Quantum algorithms like VQE and QAOA were used to analyze the impact of COVID-19 on optimal portfolio selection across different industries. Three time periods were considered - pre-COVID, during COVID, and post-COVID. Results found that COVID disrupted optimal portfolios, with sectors like retail, technology and automotive favored more pre-COVID, while oil/gas and airlines/hospitality favored post-COVID. Quantum algorithms provided comparable results to classical methods like Markowitz for portfolio optimization under changing market conditions from the pandemic.
2. • The quantum world exhibit unique characteristics of superposition of states,
interference and entanglement which are not to be observed in the classical
sense
• Superposition of states – if and are two states of quantum system then
+ s also an allowed state with = 1
• Quantum interference is the result of addition or subtraction of the amplitudes
arising from the wave nature of the particles
• Quantum entanglement is the non locality experienced in measuring or observing
the particles
• These are counter intuitive and strange to the humans of today
• Richard Feynman said in 1985, if we could compute using atoms we would
compute as nature computes
Unique Nature of Quantum Mechanics
3. • Moore’s law projects that the number of transistors in the integrated circuits should double
itself every two years
• Also by Morse law , the size of the transistors will be reaching that of molecules and atoms in
the near future
• And when it does, quantum effects such as tunneling, coherence will take over and
detrimentally affect the efficiency of computing
• Therefore, In order to scale, it is compelling to look for different ways of computation
Source : Prof. Christopher Monroe , Univ. of Maryland
(https://www.youtube.com/watch?v=Y3mcgq3_yEY)
Need for Quantum Computing
4. • Bit is the fundamental unit of classical computing and it can take either of the
two states, a 1 or 0, an On or Of, True or False
• Quantum bit or Qubit is the fundamental unit of quantum information
processing
• We can store bits of information in quantum computing. For eg. has 8
different probabilities or amplitudes which means 8 bits of information can
be stored for a three input process
+ + + + + +
+ , leads to quantum parallelism
• The QC derives its supremacy over CQ based on this exponential nature of
processing/computing
• When no. of Qubits N = 300 , we have more information to
store/process/compute with, than there are particles in the universe
Quantum Computing
5. • number of states is the result of superposition of states
• Allowing the interference to happen between the probability waves (by means
of manipulating the input states through quantum gates)of qubits/states and
obtaining the results as few tens to few hundreds/thousands (sparse and still
being dependent on any number inputs) and repeating the experiment (shots)
to get statistical estimate /distribution harnesses the concept of quantum
interference for computation
• Quantum entanglement is the correlation between two different qubits, which
means the two members of a pair exist in a single quantum state
• Observing the state of one of the qubits instantaneously changes the state of
the other one in a predictable manner, even at a long distance
• Realization: Superconducting circuit, trapped ions, silicon quantum dot and
diamond vacancies
Quantum Mechanical Resources for Computing
Output
Input
6. Quantum mechanical states are extremely fragile and require near absolute isolation from
the environment.
Creating such conditions require temperatures near absolute zero and shielding from
radiation.
Challenge only increases with increasing the size (number of qubits and the length of time
they must be coherent).
Thus, building quantum computers is expensive and difficult.
Requires contributions from many different fields, such as the design of quantum
algorithms and error correcting codes, the architecture design of the computer itself, and
the development of more reliable quantum devices.
Development of quantum versions of devices, architectures, languages, compilers, and
layers of abstraction.
Challenges and Opportunities
7. Quantum Algorithms – General Principle
Equally weighted
Superposition of
states
Single pulse on single qubit affects works on the
massive superposition of states
Pulse leads to interference which in turn
changes the prob. Amplitude
Coupled qubit gates (CNOT) a pair of qubits.
Pulse leads to interference which in turn
changes the prob. amplitude
Prob. amp. constructively interfere
on one state only and thus the result
8. Computational Complexity
Complexity of a problem gives information about how long it takes to solve a problem.
T(n) denotes time or the number of steps required to solve a problem with n being the no.
length of no. of digits of input, there broadly exists two classes of complexity namely, P and
NP. Complexity is basically knowing how as n grows.
If the time scales as , where k and p are positive numbers, then problem can be
solved in polynomial time.
If the time scales as , where k and c are positive numbers, such that for every
value of n the problem is considered to be solvable in exponential time. Note: n is an
exponent here.
In classical computing, problems are tractable if it grows in polynomial time while intractable
if it grows exponentially and these problems are called easy and hard respectively.
If a problem can be solved in polynomial time it is considered to belonging to a class known
as P while that of exponential time it is called NP (non-deterministic polynomial).
Generally, it is believed that P is not equal to NP and that there are problems in NP but not in
P, which can be solved by quantum computers in polynomial time.
9. Motivation for Quantum Algorithms
Quantum algorithms are better than classical computers at specific tasks.
Identify what kind of problems can be more efficiently solved by QC.
Speedups: How QC performs in terms of scaling with the size of the problem. Gives an idea of how big
the speedups will be as quantum hardware improves .
Also, this metric is hardware agnostic and scales similarly across different hardware platforms such as
Ion trap or superconducting or photonic etc.
Exponential speedups can offer practical speedups even at smaller problem sizes & with small
quantum computers (NISQ ??)
Polynomial speedups may require medium to large scale QC.
Both polynomial and exponential are useful.
10. Quantum Algorithms
Grover’s algorithm can provide quadratic speedup
Square of 1 million ( 1,000,000) = 1000
SQRT(Classical Algo) = Quantum Algo
Although this is polynomial, huge reduction in time.
Provable and Heuristic Quantum Algorithms
Provable: Mathematically, can be shown to outperform its classical counterparts. Eg. Shor’s
factorization , Grover’s search. All provable QAs medium to large scale QCs or fault tolerant
QCs. Challenge is to build the QC.
Most of the originally proposed quantum algorithms require millions of physical qubits to
incorporate these QEC techniques successfully.
Heuristic: No mathematical proof that quantum can outperform, driven by intuitions. Do
not know in advance how it can perform but can be run on NISQ. Need to test with real life
data.
11. NISQ
In 2017, John Preskill coined the term Noisy Intermediate Scale Quantum Computing (NISQ) to
denote the present era of quantum computing.
Intermediate scale refers to the no. of qubits available which ranges from 50 to a few hundreds of
qubits
Noisy refers to not so robust qubit meaning the present generation qubits are more highly prone to
decoherence.
NISQ era of computing, an efficient program is required to mitigate the error to extract reliable
results.
Quantum algorithms such as Shor’s prime factorization, Deutech-Jozsca algorithms operate under the
assumption that the qubits are robust and therefore do not incorporate any error mitigation
techniques.
Fault tolerance is the property that enables a system to continue operating properly in the event of
the failure of one or more faults within some of its components.
FTQC refers to the framework of ideas that allow qubits to be protected from quantum errors
introduced by poor control or environmental interactions (Quantum Error Correction, QEC) and the
appropriate design of quantum circuits to implement both QEC and encoded logic operations in a way
to avoid these errors cascading through quantum circuits.
12. NISQ
Algorithms and tools have been developed specifically for near-term quantum computers
Variational Quantum Algorithms (VQAs): Hybrid quantum-classical approach which has potential noise
reduction. In NISQ , all known quantum algorihtms are heuristic in nature.
Quantum Error Mitigation (QEM): Techniques to reduce the computational errors and then evaluate
accurate results from noisy quantum circuits
Quantum Circuit Compilation (QCC): To transform the nonconforming quantum circuit to an executable
circuit on the target quantum platform according to its constraints
Benchmarking Protocols: To evaluate the basic performance of a quantum computer and even the
capacity to solve realworld problems.
Classical Simulation: Classical simulation of quantum circuits is one of the core tools for designing
quantum algorithms and validating quantum devices
VQAs, QEMs, QCC, and quantum benchmarking
may all require the help of classical simulation
for verification or algorithm design.
Main goal of the NISQ era is to extract the maximum quantum
computational power from current devices while developing
techniques that may also be suited for the
long-term goal of the FTQC
14. • In 2017, John Preskill coined the term Noisy Intermediate Scale Quantum
Computing (NISQ) to denote the present era of quantum computing.
• Noisy – not so robust qubits.
• Intermediate – 50s to few hundreds of qubits.
• Peruzzo, A., McClean, J., Shadbolt, P. et al. A variational eigenvalue solver on a
photonic quantum processor. Nat Commun 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213
• Fedorov, D.A., Peng, B., Govind, N. et al. VQE method: a short survey and recent
developments. Mater Theory 6, 2 (2022). https://doi.org/10.1186/s41313-021-
00032-6
Variational Quantum Eigensolver (VQE)
15. • Prepare a variational quantum circuit representing the chemical problem –Qn.
Comp
• Measure the circuit, calculate the expectation value - Qn. Comp
• Update the variational parameters by optimizing algorithm – Classical Computer
• Measure the circuit again - Qn. Comp
• If present value better than previous one, stop the process
Variational Quantum Eigensolver (VQE)
16. Quantum Computing for Finance
• Finance sector encounters several computationally challenging problems such as asset
portfolio optimization, stock market prediction, arbitrage opportunities, fraud detection,
credit scoring etc.
• In a world where hug volume of data generated per second, QC promises potential
reduction in time and memory space for the computational tasks.
• Broadly, there are three classes of problems in finance:
• Optimization: Problems that scale exponentially in time required can be best solved
using quantum optimization. Eg. portfolio optimization, arbitrage opportunity,
optimal feature selection for credit scoring.
• Machine Learning: Highly Complex data structures hinder classification or pre-
diction accuracy. The multidimensional data modeling capacity of quantum
computers may allow us to find better patterns, with increasing accuracy.
E.g. Anomaly detection, Quantum NLP for virtual agents, Risk Assessment
• Simulation: Time constraints to perform sufficient scenario tests to find the best
possible solution. Efficient sampling methods leveraging quantum computers may
require less samples to reach a more accurate solution faster.
E.g. Pricing of financial derivatives, risk analysis.
17. Algorithms can improve computational efficiency, accuracy, and addressability for
defined use case
Financial services focus areas and algorithms
Ref: Quantum Computing for Finance: State-of-the-Art and Future Prospects
Quantum Algorithms for Finance
18. Fully scaled quantum
technology is still a way off,
but some banks are already
thinking ahead to the
potential value.
Major MoUs
20. Step 1: Encode the classical data into a quantum state
Step 2: Apply a parameterized model
Step 3: Measure the circuit to extract labels
Step 4: Use optimization techniques (like gradient descent) to
update model parameters
QML Steps
31. Unraveling the Effect of COVID-19 on the Selection of Optimal Portfolio Using Hybrid
Quantum Algorithms
1Shrey Upadhyay, 2Vaidehi Dhande, 1Rupayan Bhattacharjee, 1Ishan NH Mankodi, 1Aaryav Mishra, 2Anindita Banerjee, 1Raghavendra Venkatraman
1QKrishi, 2C-DAC- India
The unforeseen COVID-19 pandemic delivered a huge blow to the global economy. This
poster elaborates the effect of COVID-19 on the portfolio optimization across different
industrial sectors retail, technology, automotive, oil & gas, airlines & hospitality.
Portfolio Optimization is to select best portfolios with an objective to maximize the return
value and minimize the risk factor. To understand the trend in Portfolio Optimization pre
covid-19 and during covid-19 three time intervals are considered and the results from
different quantum algorithms are compared with classical results. The quantum algorithms
used are Variational Quantum Eigen solver (VQE), Quantum Approximate Optimization
Algorithm (QAOA).
Outline
Covariance Graphs
Results
Conclusions
Abstract
1. Portfolio Optimization- Maximize Returns and Minimize Risk
2. Classical Algorithms- Markowitz, Numpy EigenSolver
3. Quantum Computing-VQE, QAOA
4. Impact of Covid-19 on portfolio optimization
Pool Non-COVID 1
(Jan ‘16-Dec ‘17)
Non-COVID 2
(Jan ‘18-Dec ‘19)
COVID
(Jan ‘20-Dec ‘21)
Retail
Technolog
y
Automoti
ve
Oil & Gas
Airlines &
Hospitalit
y
Pool
Non-COVID 1
(Jan ‘16-Dec ‘17)
Non-COVID 2
(Jan ‘18-Dec ‘19)
COVID
(Jan ‘20-Dec ‘21)
Retail
Technology
Automotive
Oil & Gas
Airlines &
Hospitality
Impact of Covid
Pool
Non-
COVID1
Non-
COVID2
COVID Reason
Retail
(Costco,
Amazon, Target,
Walmart)
COST TGT COST
COST & TGT are major
market share holders and as
they open new stores to at
more locations and while
offering the products at
affordable prices, drives the
growth of COST.
Technology
(Google, IBM,
Intel, Microsoft)
GOOG GOOG MSFT
GOOG remains the most
used IT service in the world
in terms of apps and
browsers. MSFT also control
majority of the OS used
worldwide, while launching
its own hardware products.
Automotive
(General
Motors,
Mercedes,
Tesla, Ford )
GM TSLA TSLA
GM owned a large market
cap in automotive around
2016, but as people accept
EV as a better alternative to
gas powered engines, and
look for greener ways of
transport which is also more
technology wise advanced,
TSLA soars after 2017.
Oil & Gas
(Shell, Conoco
Phillips,
Marathon Oil,
Chevron Corp.)
CVX COP CVX
CVX & COP control majority
of gas and oil extraction in us
and also in some parts of the
world. As they continue to
innovate and expand in the
hydrocarbon fuel markets.
Airlines &
Hospitality
(Marriott Int,
Choice Hotels,
LTC Properties,
Alaska Air)
MAR CHH MAR
MAR and CHH remains
people’s first choice. As they
continue to grown and make
newer and more luxurious
properties. The in them
considerably increases with
time
Main objective of portfolio optimization is:
1. The investor’s goal is to maximize return for low level of risk
2. Risk can be reduced by diversifying a portfolio through individual, unrelated securities
Initially, the problem of portfolio optimization is translated into the form of variation
circuit called ansatz to enable the quantum computer to perform optimization on the
objective function.
VQE is Hybrid Quantum-classical algorithm. VQE which is developed on Variational
Principle calculates the lowest energy which corresponds to the optimal portfolio
It aims to find an upper bound of the lowest eigenvalue of a given Hamiltonian.
Methods
VQE has two fundamental steps:
1. Prepare the quantum state |Ψ(θ)⟩
2. Measure the expectation value ⟨Ψ(θ)|H|Ψ(θ)⟩
3. Optimize the parameter θ on classical computer and generate the updated wavefunction
4. Calculate the expectation value again for the updated wavefunction
5. Iterate until convergence criteria is met
QAOA is widely popular method for solving combinatorial optimization problems. The VQE algorithm applies
classical optimization to minimize the energy expectation of an ansatz state to find the ground state energy.
Methods Cont..
[0 1 0 0], -
0.0012
[0 1 0 0],-
0.0012
[0 1 0 0], -
0.0012
[0
1 0
0]
[0 0 1 0], -
0.0014
[1 0 0 0], -
0.0014
[1 0 0
0]
[0 0 1 0], -
0.0014
[1 0 0 0], -
0.0014
[1 0 0 0]
[1 0 0 0], -
0.0014
[1 0 0 0],-
0.0014
[0 0 0 1], -0.001 [0 0 0 1]
[0 0 0 1], -0.001 [0 0 0 1] , -
0.001
[0 0 0 1},-
0.0013
[0 0 0 1]
[0 0 0 1] , -
0.0013
[0 0 0 1] , -
0.0013
[0 0 0 1] , -
0.0015
[0 0 0 1] , -
0.0015
[0 0 0 1]
[0 0 0 1] , -
0.0015
[0 0 1 0] , -
0.007
[1 0 0 0]
[1 0 0 0] , -
0.006
[0 0 1 0] , -
0.007
[0 0 1 0] , -
0.005
[0 0 1 0]
[1 0 0 0] , 0.001 [0 0 1 0] , -
0.005
[0 0 1 0] , -
0.005
[0 0 1 0]
[0 0 0 1], , -
0.0016
[0 0 1 0], , -
0.005
[1 0 0 0] , -
0.001
[1 0 0 0]
[1 0 0 0] , -
0.001
[1 0 0 0] , -
0.001
[0 1 0 0] , -
0.0004
[0 1 0 0]
[0 1 0 0] , -
0.0004
[0 1 0 0] , -
0.0004
[0 0 1 0] , -
0.0005
[0 0 1 0]
[0 0 1 0] , -
0.0005
[0 0 1 0] , -
0.0005
[1 0 0 0] , -
0.0015
[1 0 0 0] [0 1 0 0] , -
0.0006
[1 0 0 0] , -
0.0015
[1 0 0 0] , -
0.0015
[0 1 0 0]
[0 1 0 0] , -
0.0006
[0 1 0 0] , -
0.0006
[0 1 0 0] , -
0.0008
[0 1 0 0]
[0 1 0 0] , -
0.0008
[0 1 0 0] , -
0.0008
References
• Egger, D.J., Gambella, C., Marecek, J., McFaddin, S., Mevissen, M.,
Raymond, R., Simonetto, A., Woerner, S. and Yndurain, E. (2020).
Quantum Computing for Finance: State-of-the-Art and Future
Prospects. IEEE Transactions on Quantum Engineering, 1, pp.1–24.
doi:10.1109/tqe.2020.3030314.
• Herman, D., Googin, C., Liu, X., Galda, A., Safro, I., Sun, Y., Pistoia,
M., Alexeev, Y. and Chase Bank, J. (2022). A Survey of Quantum
Computing for Finance. arxiv:2201.02773
Classical
VQE
Classical
VQE
QAOA
Classical
VQE
QAOA QAOA
Classical
VQE
Classical
VQE
QAOA
Classical
VQE
QAOA QAOA
32. Portfolio Optimization results using quantum algorithms(Work
done by Qkrishi Scientists)
Quantum based Portfolio Optimization
33. Qkrishi Projects
Forex optimization
Post quantum cryptography
Product recommendation
Electricity theft using QML
Protein folding and drug discovery
Computational chemistry and material science
36. • Prabha Narayanan – Founder Qkrishi
• Prof. Monika Agarwal - Founder Qkrishi
• Qkrishi Colleagues: Chetan, Sree, Sangram
• JR: Ragavan
• Other experts from the field
• We are also open to joint proposal/collaboration,
skilling, internship!!!
Acknowledgement
