Introduction to Quantum Computing

Chapter 1 of 7

Overview of Quantum Technology

Written by Elias Lehman · In partnership with Quantum Computing at Berkeley

In this chapter, we will skim the surface of the most popular technologies, keeping in mind their physical foundations, and looking to see what impact they might have on our lives in the near future.

1.1 Introduction

Quantum mechanics is the term we give to the behavior of reality on the most minute scales, nanoscopic to be precise. At the fundamental level, almost nothing works as you would expect it to – particles go through walls and information appears to teleport, all because the world is inherently probabilistic. Advancing the frontier in many fields will require scientists in those research areas to know quantum mechanics. Developing MRIs, car batteries, and computers; identifying molecular structures and chemical properties; searching for dark matter, and measuring gravitational waves are just a few fields that are rooted in understanding these abstract physical phenomena. The beauty of physics is that it will sometimes break your intuition and then provide a clearer picture of the world. This is what makes quantum physics arguably the most fascinating field of all. The beauty of engineering is in applying these unimaginable behaviors to create novel technologies.

1.2 Quantum Information and Computing

Since the mid-20th century, quantum physics has arguably paved the way for many technologies we interact with daily, the computer being a prime example. The earliest advancements came with the discovery of the physical properties of semiconductors, like silicon, which led to the invention of the transistor in the early 1950s. Transistors are the technology responsible for bits, the unit of information in classic computers. Billions of transistors are at the core of all computer chips, like those in your cell phone, laptop, and other smart devices. Light emitting diodes (LEDs) like those on your computer screen, similar electroluminescent technology, such as digital cameras, and photovoltaic technologies like solar cells can be accredited to discovery in the field of quantum mechanics.

The history of quantum and classical computers are very intertwined. Early modern computers were developed at labs and universities around the world and given early access, physicists were among the first to use computers to perform rigorous calculations. However, quantum physicists came to realize that classical computers were limited. Famous theorist Richard Feynman postulated that no classical system could be used to understand the behavior of even slightly complex quantum systems–so classical computers, even supercomputers, could only be used to advance quantum technologies to a certain extent.

Using new perspectives on two-level quantum systems, theorists constructed new notations and diagrams to capture information in a format that enabled it to behave quantum mechanically. The unit of this information is a quantum bit, frequently called a qubit, in contrast to the classical bit. We will explore the qubit and many interesting properties in Chapter 2.

Quantum computing provides a possible solution for some critical and intractable problems in several industries. Unfortunately, the implementation of effective quantum computation has proven to be extremely difficult. Two primary challenges have emerged on the hardware side:

1. Qubits are fragile–small external contributions to the energy of a qubit, such as fluctuations in temperature or exposure to radiation, can corrupt its state. Because quantum algorithms often rely on the relationship, or entanglement, between multiple qubits, one errored qubit can ruin a computation. Error correction is the practice of reducing the effect of corrupted qubits.

2. For most real-world problems, millions, sometimes even billions, of high-quality qubits are expected to be required to provide a trustworthy solution. This sets the date for advantageous computations using quantum computers, sometimes called quantum supremacy or practical quantum advantage, potentially very far in the future.

On the software side, quantum algorithms with advantages over classical computation are still in development. They are stumped by a primary issue as a result of lacking hardware – however, as we discuss in Chapter 5, classical simulations of quantum computers are currently mitigating this issue.

Until recently, these barriers seemed impossible to overcome. Over time, dozens of new approaches to implementing qubits have been proposed, tested, and reevaluated. Today, there is a set of widely accepted approaches that offer a variety of pros and cons which we will explore in Chapter 3. Some methods require cooling atoms to temperatures colder than deep space, others use the quantum nature of light, and qubits can even be made from superconducting electronic circuits.

1.3 Introduction to Quantum Communication, Cryptography, and Networks

The internet can be simplified to a network of interconnected computers that transmit data. We layer security protocols to multiple parts of the transmission process to ensure that data is exchanged effectively and securely. In this sense, quantum information is quite analogous to classical information, in that we would need a similar system, a quantum internet, to move qubits between quantum computers. Rather than quantum internet, most experts prefer the term quantum networks. While quantum computers perform calculations on quantum information, quantum networks transport quantum information. The primary incentive to build a quantum network is parallel computation. If we can successfully exchange data between quantum computers, while maintaining the quantum state, we can run quantum calculations in parallel, further improving computation speeds.

The implementation of a quantum network is still a highly debated topic. Today’s internet is physically represented by local machines, like your PC, connected by a collection of fiber optics cables that spread across whole nations and span thousands of kilometers, connecting continents.

It’s currently infeasible to transport quantum information this far and wide for several reasons. Despite research coming out of China showing the capability of today’s fiber optic technology to transmit qubits locally, today’s internet scheme is far too noisy to transmit over far distances. The state-of-the-art is to connect a few relatively close quantum computer centers via a higher class of optical fiber, as national labs in the US have demonstrated. In the future, we’re likely to see quantum computers concentrated a short distance apart, and access to their clusters of computers over the classical internet via cloud computing.

By drawing a comparison to the classical internet protocol, we can expand on quantum networks and into communication. Sending qubits over a quantum network to represent messages is quantum communication. A major subfield of quantum communication is quantum encryption – two quantum mechanical properties play a part in making quantum communication theoretically the most secure form of communication possible.

The no-cloning theorem, stating that an arbitrary quantum state is unique, and the collapse of a wave function upon observation, allow two parties to immediately identify if data sent between them is being observed en route–in other words, if somebody is eavesdropping. In addition to potentially securing online data, quantum technology can break modern encryption. Standard encryption methods today are based on mathematical calculations that are easy to compute in one direction, but very challenging to reverse. The most popular example is RSA encryption which uses the multiplication of two large prime numbers. If the two numbers are large enough, factoring the product is nearly impossible over a reasonable time period for a classical computer.

However, certain problems like these are theoretically solvable by quantum computers, implying that one day, quantum computers will be capable of breaking modern encryption techniques. In response to this threat, governments and companies alike have begun the shift to new encryption techniques based on algorithms that are just as hard for quantum computers as they are for classical ones. Federal financing, such as the 2021 bill passed by US President Joe Biden, is turning heads toward one of the earliest applications of quantum cybersecurity. A field titled post-quantum cryptography will be the theme of the latter half of Chapter 6.

1.4 Introduction to Quantum Simulation

Revisiting the potential solutions that quantum computing offers, recall the original reason for developing quantum computers. Nanoscale particles are governed by quantum mechanics, allowing them to be in a variety of states at once. To model this system, one would have to track every possible configuration of all of the particles’ states. This is a problem that grows exponentially, so modeling 400 particles would require more bits than there are particles in the universe. This is the case for a classical computer. Quantum simulation builds off classical simulation directly–a quantum system, such as a molecule, is mapped to a set of qubits, then operations are performed on the set that approximates the intended solution, such as the energy produced from a chemical reaction. The quantum mechanical properties of qubits allow these simulations to model quantum systems without the exponential complexity. Basic simulations can be executed on a quantum computer today. Although there are few published examples of extraordinary or novel results extracted using quantum computation, breakthroughs in this field are always making headlines.

Quantum computers promise a reduction in computational steps required to solve simulation problems, translating to a computational speedup in theory. Intrigued by this, natural and computer scientists, mathematicians, and economists are finding potential applications for quantum computers in their respective fields. Specifically, scientists search for a speedup to solve intractable problems that take on exponential complexity. So, in addition to simulating complicated quantum mechanical systems, the applications of quantum computing extend to industrial problems like optimizing combinatorial systems in a factory or simulating financial systems by making analogies to physical ones. For example, some modern research compares financial portfolio risk to energy levels, which can be optimized using quantum computation.

Further applications exist in simulations for biotechnology and pharmaceuticals. In the drug development world, knowing the electron orbitals of the atoms in a drug’s active chemical can determine exactly how it will behave when interacting with different proteins in the body. Simulation of these chemicals and their interactions opens the door for drug design in a way that has never been seen before. Instead of relying on intensive classical calculations, followed by repetitive trial and error, drugs could be designed for an application, drastically reducing the iterations required to find a successful product. But, as aforementioned, quantum computers have challenges to overcome before any of these promises become a reality.

1.5 Introduction to Quantum Sensing and Metrology

Quantum technology goes beyond computation, having impacts on the way other technologies interact with the world. Sensing is a general term used to describe the process of collecting digital information from an environment. Autonomous cars sense the environment using 3-D laser scanning, while aircraft use gyroscopes to sense their orientation in the sky. We term the use of quantum properties, distinct from classical properties like temperature and pressure, quantum sensing. Quantum properties, measured at the atomic level, include entanglement, quantum interference, and more. Utilizing these properties, as opposed to classical ones, allows for new forms of ultra-precise sensors, as is covered in Chapter 7.

Metrology, the study of measurements, is a science with applications primarily in scientific instrumentation. High-quality measurement devices like atomic clocks, magnetic resonance imaging, and electron microscopes fall under the umbrella of metrology, and all stem from discoveries in quantum physics.

If these technologies sound inapplicable to most people, you would be mistaken. Atomic clocks are responsible for not just maintaining the accuracy of high-frequency trading and major stock markets, but also allowing for GPS and other modern navigation systems. Magnetic imaging is a life-saving technology in a hospital’s MRI machine. These devices are currently benefiting from research in the field of quantum sensing.

1.6 Timescales

Different timescales associated with the sectors of quantum technology are often loosely defined, given that it’s nearly impossible to predict when a practical scientific discovery will occur. Still, the industry has standardized terms for the near future of quantum computing, which plays a key role in the timelines of other quantum technologies. Today, we are in the midst of the Noisy Intermediate-scale Quantum (NISQ) era. This name encapsulates the state of today’s quantum technology–not at a large enough scale to make a revolutionary impact, and highly exposed to noise from the environment.

With the current state of quantum computing incapable of solving problems complex enough to have real-world implications, researchers are limited to proof-of-concept demonstrations. For example, we can use quantum computers to calculate the lowest energy of the well-studied hydrogen atom, but this doesn’t offer very much commercial value. Similarly, quantum information has been securely sent over one hundred kilometers, while one thousand kilometers remains an unreachable milestone. Still, it’s important to recognize and appreciate how far science has come. Just 40 years ago these technologies only existed on a blackboard.

Since the concept was postulated, quantum computing has inched closer to reality. With each iteration, new technologies and computational techniques have been engineered to mitigate the field’s most pressing issues, particularly in dealing with error-prone qubits, and optimizing computations so theoretical algorithms will have application in this era. Now, the quantum computing industry is on a roll, hoping to provide solutions to challenging problems down the road.

Chapter 2 of 7

Quantum Information and Computing

Written by Raiyan Rizwan, Alexa Ramirez, and Elias Lehman · In partnership with Quantum Computing at Berkeley

2.1 Classical vs. Quantum Computing

Headlines about the fascinating problems that quantum computers are capable of solving hint at the fundamental difference between these computers and their classical counterparts. Quantum and classical computing are two different approaches to processing and manipulating information–they are intended to do the same things, like simulations, arithmetic, and analysis. The difference is in the unit of information that is processed on said computer. Classical computing refers to the traditional approach to computing that relies on classical physics and the manipulation of bits. Quantum computing, on the other hand, is a newer field that involves the use of quantum-mechanical phenomena, such as superposition and entanglement, to perform computing.

The key distinction between classical and quantum computing, beyond the physics governing their units of information, is how they process information.

In classical computing, bits are represented by one of two values: 0 or 1. These are then manipulated using logical operations, such as AND, OR, and NOT, to perform various tasks. Quantum computing, on the other hand, uses quantum bits, or qubits, which can represent both 0 and 1 with some probability of each through the principle of superposition, as we will explore in section 2.2. But, the probabilistic character of qubits alone is not at the root of quantum computing applications – it’s the ability to create dependencies, formally known as entanglement, between quantum states which allows for the elimination of possible outcomes faster. We call the elimination of a state’s probability interference. The advantage of quantum computers is leveraged strongest when these principles of quantum mechanics are working their hardest.

Another difference between classical and quantum computing is how they handle errors.

Transistors, the physical objects representing classical bits, are resilient to errors such as environmental interference which could incorrectly flip one from 0 to 1. Classical computers use error-correction techniques, which have been tested over many years, to detect and correct errors that may occur during computing. Quantum computers are more susceptible to errors because quantum-mechanical systems can be disturbed by external forces such as light radiation and temperature. As a result, quantum computers require specialized techniques, referred to as quantum error correction, to handle errors and maintain the integrity of the computing.

In summary, classical and quantum computing are two different approaches to processing information, with classical computing relying on classical physics and the manipulation of bits, and quantum computing using quantum-mechanical phenomena and the manipulation of qubits. While both approaches have their strengths and limitations, quantum computing has the potential to revolutionize the field of computing by allowing for faster and more efficient computing in certain cases, once their chronic errors have been mitigated.

2.2 Key Ideas

• Superposition
• Measurement
• Entanglement
• Probability

Superposition

Quantum procedures are not a wildly different concept than classical forms; meaning is extracted from and superimposed onto the physical universe to solve (often computationally heavy) problems. Thus, representation is of key importance. While electrical signals are deterministic (repeatable, set outcome), the study of quantum mechanics tends to be centered around probability. This implies a natural randomness to every measurement of a quantum circuit.

The physical property of quantum bits, qubits, (of any form) that most interests us is superposition; the nature of quantum particles to be in a distinct but unknown state. To clarify, it is not quite the case that a particle is physically in two states at once (as often misconstrued in the media), but rather that we simply don’t know which of two states it is in. We thus portray qubits as existing in a kind of transient, pre-measurement middle ground.

The two fundamental, stable quantum states are written as |0> and |1>. To an algorithms researcher, these basis kets (|>) have an arbitrary meaning that is directly dependent on the physical implementation of the computer. |0> and |1> could refer to electron spins or to the direction of current in a superconducting qubit; either way, they are well-defined, physical, and mathematical bases for the system. There are infinite other states that relate to particles in superposition. These “in-between” states exist mathematically in a Hilbert, or complex, vector space and are further conveniently depicted along the surface of a geometry known as the Bloch sphere.

The Bloch Sphere is a mathematical construct that exists in one complex and two real dimensions. Consider the unit circle in a flat plane; if we were to pull a vector along the span of its perimeter in or out of the page without changing its magnitude, we would have a sphere. This pulling corresponds to a rotation in the complex plane and adds a dimensionality that allows us to describe any point along the surface of this pseudo-sphere using only two coordinates, each complex. The Bloch Sphere is an example of a Hilbert Space that accommodates stable and superposition states, phases, and other single-qubit characteristics. We use these tools to visually understand how a qubit leans toward one or the other basis, and they can be mathematically manipulated to mirror physical processes.

Superposition is a unique property of qubits, and linear algebra is a choice of mathematical representation. An in-depth discussion of the choice of complex numbers to architect quantum mathematics is beyond the scope of this paper, though fascinating. The power of these choices and properties will become apparent in section 2.3.

Measurement

Quantum data is single-use. As soon as we make a measurement, all probabilities relating to superposition or entanglement (often denoted via the wave function ψ) “collapse” into a possible state. Notably, this makes copying quantum states through purely quantum methods (for individual qubits and larger systems) impossible, because any copies will have identical probability mappings rather than the same measured random outcome.

Read section 3 before this. An important distinction must be made concerning the measurement of entangled quantum particles. A thought experiment by John Bell in 1964 proved that an entangled quantum particle is not bound to a definite value until it or its counterpart is measured. The 2022 Nobel Laureates in Physics experimentally confirmed this. Thus, entangled particles do exist in a true ambiguity of states until one or both are measured.

Entanglement

Blind to the separation of time and space, entangled quantum particles are fundamentally linked. Once their exact relationship is determined, the state of one particle can be known with 100% certainty after measuring that of its entangled partner.

“I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”

Erwin Schrödinger

To the sci-fi-inspired reader, entanglement may ignite visions of instantaneous communication at a potentially boundless distance but unwavering, nature does not allow faster than light (FTL) communications. The inability to force an entangled quantum particle into a particular state ruins our chances of building a transmitter. Moreover, there is no known method of simply observing changes to a particle’s state without measuring it and consequently, collapsing its entanglement, ruining our chances of building an unbiased receiver.

The upshot of entanglement is its utility in quantum algorithms and potentially, cybersecurity since tampering is unavoidably easy to detect in entangled quantum particles (due to the fused nature of observation and measurement). Another useful process, quantum teleportation, uses the phenomena in tandem with classical information transfer to indirectly “copy” an unknown quantum state without directly needing to transplant that particle, and over long distances (though not FTL).

Probability

Much of the baseline quantum computing ideas we study today hover around what we call quantum gates.

The X-gate functions as a classical NOT which flips a classical bit: X|0> → |1>. The Hadamard (H) gate was designed to create an even superposition, wherein the output qubit has a 50-50 chance of being in the |0> or |1> basis. The |+> state is an example of one such output qubit.

Most quantum states are nothing more than a linear combination of the basis states, along the lines of α|0> + ß|1>, where the weights α and ß are called probability amplitudes, designed to depict the probability of our qubit being in the |0> or |1> state. For the |+> state, α = ß = √½. Taking associated amplitude to the power of 2 gives the probability of being in that state. Thus, the |+> state depicts a ½ probability of the qubit being |0> or |1>.

A mathematical operation called a tensor product is used to represent joint qubit states. For example, the two-qubit system |+0>, is the tensor product of |+> and |0>. The beauty (and utility) of tensors is that the |+0> state mathematically decomposes into √½|00> + √½|10>, characterizing a system that has a ½ probability of being in the |00> or |10> states (notice that the right qubit, which was never in superposition, is always in the |0> state; whereas the left qubit, the |+>, exists evenly between |0> and |1>).

This notation visually clarifies probability analysis for multi-qubit systems.

As a final example, the controlled-NOT, or CNOT gate is a useful application for entanglement wherein two qubits are joined such that if the control qubit is in the |1> state, the state of the target qubit is flipped. As an example, let’s pass a |+> control and a |0> target qubit through a CNOT gate, entangling them. The control qubit has a 50-50 probability of being |0> or |1>. Thus, there is a 50-50 probability of the target qubit being flipped. When we measure the target qubit we will know with 100% certainty the state of the unknown control qubit: |0> if the target hasn’t changed and |1> if the target has, all without interacting with the control, and even at a physical separation of thousands of lightyears!

As a final example, the controlled-NOT, or CNOT gate is a useful application for entanglement wherein two qubits are joined such that if the control qubit is in the |1⟩ state, the state of the target qubit is flipped. As an example, let’s pass a |+⟩ control and a |0⟩ target qubit through a CNOT gate, entangling them. The control qubit has a 50-50 probability of being |0⟩ or |1⟩. Thus, there is a 50-50 probability of the target qubit being flipped. When we measure the target qubit we will know with 100% certainty the state of the unknown control qubit: |0⟩ if the target hasn’t changed and |1⟩ if the target has, all without interacting with the control, and even at a physical separation of thousands of lightyears!

2.3 Quantum Algorithms and Applications

Specific problems in the topic of quantum computation include the understanding of finding a step-by-step solution that can be later on performed by a quantum computer. This chapter will demonstrate how an algorithm is an introduction to a range of fields that involve the basics of physics, chemistry, and machine learning.

Quantum algorithms are designed to take advantage of the unique capabilities of quantum computers to solve problems more efficiently than classical computers. One well-known quantum algorithm is Shor’s algorithm for integer factorization, which can be used to find the prime factors of a given integer. This specific algorithm is an important development in the field of cryptography to break widely used public key cryptographic systems. Another notable quantum algorithm would be Grover’s algorithm, which can be used to search an unsorted database more quickly than other algorithms — including the quantum Fourier Transform — due to how mathematical operations are being performed more efficiently.

Shor’s Algorithm

Shor’s algorithm is a quantum algorithm for integer factorization, which can be used to find the prime factors of a given integer. ​​The algorithm works by using a quantum computer to find the period of a particular function. The function used in the algorithm is the modular exponentiation function, which takes two inputs, a base and an exponent, and returns the remainder when the base is raised to the exponent and divided by a given number.

The key insight of Shor’s algorithm is that the period of this function is related to the factors of the number being factored. By finding the period using a quantum computer, the algorithm can then use classical methods to determine the factors of the number in question.

This algorithm has the potential to solve problems that are believed to be intractable on classical computers. For instance, the problem of integer factorization is believed to be difficult for classical computers, as the time required to find the prime factors of a large integer grows exponentially with the size of the integer, meanwhile, this specific algorithm can solve this problem in polynomial time. Overall, Shor’s algorithm is an important development that has the potential to significantly impact many different fields.

Grover’s Algorithm

Grover’s algorithm — which was developed in 1996 by physicist Lov Grover — is a quantum algorithm that can be used to search an unsorted database more quickly than other known algorithms. This algorithm works by using a quantum computer to search a database of N items in ON time, which is significantly faster than the O(N) time required by classical algorithms. Leveraging the ability to put qubits into a superposition of states, and operations that combine or interfere with probabilities of these states, Grover’s algorithm highlights the correct state quicker than classical means. In a real system, this correct state can be associated with a single item hidden in an unsorted list, like a marble under a cup for example. Unraveling the apparent mysteries of Grover’s algorithm requires deep mathematical analysis, but it can become much more intuitive if one understands amplitude amplification, as is explained below.

To perform a search using Grover’s algorithm, the quantum computer must be initialized with a quantum state that encodes the search criteria. The algorithm then iteratively applies a series of quantum operations to this state, each of which performs a step in the search process. These operations are designed to amplify the amplitude of the quantum state corresponding to the search result, while this will simultaneously decrease the amplitude of other states. After a sufficient number of iterations, the quantum state will be highly concentrated around the search result, and the result will be able to be determined by the measuring state, as the number of iterations required depends on the size of the database and the accuracy of the search result desired.

Grover’s algorithm is an important theoretical development in the field of quantum computing, as it demonstrates that quantum computers can perform certain tasks more efficiently than classical computers, which has stimulated further research into the development of many applications, with the inclusion of database search, optimization, and machine learning.

Hamiltonian Simulation

The Hamiltonian Simulation is a task of approximating the time evolution of a given quantum system according to its mathematical representation of the system’s energy, known as its Hamiltonian. This task is essential because it can be used to simulate the behavior of quantum systems, which can help researchers understand and predict the properties of these systems.

Several algorithms can be used to perform a Hamiltonian simulation by using quantum-mechanical phenomena, as they will include the quantum phase estimation algorithm and the quantum approximate optimization algorithm. These algorithms usually work by preparing a quantum state that represents the initial state of the system and then applying a series of quantum operations to this state to simulate the time evolution of the system. As the Hamiltonian simulation has many potential applications, the result of this simulation can be determined by measuring the final quantum state, which will exhibit complex and counterintuitive behaviors that are difficult to understand in a classical computer. This is an important theoretical development in quantum computing, as it can stimulate the behavior of quantum systems more efficiently to impact different fields.

Fujitsu Algorithm

On December 22, 2022, the financial institution Kutxabank collaborated with the company Quantum Mads and the open innovation platform INNOLAB Bilbao to develop the improvement of allocation assets of Kutxabank’s investment portfolio by a digital annealer — a Fujitsu quantum-inspired solution. This project is based on two phases, which will be concluded by December 2023, as it will aim to optimize functionalities by interacting quantitatively with the model in a real and current environment. This project is mainly to create a prototype capable of detecting needs and opportunities that are covered by Kutxabank’s financial institution, which will include a technological solution in a real environment.

Quantum algorithms have the potential to revolutionize topics involving materials science, machine learning, and drug discovery, due to the potential to significantly improve the performance of learning systems by leveraging the power of quantum computers to perform computation more efficiently. However, quantum computers are still being improved and developed, making them not widely available; as such, the development and implementation of quantum algorithms is currently an active area of research.

Chapter 3 of 7

Building a Quantum Computer

Written by Mumtaz Gababa, Karthik Narayanan, Elias Lehman, and Alexa Ramirez · In partnership with Quantum Computing at Berkeley

3.1 Universal Quantum Computing: DiVincenzo’s Criteria

To build a gate-based quantum computer, a set of criteria must be achieved. DiVincenzo’s criteria, proposed by theoretical physicist David P. DiVincenzo, explains the physical implementation requirements for quantum computers, which include a set of conditions that have to be met to achieve a practical computation. The criteria are the following:

1. A scalable physical implementation of the qubits without significantly increasing the complexity of the hardware.
2. Quantum mechanical systems with two well-defined stable states are used as qubits.
3. Long coherence times to maintain the quantum state of the system for a sufficient period to perform specific computations.
4. A universal set of quantum gates.
5. A qubit-specific measurement capability to obtain the state of the qubit with a low error correction.

To meet these criteria, a quantum computing architecture must be able to manipulate and control individual qubits, perform operations on them using universal gates, correct errors that may occur during computation, and be able to scale up to large numbers of qubits by suitably connecting them. Meeting these criteria is the primary challenge for researchers, forcing them to develop novel approaches that can achieve these requirements.

3.2 Different Ways to Quantum Computers

There are several methods for building a quantum computer, each with its advantages and challenges. A common feature of these methods is that they are all designed around the use of qubits. The most suitable approach to building a quantum computer depends on the specific goals of the manufacturer, which determines what form of qubits to use, and thus the overall structure.

Several leading quantum computer manufacturers use trapped ions. These are charged atoms that are held in place by an electric or magnetic field and can be controlled using lasers. This method provides a high level of control over the qubits. The advantage of using trapped ions is that they can remain coherent, or in a state of superposition, for long periods of time. Long coherence times are required to leverage the quantum mechanics of the qubits, perform many operations, and take accurate measurements. However, trapped ions are sensitive to noise, making them prone to decoherence if they interact with the outside world. As a result, much of the research to develop trapped ion quantum computers is in isolating the qubits.

Another leading manufacturer and electronic giant IBM uses superconducting qubits, which are based on circuits that can conduct electricity with zero resistance. These qubits are easy to scale but are sensitive to noise and can be difficult to control, particularly without cryogenic cooling. One advantage of superconducting qubits is that they can achieve quick gate operations–so long as the operation preserves the information with high fidelity.

That is, the gate operations must produce the intended result. This is a leading sector of development in this subfield of quantum computing. Unfortunately, superconducting qubits have relatively short lifetimes compared to other types of qubits.

Semiconductor materials such as enriched silicon and germanium are well-studied materials that are also being developed into qubits by tech conglomerates such as Intel. These qubits can be controlled using electric and magnetic fields, and have the advantage of being relatively easy to fabricate using existing semiconductor technology. They also operate at higher temperatures than superconducting qubits. However, they are prone to decoherence from external fields.

There are other methods for designing qubits, including using the quantum properties of polarized light, the controllability of neurally charged atoms, and the fascinating study of topology.

Ongoing research is focused on reducing noise and improving error rates, to develop efficient, fault-tolerant quantum computers just over the horizon.

The choice of which method to use depends on how the trade-offs between the various characteristics of each method interact with the application the computer is designed for, ie. the manufacturer’s goals. Ultimately, the success of a quantum computer depends on its ability to maintain high gate fidelity and connectivity between qubits, especially while scaling.

3.3 Electron Spins and Semiconductors

The prototypical qubit is based on the electron spin. When an electron is placed in a magnetic field, the well-known Zeeman effect splits the quantum mechanical spin state into two, known as spin up and spin down. These states can be used as the basis for defining qubit states. Under certain conditions, by applying specific magnetic fields, we can place an electron in a superposition of the up and down states.

One way to isolate and control electrons with the precision needed for quantum computing is to use solid-state systems. In a semiconductor, for example, the energy states of the electrons are organized into bands separated by gaps. The conduction band is the high-energy band, above the gap, and the valence band is the low-energy band, below the gap. Electrons in the conduction band are charge carriers–conductive materials have many of these, while insulators have very few. We say that semiconductors are “doped” with impurities to alter their electrical properties, and the resulting p-type and n-type semiconductors can be used to create devices such as transistors. These principles are used to create quantum dot systems, in which electrons are confined in a small region on the lattice of the solid-state semiconductor, and their spin states can be controlled and manipulated for use as qubits.

Electron spin quantum computers are a promising method, especially because of their scalability using today’s semiconductor manufacturing technology. By using a semiconducting material such as gallium arsenide, electrons can be isolated and controlled as needed to make a qubit.

However, gallium arsenide has an abundance of nuclear spins which can interact with the electron spin qubit and result in an unknown phase. To perform high-fidelity qubit operations, it is important to maintain precise control over the phase of the qubit, as any lost information will result in decoherence. To reduce this effect, materials such as silicon and germanium are being studied, as they contain isotopes with an even amount of nuclear spins, such that there is no net spin interaction. Silicon and germanium can also be purified by industry, using the same techniques applied to make wafers for electronic applications.

Purified silicon and germanium can be used to build qubits which are then operated using a magnetic field of order one Tesla. This magnetic field results in an energy of micro electron volt, which corresponds to a frequency of GHz. This frequency is the Larmor frequency and is used to keep track of the phase of the qubit. Again, keeping track of the phase, preventing the loss of information, is important for executing quantum algorithms and is done by controlling the parameters that determine the interaction between the electron and the lattice, such as separation between atoms. By controlling these parameters, the properties of the electron can be engineered to build optimal qubit parameters.

3.4 Superconducting

An LC oscillator is an electrical circuit containing an inductor and capacitor in series. When brought to low enough temperatures such that the wires begin to conduct electricity without dissipating energy, better known as superconductivity, this circuit exhibits quantized energy levels. Introducing a unique component called the Josephson junction spaces the levels unevenly, giving the circuit a quality is called anharmonicity which is critical for creating controllable energy levels in any qubit. This device composed of a capacitor and Josephson junction at superconducting (extremely low) temperatures is the general scheme of a superconducting qubit. By customizing the properties of each component in the circuit, researchers are able to create energy levels and spacing however they please, like an artificial atom, in contrast to being restricted to the inherent energy levels of a natural atom. Major players in the classical computing industry such as IBM and Google, in addition to growing names in quantum like Oxford Quantum Circuits and Rigetti Computing, have adopted this device as their qubit of choice for its variability and relevance to their prior work manufacturing circuits on computer chips.

3.5 Trapped Ion

One approach to building a quantum computer is using trapped ions, a method initially developed by Wolfgang Paul in 1953. A trapped ion is an electrically charged atomic particle that is confined or trapped in a specific location by an external electric field or electromagnetic field. Ideally, ions would be trapped at the minima of an electric potential surface where the potential is essentially the barrier from which the ion cannot escape. However, ions cannot be trapped in three dimensions by electrostatic forces, rather, experimentalists produce saddle-shaped potentials where the ion can be contained in one direction and escape in the perpendicular direction. To circumvent this, a time-dependent electric field is applied oscillating at radio frequency, allowing the saddle potential to spin about the saddle point at the same frequency of the electric field, and with the right tuning of parameters, the ion is effectively trapped at the saddle point.

However, ions cannot be trapped in three dimensions by electrostatic forces, rather, experimentalists produce saddle-shaped potentials where the ion can be contained in one direction and escape in the perpendicular direction.

To circumvent this, a time-dependent electric field is applied oscillating at radio frequency, allowing the saddle potential to spin about the saddle point at the same frequency of the electric field, and with the right tuning of parameters, the ion is effectively trapped at the saddle point.

To fulfill DiVincenzo’s second criterion, we need to use these ions to create qubits. To prepare the ground state, light is used to pump electrons from a stable state to an excited state that decays to a ground state where the laser cannot couple to, therefore remaining in that state. This process, known as optical pumping, continues until all the electrons are pumped into the ground state, creating an optical qubit. In some ions, there is a hyperfine structure where the ground and excited states have a small energy gap that cannot be accessed with light. As a result, an electron essentially has an infinite lifetime in the excited state, allowing for longer coherence times. A process similar to that of optical pumping can be used to reach these excited states to create a hyperfine qubit; however, this process poses some challenges in gate implementation. To measure the qubit, a photomultiplier tube or a charge-coupled device can collect photons that are released as a result of an electron decaying to a ground state that is not coupled to the laser.

We now have our ground and excited states, how do we make a superposition of them? For optical qubits, we can utilize electric quadrupole transitions and for hyperfine qubits, we can utilize magnetic dipole or stimulated Raman transitions to achieve a superposition of qubit states. For two-qubit quantum gates such as the controlled-NOT (CNOT) gate, we need to scale the system by placing trapped ions in a 1D array or an ion chain. From there, laser pulses can be applied to the chain to give a superposition of states until the system is maximally entangled or two qubits are correlated in such a way that the state of one qubit cannot be described independently of the state of the other qubit.

The trapped ion method is the most widely used as the ions can be isolated from their environment, which makes them useful for precision measurements and other applications that require high levels of control and stability. It also possesses long coherence times where the qubits can be in a superposition state for a while before it becomes decoherent. However, it can be difficult to scale up to larger numbers of qubits because of decoherence within the ion chain. As soon as a hundred ions are used, the frequency values needed to excite the electrons become indistinguishable. As a result, a very precise laser frequency is needed to excite the electrons or it could end up in the wrong quantum state.

Regarding DiVincenzo’s Criteria, the trapped ion model does theoretically fulfill all criteria; however, in reality, scalability remains an issue, undermining some of the criteria. Yet, the combination of control, stability, scalability, and long coherence time makes the trapped ion model a promising platform for building large-scale quantum computers.

3.6 Photonic

Photons, or particles of light, exhibit quantum behaviors of their own allowing them to be utilized for qubits. Unlike trapped ions and neutral atoms that require cryogenic or laser cooling, photonic qubits can operate at room temperature, making them a promising candidate for building large-scale quantum computers. A leading manufacturer of photonic quantum computers, Psi Quantum, refrigerates the systems to about 4 Kelvin using liquid Helium to optimize the fault tolerance of its photodetectors, whereas competitor Xanadu does not cool the qubits, just the photodetectors to less than 100mK.

The main method of building photonic quantum computers is the KLM protocol developed in 2000 where a light source that produces as few coherent photons as possible travels through fiber optic cables where the qubit states are based on the photon’s modes of polarization. In general, photons can exhibit infinite modes of polarization; however, in the KLM protocol, the photons are either in vertical or horizontal modes as the probability of a photon occupying more than one mode is zero. Therefore, a qubit state can be represented by a photon being in either of these two modes where these states are known as Fock states.

To implement qubits and gates, linear optical elements such as mirrors, beam splitters, and phase shifters are used. Beam splitters can split a beam of photons into two or more separate beams, leading to superposition, and phase shifters can change the phase of the photons which becomes important when entangling them. The combinations of beam splitters, mirrors, and phase shifters lead to a wide variety in the creation of multi-qubit quantum gates. Creating entanglement is more difficult as photons do not interact with each other; however, through a series of phase shifters, photons traveling through different fiber optic cables, entangled states can be created where many of the states that do not become entangled are filtered out.

To determine these final states, photodetectors are used where in a given elementary quantum gate, the photon in the top mode channel corresponds to a state of 0, while a photon in the bottom channel corresponds to a state of 1. As the combination of beam splitters, mirrors, and phase shifters can lead to non-deterministic gates, measuring the fidelity or accuracy of these gates can be very difficult as the operations would have to be performed repeatedly and can become exponentially larger as the number of gates increases. To circumvent this, researchers have used a phenomenon known as quantum teleportation to decrease the time and required resources to determine the fidelity of these systems. Instead of doing these operations successively, each gate would be operated offline and the event signal would be sent back to the quantum circuit using quantum teleportation.

By using photons, these quantum computers can operate at room temperature and through a combination of a few linear optical tools, a universal set of quantum gates can be created. Although using photons as the source of qubits is a seemingly robust system, photons can escape to their environment like any system. However, this is a problem as photons are in a superposition of Fock states, photons escaping can change the state of the photons within the circuit. As phase shifters are used to change the phase of photons, phase decoherence can occur where the phase of the state can change due to factors such as including noise in the control signals that are used to manipulate the quantum system, imperfections in the physical components of the system, and other sources of noise and perturbation. This can become problematic, especially when it can lead to unentangled pairs or affect successive quantum gates in a circuit.

Regarding DiVincenzo’s Criteria, the photonic quantum computer again does theoretically fulfill all five criteria; yet, photon loss, phase decoherence, and efficiency of the quantum gates remain an issue. Simpler methods of photonic quantum computers are being developed to address these issues, hopefully leading to a scalable, fully functional quantum computer.

3.7 Neutral Atom

Another method of building a quantum computer is using neutral atoms or atoms that have no net electrical charge. Although ions are easy to trap due to their charge, neutral atom quantum computers are less sensitive to stray electric fields, making them a promising candidate for quantum processors. Not just any neutral atom can be used to create these quantum processors, but group 1 or 2 alkali metals such as rubidium are mainly used as its one valence electron gives an electronic structure similar to that of hydrogen.

The method of trapping neutral atoms is not the same as ions since they do not respond to electric fields as well due to their neutral charge; however, they can be trapped by light using a method such as optical tweezing (magneto-optical trap). It uses the optical gradient force of a highly focused laser beam to trap and manipulate small particles or in this case neutral atoms. The particles are cooled to very low temperatures and are trapped in the focus of the laser beam, which creates a strong attractive force that suspends the particles in place in an ultrahigh vacuum. Then, a second laser passes through aperture lenses, creating a strongly focused beam that can act as a “tweezer” where it can control the path and motion of these neutral atoms.

From the optical tweezing method, creating qubits is very similar to the trapped ion case where two electronic levels of a neutral atom can be manipulated and controlled using lasers to create an optical qubit. As there are infinite discrete energy levels in an atom, there can be many configurations of the two electronic states used. Some alkali metals present with a hyperfine structure as rubidium where these hyperfine states are separated by microwave frequencies. By creating hyperfine qubits, the small energy gap between hyperfine states allows for an electron to stay in an excited state for a very long time before decoherence occurs. Therefore, hyperfine qubits have longer coherence times than their optical counterparts. However, these qubits are still susceptible to decoherence as their unpaired electron can interact with unwanted magnetic fields or unfavorably couple to the tweezer light.

To scale up to a two-qubit system or more, the qubits are stored in an array of optical traps, which are created by an incident beam being split up into many beams that can be focused through aperture/objective lenses. Because neutral atoms can only interact with one another through contact collisions, the only way these atoms can interact is by dipole forces if they present with a large electric dipole moment.

To create these sizeable electric dipole moments between two atoms, one of the atoms is targeted by a laser pulse that can drive the atom into a highly excited electronic state known as a Rydberg state. The Rydberg state causes a phenomenon known as a Rydberg blockade where a strong electric dipole moment is created, preventing the other atom from excitation. As such, these two atoms are entangled, a key feature needed for the implementation of quantum algorithms and protocols.

To measure the qubits, fluorescence images can be collected (through an electron-multiplying charge-coupled device) where qubit state 0 will appear as bright while 1 will appear as dark. Other methods to measure qubits include state-selecting imaging or electron shelving by resonant absorption imaging or electromagnetically induced transparency.

With the neutral atom method, some computers have scaled to over one hundred qubits. Unlike trapped ion qubits, these neutral atoms weakly interact with their environment, becoming less susceptible to stray electric or magnetic fields. Therefore, scaling neutral atom quantum computers is primarily limited by the system that generates the magneto-optical trap. Yet, scalability remains an issue due to the large experimental setup required to trap these atoms.

The method of using optical tweezers is a powerful tool in trapping neutral atoms, but there are factors such as trapping efficiency and spatiotemporal resolution that limits its ability to trap these atoms effectively. Optical tweezers have a low trapping efficiency, which means that not all particles that are exposed to the laser beam will be trapped. This can be particularly challenging when trying to trap large numbers of particles. The spatial resolution of optical tweezers is limited by the diffraction limit, which means that the position of the trapped particle can only be determined with a certain level of uncertainty. This can limit the precision of measurements and manipulations that are performed using optical tweezers. The temporal resolution of optical tweezers is limited by the laser pulse duration and other factors, which can limit the speed at which measurements and manipulations can be performed.

Regarding DiVincenzo’s Criteria, the neutral atom again does theoretically fulfill all five criteria; yet, scalability remains an issue due to the complicated setup to trap these atoms.

Overall, neutral atom quantum computers are a promising approach to quantum computing and have the potential to enable the development of powerful quantum algorithms and protocols.

3.8 Topological

Models of topological QC rely on a quasi-particle called an anyon. Crucially, these particles live on a 2-dimensional surface and are an emergent property of the underlying physics of a system. Consider 2 anyons, anyon A and anyon B, restricted to move on a 2D surface (like a sheet of paper). Keep anyon A still, and move anyon B in a loop around anyon A, bringing it back to its original position. Although we are back where we started, in 2D this path is topologically different from doing nothing at all, since we cannot mold a path that does go around anyon A into a path that does not, without hitting against anyon A. In 3D, we can simply take the loop under anyon A, and mold this path into a dot without hitting against anything, as we have one more dimension to work with.

There are several challenges involved in the physical realization of topological QC models. Chief amongst them is verifying whether systems thought to exhibit anyonic behavior do exhibit the desired properties. Potential candidates include so-called Majorana zero modes in superconducting materials, Majorana zero modes in chiral p-wave Fermi gasses, and vortex-antivortex pairs in Bose–Einstein condensates.

So far, the only system claimed to have been observed is Majorana zero modes in superconductors, in a preprint by Microsoft researchers in July 2022. Majorana zero modes have yet to be robustly proven viable for topological quantum computing.

Topology is a field of mathematics that studies properties of geometrical objects that are invariant under continuous transformations. Think of molding a piece of clay, without tearing it, creating holes in it, or sticking it to another piece. To a topologist, a doughnut is the same as a coffee cup: both have the same number of holes and can be continuously (in the above sense) molded from one to another. The idea is this: if the information on a quantum computer were encoded in a material’s topological properties, then small disturbances to the system (molding the clay), will not destroy this information. It would be topologically protected!

So for a 2D physical system where topology matters, simply taking one particle in a loop around another has the potential to induce a physical change to the system, as it is topologically different from doing nothing. This is the essence of topological QC. Models of such anyonic systems include Fibonacci anyons and Ising anyons. In these models, looping particles around each other (called ‘braiding’) induces unitary transformations, which can be combined to form quantum gates such as NOT, CNOT, and the Hadamard gate. Information is topologically protected because it is manipulated and stored in the bulk of the system and not in any one particle.

Currently, systems are not being commercially engineered using this approach.

The difficulty in verifying anyonic behavior, as evidenced by Microsoft’s earlier struggles with Majorana zero modes is far outweighed by the improvements in decoherence times and inherently lower error rate due to topological QC, which has a real chance of being the preferred method of quantum computing soon.

3.9 Industry Timeline

Unlike other deep technology industries, which often advance primarily in national labs and universities, the frontier of quantum computing is being pushed in the private sector. The only computers available to the public are commercially produced and they are quickly evolving. The Quantum Insider’s intelligence platform features a qubit roadmap to best understand the trajectories in which this industry is heading.

Chapter 4 of 7

NISQ and Error Correction

Written by Elias Lehman · In partnership with Quantum Computing at Berkeley

4.1 The NISQ Era

The noisy intermediate scale quantum (NISQ) era is a term used to describe the current state of quantum computing, in which quantum computers can perform certain tasks that are beyond the capabilities of classical computers but are not yet able to perform a wide array of quantum algorithms with high precision and low error rates. Leaving the NISQ era will be marked with milestones in the form of research supporting quantum supremacy. Claims of this tend to make headlines so it’s no surprise that Google’s 2019 publication, one of the first papers claiming a quantum computer could solve a classically intractable problem, was met with skepticism.

The NISQ era is characterized by the fact that quantum computers have a limited number of qubits and a high error rate. As a result, NISQ quantum computers are not yet able to perform certain tasks, such as factoring large numbers or simulating quantum systems with high accuracy, that are believed to be within the capabilities of fully-fledged quantum computers. The NISQ era is considered to be a transitional stage in the development of quantum computing, as researchers and engineers continue to work on developing larger, more powerful quantum computers that can perform more complex tasks with greater accuracy. It is hoped that the NISQ era will eventually give way to a more applicable quantum computing era, in which quantum computers will be able to perform a wide range of tasks with high precision and low error rates.

Despite these limitations, NISQ quantum computers are already being used for a variety of applications, including machine learning, optimization, and chemistry. These applications are possible because NISQ quantum computers can perform certain tasks that are difficult or impossible for classical computers to perform efficiently, such as finding the ground state of a quantum system or searching through a large dataset for specific patterns. Furthermore, there are numerous approaches to optimizing NISQ-era quantum computers such that they can achieve tasks on the frontier of today’s technology. These optimization techniques are in the form of addressing and relinquishing the high errors that are native to quantum computing.

Quantum error correction (QEC) and error mitigation (EM) are two techniques that are used to improve the reliability and accuracy of quantum computations. Both techniques aim to reduce the effects of noise and errors that can occur during the execution of quantum computation. However, they approach this problem in different ways.

4.2 Error Mitigation

Error mitigation involves post-processing the results of a computation to reduce the impact of errors. This is typically done by applying statistical techniques to the results of the computation, such as extrapolation or bootstrapping, to reduce the noise and errors present in the data. Extrapolation involves fitting a model to the data and using it to estimate the true value of a quantity. Bootstrapping involves resampling the data and using the resampled data to estimate the uncertainty in the measurement.

One modern extrapolation approach, called zero-noise extrapolation (ZNE), involves the purposeful application of noise to a quantum circuit. A researcher may add excessive noise, followed by a measurement of how a large amount of noise is affecting the circuit, and working in reverse, extrapolating away the small amount of noise that is naturally present. This is like taking the limit of the noise to an extreme, to see the patterns that it makes in the data, and expecting those patterns to be present on the minute level as well.

Another way to mitigate errors is to rely on the resilience of classical computers. Hybrid quantum-classical algorithms combine quantum and classical computations to perform tasks that are beyond the capabilities of either approach alone. Hybrid algorithms can be used to perform EM by using quantum computation to generate data, and classical computation to process and correct the errors present in the data.

More advanced extrapolation utilizes machine learning to learn the noise and error patterns present in quantum computation and uses this information to correct or mitigate the errors. This approach often uses the hybrid quantum-classical design. As you can imagine, a NISQ computer cannot accurately process a machine learning model to learn its noise, so instead, we rely on classical computers.

4.3 Error Correction

While EM is centralized around post-processing, quantum error correction (QEC) involves actively detecting and correcting errors as they occur during the computation. A common approach goes by the technical name “quantum subspace expansion”. This is achieved by using additional qubits, known as “ancilla” qubits, to perform error detection and correction operations. The ancilla qubits are used to encode the state of the computation redundantly, such that errors can be detected and corrected by applying appropriate operations to the ancilla qubits.

The surface code is a topological quantum error-correcting code that encodes quantum information in the state of a two-dimensional array of qubits. The qubits are arranged in a grid, with each qubit interacting with its neighbors through a series of controlled Z gates.

Additional methods include variational QEC and tensor networks. Do not be daunted by the jargon, these methods are more intuitive than they appear. For example, variational QEC uses a parametrized quantum circuit, a quantum circuit whose gates can be adjusted, to learn the error patterns present in the computation. Fluctuating parameters to reduce noise but maintain the state of the system is at the core of this technique.

The surface code can detect and correct for a single quantum error that occurs within a certain region of the grid, known as a “face” or a “plaquette.”

The color code encodes quantum information in the state of a three-dimensional array of qubits. The qubits are arranged in a lattice, with each qubit interacting with its neighbors through a series of controlled Z gates. The color code can detect and correct for multiple quantum errors that occur within a certain region of the lattice, known as a “cell.”

While the surface code and the color code require several qubits to achieve the redundancy necessary for error correction, bosonic codes try to reduce errors by leveraging the properties of a single quantum system. They are based on quantum harmonic oscillators with an infinite number of energy levels. By encoding information in a combination of these energy levels, redundancy is possible with a single physical system. The most common bosonic codes are:

– GKP code: named after its inventors Gottesman-Kitaev-Preskill, this code corrects for both bit-flips and phase-flips by using grid states of the oscillator.

– Cat code: named after Schrödinger’s cat, this code can exponentially suppress bit-flip errors at the expense of a linear increase in phase-flip errors, as the number of photons in the oscillator increases.

Bosonic codes alone cannot bring the error rates down to a useful level, so they are usually implemented in addition to error correction codes such as the surface code. But their natural robustness against errors greatly reduces the number of qubits required to reach useful logical error rates.

Both the surface code and the color code have been proposed as potential candidates for implementing fault-tolerant quantum computation, which would allow quantum computers to perform large-scale computations with high accuracy. However, both codes have their strengths and limitations, and further research is needed to determine their practicality for use in real-world quantum computing applications.

4.4 Challenges to Realizing Fault Tolerance

In summary, QEC and EM techniques have a wide range of applications in NISQ computing. Error mitigation can be used to improve the accuracy of measurements and simulations performed on NISQ computers, which are currently limited by noise and errors. Error correction, on the other hand, is essential for building large-scale, fault-tolerant quantum computers, which will be able to perform quantum computations with high reliability and accuracy. Despite recent progress, there are some significant barriers standing between today’s NISQ computers and tomorrow’s reliable quantum computers. For one, quantum errors can be difficult to detect. Errors are due to a variety of factors, such as noise in the quantum system or environmental interference. These errors often leave no trace except for their effect on the outcome of the computation.

QEC is resource-intensive because it typically requires a large number of qubits and gates to encode and decode the quantum information in a way that allows errors to be detected and corrected. Currently, one “logic” qubit must be composed of several qubits and their ancilla components plus many operations.This can be a challenge, as current quantum computers are limited in the number of qubits they can manipulate. With respect to the operations required to encode and decode qubits, they also have limited gate fidelity (i.e. the accuracy of the gate) and are prone to errors of their own. Most QEC and EM techniques require a high degree of control over the quantum system, as even small errors in the control can result in errors in the computation.

All of these techniques offer promising approaches for improving the accuracy and reliability of quantum computations. However, this is a rapidly developing field, and many of the approaches that have been proposed are still being researched and refined. As a result, it is difficult to predict which will ultimately prove to be the most effective for realizing fault-tolerant quantum computation.

Chapter 5 of 7

Quantum Annealers and Simulators

Written by Elias Lehman · In partnership with Quantum Computing at Berkeley

5.1 Quantum Annealers

A quantum annealer is a type of quantum computer that is designed to solve optimization problems in which the goal is to find the maximum or minimum value of a certain function. These types of problems are important in many fields, including finance and industrial operations, and logistics.

During the quantum annealing process, the qubits are gradually “annealed,” or cooled, from a high-energy state to a low-energy state. This is done by applying a series of quantum operations to the qubits, which control their evolution. The goal of this process is to reach the ground state of the system, which is the state that minimizes the energy of the system. In the case of an optimization problem, the ground state corresponds to the optimal solution to the problem. This is because the energy of the system is encoded by the function being optimized, such that the minimum energy state corresponds to the minimum value of the function.

The adiabatic theorem is an important quantum mechanical concept in the operation of quantum annealers. It states that if a quantum system is evolved slowly enough, it will approximately remain in the same state at all times. This means that if the qubits in a quantum annealer are initialized in the ground state of a higher energy system, then they are allowed to evolve slowly enough to a lower energy system, they will remain in the ground state, which minimizes the energy of the system. This is a careful process – if the qubits are evolved too quickly, they may become excited and reach higher energy states, which can lead to errors in the solution.

Quantum annealers have the potential to revolutionize many fields by providing faster and more efficient solutions to optimization problems. However, they also come with some challenges. For example, quantum annealers rely on a specific sequence of quantum operations to reach the ground state of the system. Any noise or errors that occur during the quantum annealing process can lead to incorrect solutions. Quantum annealers are specifically designed to solve optimization problems and are not as versatile as other types of quantum computers, which can be used to perform a wider range of quantum algorithms. This means that quantum annealers are limited to a specific set of applications. These challenges can limit the ability of quantum annealers to solve complex problems and may limit their widespread adoption. Still, quantum annealers are an exciting development in the field of quantum computing and have the potential to have a significant impact.

5.2 Quantum Simulators

While we wait for quantum supremacy to address intractable computational challenges, we need to study the potential impact of quantum computers and which approaches will be best applied to which of these problems. To do so, researchers are utilizing advanced classical computers, a combination of high-power GPUs and software, to replicate quantum computation. These quantum simulators do not avoid the limitations that classical computers face when attempting to perform algorithms of exponential complexity. Rather, we can apply these simulations to classically solvable problems or run primitive quantum algorithms and protocols to test the performance of a particular quantum computing architecture. Furthermore, we have the ability to control the amount of noise that our simulations experience, giving researchers the ability to test how accurately certain problems will be solved on quantum computers across different generations with varying levels of noise.

To take early first steps towards practical uses for quantum computing, we experiment using classical simulators that are designed to mimic the real behavior of noisy quantum computers. These give insight into which quantum computer design is best suited for which problems. Take, for example, a problem in quantum chemistry: simulating the quantum mechanical effects of molecules. Quantum computers perform operations on quantum states, allowing them to accurately mimic the behavior of other quantum systems without classical limitations. These purely quantum simulations can be used to predict the properties of atoms and molecules, which is valuable for understanding chemical reactions, designing new materials, or discovering new drugs. Unfortunately, this potential is unachievable until we elevate from the NISQ era, but given a simple enough simulation, we can emulate the effectivity that today’s quantum computers, and those to come in the future, have in solving these types of problems.

When simulating a quantum computer classically, one uses a classical computer to mimic a quantum system’s behavior. This is primarily because of accessibility – despite their high cost, even the most advanced CPUs and GPUs are much more widely available and easier to use than quantum computers. Classical computers are also much more reliable and far less prone to errors than quantum computers, as we saw in Chapter 4. State-of-the-art simulators are possessed by leaders in quantum computing, as well as classical computing, and national research facilities.

Quantum computing companies, such as Xanadu, and IBM offer specialized simulators designed around the specifications of their quantum devices, whereas Amazon and Microsoft offer a suite of simulations that replicate various types of quantum computers and run on in-house classical hardware.

Leading GPU manufacturer, NVIDIA has pivoted its focus toward quantum computing in recent months, curating a hybrid computing platform with an emphasis on quantum simulations.

Currently, classical simulators allow us to study the behavior of quantum systems at much larger scales than is currently possible with quantum computers. They can be useful for studying the performance of quantum algorithms and protocols, or for testing the accuracy of quantum simulations performed on a quantum computer. However, it’s important to note that classical simulations have a ceiling as classical computers struggle to simulate complex quantum systems. Gradually, quantum computers will begin to achieve supremacy in the simulations of quantum systems.

Chapter 6 of 7

Quantum Communication, Cryptography, and Networks

Written by Alexa Ramirez and Elias Lehman · In partnership with Quantum Computing at Berkeley

• Shor’s Algorithm
• Quantum Key Distribution
• Post-Quantum Cryptography
• Quantum Networks

6.1 Shor’s Algorithm and the Threat It Poses

In 1994, mathematician Peter Shor developed Shor’s algorithm focusing on finding the prime factors of a large composite integer. This algorithm works by using quantum mechanical principles — such as superposition and entanglement — to perform a series of operations that can factorize a large integer on a quantum computer.

The main idea of Shor’s algorithm is to use a quantum computer to measure the periodicity of modular exponentiation, which is a mathematical operation that calculates the remainder when an integer is raised to a power and then divided by a positive integer in the modulus. This operation is used to find the period of a function, which is in turn used to determine the prime factors of a composite integer. The algorithm works by computing the period of the smallest positive integer (the “modulo”) that equals one, using a unitary operation that implements the modular multiplication function. In essence, Shor’s algorithm exploits the unique properties of quantum mechanics to quickly solve a problem that is classically difficult.

Although Shor’s algorithm is a powerful factorization method, its implementation can be challenging due to several limitations. One of these is that the algorithm requires a quantum computer to run, which is currently less reliable and powerful than classical computers when it comes to performing such tasks. Moreover, Shor’s algorithm is probabilistic, which means that it cannot find all the prime factors of a given integer with absolute certainty. Instead, the algorithm provides a probability of finding the prime factors, and it may require multiple runs to obtain the correct results for large composite integers. The probability of success decreases as the size of the integer grows.

Shor’s algorithm is an active research area in quantum computing due to its potential to transform various fields of science and technology, particularly cryptography. However, the widespread implementation of this algorithm would have negative consequences for communication security and data storage. This is because Shor’s algorithm can efficiently factor large numbers, which would render many existing public-key cryptographic systems vulnerable to attacks. As a result, there is a need to develop new cryptographic systems that are resistant to quantum attacks, a field of research known as post-quantum cryptography.

6.2 Quantum Key Distribution

Quantum key distribution is a method known for secure communication that uses the principles of quantum mechanics to transmit a secret key between shared parties. This key is being used to encrypt and decrypt messages transmitted over the internet having no risk of the key being intercepted by an eavesdropper. These parties share a random sequence of quantum states — typically carried by photons — that are transmitted over a quantum channel, which is known as a communication channel that preserve the quantum state of the particles being used to transmit the key.

In quantum key distribution there’s an exchange of quantum states over a communication channel by the sender and the receiver. These quantum states, which are typically photons, are encoded with the key, to encrypt and decrypt the message. As the key is randomly generated, the sender and the receiver use a series of measurements and communications to verify the key and ensure that it hasn’t been compromised.

For instance, in 1984, Charles Bennett and Gilles Brassard developed a protocol called BB84, also known as the Bennett-Brassard 1984 protocol. This protocol is the original quantum cryptography protocol in which it uses two non-orthogonal states to encode the key and relies on the Heisenberg uncertainty principle to detect the presence of an eavesdropper. In this case, the sender is known as Alice, the receiver as Bob, and the eavesdropper as Eve. The non-orthogonal states that are being employed are states that are not mutually exclusive and cannot be perfectly distinguished from one another. Here, Alice randomly selects one of these states to decode the key, while Bob measures the state to decode the key, which then will disturb the quantum states if an eavesdropper intercepts the key.

This can be detected by error correction which increases privacy techniques. The quantum key distribution differs from the classical communication methods by its increased security and privacy of information because the principles of quantum mechanics make it impossible for an eavesdropper to intercept the key without being detected. Thanks to these principles, this makes the key features extremely secure against adversaries with unlimited resources. Heisenberg’s uncertainty principle, in particular, is responsible for the inability to learn the key without conducting a measurement and being detected.

Quantum cryptography research is focused on developing secure communication protocols over our existing networks. Researchers have developed various protocols that enable the establishment of a shared secret key between parties in a secure and efficient manner. The primary application of these protocols is in contexts where secure communication is critical, such as military and governmental communications. Quantum cryptography represents a promising new direction in the field of secure communication, given its potential to provide security that is nearly impossible to breach without detection.

6.3 Post-Quantum Cryptography

Post-quantum cryptography refers to cryptographic algorithms that are designed to be secure against attacks by quantum computers. As exhibited by Shor’s algorithm, quantum computers have the potential to break cryptographic algorithms that are currently widely used to secure communication, such as the RSA cryptography algorithm which is used to secure communication over the internet. Cryptography specialists theorize that attackers can collect encrypted data packets, and in the future may decrypt them using quantum computers – a so-called steal now, hack later approach. In response to this threat, cryptographers in the private sector, academia, and government labs alike have sought new cryptography methods that are resistant to hackers using both classical and quantum computers.

Quantum-resistant cryptographic algorithms rely on mathematical problems that are hard to solve on both classical and quantum computers. Some algorithms employ the learning with errors (LWE) problem and its variant, ring learning with errors (ring-LWE) problem for encryption and decryption. Another example of what is being used is code-based cryptography, based on the difficulty of decoding error-correcting codes.

The sensitivity of government data, and thus the importance of secure transmission of information, prompted the National Institute for Standards and Technology (NIST) to seek the best PQC algorithms for standardizing modern communications. In 2017, NIST launched a competition to select the most secure algorithm. In July 2022, after three filtration rounds, NIST announced their top four finalists, ready for implementation in today’s communication systems. The algorithms were divided into two categories – those for key-establishment, and for digital signatures. NIST recommends two primary algorithms to be implemented for most use cases: CRYSTALS-KYBER (key-establishment) and CRYSTALS-Dilithium (digital signatures). As alternative signature schemes, FALCON and SPHINCS+ were also be standardized.

There are four additional runner-up algorithms currently in a fourth round of evaluation before being considered for standardization. Among these runner-ups was the algorithm SIKE, which in August 2022 was broken by a classical, single core processor and is thus no longer in the competition.

Post-quantum cryptography is an area of research that proposes important factors to these algorithms by preparing information security systems to be able to resist quantum computing attacks. As there is no longer a debate over when we should transition to quantum-secure algorithms – because the risk is evident – there is an ongoing effort to establish which algorithms will replace today’s cryptography protocols.

6.4 Quantum Networks

An interesting network, still in the theoretical stage, is a quantum oscillator network. A quantum oscillator network is a system of quantum oscillators that are connected and interact with each other through their quantum states. As they can be modeled by different physical systems such as photons in a cavity or atomic states in a solid, these networks invoke the transmission of information using the principles of quantum mechanics. This specific network can leverage widespread entanglement which can support the lifetime of quantum states of the oscillators and the complexity of modeling and simulating similar systems.

Quantum networks transmit and process quantum information, analog to classical computer networks such as the internet. As we have examined before, due to the unique properties of quantum systems, quantum networks can be more secure and faster at certain tasks than classical networks. In a quantum network, particles such as photons or atoms are used to encode and transmit information.

An ideal quantum communication network transmits information securely between two parties. This type of network can use multiple techniques, such as QKD, to establish communication channels that are resistant to eavesdropping. Another type of network involves a quantum computing network, which uses quantum computers as nodes connected over a network, typically optical fibers that transmit photons, to perform distributed quantum computations.

On December 23, 2022, NTT Research, Inc. developed an application involving the quantum oscillator network through the cyber coherent Ising machine (cyber-CIM), a quantum algorithm that can be implemented on a modern digital platform or hybrid quantum-classical machine in the future. This application mainly focused on how the quantum oscillator network can solve combinatorial optimization problems, which were proposed by evaluating the performance of the cyber-CIM algorithm. Researchers are continuously working to develop new strategies for these types of technologies, to create scalable and practical applications.

Chapter 7 of 7

Quantum Sensing and Metrology

Written by Daniel Christensen, Karthik Narayana, and Mumtaz Gababa · In partnership with Quantum Computing at Berkeley

This chapter focuses on Quantum sensing and Metrology. Quantum sensing is the use of quantum systems to make atomic-scale measurements. Quantum sensing relates to Quantum Metrology which seeks to create highly sensitive and high-resolution measurements of physical conditions using quantum theory.

7.1 Introduction

Quantum sensing and metrology have become very useful in scientific research due to the advances made in technology within the 21st century. One field that greatly benefits from the advancements of quantum sensing and metrology is the biomedical field. The biomedical field is an interdisciplinary field that studies disciplines such as biology and chemistry to discover and solve problems in the healthcare field. In the Biomedical field, many problems require the application of state-of-the-art technology such as nanowire array-based cameras/sensors, MRIs with the use of SQUID, superconducting quantum sensors, and bioluminescence imaging to research problems on an atomic scale requiring the utmost sensitivity in the technology.

7.2 Biomedical

Nanowire array-based sensors with quantum efficiency make use of nanomaterials to gain high precision. Sensors that are nanowire array-based consist of nanowires that are made from metals and alloys. These nanowires are also known as quantum wires because the physics of the wires experience quantum mechanical effects. They are very thin with a diameter of less than or about 100 nanometers generally. One nanometer is 1×10-9 (0.000000001) meters. The use of nanowires in sensors allows researchers the ability to gather data to solve problems on an atomic scale for many applications such as drug discovery, cancer treatment, and DNA detection. These fields are currently at the forefront of research within the field. Nanowire array-based cameras (cameras that incorporate nanowire arrays) are made to be highly sensitive to specific stimuli to allow for high-resolution imaging. This high-resolution imaging allows researchers to observe phenomena and research at an atomic scale. The use of nanomaterials to create highly sensitive sensors has had a positive impact on the Biomedical field due to being able to observe phenomena on a quantum scale.

Magnetic Resonance Imaging (MRI) is the use of magnetic fields and radio waves to provide details of body parts with high precision. An MRI makes use of quantum principles such as radio waves. SQUID stands for Superconducting Quantum Interference Device which has been used to advance imaging technology in MRI machines. SQUID is an example of technological development in quantum sensing and metrology. The use of SQUID in the Biomedical field has allowed researchers to gather high-resolution images of the human brain and various other samples. The use of the ultrasensitive detector present in SQUID allows researchers in the biomedical field to map out anatomical features with high resolution and could even be applied to other fields of research outside of the biomedical field.

Superconducting quantum sensors make use of nanomaterials to allow for accurate data to be gathered. SQUID could be considered a superconducting quantum sensor. The use of superconducting quantum sensors has many applications in protein-biophysics which contribute to biomedical research. Protein-biophysics is the use of methods in physics to explain biological facts specifically relating to proteins. Superconducting quantum sensors are used in devices used in fluorescence spectroscopy which is the use of a beam, like ultraviolet light, to excite electrons causing them to emit light. Fluorescence spectroscopy is used in protein biophysics to find information about the structure and function of proteins. Having devices that can reveal information about the proteins down to an atomic level due to having highly sensitive devices with high-resolution images at such a scale contributes tremendously to biomedical research.

In cancer treatments such as chemotherapy, patients are given doses of drugs based on factors such as body size, age, and body surface. This method does not entirely take into account internal variables within the patient’s body such as metabolism. Bioluminescence imaging (BLI) devices allow observations of ongoing biological processes due to the natural light living organisms emit. BLI devices make use of a visible light camera where the use of superconducting nanowires in the camera allows for higher resolution imaging. These BLI devices allow for the imaging of responses to cancer therapy treatments, cancer metabolism, and much more. Based on the imaging results for a given parameter, a doctor could increase or decrease drug dosage and monitor its effect on the patient. The BLI devices allow researchers to observe cancer and cancer treatment on an atomic scale, which is promising in ongoing cancer research. With more advances in imaging technology with quantum sensing and metrology, drug monitoring for cancer treatment using bioluminescence imaging could become vastly more effective.

The use of nanomaterials to create nanostructures such as nanowires to advance technology that advances observations made at an atomic scale has considerably improved research within the biomedical field. These advancements made with the use of nanowire array-based cameras/sensors, MRIs, superconducting quantum sensors, and bioluminescence imaging are only a few examples of recent and ongoing research in the Biomedical field that makes use of Quantum sensing and Metrology. Soon, this technology might cause breakthroughs in the healthcare field and related fields.

7.3 Computing and Communication

A quantum computer whose output cannot be read is useless. For superconducting qubits, that’s where quantum technology comes in, in the form of low noise amplifiers for qubit readout. Where typical semiconductor-based amplifiers produce noise that is 10-20 times larger than the qubit signal, quantum-limited amplifiers produce only the minimum noise required by the laws of quantum mechanics. Examples of such detectors that are frequently used (eg. by IBM Quantum) include Josephson Traveling Wave Parametric Amplifiers (JTWPAs) and Josephson Parametric Converters (JPCs) which can deliver amplification gains up to a factor of 100.

Quantum technology also plays an important role in the readout of quantum information in the field of quantum communication. For example, superconducting nanowire single-photon detectors (SNSPDs) are more efficient at detecting photons and produce less noise than their semiconductor-based counterparts. When a photon hits an SNSPD, the nanowires stop superconducting and enter a resistive state.

This property is used to detect when a photon has been absorbed by the detector. These detectors can be designed to be receptive to photons of different wavelengths that match the source being detected. The development of nano-scale quantum devices requires nano-scale analysis tools. The near-field scanning microwave microscope (NSMM) allows for this by probing samples with microwave frequency radiation. Probing solid-state quantum systems such as superconducting circuits without collapsing quantum superposition is challenging because the interactions must necessarily be quantum coherent. One way to achieve this is by using NSMMs that operate at cryogenic temperatures so that the interaction with the quantum system being measured is quantum coherent. NSMMs are typically a combination of an atomic force microscope (AFM), which maps the deflection of a probe due to atomic forces from the sample, with a microwave detector/emitter at the tip. Changes in the phase, amplitude, and frequency of the received microwave signal convey key parameters of the sample such as conductivity, polarization, and dielectric constant.

An important challenge in the development of quantum computers is to measure applied power to quantum circuits and calibrate microwave lines, all of which operate at millikelvin temperatures. Room-temperature characterization devices (such as power meters) built using semiconductors do not function well in this regime as their transfer functions change, making them ineffective. Quantum mechanics presents a novel solution to the problem: use a two-level quantum system, and measure the rate of photon absorption by this system. This rate directly depends on the voltage amplitude of the microwave signal we wish to measure, due to the coupling of the electromagnetic radiation with the two-level system. Such a sensor is called an “absolute power quantum sensor” and can be used to study the characteristics of superconducting microwave components. As seen in these examples, quantum technologies present critical solutions to challenges in the quantum communication and computation industry.

7.4 Fundamental Physics

By measuring the state of photons using superconducting qubits in an aluminum cavity, scientists know that these measurements are a result of dark matter interacting with these magnetic fields rather than experimental noise due to the sensitive nature of the measurement. By improving the material of the cavity to strengthen superconductivity and to detect multiple wavelengths of photons emitted, scientists hope to find direct evidence of the existence of dark matter.

Quantum sensing techniques can also be used to measure the phase of a superconducting current in coherent quantum phase slip (CQPS) junctions. CQPS junctions are quantum devices that are designed to measure the phase of a superconducting current and are based on the phenomenon of quantum phase slip (QPS). To measure the phase of the superconducting current, a method such as quantum interferometry can be used that involves splitting the superconducting current into two or more paths, manipulating the phase of one of the paths, and then recombining the paths to measure the phase difference. Another method is using superconducting qubits that can couple to the superconducting current, allowing for the detection of small changes in the phase of the current. Using these methods can allow a deeper understanding of quantum phase slips and have the potential to enable the development of advanced technologies and applications.

Quantum Sensing and Metrology can also be used for fundamental physics applications such as Gravitational wave detectors, axion dark matter experiments, coherent quantum phase slip junctions, and atomic clocks.

Since its founding in 2015, the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) has successfully detected gravitational waves from black hole and neutron star mergers. But researchers have a more challenging task: to detect gravitational waves from the beginning of the universe. This requires enhanced detectors that can detect very low frequencies, which are not accessible by sensors that use laser interferometers. As atoms contain atomic transitions that have microwave frequencies or even lower, why not build quantum sensors that are based on atom interferometry or atomic clocks? Scientists have done just that. By using these methods to create entangled pairs, these sensors can be used to detect the extremely small changes in distance that are caused by the passage of gravitational waves. Dark matter has been a mystery to scientists ever since its discovery although it makes up 27% of the total matter in the universe. Researchers have found a way to apply the methods of quantum metrology by using qubits to detect photons that are emitted as a result of dark matter or axions interacting with strong magnetic fields.

In Chapter 3, we learned that trapped ions can be a source of particles for a quantum computer, but also can be used as atomic clocks as they have several properties that make them suitable for this application. One of the main advantages of using trapped ions as atomic clocks is that the energy levels of ions are highly stable and can be used as a reference for time measurement. This is because the energy levels of ions are determined by their electronic and nuclear structure, which is relatively insensitive to external perturbations such as temperature, pressure, and magnetic fields. Recently, scientists have been trying to create a quantum sensor that is essentially an atomic clock with the feature of entanglement to determine and improve the accuracy of atomic clocks. By creating a new generation of atomic clocks using these quantum sensors, scientists can improve navigational systems to detect subtle gravitational waves or even detect gravity on its smallest scale, possibly reconciling quantum mechanics and gravity.

7.5 Energy Applications

Quantum technologies have great potential in the field of energy exploration. Two future mission-critical technologies will be featured here: the quantum gradiometer and quantum LIDAR.

Gradiometers measure the gradient of the gravitational field strength at a particular location. The gravitational field strength is the value of g (gravitational acceleration) at a point in space. On earth, we know the value of g to be approximately 9.81 meters per second, but this can change from place to place. As one goes higher up in altitude, the value of g generally decreases, and when one is near a massive object (say, a mountain), the value of g is higher since mass generates gravity. Measuring the gradient of g is just another way of measuring changes in nearby mass density, without having to calibrate the sensor to know the absolute value of g at that location.

Gradiometers built using quantum mechanical principles (Quantum gradiometers) are precise enough to detect subterranean tunnels, since there will be a small change in the gradient of g. Conventional measurements of gravity cannot be done as precisely, and are therefore less useful at spotting these hidden underground features.

Quantum gradiometers work on the operating principle of an atom interferometer. Trapped and cooled atoms are allowed to free fall in a (portable) vacuum chamber, where one-half of the atom population takes one route, and the other takes another route. The tiny differences in gravity experienced by these two atom populations result in the atomic wavefunctions obtaining slightly different phases so that when the atoms are recombined, there is interference depending on the difference in phase (just like waves in water which can reinforce or cancel each other out). This phase difference is used to determine the gravitational gradient. Recent efforts by research groups and startups have compiled this system into one portable unit.

The highly sensitive nature of quantum gradiometers makes them ideal for a variety of tasks, especially in the natural resource exploration sector where prior verification of the locations of underground mineral deposits, for example, greatly reduces uncertainty in mining yield.

A similar quantum improvement comes to the LIDAR sensor, which is the light-powered version of radar. LIDAR uses the signals received from light bouncing off objects to generate 3-dimensional information about the object or surroundings (unlike a photo which only has 2 dimensions). Quantum LIDAR takes this one step further by increasing accuracy to unprecedented levels. Single-photon quantum LIDAR systems being developed for commercial use can detect small gas molecules underwater to facilitate oil discovery, or detect gas leaks in industrial plants.