Quantum Computing for Everyone (2019) by Chris Bernhardt

According to mathematician Chris Bernhardt, quantum computing represents “a beautiful fusion of quantum physics and computer science.” His book, Quantum Computing for Everyone (2019), offers a remarkably accessible primer on a technological revolution that now seems just over the horizon. Bernhardt focuses not on the engineering challenge of building quantum computers, but on the theory and software concepts underlying how they work and how they may eventually be used. Readers should be warned, however, that quantum computing is filled with ideas that are both novel and profoundly counterintuitive, often resisting ordinary verbal explanation. In the end, Bernhardt argues, the only truly precise language for describing quantum systems is mathematics – especially linear algebra, the study of vectors and their transformations, which forms the conceptual foundation of quantum computing itself. The book ultimately advances a striking proposition: that quantum computing is not merely a new form of computation, but rather a deeper discovery about the true nature of computation itself.

The book focuses on explaining the three main mechanical ideas behind quantum mechanics: superposition, measurement, and entanglement. It does so in nine concise chapters, each roughly fifteen to twenty pages long, with nearly every page dense with diagrams, equations, and conceptual illustrations. Chapter One, “Spin,” introduces the reader to the measurement of a qubit – short for quantum bit – the fundamental unit of quantum computing. Spin occupies a role in quantum computing analogous to that of the switch in classical computing. Yet while a classical bit can be intuitively understood as an ordinary on-off light switch, quantum spin is tied to the angular momentum of electrons or the polarization of photons, concepts that are both less familiar and far more difficult to visualize. 

Bernhardt traces the origins of these ideas to the famous Stern-Gerlach experiment conducted by Otto Stern and Walther Gerlach at the University of Frankfurt in 1922. The experiment revealed the startling insight that the act of measurement itself alters the outcome being measured. In Bernhardt’s simplified formulation, if the same question is asked three times, the same answer will be returned each time. But if a different question is inserted between the first and third measurements, the answer to the original question may change. This marks a profound break from classical mechanics. 

In the classical world – whether calculating the arc of a thrown baseball or predicting the outcome of a coin toss – the act of observation does not itself alter the result. In principle, even a coin flip could be predicted with perfect accuracy if every initial condition were measured precisely enough, a phenomenon known as sensitive dependence on initial conditions. Quantum mechanics, by contrast, incorporates both unavoidable randomness and the disruptive effects of measurement into the fabric of reality itself. When spin is measured, it collapses into one of only two possible outcomes. Bernhardt compares this to asking the time on an unseen analog clock: if you ask whether it is three o’clock, the answer can only be “yes” or “no” — where “no” effectively corresponds to the opposite state, nine o’clock. The analogy is helpful, though not entirely satisfying, since it leaves the intuitive reader wondering why a third possibility – neither three nor nine – cannot also exist.

Chapter Two serves as an introduction to linear algebra, the mathematical foundation of quantum computing. While advanced quantum mechanics relies heavily on complex numbers, Bernhardt limits the discussion to real numbers, arguing that they are sufficient for understanding the core concepts. The chapter focuses primarily on vectors, including how to calculate their length and determine whether they are perpendicular to one another.

Much of the terminology initially makes the subject sound more intimidating than it really is. In practice, a vector is simply a list of numbers, and the dimension of a vector is just the number of entries in that list. Horizontal vectors are called “bras,” while vertical vectors are known as “kets” – together forming the “bra-ket” notation. Ordinary numbers are often referred to as scalars, while “perpendicular” becomes orthogonal, a critical concept in quantum computing because orthogonality determines whether quantum states are fully distinct from one another.

Bernhardt also introduces the concept of orthonormality, another foundational idea in quantum mechanics. Orthonormal states are both orthogonal (completely distinct from one another) and normalized (scaled so their total probability equals one). Without orthonormality, the mathematics of quantum probabilities and measurement would not function correctly. Different orthonormal bases, meanwhile, correspond to different orientations for measuring quantum properties such as spin.

The chapter also introduces matrices – rectangular arrays of numbers – with special attention given to square matrices, which contain the same number of rows and columns. Finally, Bernhardt explains two of the most important operations in quantum computing: the Hadamard gate and the CNOT (Controlled-NOT) gate. These gates are fundamental “moves” in quantum systems. The Hadamard gate places a qubit into superposition, allowing it to exist in multiple states simultaneously, while the CNOT gate operates on two qubits to create entanglement, the uniquely quantum phenomenon in which qubits become linked together in ways impossible in classical computing. Together, superposition and entanglement form the foundation of quantum computing’s extraordinary potential.

Chapter Three, “Spin and Qubits,” ties together the first two chapters to define what a qubit is and explain what happens when it is measured. The mathematical model describing quantum spin relies on both probabilities and vectors. One of the central ideas in quantum mechanics is that the act of measurement changes the vector describing the quantum state.

A unit vector in quantum computing is a vector with a length of exactly 1. This concept is important because vectors are used to represent the state of a quantum system, such as the state of a qubit, which Bernhardt defines as any unit ket in R². Requiring the vector to have a length of 1 ensures that the probabilities encoded in the quantum state add up to 100 percent.

To extract information from a qubit, it must be measured. This is done using an ordered orthonormal basis, meaning the vectors appear in a specific sequence and the order matters. Each vector has a length of 1 (normalized), and every pair of different vectors is perpendicular (orthogonal). Together, the vectors can describe every possible vector in the space. A qubit can then be represented as a linear combination of these basis vectors, known as a superposition. Finally, Bernhardt introduces the BB84 Protocol, a simple cryptographic scheme that uses two ordered orthonormal bases to securely exchange information.

Chapter Four, “Entanglement,” introduces another core concept in quantum theory that defies simple verbal explanation. The mathematical foundation of entanglement is the tensor product, another concept drawn from linear algebra. Tensor products make it possible to capture all the possible relationships between qubits rather than describing each qubit independently. This becomes essential for understanding entanglement because entangled qubits cannot be fully described on their own. Instead, their states become linked together as part of a single unified system, even when the individual qubits no longer possess separate, well-defined states. Albert Einstein famously referred to entanglement as “spooky action at a distance.”

Bernhardt further explains how tensor products demonstrate that superluminal communication – communication faster than the speed of light – is impossible. Communication involving entangled qubits ultimately requires measurement, and the act of measurement itself changes the quantum state. In practice, it becomes impossible to determine which qubit was measured first. Much of this stretches the limits of intuitive explanation, though the author repeatedly emphasizes that the relatively simple mathematics underlying quantum mechanics tells a precise and unambiguous story.

Chapter Five, “Bell’s Inequality,” examines Einstein’s objections to quantum entanglement and his hope that a hidden-variable theory might preserve the principle of local realism – the idea that an object can be influenced only by events occurring in its immediate surroundings. Classical theories such as gravity operate according to this principle. If a hidden-variable theory could be found, it would eliminate quantum randomness and provide a more intuitive explanation for entanglement, resolving what Einstein famously dismissed as “spooky action at a distance.” Yet the history of physics ultimately favored the counterintuitive Copenhagen Interpretation championed by Niels Bohr over the objections of Einstein and Erwin Schrödinger. The decisive breakthrough came from the Irish physicist John Stewart Bell, who devised a mathematical test capable of distinguishing between the competing interpretations and determining which description of reality was correct.

When a pair of entangled qubits is measured, the quantum state of the system changes immediately, even when the qubits are separated by great distances. Bohr maintained that there was no deeper underlying mechanism to explain this phenomenon; the strange behavior predicted by quantum mechanics was simply a fundamental feature of nature. Einstein disagreed. Along with Boris Podolsky and Nathan Rosen, he argued that information cannot travel faster than the speed of light. Yet the Copenhagen Interpretation appeared to imply that changes in one member of an entangled pair were reflected instantaneously in the other, regardless of the distance separating them. This apparent contradiction became known as the EPR Paradox, named after the initials of the three physicists who first articulated it. Einstein believed the paradox pointed to an incompleteness in quantum theory and suggested that a deeper, hidden-variable explanation remained to be discovered.
In the classical view of physics, the universe is fundamentally deterministic: if all relevant variables are known with complete precision, future events can, in principle, be predicted exactly. This was the perspective Einstein had in mind when he famously remarked that “God does not play dice with the universe.” To him, the probabilistic nature of quantum mechanics suggested not that reality itself was random, but that the theory was incomplete. Einstein believed that hidden variables and a deeper underlying framework would eventually restore the determinism that characterized classical physics.

Yet, as Bernhardt explains, subsequent developments in quantum theory have largely vindicated the probabilistic interpretation and undermined Einstein’s hopes for a hidden-variable explanation. In essence, “everything is deterministic until we take a measurement, and then it jumps to one of the basis vectors.” At the moment of measurement, the mathematical description of a quantum system shifts from deterministic evolution to probabilistic outcomes. Precisely why and how this transition occurs remains one of the central mysteries of quantum mechanics. Indeed, Bernhardt notes that even the concept of a “measurement” is surprisingly elusive and may be better understood as a form of observation or interaction with the surrounding world.

Chapter Six, “Classical Logic, Gates, and Circuits,” is essentially Bernhardt’s bridge from ordinary computers to quantum computers. After spending the first five chapters developing quantum mechanics, qubits, entanglement, and Bell’s inequality, he temporarily steps back and asks: How does a conventional computer actually compute? He introduces bits, Boolean logic, logic gates, circuits, memory, and finally reversible computation.

Bernhardt begins with the bit, which can have one of two values, 0 or 1. Computation consists fundamentally of manipulating these bits according to the rules of Boolean logic. Operations such as NOT, AND, and OR can be embodied in physical logic gates: NOT reverses a bit, AND produces 1 only when both inputs are 1, and OR produces 1 when at least one input is 1. Connecting gates together produces circuits, and increasingly complicated circuits can perform increasingly sophisticated computations.

An important idea is functional completeness. You do not actually need a huge assortment of different gates to construct a computer. Bernhardt shows, for example, that NAND is a universal gate: combinations of NAND gates can reproduce the behavior of all the other basic logical operations. In principle, therefore, an entire classical computer could be constructed using nothing but NAND gates.

He then introduces a concept that becomes crucial for quantum computing: reversible computation. Most familiar classical gates are irreversible. If an AND gate outputs 0, for example, you cannot determine from the output whether its original inputs were 00, 01, or 10. In other words, the computation has destroyed information. A reversible gate, by contrast, preserves enough information that its inputs can always be reconstructed from its outputs.

Bernhardt uses the work of physicist and computer scientist Edward Fredkin to demonstrate that reversible computation is not merely a curiosity. Both the Fredkin gate and the related Toffoli gate are universal: in principle, an entire computer can be constructed exclusively from either type of reversible gate. This matters because quantum mechanics itself is fundamentally reversible between measurements, so quantum gates must be reversible. Chapter Six is therefore quietly preparing the conceptual ground for the quantum circuits of Chapter Seven.

Bernhardt ends with Fredkin’s wonderfully imaginative billiard-ball computer. Instead of electricity and transistors, imagine perfectly elastic billiard balls traveling along predetermined paths. The presence or absence of a ball represents a bit, while collisions between balls perform logical operations. Because the balls collide without losing energy and their trajectories can theoretically be run backward, the machine illustrates how computation can be both physical and reversible. Bernhardt acknowledges that the example is not necessary for understanding quantum computing, but includes it because of its ingenuity.

The larger point of Chapter Six is that computers are ultimately machines for transforming information according to logical rules. Classical computers accomplish this by passing bits through gates arranged into circuits. Quantum computers will do essentially the same thing, but replace bits with qubits and classical gates with reversible quantum gates.

The final three chapters of Chris Bernhardt’s Quantum Computing for Everyone move from how quantum computers operate, to why their algorithms can outperform classical ones, and finally to why any of this matters.

Chapter Seven, “Quantum Gates and Circuits,” turns the quantum mechanics developed earlier in the book into an actual model of computation. Just as an ordinary computer manipulates bits through logic gates, a quantum computer manipulates qubits through quantum gates. The crucial difference is that qubits can exist in superpositions and become entangled, while quantum operations must be reversible. Bernhardt introduces gates such as CNOT and shows how gates can be assembled into quantum circuits.

The chapter then demonstrates what these strange properties make possible. Superdense coding allows two classical bits of information to be communicated by transmitting a single qubit when the parties already share entanglement. Quantum teleportation allows the quantum state of one qubit to be transferred to another distant qubit using entanglement and ordinary classical communication, without physically transporting the original qubit – clearly an example of Einstein’s “spooky action at a distance.” At the same time, the no-cloning theorem establishes that an unknown quantum state cannot simply be copied, an important distinction from classical information. Bernhardt closes with quantum error correction, explaining how information can nevertheless be protected against errors despite the impossibility of simply making backup copies of qubits.

Chapter Eight, “Quantum Algorithms,” tackles the central question: Can quantum computers actually solve problems faster than classical computers? To answer it, the author first introduces computational complexity, including the familiar classes P and NP, and then focuses on query complexity; essentially, how many times an algorithm must interrogate a function or “black box” to determine an answer.

Bernhardt then works through three increasingly sophisticated examples: Deutsch’s algorithm, the Deutsch–Jozsa algorithm, and Simon’s algorithm. All demonstrate circumstances in which a quantum algorithm requires fewer queries than its classical counterpart. But Bernhardt emphasizes an important point: quantum speedup is not simply the result of trying every possible answer simultaneously through superposition. The real power comes from cleverly manipulating probability amplitudes so that interference suppresses incorrect possibilities and reinforces information revealing the correct answer. Quantum algorithms therefore exploit the mathematical structure of a problem, using superposition, interference, and entanglement in carefully orchestrated ways.

The final chapter, “The Impact of Quantum Computing,” shifts to the two famous examples that demonstrate quantum computing’s potential practical significance: Shor’s algorithm and Grover’s algorithm. Shor showed that a sufficiently powerful quantum computer could factor very large integers dramatically more efficiently than known classical methods. Because widely used public-key cryptography has historically relied on the practical difficulty of problems such as factoring large numbers, Shor’s discovery demonstrated that quantum computing could have enormous consequences for information security.

Grover’s algorithm addresses a different problem: searching an unstructured collection. A classical computer may need to examine roughly N possibilities in the worst case, whereas Grover’s quantum algorithm can find the desired item using on the order of √N operations. That is not the spectacular exponential-type advantage associated with Shor’s factoring algorithm, but it is still a substantial quadratic speedup for sufficiently large search spaces.

Bernhardt also discusses the formidable challenge of actually building quantum computers. Qubits are extraordinarily fragile: interactions with their environment cause decoherence, while errors accumulate rapidly during computation. Consequently, useful large-scale quantum computers require sophisticated error correction and enormous technical control. His point is therefore not that quantum computers are simply faster replacements for conventional computers. They are fundamentally different machines whose advantages appear for particular classes of problems.

The larger message of these chapters – and really of Bernhardt’s entire book – is that quantum computing should not be understood as conventional computing made mysteriously faster. Quantum mechanics changes the underlying rules by which information can be represented and manipulated. Qubits, superposition, entanglement, interference, and measurement create computational possibilities that bits alone cannot reproduce efficiently. Bernhardt ultimately wants the reader to see classical computing as a special case of the deeper quantum-mechanical theory of information and computation.