Category: Simple Quantum Computing

The Standard Computational Bases

In the previous article, I covered the Bloch sphere, one of the most helpful visual tools for understanding qubits. The Bloch sphere allows us to represent a qubit’s state as a point in three-dimensional space. While this point represents the qubit’s state, it’s often even more useful to consider its position in relation to one of the three primary axes on the sphere – x, y, and z. These axes are not arbitrary; each one is linked to a unique set of orthonormal basis vectors, commonly referred to as computational bases.

In quantum computing, a computational basis provides a standardised framework to measure or affect a qubit’s state. If you’re unfamiliar with the term “orthonormal basis vectors,” consider reviewing this helpful Wikipedia page on orthonormality. These bases are “computational” because they lay the groundwork for every computation we perform in quantum computing.

While any set of orthonormal basis vectors could theoretically serve as computational bases in quantum computing, there are three primary ones that are used most frequently: the computational basis, the Hadamard basis, and the circular basis. Each of these serves a distinct purpose in interpreting and manipulating qubit states.

The Computational Basis

The computational basis, aligned with the z-axis of the Bloch sphere, is the foundation of quantum measurement. It’s the most common basis for reading the results of quantum computations because it directly corresponds to binary classical states, making it easier to translate quantum states into familiar classical outputs. The computational basis consists of two column vectors {"aid":null,"id":"1-0","backgroundColorModified":false,"type":"$$","code":"$$\\begin{bmatrix}\n{1}\\\\\n{0}\\\\\n\\end{bmatrix}$$","backgroundColor":"#ffffff","font":{"size":11,"color":"#000000","family":"Arial"},"ts":1730705283029,"cs":"htX4ZvSp/vZU5LxOdr4lKg==","size":{"width":17,"height":40}} and {"id":"1","font":{"color":"#000000","family":"Arial","size":11},"backgroundColorModified":false,"backgroundColor":"#ffffff","aid":null,"type":"$$","code":"$$\\begin{bmatrix}\n{0}\\\\\n{1}\\\\\n\\end{bmatrix}$$","ts":1730705302677,"cs":"KT/yysX+vZcUfh8RBD5pbQ==","size":{"width":17,"height":40}}, often written in Dirac notation as |0⟩ and |1⟩. These two states represent the “north” and “south” poles of the Bloch sphere, and they correspond to the conventional binary states, 0 and 1. When a qubit is measured in this basis, it “collapses” to one of these two poles, giving us a straightforward binary result.

The Hadamard Basis

Next, the Hadamard basis is associated with the x-axis of the Bloch sphere, which represents a state along the sphere’s equator rather than at its poles. This basis is one of the most powerful tools in quantum computing because it allows us to use any real number as a qubit’s state, not just the discrete values of 0 or 1. For a deeper dive into why this is significant, check out my article on quantum vs classical computing. The Hadamard basis enables superdense coding, a technique that leverages quantum superposition to represent classical bits in more complex ways. 

The Hadamard basis consists of two vectors {"backgroundColor":"#ffffff","id":"4-0","code":"$$\\begin{bmatrix}\n{\\frac{{\\sqrt[]{3}}}{2}}\\\\\n{\\frac{1}{2}}\\\\\n\\end{bmatrix}$$","aid":null,"type":"$$","font":{"family":"Arial","color":"#000000","size":11},"backgroundColorModified":false,"ts":1730791999404,"cs":"hTWAQA6GbNSUcVtenr2D2A==","size":{"width":34,"height":52}} and {"code":"$$\\begin{bmatrix}\n{\\frac{1}{2}}\\\\\n{-\\frac{{\\sqrt[]{3}}}{2}}\\\\\n\\end{bmatrix}$$","backgroundColor":"#ffffff","type":"$$","font":{"size":11,"color":"#000000","family":"Arial"},"aid":null,"backgroundColorModified":false,"id":"4-1","ts":1730792031290,"cs":"/1LehwHPaq45D7XFsEbg1g==","size":{"width":48,"height":52}}, denoted in Dirac notation as |+⟩ and |-⟩. These states lie along the equator of the Bloch sphere, representing the ability of qubits to exist in a superposition of 0 and 1 simultaneously. This property underlies many of quantum computing’s most promising applications, such as quantum parallelism, where the system explores multiple possibilities at once, and quantum encryption, where superpositions contribute to secure data encoding.

The Circular Basis

Finally, the circular basis aligns with the y-dimension of the Bloch sphere and introduces an essential concept called phase. In quantum mechanics, a qubit’s phase describes the relative angle or orientation of its wave function in its quantum state. This phase does not impact the probability of measuring the qubit in a particular state; rather, it influences how qubits interact with each other, particularly in superpositions and entangled states. 

This phase-based interference is foundational for many quantum algorithms and gate operations, such as the Hadamard gate and controlled-phase gates, which rely on phase manipulation to perform complex calculations. The circular basis uses two vectors {"backgroundColorModified":false,"backgroundColor":"#ffffff","font":{"color":"#000000","family":"Arial","size":11},"type":"$$","id":"5","code":"$$\\begin{bmatrix}\n{\\frac{1}{{\\sqrt[]{2}}}}\\\\\n{\\frac{1}{{\\sqrt[]{2}}}}\\\\\n\\end{bmatrix}$$","aid":null,"ts":1730792529020,"cs":"7EktR3+oTgrPSWxbYh2kZg==","size":{"width":36,"height":56}} and {"code":"$$\\begin{bmatrix}\n{\\frac{1}{{\\sqrt[]{2}}}}\\\\\n{-\\frac{1}{{\\sqrt[]{2}}}}\\\\\n\\end{bmatrix}$$","type":"$$","backgroundColor":"#ffffff","font":{"color":"#000000","family":"Arial","size":11},"id":"5","backgroundColorModified":false,"aid":null,"ts":1730792547294,"cs":"uBfxjoFz/ylEchizDQptRw==","size":{"width":48,"height":56}}, usually written as |i⟩ and |-i⟩ in Dirac notation, to describe these states. This notation represents the relative phase of the qubit, allowing for fine-tuned control over quantum states and facilitating advanced computational processes like error correction and phase estimation.

Conclusion

Understanding the computational bases—computational, Hadamard, and circular—is essential for anyone delving into quantum computing. These bases provide different frameworks for interpreting and manipulating qubit states, whether through straightforward binary measurement, superposition, or phase manipulation.

Each basis has a unique role: the computational basis is the go-to for classical readouts, the Hadamard basis introduces superpositions that allow for more complex computations, and the circular basis enables control over phase, crucial for advanced algorithms and gate operations.

By mastering these foundations, we gain a toolkit to understand and perform quantum operations more effectively, unlocking the full potential of qubits for quantum computing tasks. Whether you’re analysing measurement outcomes or engineering interference patterns, these computational bases form the core of your journey through quantum mechanics and computation.

Understanding the Bloch Sphere: A Visualisation of Qubits in Quantum Computing

In my previous blog post I briefly described the fundamental advantages of quantum computing, such as superposition and entanglement. The key player behind these powerful features is the qubit, which serves as the fundamental unit of quantum information. Unlike classical bits that represent only 0s and 1s, qubits can exist in multiple states at once due to the principles of superposition and entanglement. This allows quantum computers to solve certain problems much faster than classical computers.

But how do we actually represent a qubit? One of the most useful ways to visualise qubits is through the Bloch sphere, which is what I will explain in this post. The Bloch sphere gives us a geometrical representation of a qubit’s state, helping us better understand its behaviour and how it can be manipulated during quantum computations.

Qubit States on the Bloch Sphere

The Bloch sphere is a geometrical representation of a qubit’s state space. Named after physicist Felix Bloch, who contributed to nuclear magnetic resonance (NMR) theory, and was awarded a Nobel prize for this work, it is essentially a unit sphere in a three-dimensional space. Each point on this sphere represents a possible state of a qubit, with the surface of the sphere encoding all possible superpositions of the quantum states {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} and {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}.

In the traditional representation of a qubit, its state can be written as:

{"font":{"color":"#000000","family":"Arial","size":11},"aid":null,"id":"1","type":"$$","code":"$$a\\lbrack0\\rangle \\,+\\,b\\lbrack1\\rangle $$","backgroundColor":"#ffffff","backgroundColorModified":false,"ts":1728628958459,"cs":"pzD7Y3hCvVkJmrkKQ+sJwg==","size":{"width":80,"height":16}}

where {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} and {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}} are the basic states (think of them as the quantum version of 0 and 1), and a and b are complex numbers that describe the probabilities of the qubit being in state {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} or {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}.

The coefficients a and b can be written using complex numbers, which introduce both magnitude and phase. But rather than get bogged down in complex numbers, we can think of the Bloch sphere as a way of mapping these values into three dimensions that correspond to specific aspects of the qubit’s state.

The Three Dimensions of the Bloch Sphere

The Bloch sphere has three key axes that correspond to different aspects of a qubit’s state:

Z-axis (Computational Basis)

This dimension is the simplest to understand and relates to the “classical” states of a qubit: {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} and {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}. 

  • If the qubit is at the north pole of the Bloch sphere, it represents {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}}.
  • If the qubit is at the south pole, it represents {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}.

This is why the z-axis is often referred to as the computational basis, since these states are the fundamental building blocks of classical computing.

X-axis (Superposition on the Equator)

This dimension captures superposition, where the qubit exists in a state between {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} and {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}. A qubit in superposition might lie on the equator of the Bloch sphere, representing a mix of both classical states.

For example, the states {"code":"$$\\lbrack+\\rangle $$","id":"2-1","backgroundColorModified":false,"aid":null,"font":{"color":"#000000","family":"Arial","size":10},"backgroundColor":"#ffffff","type":"$$","ts":1728633544861,"cs":"G8l4wtC/5G/z3f7Vve0pUw==","size":{"width":18,"height":14}} and {"type":"$$","font":{"size":10,"color":"#000000","family":"Arial"},"backgroundColorModified":false,"id":"2","code":"$$\\lbrack-\\rangle $$","backgroundColor":"#ffffff","aid":null,"ts":1728633575159,"cs":"ZivHJVPxJ6Z6UfnmbbDmcg==","size":{"width":18,"height":14}} lie on the equator. These states are key to many quantum algorithms because they describe qubits that are equally likely to be measured as {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} or {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}.

The x-axis helps us understand how qubits are manipulated during quantum computations, particularly when using gates like the Hadamard gate, which create superposition.

Y-axis (Phase)

The y-axis deals with the phase of a qubit. Phase is a bit tricky—it’s not something we can directly measure, but it plays a critical role in quantum computations. Phase helps differentiate states that might have the same probabilities of being {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} or {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}, but which behave differently when subjected to certain quantum gates.

For example, the qubit can have a positive or negative phase ({"type":"$$","code":"$$\\lbrack i\\rangle $$","backgroundColor":"#ffffff","backgroundColorModified":false,"id":"10","aid":null,"font":{"family":"Arial","color":"#000000","size":12},"ts":1728633607114,"cs":"TKiTByTL2a/pmLuqdBhKyg==","size":{"width":13,"height":17}} or {"backgroundColorModified":false,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"aid":null,"backgroundColor":"#ffffff","code":"$$\\lbrack -i\\rangle $$","id":"8","ts":1728633631336,"cs":"r9QBSnt+6me+TR5iRpxvCA==","size":{"width":28,"height":17}}), which describes how the qubit oscillates over time. This oscillation (or rotation on the sphere) affects how qubits interfere with each other during quantum operations.

Putting It All Together

At any given moment, a qubit’s state can be represented by a point on the surface of the Bloch sphere. This point is determined by two angles:

  • θ (theta): This angle tells us how far the state is from the north or south pole (i.e., the computational basis). A qubit at the poles is in a classical state, while a qubit on the equator is in a superposition.
  • φ (phi): This angle defines where on the equator the qubit lies, describing the phase of the superposition.

We can mathematically represent any qubit state ({"font":{"color":"#000000","size":12,"family":"Arial"},"backgroundColor":"#ffffff","code":"$$\\lbrack\\psi\\rangle $$","id":"12","backgroundColorModified":false,"type":"$$","aid":null,"ts":1728633682411,"cs":"O46rls3dSzxr1042dHPIEw==","size":{"width":20,"height":17}}) using these two angles:

{"type":"$$","backgroundColorModified":false,"backgroundColor":"#ffffff","id":"13","aid":null,"font":{"size":11,"family":"Arial","color":"#000000"},"code":"$$\\lbrack\\psi\\rangle \\,=\\,\\cos\\left(\\frac{\\theta}{2}\\right)\\lbrack0\\rangle +\\,\\sin\\left(\\frac{\\theta}{2}\\right)e^{\\varphi^{i}}\\lbrack1\\rangle $$","ts":1728630633800,"cs":"LVHfevq05WQXYay9pf45Pg==","size":{"width":264,"height":40}}

In this equation, θ and φ map the qubit’s state to a specific point on the Bloch sphere, where we use trigonometry to describe its position. And since we have come to the point where we are talking about trigonometry, we can finally plot the Bloch sphere.


This model gives us an intuitive way to visualise qubits and the operations we perform on them. For example:

  • Quantum gates can be thought of as rotations of the qubit on the Bloch sphere. When we apply a quantum gate, we essentially rotate the point that represents the qubit’s state.
  • Superposition is clearly seen when a qubit lies on the equator, halfway between {"id":"2-1","code":"$$\\lbrack0\\rangle $$","aid":null,"backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#000000","family":"Arial","size":12},"ts":1728633312005,"cs":"54cxfABlUHryeVsbIcegtg==","size":{"width":16,"height":17}} and {"code":"$$\\lbrack1\\rangle $$","backgroundColorModified":false,"id":"3","aid":null,"type":"$$","font":{"size":12,"color":"#000000","family":"Arial"},"backgroundColor":"#ffffff","ts":1728633332948,"cs":"dfJUZHYoh6Y8oqfTT3WFjQ==","size":{"width":16,"height":17}}.
  • Phase changes are rotations around the z-axis, altering the angle φ but keeping the qubit’s probabilities the same.

Most importantly, the Bloch sphere shows us how quantum operations work in three dimensions, even though we might only measure the final state in one dimension (the z-axis). It helps explain why quantum algorithms can achieve speedups by harnessing the full power of quantum mechanics.

Conclusion

The Bloch sphere is an incredibly useful tool for visualising the state of a qubit and understanding how quantum operations work. By representing qubit states as points on a sphere, we can grasp key quantum properties like superposition and phase, which are crucial for quantum computations. As we continue exploring quantum computing, the Bloch sphere will serve as a fundamental concept for understanding how qubits behave and how quantum gates manipulate them.

In the next post, I’ll dive deeper into the computational basis and how it’s used in quantum algorithms. If you’re curious about the notation used to represent qubit states, check out this article on bra-ket notation until then!

Quantum vs Classical Computing: How Different Are They?

Computation is a core aspect of modern technology, driving innovations across science, engineering, entertainment, and communication. But what exactly is a computation, and how does it differ in classical and quantum contexts?

What is Computation?

At its most fundamental level, computation refers to using technology to perform tasks, solve problems, and manipulate information. Whether it’s simulating physical systems, developing intelligent algorithms, or managing vast data sets, computation is the bedrock of all digital advancements. Classical computing, which powers most of today’s digital systems, has long been the standard. However, quantum computing, a revolutionary paradigm, promises to redefine what’s possible in the field.

Classical vs. Quantum Computation

Classical Computing

Classical computing is rooted in binary logic. The smallest unit of information is a bit, which can exist in one of two possible states: 0 or 1. Every computation a classical computer performs is deterministic, meaning the result is always predictable based on the input. Operations are executed using logic gates, and bits can be copied, stored, and manipulated across various systems and memory registers without loss of information.

To illustrate, consider a light switch. A light switch can only be ON or OFF just like a bit can only be 1 or 0. Classical computers, no matter how complex, are simply massive collections of these switches, manipulating them to perform calculations. While highly effective for many applications, classical computers are limited by their binary nature.

Quantum Computing

Quantum computing, on the other hand, is a game changer because it relies on principles from quantum mechanics, which describe the behaviour of particles at extremely small scales. Instead of bits, quantum computers use qubits as the fundamental unit of information. Unlike classical bits, qubits can exist in an infinite number of states.

The state of a qubit can be visualised using a Bloch sphere. Imagine a globe, with 0 at the South Pole and 1 at the North Pole. While a classical bit is like a light switch, only toggling between ON (1) or OFF (0), a qubit can be any point on the surface of this globe. It can blend 0 and 1 in a continuum of possibilities, opening up a much larger space for computation.

Moreover, quantum operations exploit unique quantum phenomena, such as entanglement and superdense coding, which enable powerful new methods of processing information that classical systems cannot easily replicate.

What Makes Quantum Computation Different?

Quantum computing fundamentally differs from classical computing in several key ways:

  1. Superposition and Infinite States: A single qubit can encode an infinite number of possible states, as opposed to the strict binary options of classical bits. This means quantum computers can process much more information simultaneously.
  2. Measurement and Probability: While qubits can exist in an infinite number of states, any attempt to measure them translates their quantum state into a classical state (either 0 or 1). This process is probabilistic. The quantum state does not yield a fixed answer but rather gives a probability distribution over possible outcomes, which makes quantum computing inherently different from the deterministic operations of classical systems.
  3. No Cloning: A vital distinction between classical and quantum systems is that qubits cannot be copied. In classical computing, data can be duplicated as needed, but in quantum systems, copying qubits requires measurement, which essentially destroys their quantum states. This limitation introduces significant challenges in designing quantum memory and other hardware.
  4. Quantum Speedup: While quantum computers still rely on classical control processors, they can solve certain types of problems far more efficiently. The sheer complexity of operations performed on multiple qubits in superposition provides a level of computational parallelism that classical systems can’t achieve. As a result, quantum computers can solve some problems exponentially faster than classical computers.

Classical vs Quantum: Two Types of Computation

In formal terms, classical computing is a subset of quantum computing. Quantum systems are, by their nature, a more general form of computation. A quantum computer can theoretically perform any task a classical computer can, but the reverse isn’t true.

While classical systems excel at deterministic, straightforward calculations and are still the most practical solution for everyday computing needs, quantum computers promise breakthroughs in fields like cryptography, materials science, and complex simulations. However, the full potential of quantum computing is still in its early stages, with numerous technical challenges remaining.

Conclusion

The advent of quantum computing represents a paradigm shift in how we think about computation. While classical computing remains essential for most of today’s technology, quantum computing opens the door to unprecedented computational power. By leveraging the principles of quantum mechanics, future quantum computers will tackle problems once deemed unsolvable, pushing the boundaries of science, technology, and innovation.

The light switch analogy offers a simple glimpse into the complexity of these two systems: while classical bits are limited to being ON or OFF, qubits, like points on the surface of a sphere, reveal an infinite range of possibilities. This difference is the key to quantum computing’s extraordinary potential. The future of computation lies in the quantum realm, where the probabilistic nature of the universe is harnessed for revolutionary breakthroughs.

Why is Quantum Computing So Hard to Understand?

Preface

As a software engineer, I recently embarked on a journey to learn quantum computing. The experience has been both fascinating and frustrating, as I found it incredibly difficult to come across simple explanations for this complex field. In light of this, I decided to start a “Simple Quantum Computing” series. Through this series, I aim to demystify quantum computing for myself and for you, dear reader, as we explore what could very well be the most revolutionary computing technology of the 21st century.

Introduction

Quantum computing represents the bleeding edge of technology. As software engineers, it is one of the most advanced areas we can dive into. Its potential impact on the world is enormous, particularly when it comes to cryptography. Quantum computers have already shown that they can break modern cryptographic algorithms with relative ease.

Despite its prospect, quantum computing has not yet gained widespread popularity. Unlike artificial intelligence, which saw a surge in interest with the advent of models like ChatGPT, quantum computing remains a niche area. This is partly due to the limited job market demanding quantum skills, but it is also due to the steep learning curve associated with the technology.

So, why exactly is quantum computing so hard to understand?

Analog Computational Values

As software engineers, we are used to binary computational values—0s and 1s. These discrete values form the foundation of classical computing, with processor words built from them, and every computation within a CPU is based on these simple, binary states.

Quantum Processing Units (QPUs), however, operate on a completely different principle. They use analog, or continuous, values instead of discrete ones. There is no clear-cut 0 or 1 in quantum computing; instead, everything is represented by complex numbers that can take on any value between the quantum states |0⟩ and |1⟩. These states are known as kets, a concept I will delve into further in a later post.

This shift from binary to analog values presents a significant challenge. As developers, we are not trained to work with these continuous values. For example, there is no straightforward equivalent of a boolean in quantum computing—no simple 0 or 1 to latch onto.

Moreover, modern QPUs must be controlled by classical CPUs. This means that the analog values computed by a QPU have to be translated into discrete values for the CPU to process. This translation introduces a loss of information and a level of unpredictability, as the process is not deterministic. The notion of working with probabilities—something that quantum computing heavily relies on—is foreign to most of us in the field of computer science, where even pseudo-random numbers are not truly random.

Overall, the different computational bases and the need to convert between them add a significant layer of complexity that we are simply not accustomed to dealing with.

Complex Mathematics

Another major hurdle in understanding quantum computing is the complex mathematics involved. To grasp the concepts of quantum computing, one must be comfortable with advanced mathematical theories and methods that go beyond basic arithmetic or algebra.

Some of the key mathematical areas used in quantum computing include:

  • Complex Numbers: One of the fundamental concepts is the extension of the real number system to include imaginary numbers. A complex number is expressed as a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying i2 = −1.
  • Complex Analysis: This is the study of functions that operate on complex numbers. It includes topics such as analytic functions, complex integration, and conformal mappings.
  • Abstract Algebra: This area involves the study of algebraic structures like groups, rings, and fields. These structures are used to generalise algebraic concepts and solve equations in more abstract settings.

While it is possible to create simpler mathematical models by abstracting parts of this complex mathematics, understanding the original concepts is still crucial, and may be difficult (definitely for me!).

Programming a Processing Unit

Beyond the challenge of analog values and complex mathematics, there is the task of actually programming quantum processing units. Programming modern QPUs is reminiscent of programming early CPUs in the 1950s. If you compare quantum programming frameworks like Qiskit to early high-level programming languages like Autocode, you will notice striking similarities.

Both require you to think in terms of bits (or qubits), registers, memory, and gates. It is a very low-level form of programming that most modern software engineers are unaccustomed to. We lack the high-level frameworks and abstractions that we are familiar with in classical computing, making it even harder to get started in quantum computing.

Conclusion

Quantum computing represents a paradigm shift in how we approach computation, challenging many of the fundamental principles that have guided software engineering for decades. The complexity of analog computational values, the necessity to grasp advanced mathematics, and the low-level nature of quantum programming make it a field that is difficult to understand and master. 

However, these very challenges are what make quantum computing so promising and exciting. Just as classical computing evolved from low-level machine code to high-level languages and user-friendly frameworks, quantum computing will likely undergo a similar evolution. We are simply not there yet.

In this post series – simple quantum computing – I plan to write about different aspects of quantum computing that I found difficult to learn, and do my best to explain them in much simpler terms. I hope it will help both me and you understand what I believe to be the most promising computational technology of the XXI century.