1. What is Quantum Computing?
“A classical computation is like a solo voice – one line of pure tones succeeding each other. A quantum computation is like a symphony – many lines of tones interfering with one another”. 
– Seth Lloyd, professor of mechanical engineering and physics at the Massachusetts Institute of Technology (MIT)
1.0 An introduction
With technology giants Google, Amazon, and IBM racing to find the next big breakthrough in quantum computing, it appears that we will soon be in the quantum age. From the very first quantum computing speculations made by American physicist Richard Feynman in 1959 , to the current prototypes, the potential power of quantum computers has fascinated society for decades.
Indeed, while classical computers have proven sufficient in solving most problems, quantum computers are designed to decode more complex issues, such that what might take classical computers a week to solve will only take quantum computers one second . This makes quantum computers highly desirable for companies that deal with big data and need models that can factor in more real-world variables.
A simplified example of a real-world problem is finding an optimal seating plan for 10 people at a dinner party. This problem alone generates over 3.5 million possible combinations, and if the number of dinner guests were increased, classical computers would struggle to find all possible solutions due to their limited working memory and inability to analyse multiple combinations at once . As we will see, the inherent difference between classical and quantum computers lies in the basic units of data that they can process.
1.1 Bits and Qubits
Classical computing, also known as binary computing, processes data through bits, which are streams of electrical pulses that can only exist in one of two states (0 or 1). Every element of classical computing is coded in long strings of 0s and 1s, where information is analysed sequentially . On the other hand, quantum computers use qubits that can represent both states simultaneously, in what is known as a quantum superposition. For example, a qubit can exist as a 0, 1 or both 0 and 1 at once. Now, imagine a coin that can land either heads or tails when flipped. This represents the properties of bits in a classical computer. However, if you spin the coin on a table, it now represents a qubit in superposition. Superposition is fundamental to quantum mechanics  and enables quantum machines to crunch through many calculations at once . Some may recall Schrödinger’s cat, a famous thought experiment which illustrates the paradox of quantum superposition.
Schrödinger’s cat involves a cat and a box. Inside this box is a hammer, the cat, a vial of poison, a Geiger counter, and radioactive material. The box is rigged up so that if the material decays (a completely random event), the Geiger counter detects it, causes the hammer to swing and break the poison, and the cat dies. If the box containing the cat is shut, there are no observers. Therefore, until someone opens the box, nobody knows for sure whether the cat is alive or dead. The thought experiment proposes that, until the box is opened, the cat is both alive and dead at the same time.
To further demonstrate the differences between bits and qubits, consider a sphere with a pole at opposite ends. A bit can only exist at one of the poles, while a qubit can be at any point on the sphere. This allows quantum computers to store more data and store it more efficiently compared to classical computers . In theory, an N-qubit quantum system is able exist in a superposition of 2^N different states at any given time.
1.2 Quantum entanglement
It is simply impossible to discuss quantum computers without acknowledging quantum entanglement and the profound impact it has on this technology. The qubits within quantum computers exist in entanglement, where they are intertwined and interdependent in nature, such that changing the state of one qubit will simultaneously change another. A simple way to consider this is imagining that you are studying with a large group of students for an economics exam. Clearly, a quiet area is where you will all be the most productive. In a noisy environment, one of your peers could be distracted, and in turn negatively impact others. Likewise, qubits cannot complete their tasks in ‘noisy’ environments as the slightest disturbances like vibration, radiation or change in temperature can cause one qubit, then all the qubits to fall out of superposition before their job has been completed.
While quantum entanglement is a barrier to unleashing the full potential of quantum computers, it is also what enables the processing power of these devices to increase exponentially with each extra qubit. Conversely, classical computing’s processing power increases at a linear 1:1 rate to bits .
1.3 Quantum supremacy
With 100 qubits able to store 1,267,650,600,228,229,401,496,703,205,375 different numbers – trillions of times the storage capacity of all computers ever made  and 300 qubits able to store more possible configurations than atoms in the universe , there is no denying the disruptive potential of quantum computers. The most prominent recent breakthrough in quantum computing occurred in October 2019, when Google announced that it had achieved quantum supremacy . For this to be achieved, a quantum computer must be able to solve a problem no classical computer can in a feasible amount of time. Google reported that their quantum computer, Sycamore, was able to sample the output of a pseudo-random quantum circuit in 200 seconds , whilst the world’s fastest supercomputer, Summit, would have taken 10,000 years to complete it . To further illustrate the implications of this breakthrough, Sycamore had 53 qubits when it achieved this feat. If its number of qubits increased to 60, thirty-three classical Summit supercomputers would be needed to achieve the same processing power. Likewise, at 70 qubits, the Summit supercomputer would have to extend to the size of a city to keep up . While critics such as IBM dispute Google’s claim it would take 10,000 years to solve this problem with a classical computer , it is nonetheless an important step towards the future of quantum computing, and in the words of IBM, “the best is yet to come”.
Of course, while this is an important milestone in quantum computing, the significant issue of quantum error remains at large. The difference between a random noise generator and the world’s most powerful computer will ultimately lie in the ability of developers to mitigate the fragile, and error prone state of the entangled qubits in quantum computers .
1.4 Types of quantum computers
Currently, there are three quantum computing designs, based on their qubit count. From weakest to most powerful, the three types are :
1.5 Types of qubits
When it comes to the way qubits themselves are constructed, quantum scientists have explored several approaches. Some include:
“Mixed reality, artificial intelligence and quantum computing are the three path-breaking technologies that will shape the world in the coming years”. 
– Satya Nadella, Chief Executive Officer of Microsoft
2.1 Quantum computing and dynamic portfolio optimization
For a given risk, there is one portfolio of assets that maximizes the return. Conversely, for a given return, there is one portfolio that minimizes the risk . One of the most important questions in finance is how to construct such a portfolio, and how to modify it to account for the conditions of the market.
In optimization problems, you search for the best of many possible combinations. One example of this is the famous ‘Travelling Salesperson Problem’ . In this problem, given a list of cities and the distances between each city, you must find the shortest possible route that visits each city exactly once. In physics, optimization problems can be reframed as energy minimization problems. A fundamental rule of physics is that everything tends to seek a minimum energy state. Objects slide down hills and hot things cool down over time . This behaviour also applies to quantum physics. Quantum computers can exploit this through a process known as quantum annealing. Here, a quantum computer uses algorithms to find the low-energy states of a problem and therefore determine the optimal or near-optimal combination of variables.
One small-scale study  found that a D-Wave  quantum annealer (less powerful quantum computer) was able to achieve high success rates when solving a difficult multi-period portfolio optimization problem. By fine tuning the quantum annealing process, researchers were able to achieve both portfolio optimization results and a solution time comparable to those achieved by classical computing technologies.
In their tests, the researchers imagined an asset manager wishing to invest K dollars in a set of N assets with an investment horizon divided into T time steps. Considering a forecast of future returns and the risk of each asset at each time step, the asset manager must decide how much to invest in each asset at each time step, while taking into account transaction costs, including permanent and temporary market impact costs .
The right side of the tables below displays the success rate of D-Wave’s devices when solving this optimization problem with various inputs. S(a%) represents the accuracy of the quantum computer for different margins of error. The success rate was calculated by comparing the quantum annealer’s result to that achieved by classical technologies like an exhaustive integer solver, or for larger problems a heuristic solver run many times .
Table IV illustrates the first attempts at a set of portfolio optimization problems using D-Wave’s quantum annealer.
Table VI records the results of a more powerful, finely tuned D-Wave quantum annealer. Comparing portfolio optimization problems with equivalent inputs of N, T, and K (as highlighted above), the fine-tuned device produced solutions with higher success rates. Ironically, configuring quantum computers appears to be an optimization problem in itself!
It must also be mentioned that the D-Wave’s quantum annealer is not a universal quantum computer. It is not digital, nor error-correcting, nor fault tolerant . Therefore, this device is not a good representation of all quantum computers, and instead represents one practical application of quantum computing.
2.2 Monte Carlo simulations and the Black Scholes Merton options pricing model
One of the most famous models in finance is the Black Scholes Merton (BSM) model for pricing options. This formula is particularly useful for brokers need to assign a fair price to the derivatives from the state of the market. The BSM model relies on a small number of input parameters such as the current stock price of the underlying asset, exercise price of the option, risk-free rate of return, expected dividends, time to expiration of the option and expected volatility of the underlying asset .
Another widely used instrument in finance is the Monte Carlo simulations. Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results . This model is a popular method for traders calculating risk positions but is also used widely in engineering, physical sciences and even law. Monte Carlo simulations play a major role in managing the risk that financial institutions are exposed to. Especially after the financial crisis of 2008-9, sophisticated risk management in banks is increasingly important and required by government regulators . As quantum technology is best applied to algorithms with a small number of inputs and large number of outputs , both the BSM and Monte Carlo are prime candidates to be computationally sped up by quantum computers.
Recently, researchers from JP Morgan, IBM and ETH Zurich demonstrated how a quantum computer could compute both the Black Scholes Merton model and Monte Carlo simulations . The team ran the BSM model using amounts between three and nine qubits, with the accuracy of their estimated option price increasing as qubit count rose. They also computed a Monte Carlo simulation using seven qubits. While these results are promising, the researchers cited that until fault tolerant quantum computer are widely available, it will be difficult to achieve consistent results.
By unleashing the potential of quantum superpositions, researchers hope quantum computers will compute certain financial models exponentially faster than classical computers with similar, if not improved accuracy. In the future, financial institutions with well calibrated quantum computers may realise huge advantages over their competitors. Not only can quantum computers identify anomalies in the market more quickly, but they can also reveal opportunities that previously would have been overlooked. When asked about quantum computing, Paul Buchard, a senior researcher at Goldman Sachs suggested there is a possibility this becomes a ‘critical technology’ . Meanwhile, a team at JPMorgan Chase recently conducted experiments using Honeywell’s quantum computer , part of their strategy to build a ‘Quantum Culture’ within the firm . Ning Shen, managing director at JPMorgan Chase, said some experts believe quantum computing could speed up financial computations by more than a thousand times , however it may take time before these devices live up to their expectations .
2.3 Quantum computing and medicine
Since the age of modern computing, models and algorithms have been routinely used to inform and extract information from biological experiments. Drug discovery is the process of developing a drug from an initial hypothesis to a fully commercialized product. This process can often take more than a decade and billions of dollars in expenditure before a molecule can be recognized as a drug . In the present, the most powerful classical supercomputers can only simulate relatively simple biological models and are still very limited in their capabilities to model more complex molecules .
Even modelling the molecular structure of an everyday drug like penicillin is an insurmountable task for current computing technologies. Penicillin has 41 atoms at ground state, requiring a classical computer with some 10^86 bits to model its structure . A classical computer achieving this is a physical impossibility. But for quantum computers, this type of simulation is well within the realm of possibility, requiring a processor with 286 qubits.
The key issue with modelling these drugs, is that as molecules grow more complex, the possible configurations also grow exponentially. For example, the most efficient algorithm to compute the energy of a system of electrons, full configuration interaction (FCI), scales exponentially with the number of electrons . Therefore, material development and drug discovery are constrained by the combinatorial challenges simulating more complex molecules creates. In fact, throughout medical history many material discoveries and drug developments were heavily reliant on luck, for example the discovery of Ivermectin which acts as a heartworm treatment in animals .
There are many ways that quantum computing may be able to reduce current inefficiencies found in the drug discovery process. For example, predicting protein structures from the knowledge of amino acid sequences, by simulating the protein folding process . In a similar way to how it is applied to financial models, the approach of “quantum annealing” can also be applied to drug discovery. Pioneered by D-Wave Systems, this process explores optimization problems by exploiting the tendency of quantum mechanics to tunnel through barriers between different possible solutions. Following IBM Q’s simulation of beryllium hydride in 2017, scientist Alan Aspuru-Guzik suggested that if quantum computers can carry out chemical simulations in a numerically exact way, this may lead to discovery of new small-molecule drugs or organic materials .
So, while there are classes of molecules too complex to simulate with classical computing methods, in the future we may see more quantum driven analysis of larger-scale molecules and combinations of atoms.
3. Limitations of quantum computers
The challenges associated with quantum computing can be summarised by MIT professor Isaac Chuang’s statement that quantum computing is “no longer a physicist’s dream – it is an engineer’s nightmare”. At the forefront of this technology’s drawbacks is the sensitive nature of qubits. The entanglement they experience allows outside forces to easily “decohere” qubits from a quantum state, back to a traditional computing state of being just a one or a zero . The solution so far has been to operate quantum machines at extremely cold temperatures near absolute zero. For example, the D-Wave 2X processor operates at a temperature of 15 millikelvin, which is approximately 180 times colder than interstellar space . However, even at near absolute zero temperatures, qubits can decohere quickly to produce errors that must be compensated for.
Equally concerning is the issue of this technology’s scalability. As more qubits are added to a system, keeping them in a quantum superposition (also known as ‘coherence’) becomes more difficult . This begs the question, will the exponential speed up provided by quantum computers be cancelled out by the exponential complexity needed to protect the system from crashing? While companies  and universities  are developing new quantum error correction methods, this technology will continue to remain more of a dream than reality until qubit related errors are largely controlled.
4. Ethical concerns
Though quantum computers boast potential boons to the economy, we must recognise that they pose risks to society. One of the most common encryption methods for our personal, medical and government data is based around the factorization of large numbers. Consequently, current standards of digital security rely on classical computers being unable to quickly determine ‘key factors’ of a large unique number. With the advent of quantum computers however, this process can and has been trivialised (i.e., Shor’s Algorithm, Variational Quantum Factoring ) as Analog and Universal quantum computers will be able to find these factors quickly and access encrypted information.
While the world may migrate to alternative encryption methods in the future, it does come with great challenges. Factorization encryption is deeply imbedded into our society, and the cost required to transition away from this method limits our ability to respond to the threat Quantum Computers pose. Additionally, at any given moment one can store factorization-encrypted data and decrypt it once quantum computers have evolved to solve these algorithms in a feasible time . Therefore, it is apparent that the ethical concerns surrounding this technology must be discussed now, and not once quantum computers are widely commercially viable.
One proposed countermeasure against these threats is the implementation of quantum-based countermeasures such as Quantum Key Distribution (QKD) . The security of QKD is based on a fundamental characteristic of quantum mechanics: The act of measuring a quantum system disturbs the system. Therefore, an eavesdropper trying to intercept a quantum exchange will inevitably leave detectable traces . The legitimate exchanging parties can then decide to either discard the corrupted information or reduce the information available. While Quantum Key Distribution provides average people with an awareness of surveillance, it does not change the fact that someone has access to their data.
Another concern is that realistically the only parties able to afford quantum computers are governments, large corporations, and super wealthy individuals. This fact alone leaves the public vulnerable, as average person will be able to challenge this technology if it is misused?
“I’m not happy with all the analyses that go with just the classical theory, because Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy. It’s not a Turing machine, but a machine of a different kind”. 
– Richard Feynman, awarded the Nobel Prize in physics 1965
Fundamentally, quantum computers are a new technology, and present exciting opportunities for those that can harness their power. Equivalently, these devices have the potential to harm society greatly if misused. Bringing large numbers of qubits into controllable states remains an immense challenge for scientists. While the ‘imagined’ commercially viable quantum computers require at least thousands of qubits (with the plan to increase to one million qubits eventually ), current models are far behind this target and are extremely power hungry . Like all disruptive and exciting technologies, there are enormous expectations surrounding quantum computing. However, with great hype, the evolution of these devices will inevitably test our patience. And so, we must remember, the best is yet to come.
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[Cover image] Garlinghouse, T. (2020, January 21). Quantum Computing: Opening New Realms of Possibilities. Princeton University. https://www.princeton.edu/news/2020/01/21/quantum-computing-opening-new-realms-possibilities
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[Bit and Qubit] Moe, M. (2017, July 24). Quantum Leap. Equities. https://www.equities.com/news/quantum-leap
[Chimera graph and Tables] G. Rosenberg., P. Haghnegahdar., P, Goddard., P. Carr., K. Wu., & M. L. de Prado. (2016). Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer. IEEE Journal of Selected Topics in Signal Processing, 10(6), 1053-1060. http://dx.doi.org/10.1109/JSTSP.2016.2574703
[Estimated option price] Stamatopoulos, N., Egger, D. J., Sun, Y., Zoufal, C., Iten, R., Shen, N., & Woerner, S. (2020). Option Pricing using Quantum Computers. Quantum, 4, 1-20. http://dx.doi.org/10.22331/q-2020-07-06-291
[Drug discovery pipeline] Cao, Y., Romero, J., & Aspuru-Guzik, A. (2018). Potential of quantum computing for drug discovery. IBM Journal of Research and Development, 62(6), 6-1. https://ieeexplore.ieee.org/document/8585034
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