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Quantum computational finance: quantum algorithm for portfolio optimization

In this tutorial, we will work through a concrete example of portfolio optimization. Our goal is to determine how to allocate capital across several assets so that we control risk while still aiming for good returns. To do this, we use historical stock price data to construct a portfolio, that is, the allocation ratios of the assets. The approach is motivated by the methods proposed in References [1,2], where portfolio optimization is identified as a problem that can potentially be accelerated using the HHL algorithm. As a simple practical example, we consider an investment in four major technology stocks and examine how to distribute capital among them in an effective way. first we need to install some out package to collect stock data. 
# !pip install yfinance
# !pip install pandas_datareader


# import datetime
# import yfinance as yf


# codes = ['NVDA', 'AAPL', 'TSLA', 'AMZN']


# start = datetime.datetime(2024, 1, 1)
# end   = datetime.datetime(2024, 12, 31)

# data = yf.download(codes, start=start, end=end, auto_adjust=False, progress=False)


# df = data['Adj Close']
# df.to_csv("asset_data.csv")
# display(df.head(10))


import datetime

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

from classiq import *

​
Loading Dataset of Stock

We define csv_data which was generated from yfinance. We use the data from 2024-01-02 to 2024-12-30. 
import numpy as np
import pandas as pd

stock_data = {
    "Date": [
        "2024-01-02",
        "2024-01-03",
        "2024-01-04",
        "2024-01-05",
        "2024-01-08",
        "2024-01-09",
        "2024-01-10",
        "2024-01-11",
        "2024-01-12",
        "2024-01-16",
        "2024-01-17",
        "2024-01-18",
        "2024-01-19",
        "2024-01-22",
        "2024-01-23",
        "2024-01-24",
        "2024-01-25",
        "2024-01-26",
        "2024-01-29",
        "2024-01-30",
        "2024-01-31",
        "2024-02-01",
        "2024-02-02",
        "2024-02-05",
        "2024-02-06",
        "2024-02-07",
        "2024-02-08",
        "2024-02-09",
        "2024-02-12",
        "2024-02-13",
        "2024-02-14",
        "2024-02-15",
        "2024-02-16",
        "2024-02-20",
        "2024-02-21",
        "2024-02-22",
        "2024-02-23",
        "2024-02-26",
        "2024-02-27",
        "2024-02-28",
        "2024-02-29",
        "2024-03-01",
        "2024-03-04",
        "2024-03-05",
        "2024-03-06",
        "2024-03-07",
        "2024-03-08",
        "2024-03-11",
        "2024-03-12",
        "2024-03-13",
        "2024-03-14",
        "2024-03-15",
        "2024-03-18",
        "2024-03-19",
        "2024-03-20",
        "2024-03-21",
        "2024-03-22",
        "2024-03-25",
        "2024-03-26",
        "2024-03-27",
        "2024-03-28",
        "2024-04-01",
        "2024-04-02",
        "2024-04-03",
        "2024-04-04",
        "2024-04-05",
        "2024-04-08",
        "2024-04-09",
        "2024-04-10",
        "2024-04-11",
        "2024-04-12",
        "2024-04-15",
        "2024-04-16",
        "2024-04-17",
        "2024-04-18",
        "2024-04-19",
        "2024-04-22",
        "2024-04-23",
        "2024-04-24",
        "2024-04-25",
        "2024-04-26",
        "2024-04-29",
        "2024-04-30",
        "2024-05-01",
        "2024-05-02",
        "2024-05-03",
        "2024-05-06",
        "2024-05-07",
        "2024-05-08",
        "2024-05-09",
        "2024-05-10",
        "2024-05-13",
        "2024-05-14",
        "2024-05-15",
        "2024-05-16",
        "2024-05-17",
        "2024-05-20",
        "2024-05-21",
        "2024-05-22",
        "2024-05-23",
        "2024-05-24",
        "2024-05-28",
        "2024-05-29",
        "2024-05-30",
        "2024-05-31",
        "2024-06-03",
        "2024-06-04",
        "2024-06-05",
        "2024-06-06",
        "2024-06-07",
        "2024-06-10",
        "2024-06-11",
        "2024-06-12",
        "2024-06-13",
        "2024-06-14",
        "2024-06-17",
        "2024-06-18",
        "2024-06-20",
        "2024-06-21",
        "2024-06-24",
        "2024-06-25",
        "2024-06-26",
        "2024-06-27",
        "2024-06-28",
        "2024-07-01",
        "2024-07-02",
        "2024-07-03",
        "2024-07-05",
        "2024-07-08",
        "2024-07-09",
        "2024-07-10",
        "2024-07-11",
        "2024-07-12",
        "2024-07-15",
        "2024-07-16",
        "2024-07-17",
        "2024-07-18",
        "2024-07-19",
        "2024-07-22",
        "2024-07-23",
        "2024-07-24",
        "2024-07-25",
        "2024-07-26",
        "2024-07-29",
        "2024-07-30",
        "2024-07-31",
        "2024-08-01",
        "2024-08-02",
        "2024-08-05",
        "2024-08-06",
        "2024-08-07",
        "2024-08-08",
        "2024-08-09",
        "2024-08-12",
        "2024-08-13",
        "2024-08-14",
        "2024-08-15",
        "2024-08-16",
        "2024-08-19",
        "2024-08-20",
        "2024-08-21",
        "2024-08-22",
        "2024-08-23",
        "2024-08-26",
        "2024-08-27",
        "2024-08-28",
        "2024-08-29",
        "2024-08-30",
        "2024-09-03",
        "2024-09-04",
        "2024-09-05",
        "2024-09-06",
        "2024-09-09",
        "2024-09-10",
        "2024-09-11",
        "2024-09-12",
        "2024-09-13",
        "2024-09-16",
        "2024-09-17",
        "2024-09-18",
        "2024-09-19",
        "2024-09-20",
        "2024-09-23",
        "2024-09-24",
        "2024-09-25",
        "2024-09-26",
        "2024-09-27",
        "2024-09-30",
        "2024-10-01",
        "2024-10-02",
        "2024-10-03",
        "2024-10-04",
        "2024-10-07",
        "2024-10-08",
        "2024-10-09",
        "2024-10-10",
        "2024-10-11",
        "2024-10-14",
        "2024-10-15",
        "2024-10-16",
        "2024-10-17",
        "2024-10-18",
        "2024-10-21",
        "2024-10-22",
        "2024-10-23",
        "2024-10-24",
        "2024-10-25",
        "2024-10-28",
        "2024-10-29",
        "2024-10-30",
        "2024-10-31",
        "2024-11-01",
        "2024-11-04",
        "2024-11-05",
        "2024-11-06",
        "2024-11-07",
        "2024-11-08",
        "2024-11-11",
        "2024-11-12",
        "2024-11-13",
        "2024-11-14",
        "2024-11-15",
        "2024-11-18",
        "2024-11-19",
        "2024-11-20",
        "2024-11-21",
        "2024-11-22",
        "2024-11-25",
        "2024-11-26",
        "2024-11-27",
        "2024-11-29",
        "2024-12-02",
        "2024-12-03",
        "2024-12-04",
        "2024-12-05",
        "2024-12-06",
        "2024-12-09",
        "2024-12-10",
        "2024-12-11",
        "2024-12-12",
        "2024-12-13",
        "2024-12-16",
        "2024-12-17",
        "2024-12-18",
        "2024-12-19",
        "2024-12-20",
        "2024-12-23",
        "2024-12-24",
        "2024-12-26",
        "2024-12-27",
        "2024-12-30",
    ],
    "AAPL": [
        183.9,
        182.52,
        180.2,
        179.48,
        183.82,
        183.4,
        184.44,
        183.85,
        184.18,
        181.91,
        180.97,
        186.86,
        189.76,
        192.07,
        193.35,
        192.68,
        192.35,
        190.61,
        189.93,
        186.28,
        182.67,
        185.11,
        184.11,
        185.92,
        187.52,
        187.63,
        186.55,
        187.32,
        185.63,
        183.54,
        182.65,
        182.37,
        180.83,
        180.09,
        180.84,
        182.87,
        181.04,
        179.69,
        181.15,
        179.95,
        179.28,
        178.2,
        173.68,
        168.74,
        167.75,
        167.63,
        169.34,
        171.35,
        171.82,
        169.74,
        171.6,
        171.22,
        172.31,
        174.65,
        177.22,
        169.98,
        170.88,
        169.46,
        168.33,
        171.9,
        170.09,
        168.65,
        167.47,
        168.27,
        167.45,
        168.2,
        167.08,
        168.29,
        166.42,
        173.62,
        175.12,
        171.29,
        168.0,
        166.64,
        165.68,
        163.66,
        164.49,
        165.54,
        167.65,
        168.51,
        167.93,
        172.09,
        168.95,
        167.93,
        171.62,
        181.89,
        180.23,
        180.92,
        181.26,
        183.07,
        181.81,
        185.02,
        186.16,
        188.44,
        188.55,
        188.58,
        189.75,
        191.05,
        189.61,
        185.61,
        188.69,
        188.7,
        189.0,
        189.99,
        190.95,
        192.72,
        193.03,
        194.54,
        193.16,
        195.56,
        191.81,
        205.75,
        211.63,
        212.79,
        211.05,
        215.2,
        212.84,
        208.26,
        206.09,
        206.73,
        207.65,
        211.81,
        212.65,
        209.19,
        215.28,
        218.78,
        220.05,
        224.81,
        226.28,
        227.13,
        231.4,
        226.03,
        228.98,
        232.81,
        233.23,
        227.33,
        222.66,
        222.79,
        222.44,
        223.49,
        217.06,
        216.02,
        216.48,
        216.76,
        217.32,
        220.58,
        216.88,
        218.37,
        207.85,
        205.83,
        208.4,
        211.87,
        214.78,
        216.31,
        220.03,
        220.47,
        223.46,
        224.78,
        224.62,
        225.24,
        225.13,
        223.27,
        225.57,
        225.9,
        226.75,
        225.22,
        228.5,
        227.71,
        221.52,
        219.61,
        221.13,
        219.58,
        219.67,
        218.87,
        221.41,
        221.52,
        221.25,
        215.1,
        215.57,
        219.45,
        227.58,
        226.92,
        225.2,
        226.09,
        225.1,
        226.24,
        226.51,
        231.69,
        224.94,
        225.51,
        224.4,
        225.53,
        220.44,
        224.5,
        228.25,
        227.75,
        226.27,
        230.0,
        232.54,
        230.48,
        230.85,
        233.68,
        235.15,
        234.54,
        229.46,
        229.27,
        230.11,
        232.09,
        232.36,
        228.81,
        224.64,
        221.66,
        220.76,
        222.19,
        221.47,
        226.2,
        225.93,
        223.22,
        223.22,
        224.1,
        227.19,
        223.98,
        226.99,
        227.25,
        227.96,
        227.49,
        228.83,
        231.82,
        234.0,
        233.87,
        236.26,
        238.51,
        241.55,
        241.91,
        241.94,
        241.74,
        245.63,
        246.65,
        245.38,
        246.84,
        247.01,
        249.9,
        252.33,
        246.93,
        248.66,
        253.34,
        254.12,
        257.03,
        257.85,
        254.43,
        251.06,
    ],
    "AMZN": [
        149.92,
        148.47,
        144.57,
        145.24,
        149.1,
        151.36,
        153.72,
        155.17,
        154.61,
        153.16,
        151.71,
        153.5,
        155.33,
        154.77,
        156.02,
        156.86,
        157.75,
        159.11,
        161.25,
        159.0,
        155.19,
        159.27,
        171.8,
        170.3,
        169.14,
        170.52,
        169.83,
        174.44,
        172.33,
        168.63,
        170.97,
        169.8,
        169.5,
        167.08,
        168.58,
        174.58,
        174.99,
        174.72,
        173.53,
        173.16,
        176.75,
        178.22,
        177.58,
        174.11,
        173.5,
        176.82,
        175.35,
        171.96,
        175.38,
        176.55,
        178.75,
        174.41,
        174.47,
        175.89,
        178.14,
        178.14,
        178.86,
        179.71,
        178.3,
        179.83,
        180.38,
        180.97,
        180.69,
        182.41,
        180.0,
        185.07,
        185.19,
        185.66,
        185.94,
        189.05,
        186.13,
        183.61,
        183.32,
        181.27,
        179.22,
        174.63,
        177.22,
        179.53,
        176.58,
        173.66,
        179.61,
        180.96,
        175.0,
        179.0,
        184.72,
        186.21,
        188.69,
        188.75,
        188.0,
        189.5,
        187.47,
        186.57,
        187.07,
        185.99,
        183.63,
        184.69,
        183.53,
        183.14,
        183.13,
        181.05,
        180.75,
        182.14,
        182.02,
        179.32,
        176.44,
        178.33,
        179.33,
        181.27,
        185.0,
        184.3,
        187.05,
        187.22,
        186.88,
        183.83,
        183.66,
        184.05,
        182.8,
        186.1,
        189.08,
        185.57,
        186.33,
        193.61,
        197.85,
        193.25,
        197.19,
        200.0,
        197.58,
        200.0,
        199.28,
        199.33,
        199.78,
        195.05,
        194.49,
        192.72,
        193.02,
        187.92,
        183.75,
        183.13,
        182.55,
        186.41,
        180.83,
        179.85,
        182.5,
        183.19,
        181.71,
        186.97,
        184.07,
        167.89,
        161.02,
        161.92,
        162.77,
        165.8,
        166.94,
        166.8,
        170.22,
        170.1,
        177.58,
        177.05,
        178.22,
        178.88,
        180.11,
        176.13,
        177.03,
        175.5,
        173.11,
        170.8,
        172.11,
        178.5,
        176.25,
        173.33,
        177.88,
        171.38,
        175.39,
        179.55,
        184.52,
        187.0,
        186.49,
        184.88,
        186.88,
        186.42,
        189.86,
        191.6,
        193.88,
        193.96,
        192.52,
        191.16,
        187.97,
        186.33,
        185.13,
        184.75,
        181.96,
        186.5,
        180.8,
        182.72,
        185.16,
        186.64,
        188.82,
        187.53,
        187.69,
        186.88,
        187.52,
        188.99,
        189.07,
        189.69,
        184.71,
        186.38,
        187.83,
        188.38,
        190.83,
        192.72,
        186.39,
        197.92,
        195.77,
        199.5,
        207.08,
        210.05,
        208.17,
        206.83,
        208.91,
        214.1,
        211.47,
        202.61,
        201.69,
        204.61,
        202.88,
        198.38,
        197.11,
        201.44,
        207.86,
        205.74,
        207.88,
        210.71,
        213.44,
        218.16,
        220.55,
        227.02,
        226.08,
        225.03,
        230.25,
        228.97,
        227.46,
        232.92,
        231.14,
        220.52,
        223.28,
        224.91,
        225.05,
        229.05,
        227.05,
        223.75,
        221.3,
    ],
    "NVDA": [
        48.14,
        47.54,
        47.97,
        49.06,
        52.22,
        53.11,
        54.31,
        54.79,
        54.67,
        56.35,
        56.02,
        57.07,
        59.45,
        59.62,
        59.83,
        61.32,
        61.58,
        60.99,
        62.43,
        62.73,
        61.49,
        62.99,
        66.12,
        69.29,
        68.18,
        70.05,
        69.6,
        72.09,
        72.2,
        72.08,
        73.85,
        72.61,
        72.57,
        69.41,
        67.43,
        78.49,
        78.77,
        79.04,
        78.65,
        77.61,
        79.06,
        82.23,
        85.18,
        85.92,
        88.65,
        92.62,
        87.48,
        85.73,
        91.86,
        90.84,
        87.89,
        87.79,
        88.4,
        89.35,
        90.32,
        91.38,
        94.24,
        94.95,
        92.51,
        90.2,
        90.3,
        90.31,
        89.4,
        88.91,
        85.86,
        87.96,
        87.08,
        85.31,
        86.99,
        90.56,
        88.14,
        85.95,
        87.37,
        83.99,
        84.62,
        76.16,
        79.47,
        82.38,
        79.63,
        82.58,
        87.69,
        87.71,
        86.35,
        82.99,
        85.77,
        88.74,
        92.09,
        90.5,
        90.36,
        88.7,
        89.83,
        90.35,
        91.3,
        94.58,
        94.31,
        92.43,
        94.73,
        95.33,
        94.9,
        103.74,
        106.41,
        113.84,
        114.76,
        110.44,
        109.57,
        114.94,
        116.37,
        122.37,
        120.93,
        120.82,
        121.72,
        120.85,
        125.14,
        129.55,
        131.82,
        130.92,
        135.52,
        130.72,
        126.51,
        118.05,
        126.03,
        126.34,
        123.93,
        123.48,
        124.24,
        122.61,
        128.22,
        125.77,
        128.14,
        131.32,
        134.85,
        127.34,
        129.18,
        128.38,
        126.3,
        117.93,
        121.03,
        117.87,
        123.48,
        122.53,
        114.2,
        112.23,
        113.01,
        111.54,
        103.68,
        116.96,
        109.16,
        107.22,
        100.4,
        104.2,
        98.86,
        104.92,
        104.7,
        108.97,
        116.09,
        118.02,
        122.8,
        124.52,
        129.94,
        127.19,
        128.44,
        123.68,
        129.31,
        126.4,
        128.24,
        125.55,
        117.53,
        119.31,
        107.95,
        106.16,
        107.16,
        102.78,
        106.42,
        108.05,
        116.85,
        119.09,
        119.05,
        116.74,
        115.55,
        113.33,
        117.82,
        115.96,
        116.22,
        120.82,
        123.46,
        123.99,
        121.35,
        121.39,
        116.95,
        118.8,
        122.8,
        124.87,
        127.67,
        132.84,
        132.6,
        134.76,
        134.75,
        138.02,
        131.55,
        135.67,
        136.88,
        137.95,
        143.66,
        143.54,
        139.51,
        140.36,
        141.49,
        140.47,
        141.2,
        139.29,
        132.71,
        135.35,
        136.0,
        139.86,
        145.56,
        148.82,
        147.57,
        145.21,
        148.23,
        146.21,
        146.7,
        141.93,
        140.1,
        146.95,
        145.84,
        146.61,
        141.9,
        135.97,
        136.87,
        135.29,
        138.2,
        138.58,
        140.21,
        145.09,
        145.02,
        142.4,
        138.77,
        135.03,
        139.27,
        137.3,
        134.21,
        131.96,
        130.35,
        128.87,
        130.64,
        134.66,
        139.63,
        140.18,
        139.89,
        136.97,
        137.45,
    ],
    "TSLA": [
        248.41,
        238.44,
        237.92,
        237.49,
        240.44,
        234.96,
        233.94,
        227.22,
        218.88,
        219.91,
        215.55,
        211.88,
        212.19,
        208.8,
        209.13,
        207.83,
        182.63,
        183.25,
        190.92,
        191.58,
        187.28,
        188.86,
        187.91,
        181.05,
        185.1,
        187.58,
        189.55,
        193.57,
        188.13,
        184.02,
        188.71,
        200.44,
        199.94,
        193.75,
        194.77,
        197.41,
        191.97,
        199.39,
        199.72,
        202.03,
        201.88,
        202.63,
        188.13,
        180.74,
        176.53,
        178.64,
        175.33,
        177.77,
        177.53,
        169.47,
        162.5,
        163.57,
        173.8,
        171.32,
        175.66,
        172.82,
        170.83,
        172.63,
        177.66,
        179.83,
        175.78,
        175.22,
        166.63,
        168.38,
        171.11,
        164.89,
        172.97,
        176.88,
        171.75,
        174.6,
        171.05,
        161.47,
        157.11,
        155.44,
        149.92,
        147.05,
        142.05,
        144.67,
        162.13,
        170.17,
        168.28,
        194.05,
        183.27,
        179.99,
        180.0,
        181.19,
        184.75,
        177.8,
        174.72,
        171.97,
        168.47,
        171.88,
        177.55,
        173.99,
        174.83,
        177.46,
        174.94,
        186.6,
        180.11,
        173.74,
        179.24,
        176.75,
        176.19,
        178.78,
        178.08,
        176.28,
        174.77,
        175.0,
        177.94,
        177.47,
        173.78,
        170.66,
        177.28,
        182.47,
        178.0,
        187.44,
        184.86,
        181.57,
        183.0,
        182.58,
        187.35,
        196.36,
        197.41,
        197.88,
        209.86,
        231.25,
        246.38,
        251.52,
        252.94,
        262.32,
        263.26,
        241.02,
        248.22,
        252.63,
        256.55,
        248.5,
        249.22,
        239.19,
        251.5,
        246.38,
        215.99,
        220.25,
        219.8,
        232.1,
        222.61,
        232.07,
        216.86,
        207.66,
        198.88,
        200.63,
        191.75,
        198.83,
        200.0,
        197.49,
        207.83,
        201.38,
        214.13,
        216.11,
        222.72,
        221.1,
        223.27,
        210.66,
        220.32,
        213.21,
        209.21,
        205.75,
        206.27,
        214.11,
        210.6,
        219.41,
        230.16,
        210.72,
        216.27,
        226.16,
        228.13,
        229.8,
        230.28,
        226.77,
        227.86,
        227.19,
        243.91,
        238.25,
        250.0,
        254.27,
        257.01,
        254.22,
        260.45,
        261.63,
        258.01,
        249.02,
        240.66,
        250.08,
        240.83,
        244.5,
        241.05,
        238.77,
        217.8,
        219.16,
        219.57,
        221.33,
        220.88,
        220.69,
        218.85,
        217.97,
        213.64,
        260.48,
        269.19,
        262.51,
        259.51,
        257.54,
        249.85,
        248.97,
        242.83,
        251.44,
        288.52,
        296.91,
        321.22,
        350.0,
        328.48,
        330.23,
        311.17,
        320.72,
        338.73,
        346.0,
        342.02,
        339.64,
        352.55,
        338.58,
        338.23,
        332.89,
        345.16,
        357.08,
        351.42,
        357.92,
        369.48,
        389.22,
        389.79,
        400.98,
        424.76,
        418.1,
        436.23,
        463.01,
        479.85,
        440.13,
        436.17,
        421.05,
        430.6,
        462.27,
        454.13,
        431.66,
        417.41,
    ],
}

# Create DataFrame from the dictionary
df = pd.DataFrame(stock_data)
num_data = len(df)

# Process Date as index (Ensuring 1:1 match with provided data)
df["Date"] = pd.to_datetime(df["Date"])
df.set_index("Date", inplace=True)
print(df.head(10))
Output:

AAPL    AMZN   NVDA    TSLA
  Date                                     
  2024-01-02  183.90  149.92  48.14  248.41
  2024-01-03  182.52  148.47  47.54  238.44
  2024-01-04  180.20  144.57  47.97  237.92
  2024-01-05  179.48  145.24  49.06  237.49
  2024-01-08  183.82  149.10  52.22  240.44
  2024-01-09  183.40  151.36  53.11  234.96
  2024-01-10  184.44  153.72  54.31  233.94
  2024-01-11  183.85  155.17  54.79  227.22
  2024-01-12  184.18  154.61  54.67  218.88
  2024-01-16  181.91  153.16  56.35  219.91

​
Plot the stock price of selected stock

df.loc[:, ["NVDA", "AAPL", "TSLA", "AMZN"]].plot()
Output:
<Axes: xlabel='Date'>
output

​
Conversion to Daily Returns

The daily return (rate of change) yty_t of an individual stock is defined as follows, letting tt be the date: yt=Pt−Pt−1Pt−1y_t = \frac{P_t - P_{t-1}}{P_{t-1}} It is obtained using pct_change() in pandas. 
daily_return = df.pct_change().dropna()
display(daily_return.tail())
AAPLAMZNNVDATSLA
Date
---------------
2024-12-230.0030790.0006220.0369080.022681
2024-12-240.0114510.0177740.0039390.073549
2024-12-260.003190-0.008732-0.002069-0.017609
2024-12-27-0.013264-0.014534-0.020874-0.049479
2024-12-30-0.013245-0.0109500.003504-0.033012

​
Expected Return

Calculate the expected return R⃗\vec{R} for each stock. Here, the arithmetic mean of historical returns is used. R⃗=1T∑t=1Ty⃗t\vec{R} = \frac{1}{T}\sum_{t=1}^{T}\vec{y}_t 
expected_return = daily_return.dropna(how="all").mean() * num_data
print(expected_return)
Output:

AAPL    0.337606
  AMZN    0.430596
  NVDA    1.191089
  TSLA    0.718326
  dtype: float64

​
Variance-Covariance Matrix

The sample unbiased variance-covariance matrix of returns, Σ\Sigma, is defined as follows: Σ=1T−1∑t=1T(y⃗t−R⃗)(y⃗t−R⃗)T\Sigma = \frac{1}{T-1}\sum_{t=1}^{T} (\vec{y}_t - \vec{R})(\vec{y}_t - \vec{R} )^{T}
  • Let the return of each asset ii be the random variable RiR_i.
  • The covariance is:
Cov(Ri,Rj)=E[(Ri−E[Ri])(Rj−E[Rj])]\mathrm{Cov}(R_i, R_j) = \mathbb{E}\big[(R_i - \mathbb{E}[R_i])(R_j - \mathbb{E}[R_j])\big]
  • The matrix aggregating this for all assets is Σ\Sigma:
\Sigma = \begin\{bmatrix\} \sigma_\{00\} & \sigma_\{01\} & \cdots & \sigma_\{0n\} \\ \sigma_\{10\} & \sigma_\{11\} & \cdots & \sigma_\{1n\} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_\{n0\} & \sigma_\{n1\} & \cdots & \sigma_\{nn\} \end\{bmatrix\} $ Where $\sigma_{ij} = \mathrm{Cov}(R_i, R_j)$. * With $\Sigma$, the "diversification effect" is incorporated into the calculation. * If assets have a negative correlation with each other, the risk of the entire portfolio decreases. * Conversely, if there is a strong positive correlation, the risk reduction effect will not be significant even if assets are diversified. 
cov = daily_return.dropna(how="all").cov() * num_data
display(cov)
AAPLAMZNNVDATSLA
AAPL0.0502040.0211670.0298190.046741
AMZN0.0211670.0788920.0642260.056531
NVDA0.0298190.0642260.2749810.071096
TSLA0.0467410.0565310.0710960.403576

​
Portfolio Optimization

Having defined the necessary variables, we now proceed to actual portfolio optimization. We represent the portfolio as a 44-component vector w=(w0,w1,w2,w3)T\mathbf{w} = (w_0,w_1,w_2,w_3)^T. This represents the proportion of holdings in each stock; for example, w=(1,0,0,0)\mathbf{w}=(1,0,0,0) implies a portfolio where 100%100\% of total assets are invested in AAPL stock. Consider a portfolio that satisfies the following equation: O=minw12wTΣw=∑iwi2σii+∑i≠jwiwjσij\mathcal{O} = {\rm min_{\mathbf{w} }}\frac{1}{2}{\mathbf{w}^T}\Sigma\mathbf{w} = \sum_{i} w_i^2 \sigma_{ii} + \sum_{i\neq j} w_i w_j \sigma_{ij}
  • The diagonal elements σii\sigma_{ii} are the variances of each asset (intensity of risk).
  • The off-diagonal elements σij\sigma_{ij} are the covariances between assets (strength of correlation).
Here, the constraints are as follows: const.1=RTw=μ{\rm const.1} = \mathbf{R^Tw}=\mu const.2=1Tw=1{\rm const.2} = \mathbf{1^Tw}=1 Here, 1\mathbf{1} represents (1,1,1,1)T(1,1,1,1)^T. The meanings of the two constraint equations are:
  • The expected return of the portfolio (average value of returns) is μ\mu.
  • The sum of the weights invested in the portfolio is 1 (1=(1,1,1,1)T\mathbf{1}=(1,1,1,1)^T).
Under these conditions, we perform the “minimization of the variance of portfolio returns.” This means that when desiring a return of μ\mu in the future, the best portfolio is one that minimizes the fluctuation (risk) associated with it as much as possible. This type of problem setup is known as the Markowitz Mean-Variance Approach and is one of the foundational concepts of modern financial engineering.

​
Lagrange Multiplier Method in Mathematical Optimization

The mathematical optimization problem described above is an optimization problem with equality constraints. By solving this using the Method of Lagrange Multipliers, we can find candidates for the local optimal solution. The Method of Lagrange Multipliers is defined as follows: L(x,u)=f(x)+∑i=1muigi(x)(0)\mathcal{L}(x,u) = f(x) + \sum_{i=1}^m u_i g_i(x) \tag{0} Here, f(x)f(x) represents the objective function, and the second term represents the constraint functions. u∈Rdu\in R_{d} are the Lagrange multipliers. In the case of our portfolio optimization problem, if we introduce Lagrange multipliers η\eta and θ\theta, the Lagrangian is given by: L(w,η,θ)=12wTΣw+η(Rw−μ)+θ(1Tw−1)(1)\mathcal{L}(w,\eta, \theta) = \frac{1}{2}\mathbf{w}^T\Sigma\mathbf{w} + \eta(\mathbf{R}\mathbf{w}-\mu) + \theta(\mathbf{1}^T\mathbf{w}-1) \tag{1} Since the constraints are linear equations, the optimal solution is guaranteed by just the first-order derivatives. Therefore, we differentiate the Lagrangian function above with respect to the variables w,η,and θ\mathbf{w}, \eta, \text{and } \theta.

​
First-Order Optimality Conditions (KKT Conditions)

∂L∂w⃗=Σw⃗+η R⃗+θ 1⃗=0⃗(A)\frac{\partial \mathcal L}{\partial \vec w}= \Sigma\vec w+\eta\,\vec R+\theta\,\vec 1=\vec 0 \tag{A} ∂L∂η=R⃗ Tw⃗−μ=0(B)\frac{\partial \mathcal L}{\partial \eta}= \vec R^{\,T}\vec w-\mu=0 \tag{B} ∂L∂θ=1⃗ Tw⃗−1=0(C)\frac{\partial \mathcal L}{\partial \theta}= \vec 1^{\,T}\vec w-1=0 \tag{C}

​
Organizing as a System of Linear Equations

If we organize (A) through (C) into a linear equation regarding the unknowns (η,θ,w⃗)(\eta,\theta,\vec w), we get: {R⃗ Tw⃗=μ1⃗ Tw⃗=1Σw⃗+η R⃗+θ 1⃗=0⃗⟺[00R⃗ T001⃗ TR⃗1⃗Σ]⏟W[ηθw⃗]=[μ10⃗]\begin{cases} \vec R^{\,T}\vec w=\mu\\[2pt] \vec 1^{\,T}\vec w=1\\[2pt] \Sigma\vec w+\eta\,\vec R+\theta\,\vec 1=\vec 0 \end{cases} \quad\Longleftrightarrow\quad \underbrace{\begin{bmatrix} 0&0&\vec R^{\,T}\\ 0&0&\vec 1^{\,T}\\ \vec R&\vec 1&\Sigma \end{bmatrix}}_{W} \begin{bmatrix}\eta\\\theta\\\vec w\end{bmatrix} = \begin{bmatrix}\mu\\1\\\vec 0\end{bmatrix} Looking at this row by row:
  • 1st Row: 0⋅η+0⋅θ+R⃗ Tw⃗=μ0\cdot\eta+0\cdot\theta+\vec R^{\,T}\vec w=\mu (Expected Return Constraint)
  • 2nd Row: 0⋅η+0⋅θ+1⃗ Tw⃗=10\cdot\eta+0\cdot\theta+\vec 1^{\,T}\vec w=1 (Sum of Weights = 1 Constraint)
  • 3rd Row: R⃗ η+1⃗ θ+Σw⃗=0⃗\vec R\,\eta+\vec 1\,\theta+\Sigma\vec w=\vec 0 (Stationarity Condition)
This explains why “solving the matrix WW satisfies both the objective function and the constraints simultaneously.” This block matrix WW is called the KKT Matrix (symmetric and indefinite). Therefore, based on the results of the Lagrange Multiplier Method described above, we can obtain the optimal weights w\mathbf{w} that satisfy the conditions by solving the linear system of equations: W(νθw)=(μ10) , W=[00R⃗ T001⃗ TR⃗1⃗Σ]\mathbf{W} \begin{pmatrix} \nu\\\theta\\\mathbf{w} \end{pmatrix} = \begin{pmatrix} \mu\\1\\\mathbf{0}\end{pmatrix}\,,\, W = \begin{bmatrix} 0&0&\vec R^{\,T}\\ 0&0&\vec 1^{\,T}\\ \vec R&\vec 1&\Sigma \end{bmatrix} using the HHL algorithm.

​
Preprocessing for HHL

​

​
Construction of matrix WW

R = expected_return.values


Pi = np.ones(4)
S = cov.values

row1 = np.append(np.zeros(2), R).reshape(1, -1)
row2 = np.append(np.zeros(2), Pi).reshape(1, -1)
row3 = np.concatenate([R.reshape(-1, 1), Pi.reshape(-1, 1), S], axis=1)
W = np.concatenate([row1, row2, row3])

np.set_printoptions(linewidth=200)
print(W)
Output:
[[

0.         0.         0.33760624 0.43059648 1.19108918 0.71832606]
   [

0.         0.         

1.         1.         

1.         1.        ]
   [0.33760624 

1.         0.05020389 0.02116691 0.02981871 0.04674084]
   [0.43059648 

1.         0.02116691 0.07889178 0.06422647 0.05653149]
   [1.19108918 

1.         0.02981871 0.06422647 0.27498126 0.07109579]
   [0.71832606 

1.         0.04674084 0.05653149 0.07109579 0.40357553]]

​

​
Construction of Vector (μ10)\begin{pmatrix}\mu\\1\\\mathbf{0}\end{pmatrix}

mu = 0.1
xi = 1.0
b = np.append(np.array([mu, xi]), np.zeros_like(R)).reshape(-1, 1)
print(b)
Output:
[[0.1]
   [

1. ]
   [

0. ]
   [

0. ]
   [

0. ]
   [

0. ]]

​
Redefining the Matrix

Typically, the matrix formulation for HHL assumes a standard form with the following properties:
  1. The right-hand side vector b⃗\vec{b} is normalized.
  2. The matrix AA has a size of 2n×2n2^n \times 2^n.
  3. The matrix AA is Hermitian.
  4. The eigenvalues of matrix AA lie within the range (0,1)(0,1).
However, even general problems that do not meet these conditions can be solved using the following methods:

​

​
1) Normalized b

As preprocessing, normalize b⃗\vec{b} and then return the normalization factor as postprocessing. 
norm_factor = np.linalg.norm(b)
b_normalized = b / norm_factor

​

​
2) Make the Matrix AA of Size 2n×2n2^n\times 2^n

Complete the matrix dimension to the closest 2n2^n with an identity matrix. The vector b⃗\vec{b} is completed with zeros. (A00I)(b⃗0)=(x⃗0).\begin{pmatrix} A & 0 \\ 0 & I \end{pmatrix} \begin{pmatrix} \vec{b} \\ 0 \end{pmatrix} = \begin{pmatrix} \vec{x} \\ 0 \end{pmatrix}. However, our matrix is already the right size. 

## rquired number of qubits for encoding data
nbit = 3
N = 2**nbit

# padding
W = np.block([[W, np.zeros((6, 2))], [np.zeros((2, 6)), np.eye(2)]])

b = np.vstack([b, np.zeros((2, 1))])

print("W_padded:\n", W)
print("b_padded:\n", b)
Output:

W_padded:
   [[

0.         0.         0.33760624 0.43059648 1.19108918 0.71832606 

0.         0.        ]
   [

0.         0.         

1.         1.         

1.         1.         

0.         0.        ]
   [0.33760624 

1.         0.05020389 0.02116691 0.02981871 0.04674084 

0.         0.        ]
   [0.43059648 

1.         0.02116691 0.07889178 0.06422647 0.05653149 

0.         0.        ]
   [1.19108918 

1.         0.02981871 0.06422647 0.27498126 0.07109579 

0.         0.        ]
   [0.71832606 

1.         0.04674084 0.05653149 0.07109579 0.40357553 

0.         0.        ]
   [

0.         0.         

0.         0.         

0.         0.         

1.         0.        ]
   [

0.         0.         

0.         0.         

0.         0.         

0.         1.        ]]
  b_padded:
   [[0.1]
   [

1. ]
   [

0. ]
   [

0. ]
   [

0. ]
   [

0. ]
   [

0. ]
   [

0. ]]

​

​
3) Hermitian Matrix

Symmetrize the problem: (0ATA0)(x⃗0)=(0b⃗).\begin{pmatrix} 0 & A^T \\ A & 0 \end{pmatrix} \begin{pmatrix} \vec{x} \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ \vec{b} \end{pmatrix}. This increases the number of qubits by 
def to_hermitian(A):
    N = A.shape[0]
    A_hermitian = np.concatenate(
        [
            np.concatenate([np.zeros((N, N)), A.transpose().conj()], axis=1),
            np.concatenate([A, np.zeros((N, N))], axis=1),
        ]
    )
    return A_hermitian


N = W.shape[0]
b_new = np.concatenate([np.zeros((2 * N - len(b_normalized), 1)), b_normalized])
plt.matshow(b_new.transpose())
plt.title("Normalized and Padded Vector b")
plt.show()

W_hermitian = to_hermitian(W)
plt.matshow(W_hermitian)
plt.title("Hermitian Matrix W")
plt.show()
output output

​

​
4) Rescaled Matrix

If the eigenvalues of matrix AA lie within the interval [wmin⁡,wmax⁡][w_{\min}, w_{\max}], we can handle this by transforming the matrix and then reverting the result. The transformation is defined as follows: A~=(A−wmin⁡I)(1−12m)1wmax⁡−wmin⁡.\tilde{A} = (A - w_{\min} I)\left(1 - \frac{1}{2^{m}}\right)\frac{1}{w_{\max} - w_{\min}}. In this case, the eigenvalues of A~\tilde{A} fall within the interval [0,1)[0,1). The relationship with the eigenvalues of the original matrix is given by: λ=(wmax⁡−wmin⁡)λ~[11−12m]+wmin⁡,\lambda = (w_{\max} - w_{\min})\tilde{\lambda}\left[\frac{1}{1 - \frac{1}{2^{m}}}\right] + w_{\min}, Where λ~\tilde{\lambda} is the eigenvalue of A~\tilde{A} obtained by the QPE (Quantum Phase Estimation) algorithm. This correspondence between eigenvalues is used in the formula for eigenvalue inversion within the AmplitudeLoading function. 
def condition_number(A):
    w, _ = np.linalg.eig(A)
    return max(np.abs(w)) / min(np.abs(w))


QPE_RESOLUTION_SIZE = 6

assert QPE_RESOLUTION_SIZE > np.log2(
    condition_number(W)
), "condition number is too big, and QPE resolution cannot hold all eigenvalues"

w, v = np.linalg.eig(W_hermitian)
w_max = np.max(w)
w_min = np.min(w)
mat_shift = -w_min
# assures eigenvalues in [0,1-1/2^QPE_SIZE]
mat_rescaling = (1 - 1 / 2**QPE_RESOLUTION_SIZE) / (w_max - w_min)
# this is the minimal eigenvalue which can be resolved by the QPE
min_possible_w = (w_max - w_min) / 2**QPE_RESOLUTION_SIZE

W_rescaled = (
    W_hermitian + mat_shift * np.identity(W_hermitian.shape[0])
) * mat_rescaling
W_rescaled = W_rescaled.real

# verifying that the matrix is symmetric and has eigenvalues in [0,1)
if not np.allclose(W_rescaled, W_rescaled.T, rtol=1e-6, atol=1e-6):
    raise Exception("The matrix is not symmetric")

w_rescaled, _ = np.linalg.eig(W_rescaled)

for lam in w_rescaled:
    if lam < -1e-6 or lam >= 1:
        raise Exception("Eigenvalues are not in (0,1)")

plt.matshow(W_rescaled)
plt.title("Rescaled Matrix W")
plt.show()
output

​
Defining HHL Algorithm for the Quantum Solution

This section is based on Classiq HHL in the user guide, here and here. Note the rescaling in simple_eig_inv based on the matrix rescaling. 
from classiq import *


@qfunc
def simple_eig_inv(
    gamma: float,
    delta: float,
    c_param: float,
    phase: QNum,
    indicator: Output[QBit],
):
    allocate(indicator)
    assign_amplitude_table(
        lookup_table(lambda p: np.clip(c_param / ((gamma * p) + delta), -1, 1), phase),
        phase,
        indicator,
    )


import numpy as np
import scipy

exponentiation_W_rescaled = scipy.linalg.expm(1j * 2 * np.pi * W_rescaled).tolist()
b_list = np.concatenate(b_new).tolist()


@qfunc
def main(
    indicator: Output[QBit],
    res: Output[QNum],
    rescaled_eig: Output[QNum],
) -> None:
    allocate(QPE_RESOLUTION_SIZE, False, QPE_RESOLUTION_SIZE, rescaled_eig)
    prepare_amplitudes(b_list, 0, res)
    within_apply(
        lambda: qpe(
            unitary=lambda: unitary(exponentiation_W_rescaled, res),
            phase=rescaled_eig,
        ),
        lambda: simple_eig_inv(
            gamma=mat_rescaling ** (-1),
            delta=-mat_shift,
            c_param=min_possible_w,
            phase=rescaled_eig,
            indicator=indicator,
        ),
    )


MAX_WIDTH_BASIC = 18
constraints = Constraints(max_width=MAX_WIDTH_BASIC)
# preferences = Preferences(
#     optimization_level=0, optimization_timeout_seconds=90, transpilation_option="none"
# )
backend_preferences = ClassiqBackendPreferences(
    backend_name=ClassiqSimulatorBackendNames.SIMULATOR_STATEVECTOR
)
execution_preferences = ExecutionPreferences(
    num_shots=1, backend_preferences=backend_preferences
)

qmod_hhl_basic = create_model(
    main,
    constraints=constraints,
    # preferences=preferences,
    execution_preferences=execution_preferences,
)

qprog = synthesize(qmod_hhl_basic)
show(qprog)
Output:

Quantum program link: https://platform.classiq.io/circuit/38EbCHhb4OwIspsNgY9Dv5T54JR
  


execution_job_id = execute(qprog)
result = execution_job_id.result_value()


filtered_hhl_statevector = dict()
for sample in result.parsed_state_vector:
    if sample.state["indicator"] == 1 and sample.state["rescaled_eig"] == 0:
        filtered_hhl_statevector[sample.state["res"]] = sample.amplitude

states = sorted(filtered_hhl_statevector)
raw_qsol = np.array([filtered_hhl_statevector[s] for s in states])


qsol_hermitian = raw_qsol / (min_possible_w)
print(qsol_hermitian)
Output:
[-2.89639576e-01+5.95742356e-01j  3.08335503e-01-6.34196894e-01j -2.27110910e+00+4.67130877e+00j  2.75233940e+00-5.66112264e+00j -1.44012178e-01+2.96210054e-01j -3.36318572e-01+6.91753598e-01j
    2.39148259e-14-7.26657302e-14j -7.64513494e-15-3.53183012e-14j  3.90607598e-14-2.45511623e-13j  6.08553768e-15+1.83911091e-13j -6.16524881e-14+1.44844906e-13j  5.18031970e-14-1.22156005e-13j
    3.10460540e-14+8.93174264e-14j -2.39948943e-14-1.53061828e-13j  2.33085233e-14+1.36384476e-14j -2.52635402e-14+1.00776109e-14j]
  


x_classical = np.array([s[0] for s in np.linalg.solve(W_hermitian, b_new)])
x_classical = x_classical[:6].reshape(-1)


x_hhl = qsol_hermitian[:6].reshape(-1)


alpha = (x_hhl @ x_classical) / (x_hhl @ x_hhl)
x_hhl_rescaled = alpha * x_hhl

print(
    "rel error:",
    np.linalg.norm(x_hhl_rescaled - x_classical) / np.linalg.norm(x_classical),
)
Output:
rel error: 0.08021847118398923
  


print("x-HHL", x_hhl_rescaled.real)
print("x-classical", x_classical.real)
Output:
x-HHL [ -1.27904651   1.36160761 -10.02920324  12.15431315  -0.63595686  -1.48518066]
  x-classical [ -0.67035649   0.65436901 -10.36202934  12.09472133  -0.98362358  -0.74906842]
  


x_exact = np.linalg.lstsq(W_hermitian, b_new, rcond=0)[0]
x_exact = x_exact[:6].reshape(-1)


import pandas as pd

w_opt_HHL = x_hhl_rescaled.real[2:6]
w_opt_exact = x_classical[2:6]  # x_exact[2:6]
w_opt = pd.DataFrame(
    np.vstack([w_opt_exact, w_opt_HHL]).T,
    index=["GOOG", "AAPL", "FB", "AMZN"],
    columns=["exact", "HHL"],
)
w_opt.plot.bar()
Output:
<Axes: >
output The next step is to calculate the “total portfolio value for each day.” This operation involves multiplying the price of each asset by its weight and summing them up. Expressed as a formula, it looks like this: Portfolio Value=wAAPL⋅PAAPL+wAMZN⋅PAMZN+wNVDA⋅PNVDA+wTSLA⋅PTSLA\text{Portfolio Value} = w_{\text{AAPL}} \cdot P_{\text{AAPL}} + w_{\text{AMZN}} \cdot P_{\text{AMZN}} + w_{\text{NVDA}} \cdot P_{\text{NVDA}} + w_{\text{TSLA}} \cdot P_{\text{TSLA}} For example, if we assume: w=[0.4,0.3,0.2,0.1]w = [0.4, 0.3, 0.2, 0.1] Then: Portfolio Value=0.4A+0.3B+0.2C+0.1D\text{Portfolio Value} = 0.4A + 0.3B + 0.2C + 0.1D This gives you the general idea. To perform this calculation automatically for all dates, you can write it in a single line of Python code as follows: 
pf_value = df.values.dot(w_opt)
pf_value = pd.DataFrame(df.values.dot(w_opt), index=df.index, columns=["exact", "HHL"])
pf_value.head()
exactHHL
Date
---------
2024-01-02-325.764298-421.744540
2024-01-03-320.943657-410.339168
2024-01-04-344.106605-433.974406
2024-01-05-329.292530-418.664555
2024-01-08-332.896116-421.666555
pf_value["exact"] = pf_value["exact"] / pf_value["exact"].iloc[0]
pf_value["HHL"] = pf_value["HHL"] / pf_value["HHL"].iloc[0]

print(pf_value.tail())
Output:
exact       HHL
  Date                          
  2024-12-23  1.139395  1.284214
  2024-12-24  1.157931  1.350494
  2024-12-26  1.238676  1.398530
  2024-12-27  1.191926  1.328773
  2024-12-30  1.144376  1.269782
  


pf_value.plot(figsize=(9, 6))
Output:
<Axes: xlabel='Date'>
output

​
Reference

  • [1] P.
Rebentrost and S. Lloyd, “Quantum computational finance: quantum algorithm for portfolio optimization“, https://arxiv.org/abs/1811.03975
  • [2] 7-
  1. Portfolio optimization using HHL algorithm: https://dojo.qulacs.org/en/latest/notebooks/7.3_application_of_HHL_algorithm.html