Quantum Computing

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Contents

Introduction

Quantum Computers Explained

What is Quantum Computing

Quantum computing is the study of theoretical computational systems that use a variety of quantum mechanics to perform operations on data.[1] As opposed to classical transistor-based computers that encode data into binary digits, called bits, quantum computation requires data to be encoded in the form of quantum bits, or qubits. While bits are in a definitive state of either 0 or 1, qubits are in a superposition of both 0 and 1 at the same time - a quantum phenomenon called quantum superposition. Unlike classical computers where heat is produced and lost from the system during work, current quantum computers are adiabatic, meaning that the system is isolated from thermodynamic processes. This is necessary in order to retain the integrity of quantum bits and ensures that qubits remain in their unobserved state of superposition. Failure to isolate the system will result in quantum decoherence.[2] By taking advantage of quantum superposition and creating an environment to incubate these quantum systems, scientist have been able to surpass the current limits of classical computer systems.

Although quantum computing is still at its early stages of development, the potential impacts of quantum computing on our future lives is endless. Since late-stage quantum computers could theoretically perform functions and process data at an exponential rate compared to classical computers, it will allow us to solve problems that have been unfeasible based on modern-day technology. Some of its major areas of impact include: optimization, cryptography, particle physics, and artificial intelligence. Quantum computers will also change the way we create models. As a result, molecular modelling, financial modelling, and even weather forecasting will be taken to new heights.



Quantum Mechanics

In the world of quantum computing, there are two important quantum mechanical theories that we must note in order to understand the way quantum computers work. These two quantum theories are the backbone of all quantum computation today.

Schrödinger's cat: a cat, a flask of poison, and a radioactive source are placed in a sealed box. If an internal monitor (e.g. Geiger counter) detects radioactivity, the flask is shattered, releasing the poison, which kills the cat. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead. Yet, when one looks in the box, one sees the cat either alive or dead not both alive and dead.[3]

Quantum Superposition

Quantum superposition is a fundamental principle of quantum mechanics as it states that any two, or more, quantum states can be superposed and the result will be a third, valid quantum state.[4] In terms of quantum information processing using qubits, this means that while 0 and 1 and both valid states of a qubit, there is also a third quantum state where the qubit is in an unobserved superposition of both 0 and 1 simultaneously. An illustration of the states of quantum superposition and how qubits work is devised by the Austrian physicist Erwin Schrödinger in 1935 through his famous thought experiment, Schrödinger’s cat. The thought experiment presents a scenario where the cat is in a state of superposition of being simultaneously dead and alive, which is supposed to represent the same concept behind the theory of quantum superposition.

Quantum superposition is important in quantum computing because it allows computer scientists to manipulate qubits way produce and calculate information in ways that traditional bits cannot. Since a qubit is in a quantum superposition of both 0 and 1, this means that given n bits of information in a classical computer, a quantum computer can produce 2n qubits of information. It’s easy to see that using quantum superposition, the manipulation of qubits allows quantum computers to reach levels of speed and accuracy that is unfeasible in classical computers.

Another important implication of quantum superposition is its effects on transistors and thus points to the limits of classical computing. The current size of transistors is 14 nanometers. The Landaeurs Limit says that after transistors become smaller than 5 nanometers, electrons might jump the transistor barrier, thereby, making that method of computing ineffective at that scale. This is because of the effects of quantum superposition.

Quantum Coherence

In quantum mechanics, particles, such as electrons, behave like waves that can interfere with one another and result in peculiar behavior of quantum particles.[5] In a quantum computer, as long as the system is isolated from outside interference, these waves will behave in a coherent state. During this state, qubits are in its state of superposition and the system will work as intended. However, when a quantum system is not perfectly isolated and comes into contact with its surroundings (ie. thermodynamic disturbance from external heat sources), coherence will begin to decay in a process called quantum decoherence. When this occurs, quantum behavior is lost and the quantum system will cease to work.

The decoherence process is essentially the loss of information from a quantum system to its external environment. Even the slightest interference from outside waves can cause decoherence in a quantum system. Because of this, quantum systems must be heavily monitored and controlled, which is one of the reasons why incubating an environment for the operation of quantum computers is so expensive.

History of Computers

The computers that we have come to know today are built over decades of research and testing. In the early 1900s, computers were far from being feasible tools to the technological evolution of mankind. Now, computers are used in every corner of our lives as the functioning of the entire world is built on layers of computer systems and networks. As quantum computing marks a new dawn to the advancement of computation systems, let us take a brief look at the major advancements in the history of computers.

1890: The Hollerith Tabulator

Invented by Herman Hollerith, the Hollerith Tabulator is a tabulating machine designed to assist in summarizing information stored on punch cards.[6] This machine was one of the first system electromechanical system that used physical relays to perform calculations. Initially designed to help process data for the 1890 U.S. Census, later models of this system became widely used for accounting and inventory control. The tabulating machine also spawned a class of machines, as well as the entire data processing industry.

1946: ENIAC

Short for Electronic Numerical Integrator and Computer, ENIAC was one of the earliest electronic general-purpose computers ever made. It was Turing-complete and completely digital, and was able to solve “a large class of numerical problems” by simply reprogramming the system.[7] ENIAC used vacuum tubes to allow control of the electrical current inside the machines, making it one of the first leaps into the digital age.

1957: IBM 608

Known as a transistor calculator, the IBM 608 was the very first IBM product to use transistor circuits without any vacuum tubes.[8] It was also the world’s first calculator manufactured for commercial sale that used only transistors. These transistors were much smaller and more reliable than vacuum tubes because its components used less energy and generated less heat. Although the IBM 608 was withdrawn from the market only a few years after its release due to its obsolesce to newer IBM products.

1966: AGC

The Apollo Guidance Computer was a digital computer produced for the Project Apollo, the United States human spaceflight program carried out by NASA, that was installed on board each Apollo Command Module and Lunar Module.[9] The AGC was the very first computer to use integrated circuits, which allowed electrical components of the system to be put on a single miniature chip. The integrated circuits boast greater computational power and lower cost than its predecessor.

Types of Quantum Computing

Quantum Annealer

The optimization problem is represented in terms of a cost function where the solution is the lowest energy state represented on the diagram as the lowest point.[10]

Quantum annealing is a computational paradigm to search for the minimum of a cost function through a control of quantum fluctuations. This type of computation can be used mainly for combinatorial optimization problems. These include but are not limited to machine learning for pattern recognition, natural language processing, and medical diagnosis.[11]


As you can see from the diagram on the right, the probability function starts off as a flat line which means that all possible solutions are equally likely to be the most efficient. Then, a term is added to the equation to induce the quantum fluctuations mentioned above. The gradual decrease of the coefficient of this term leads to the solution as shown above.

Strengths

One of the reasons that this type of quantum computers are the closest to becoming reality is because they are less prone to decoherence compared to circuit models.[12] This is because of the former uses inherently low state matter i.e. the huge fridge that cools down the chip decreases external noise.

The way we use machine learning to create artificial intelligence allows us to represent problems as combinatorial optimization problems which let us use quantum annealing to make these systems more efficient.

The way in which quantum computers work allows them to automatically account for time evolution. Other types of quantum computers require the time evolution in the problem to be stated explicitly. However, in quantum annealing, the system evolves naturally following the Schrodinger equation.[13] This is the gradual decrease of the coefficient of the quantum term mentioned above that allows for the automatic time evolution.

Weaknesses

Compared to its peers in the quantum computing era, the quantum annealer is highly inefficient at solving several problems even in its forte; optimization. Several high-difficulty problems such as QUBO (Quadratic unconstrained binary optimization) would take an inefficient amount of time to be solved using quantum annealing. This is because the qubits inside an adiabatic quantum system are not well connected. The connectivity of qubits is what makes a quantum computer more powerful.[14]

Researchers have yet to find proven applications of this already existing technology. Combinatorial optimization problems are solved through simulated annealing in classical computers. However, there exist few problems that the quantum annealer may be more efficient at compared to simulated annealing. The practicality of using quantum annealers instead of regular supercomputers isn’t yet significant enough to warrant its use and exorbitant prices as can be seen in the section below. For example, the binary tree problem is one such problem where quantum annealers are proven be better at solving compared to classical computers. Unfortunately, this problem (and its solution’s) practical significance in the real world is limited.[15]

Real World Example: D-Wave

Their latest $15 million quantum computer is called the 2000Q. The ‘2000’ stands for the number of qubits. D-Wave Systems is the first company to have realized the hardware to conduct quantum annealing on a scale that may be of practical use to research organizations. Their impressive client list includes Google, NASA and Lockheed Martin. The key piece of technology is their chip that is kept cool at 15mK, making it one of the coldest places in the known universe. Its current applications will be discussed later.[16]

IARPA (Intelligence Advanced Research Projects Activity) is a US government funded research organization funding their own quantum annealer. They have developed a QEO (Quantum Enhanced Optimization) with the eventual goal of building a 100 qubit quantum annealer. Google has also followed suit with their announcement at AQC 2016 that they were building their own quantum annealer.[17]

Quantum Simulators (Analog and Digital)

Quantum simulation is an idea that has been at the center of quantum information science since its inception, beginning with Feynman’s vision of simulating physics using quantum computers. A quantum simulator is a device that maintains quantum coherence among its degrees of freedom over long enough timescales to extract information that is not efficiently computable using classical computers.[18]

Currently, there are two types of quantum simulators; Analog and Digital. Analog quantum simulator (AQS) uses a physical mimic of the quantum model to infer its properties. Eg. cooling down atoms so that their behavior can be observed. So, the AQS can also be referred as adiabatic quantum computers. The hardware is similar to quantum annealers but the qubits in AQS have higher connectivity. Digital quantum simulators (DQS) uses the quantum equivalent of logic gates and their qubits are based on well-defined and understood quantum states. These are used to perform simulations of the quantum model. A hybrid approach is also underway but it will not be mentioned here as it is largely theoretical as of now. [19]

Imagine that you and some friends are in a row of rooms that are joined by a set of vertical windows. If two of you are standing up, you can communicate using hand signs; if you are both lying down, you can communicate with hand signs. But, if one of you is lying down and the other is standing up, you cannot see each other's hands and no communication is possible. And, if people are in the rooms in between, they might block your view anyway. So, for you to communicate with one of your friends, you have to do two things: you have to make sure you and your friend have the same orientation. And, since the rooms are all in a row, you have to make sure that all the people in between you are in the opposite orientation. This is what decoupling sequences do.[20]

Strengths

The adiabatic quantum computer is simpler in terms of its build and so it is easier to scale.[21] See problems with scaling in weaknesses below.

The quantum models developed by a real adiabatic system can be replicated with a fair degree of accuracy in a digitized adiabatic system, a subset of DQS. To give you an idea of the complexity: each qubit in the network is, even when you don't want it to, able to talk to the rest of the network. So, setting the coupling between any two qubits varies the coupling of adjacent qubits too. Researchers have demonstrated the ability to use decoupling sequences to fix this issue on a large scale.[22] The method in terms of an analogy is mentioned below the picture on the right.

Weaknesses

However, a general purpose quantum computer (the gold standard) is only possible in an adiabatic system if there is high qubit connectivity. This level of connectivity is impractical because of the resultant high error rate. Each qubit is driven by how strongly it is coupled to every other qubit. For a fully interconnected adiabatic quantum computer, however, the weak connections between distant qubits are very sensitive to environmental noise.[23]

Digital quantum computing, which uses logic operations and quantum gates, offers the possibility of error correction. By encoding information in multiple qubits, you can detect and correct errors. Unfortunately, digital qubits are delicate things compared to those used in adiabatic quantum computers, and the ability to program and run complex problems with them is out of reach at the moment.[24]

Real World Example: Harvard Quantum Computer

Harvard has developed a 51 qubit analog quantum computer in conjunction with Russian scientists from Moscow. In 2016, a scientist from Harvard and Google were able to use this quantum computer to simulate a hydrogen molecule.[25] This complex simulation is already possible with classical computers but, this opens up the possibility of other larger molecules being simulated using Harvard’s analog quantum.

A set of atoms are kept inside special laser cells and cooled to extremely low temperatures, making it an AQS. The computer harnesses the nanoscale atomic defects in diamonds materials as mentioned above to create a quantum model capable of simulating complex molecules.[26] This computer has a wider range of practical uses when compared to quantum annealers.

Universal Quantum

A quantum Turing machine (QTM), also a universal quantum computer, is an abstract machine used to model the effect of a quantum computer. It provides a very simple model which captures all of the power of quantum computation. Any quantum algorithm can be expressed formally as a particular quantum Turing machine.[27]

One of the most famous potential uses for quantum computers is breaking up large integers into their prime factors. For classical computers, this task is so difficult that credit card data and other sensitive information are secured via encryption based on factoring numbers. Eventually, a large enough quantum computer could break this type of encryption, factoring numbers that would take millions of years for a classical computer to crack.[28]

Strengths

The universal quantum differs from the above types because of its generality. Even the 51 qubit Harvard built AQS has a very specific use to solve one particular equation. They will need to rebuild their AQS from the ground up to solve another equation. The universal computer will be able to solve many different equations.

Weaknesses

In a recent commentary in Nature, Martinis and colleagues estimated that a 100-million-qubit system would be needed to factor a 2,000-bit number—a not-uncommon public key length—in one day. Most of those qubits would be used to create the special quantum states that would be needed to perform the computation and to correct errors, creating a mere thousand or so stable “logical qubits” from thousands of less stable physical components.[29]

However, the inherent computational power of a quantum processor to solve practical problems depends on far more than simply the number of qubits. Due to the fragile nature of quantum information, increasing the computational power requires advances in the quality of the qubits, how the qubits talk to each other and minimizing the quantum errors that can occur.[30]

IBMQ Universal Quantum Computer[31]

Real World Example: IBM Q

The vision of Web-connected quantum computers has already begun to Quantum computing. In 2016, IBM unveiled the Quantum Experience, a quantum computer that anyone around the world can access online for free.[32] It only comes with 5 qubits. It has provided a way for researchers around the world to practice building quantum algorithms without access to their own quantum computer. IBM’s overall strategy is to build “a community and an ecosystem” around its technology.[33] To date, users have run more than 300,000 quantum experiments on the IBM Cloud.[34]

IBM is coming out with a new class of quantum processors: A 16 qubit processor that will allow for more complex experimentation than the previously available 5 qubit processor. It is freely accessible for developers, programmers, and researchers to run quantum algorithms and experiments, work with individual quantum bits, and explore tutorials and simulations. Their first prototype commercial processor with 17 qubits leverages significant materials, device, and architecture improvements to make it the most powerful quantum processor created to date by IBM. It has been engineered to be at least twice as powerful as what is available today to the public on the IBM Cloud and it will be the basis for the first IBM Q early-access commercial systems.[35]

Applications of Quantum Computers

The following is a list of areas that quantum computers are expected to realize tangible benefits over traditional computers. Do keep in mind that this list is not exhaustive and quantum computing could assist in unforeseeable and revolutionary areas. New applications will be developed as the technology continues to progress.

Artificial Intelligence

A primary application for quantum computing is artificial intelligence (AI). AI is based on the principle of learning from experience, becoming more accurate as feedback is given, until the computer program appears to exhibit “intelligence.” This feedback is based on calculating the probabilities for many possible choices, and so AI is an ideal candidate for quantum computation. For example, Lockheed Martin plans to use its D-Wave quantum computer to test autopilot software that is currently too complex for classical computers, and Google is using a quantum computer to design software that can distinguish cars from landmarks.[36]

Molecular Modelling

Another example is precision modeling of molecular interactions, finding the optimum configurations for chemical reactions. Such “quantum chemistry” is so complex that only the simplest molecules can be analyzed by today’s digital computers. For example, the hydrogen molecule simulation mentioned above is at the forefront of half a century worth of classical computing developments but Google has already achieved the simulation with quantum computing. The implication of this is more efficient products, from solar cells to pharmaceutical drugs, and especially fertilizer production; since fertilizer accounts for 2 percent of global energy usage, the consequences for energy and the environment would be profound.[37]

Solution to a symmetric TSP where N is 7 cities. The time complexity of the brute-force approach is O(n22n). [38]

Optimization

Due to the sheer computational power of quantum computers relative to classical computers, optimization problems is one of the first areas where quantum computing can be applied in order to improve current algorithms. Mathematical deals with finding the most optimal solution to a problem. As data becomes more complex in various fields such as economics or engineering, there is an increasing need for more efficient ways to solve optimization problems than what is currently available with classical computers. Quantum computing will allow problems that have been previously unfeasible to solve on classical computers or impractical to solve due to its requirement of computing resources to be solved much more efficiently. These quantum optimization algorithms will completely change the way optimization problems are solved.

One example of this is the Travelling Salesman Problem (TSP), which is a classic algorithm program in the field of computer science.

"The Travelling Salesman Problem describes a salesman who must travel between N cities. The order in which he does so is something he does not care about, as long as he visits each once during his trip, and finishes where he was at first. Each city is connected to other close by cities, or nodes, by airplanes, or by road or railway. Each of those links between the cities has one or more weights (or the cost) attached. The cost describes how "difficult" it is to traverse this edge on the graph, and may be given, for example, by the cost of an airplane ticket or train ticket, or perhaps by the length of the edge, or time required to complete the traversal. The salesman wants to keep both the travel costs, as well as the distance he travels as low as possible."[39]

This problem is notorious among computer and mathematical scientists because of the difficulty of brute-forcing a solution using classical computers. However, since quantum computers are able to perform large calculations very quickly, this, along with other optimization problems, can be easily solved.

Cryptography

The issue with modern-day cryptography methods is that it relies largely on difficult mathematical problems that would take decades for a classical computer to solve. The majority of these mathematical problems are concerned with either the integer the factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem.[40] Using quantum computing algortihms, these mathematical problems could be easily solved. This would undermine the security of all networks, data, and systems that currently exist today. As the future of quantum computing nears, there is an increasing need for quantum-proof cryptography methods. Thankfully, researchers have already began looking into post-quantum cryptography. As a result, they have been able to devise a few quantum-proof algorithms that are able to fend off attacks even from a quantum computer (ex. Lattice-based cryptography).

Particle Simulations

Traditional simulations of particle physics are highly complex. Great amounts of computational power and time are required when traditional computers attempt to model such a simulation, as the number of variables to consider are immense. Quantum computing can greatly ease the speed and accuracy of these models, and in fact, there has been significant progress in this area already. In June of 2016, a team of students at the University of Innsbruck utilized quantum computing to simulate particle pair activity during arithmetic functions similar to how they act within a computer.[41] Their simulation had excellent agreement compared to actual particle experiments, a strong indicator that more complex models are to come.

Financial Modelling

The field of finance often considers optimization, which quantum computers excel at. Finding the optimal combination from a selection of investments based on risk, historical, and projected return can be greatly enhanced through quantum computing. By using quantum technology to develop an optimal portfolio, one would be able to realize a more comprehensive analysis as well as a decrease in solution time.

Weather Forecasting

Despite cutting-edge measurement instruments, current weather forecasting is at best an educated guess as to what weather will manifest. Quantum computers could better analyze weather data more deeply, including historical data, giving a better insight in to when and where bad weather will strike. Bad weather has been estimated to directly and indirectly affect 30% of the US GDP[42]; creating an exact model of where bad weather will occur could yield significant benefit for areas such as transport, retail, and food production.

Conclusion

Investments into researching and developing quantum computers is at an all-time high. Microsoft, Google and IBM are deeply involved in quantum computer research. Just recently, on July 25th, Google announced a multi-million dollar investment into quantum computing with the University of Sydney [43]. The exact figure has not been disclosed to the public.

Our team does not predict that quantum computing will be changing consumer technology in the near future. Quantum computers are better suited to handle efficiencies, researching, and modelling tasks. Traditional computers will still be relevant for consumers, interested in more non-intensive tasks such as web-browsing, document creation, and gaming.

Authors

Dhruv Bhatia Sameer Parikh Jack Jiang
dbhatia@sfu.ca sparikh@sfu.ca zfjiang@sfu.ca
Beedie School of Business
Simon Fraser University
Burnaby, BC, Canada
Beedie School of Business
Simon Fraser University
Burnaby, BC, Canada
Beedie School of Business
Simon Fraser University
Burnaby, BC, Canada

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