# Quantum Finance: Quantum Computing Applications in High-Frequency Trading

*Introduction*

Computer algorithms are inherent in every aspect of our daily lives without us noticing their existence. The smartphone in one’s pocket is thousands of times far more powerful than the computer which NASA has used to land the first human on the moon. In 1982, quantum computing — a new computing paradigm — has been postulated by Richard Feynman — physicist and Nobel Prize-winner. quantum computing makes use of the laws and principles of Einsteinian quantum physics to perform unprecedented numbers of calculations in parallel. Together we shall examine how quantum computing influences the finance sector and the implications of the new interdisciplinary research field of quantum finance.

*Computational Intractability*

Computational complexity theory is a cornerstone in today’s computer science. One of the most puzzling conjectures is stated as a question whether p equals np. In plain English without jumping into mathematical jargon, there is a family of problems that are ** intractable **digital computers. This is a major problem that remains unsolved till now. The first one who comes up with a correct solution for this puzzle is entitled to a one million dollar prize by Clay Institute.

*Quantum Superposition*

Quantum computers exhibit inherent parallelism allowing them to solve an exponential number of problems simultaneously. How is that possible? The answer comes from Schrödinger’s cat thought experiment. The experiment setup examines the superposition of two states at a microscopic level and how such influences the cat at a macroscopic level. Let’s again explain these phenomena in simple words so that a reader without a background in quantum mechanics shall grasp the idea. A microscopic object such as an atom, an electron, a muon, or a photon shall exist in multiple states at the same time i.e. quantum superposition. This allows a quantum computer to solve as many problems at the same time and only the final state is observed to collect the results.

*Quantum Finance*

The finance sector has been one of the very early adopters of digital computers. However, there are so many intractable problems that are waiting to be solved by mathematicians, programmers, and economists. The existing solutions if any shall take digital computers years to solve them. Scientists from different fields such as finance, computer science, quantum computing, and quantum mechanics have come together to address these problems. Hence, the term quantum finance was coined for this new interdisciplinary field.

*Quantum Finance Applications*

Classic computers have empowered economists with technological tools. These tools range from basic spreadsheets application to complex enterprise resource planning solutions. The finance sector has benefitted substantially from such advanced computational power. These tools assisted in addressing problems related to financial modeling, complex analysis, time value of money, cash flow analysis, loan repayments, interest rates, assets valuation, future growth forecasting, business process workflow modeling … etc. Here we have only named a few applications of classic computers and the list shall go on indefinitely.

In a fast-paced economy, this still was not good enough to meet the unbounded ambitions of decision makers. Some problems are still intractable such as the optimal trading trajectory, multi-period profile optimization, risk management, regression analysis, scenario analysis, path-dependent option pricing … etc. Here comes up the notion of using a quantum computer to solve intractable financial problems. Finding scalable solutions to these problems shall empower decision makers with insights and analytics. These problems were previously expected to be solved in years, but with the advent of quantum algorithms, they shall be solved in only a matter of days if not less.

*The Flash Crash*

Let’s take for example the “Flash Crash” crisis of 2010 in the United States. It was a sudden crash of the American stock market which lasted for approximately 36 minutes and caused a loss of more than one trillion dollars. The economists spent more than five months analyzing the problem to find out the real reasons behind the crash. The most discussed causes were the U.S. Dollar/Japanese Yen exchange rates, fat-finger trade at Procter & Gamble stock or a large purchase of put options by the hedge fund “Universal Investment”. Howsoever, the final report blamed a small-time day-trader for the crash.

Five years later, a study conducted by Easley et al. proved that the high-frequency trading ( HFT ) made the modern U.S. economy subject to such incidents. HFT is a set of complex computer algorithms that are characterized by making automated buy/sell decisions after investigating investment horizons. Such horizons are not adequate to arrive at the optimal decisions. Since high-frequency traders compete by making faster decisions at fractions of milliseconds and taking advantage of slower traders.

For example, An asset manager willing to invest a certain amount has to study his investment on a horizon divided into multiple profiles. At each step, he must consider the associated risk, the price impact, and the transaction cost. This could be modeled as a maximization problem for the expected return resulting in a series of statistically optimal portfolios. As a consequence, a portfolio rebalance is executed at every step. As a result, asset managers perform a portfolio rebalance more frequently which exposes investors to a loss of fortune due to transaction costs and price impact.

Classic computers use heuristics to arrive at conclusions because they don’t have the computational power to discover every possible path. Automated decision-making process may take a few milliseconds at times and several seconds at other times. Such is called ** latency **and in an environment where HFT competition is very fierce a delay of one millisecond could translate into a loss of millions of dollars. HFT is blamed for such tremendous incidents, but the hope is that as algorithms mature and high-frequency traders fine-tune their performance then the traders shall compete and the environment shall auto-correct itself and claim its balance back.

Quantum finance carries within its folds the very solution to such repeated HFT crashes. Several other theories for the real causes of the crisis were postulated and an interested reader is advised to get acquainted with terms such as *jitter.* It’s worth noting that* 2010 Flash Crash* is ranked one of top 3 financial crises caused by HFT. Look up *Peet’s Coffee & Tea* and *Knight Capital *for similar incidents caused by HFT and their related analyses.

**Quantum Annealing**

D-Wave Systems is a Canadian venture which has built a special purpose quantum computer to perform the quantum annealing. Google, NASA, and other tech pioneers are the major clients for the quantum annealer. D-Wave has recently announced the release of a 2000 qubits quantum annealer. Qubits are the building blocks of a quantum computer. The new release which was announced in January 2017 is called 2000Q and is a major breakthrough as D-Wave has doubled the number of qubits on the previous chip 1000Q.

Quantum annealer is not a general-purpose quantum computer i.e. it can not solve all kinds of computational problems — special purpose computers solve only a defined set of problems. Quantum annealers solve the optimization problem which is the underlying core of the well-known intractable problems in machine learning, radiotherapy, and financial analysis. Financial services companies have shown great interest in 2000Q to get their hands on the cutting-edge technology. Even though 2000 qubits may sound like a large number, chief scientists of quantum computing believe that millions of qubits are needed to reach quantum annealers of adequate commercial interest. However, this number is unlikely to be realized in the foreseeable future.

D-Wave targets applications of quantum computing in outer space while VC backed Rigetti and Microsoft have different approaches to quantum computing that shall be discussed in a dedicated technical review to the subject.

*Closing Thoughts*

Finance sector key-role players should be acquainted with quantum finance and the potentials it has for them. Just like how digital computers changed our today’s economy, quantum computers shall be a disruptive technology to the finance sector and all the rules of the games will be effected subsequently. Quantum finance will be an inevitable must-have qualification for tomorrow’s economists.