame and set the first column as the index
    df_list = [pd.read_csv(f).set_index("Unnamed: 0") for f in csv_files]

    # Rename columns in each DataFrame to the base file names
    for df, name in zip(df_list, file_names):
        df.columns = [name]

    # Combine all DataFrames into one by merging them side by side
    combined_df = pd.concat(df_list, axis=1)
    return combined_df

file_pattern = "route/to/folder/with/assets/data/*.csv"
assets = read_and_join_csv(file_pattern).to_dict()

For this example, we have used the assets 8801.T, CLF, GBPJPY, ITX.MC, META, TMBMKDE-10Y, and XS2239553048. The following figure illustrates the data used in this example, showing the daily closing price evolution of the assets from 1 January to 1 September in 2023.

In this example, to ensure uniformity across dates, we have filled non-trading days with the closing price from the previous available date. We apply this step because the selected assets come from different markets with varying trading days, making it essential to standardize the dataset for consistency.

2. Define the problem.

Define the specifications of the problem by configuring the parameters in the qubo_settings dictionary.

qubo_settings = {
    "nt": 4,
    "nq": 4,
    "dt": 30,
    "max_investment": 25,
    "risk_aversion": 1000.0,
    "transaction_fee": 0.01,
    "restriction_coeff": 1.0,
}

3. Define the optimizer and ansatz settings. (Optional)

Optionally define specific requirements for the optimization process, including the selection of the optimizer and its parameters, as well as the specification of the primitive and its configurations.

For the Tailored Ansatz, the chosen population size was based on previous experiments showing that this value yields stable and efficient optimization.

In the case of the Real Amplitudes Ansatz, you can follow a linear relationship between the population_size and the number of qubits in the circuit. As an approximated rule of thumb, it is recommended to use a minimum population_size ~ 0.8 * n_qubits for the real_amplitudes ansatz.

It is expected that the Optimized Real Amplitudes will have a better optimization performance than the Real Amplitudes ansatz. However, the number of variables to optimize in this ansatz increases much faster than in the Real Amplitudes case (see the manuscript). Thus, for large problems, the Optimized Real Amplitudes requires more circuit executions. The Optimized Real Amplitudes is likely to be useful for problems requiring up to 100 qubits, but it is recommended to be mindful when setting the population_size parameters. As an example of this scale-up in population_size, the previous table shows that for an 84-qubit problem, the Optimize Real Amplitudes requires 120 population_size, while for a 56-qubit problem, a population_size of 40 is enough.

optimizer_settings = {
    "de_optimizer_settings": {
        "num_generations": 20,
        "population_size": 90,
        "recombination": 0.4,
        "max_parallel_jobs": 5,
        "max_batchsize": 4,
        "mutation_range": [0.0, 0.25],
    },
    "optimizer": "differential_evolution",
    "primitive_settings": {
        "estimator_shots": 25_000,
        "estimator_precision": None,
        "sampler_shots": 100_000,
    },
}

It is also possible to choose a specific ansatz. The following uses the 'Tailored' ansatz.

ansatz_settings = {
    "ansatz": "tailored",
    "multiple_passmanager": False,
}

4. Run the problem.

dpo_job = dpo_solver.run(
    assets=assets,
    qubo_settings=qubo_settings,
    optimizer_settings=optimizer_settings,
    ansatz_settings=ansatz_settings,
    backend_name="<backend name>",
    previous_session_id=[],
    apply_postprocess=True,
)

5. Retrieve results.

As mentioned in the Output section, the function returns a dictionary with the investment trajectories ordered from lowest to highest according to their objective function value. This set of results allows for the identification of the trajectory with the lowest cost and its corresponding investment evaluations. Additionally, it provides for the analysis of different trajectories, facilitating the selection of those that best align with specific needs or objectives. This flexibility ensures that choices can be tailored to suit a variety of preferences or scenarios. Begin by presenting the resulting strategy that achieved the lowest objective cost found during the process.

# Get the results of the job
dpo_result = dpo_job.result()

# Show the solution strategy
dpo_result["result"]

{'time_step_0': {'8801.T': 0.11764705882352941,
  'ITX.MC': 0.20588235294117646,
  'META': 0.38235294117647056,
  'GBPJPY=X': 0.058823529411764705,
  'TMBMKDE-10Y': 0.0,
  'CLF': 0.058823529411764705,
  'XS2239553048': 0.17647058823529413},
 'time_step_1': {'8801.T': 0.11428571428571428,
  'ITX.MC': 0.14285714285714285,
  'META': 0.2,
  'GBPJPY=X': 0.02857142857142857,
  'TMBMKDE-10Y': 0.42857142857142855,
  'CLF': 0.0,
  'XS2239553048': 0.08571428571428572},
 'time_step_2': {'8801.T': 0.0,
  'ITX.MC': 0.09375,
  'META': 0.3125,
  'GBPJPY=X': 0.34375,
  'TMBMKDE-10Y': 0.0,
  'CLF': 0.0,
  'XS2239553048': 0.25},
 'time_step_3': {'8801.T': 0.3939393939393939,
  'ITX.MC': 0.09090909090909091,
  'META': 0.12121212121212122,
  'GBPJPY=X': 0.18181818181818182,
  'TMBMKDE-10Y': 0.0,
  'CLF': 0.0,
  'XS2239553048': 0.21212121212121213}}

Afterwards, using the metadata, you can access the results of all the sampled strategies. You can thereby further analyze the alternative trajectories returned by the optimizer. To do this, read the dictionary stored in dpo_result['metadata']['all_samples_metrics'], which contains not only additional information about the optimal strategy, but also details of the other candidate strategies evaluated during the optimization.

The following example shows how to read this information using pandas to extract key metrics associated with the optimal strategy. These include Restriction Deviation, Sharpe Ratio, and the corresponding investment return.

# Convert metadata to a DataFrame
df = pd.DataFrame(dpo_result["metadata"]["all_samples_metrics"])

# Find the minimum objective cost
min_cost = df["objective_costs"].min()
print(f"Minimum Objective Cost Found: {min_cost:.2f}")

# Extract the row with the lowest cost
best_row = df[df["objective_costs"] == min_cost].iloc[0]

# Display the results associated with the best solution
print("Best Solution:")
print(f"  - Restriction Deviation: {best_row['rest_breaches']}%")
print(f"  - Sharpe Ratio: {best_row['sharpe_ratios']:.2f}")
print(f"  - Return: {best_row['returns']}")

Minimum Objective Cost Found: -3.78
Best Solution:
  - Restriction Deviation: 40.0
  - Sharpe Ratio: 24.82
  - Return: 0.46

6. Performance analysis

Last, analyze the performance of your optimization application. Specifically, compare your results, obtained in the previous example, against a random baseline to assess the effectiveness of our approach. If the quantum algorithm demonstrably and consistently produces results with lower cost values, it indicates an effective optimization process.

The figure presents the probability distributions of the objective costs. To generate these distributions, take the list of objective costs from the function result and count the occurrences of each cost value (values rounded to the second decimal place). Then, update the count column accordingly by joining counts of identical rounded values. Note that, for better visual comparison, the occurrence counts have been normalized so that each distribution is displayed between 0 and 1. As shown in the figure (blue solid line), the cost distribution for our Variational Quantum Eigensolver (post-processed with SQD) approach is sharply concentrated at lower objective cost values, indicating good optimization performance. In contrast, the noisy baseline exhibits a broader distribution, centered around higher cost values. The gray dashed vertical line represents the mean value of the random distribution, further highlighting the consistency of the function in returning optimized investment strategies. For an additional comparison, the black dashed line in the figure corresponds to the solution obtained with the Gurobi optimizer (free version). All these results are further explored in the benchmarks below for the "Mixed Assets" example evaluated with the "Tailored" ansatz.

Benchmarks

This function was tested under different configurations of resolution qubits, ansatz circuits, and groupings of assets from various sectors: a mix of different assets (Set 1), oil derivatives (Set 2), and IBEX35 (Set 3). See more details in the following table.

Set	Date	Assets
Set 1	01/01/2023	8801.T, CL=F, GBPJPY=X, ITX.MC, META, TMBMKDE-10Y, XS2239553048
Set 2	01/06/2023	CL=F, BZ=F, HO=F, NG=F, XOM, RB=F, 2222.SR
Set 3	01/11/2022	ACS.MC, ITX.MC, FER.MC, ELE.MC, SCYR.MC, AENA.MC, AMS.MC

Two key metrics were used to evaluate solution quality.

1. The objective cost, which measures optimization efficiency by comparing the cost function value from each experiment with results from Gurobi (free version).
2. The Sharpe ratio, which captures the risk-adjusted return of each portfolio, offering insight into the financial performance of the solutions.

Together, these metrics benchmark both computational and financial aspects of the quantum-generated portfolios.

Example	Qubits	Ansatz	Depth	Runtime Usage (s)	Total usage (s)	Objective cost	Sharpe	Gurobi objective cost	Gurobi Sharpe
Mixed Assets (Set 1, 4 time steps, 4-bit)	112	Tailored	83	12735	13095	-3.78	24.82	-4.25	24.71
Mixed Assets (Set 1,4 time steps, 4 time steps, 4-bit)	112	Real Amplitudes	359	11739	11903	-3.39	23.64	-4.25	24.71
Oil Derivatives (Set 2, 4 time steps, 3-bit)	84	Optimized Real Amplitudes	78	6180	6350	-3.73	19.13	-4.19	21.71
IBEX35 (Set 3, 4 time steps, 2-bit)	56	Optimized Real Amplitudes	96	3314	3523	-3.67	14.48	-4.11	16.44

Results show that the quantum optimizer, with problem-specific ansatzes, effectively identifies efficient investment strategies across various portfolio types. Below we detail both the population size and the number of generations specified in the optimizer_options dictionary. All other parameters were set to their default values.

Example	population_size	num_generations
Mixed Assets Portfolio	90	20
Mixed Assets Portfolio	92	20
Oil Derivatives Portfolio	120	20
IBEX35 Portfolio	40	20

The number of generations was set to 20, as this value was found to be sufficient to reach convergence. Additionally, the default values for the optimizer's internal parameters were left unchanged, as they consistently provided good performance and are generally recommended by the literature and implementation guidelines.

Get support

If you need help, you can send an email to qpo.support@globaldataquantum.com. In your message, provide the function job ID.

Next steps

Recommendations

• Read the associated research paper.
• Request access to the function by filling in this form.
• Try the Dynamic Portfolio Optimization tutorial.