Introduction to Quantum Computing

Chapter 3
Quantum-Mechanical Foundations

Diploma in Quantum Computing
Introduction to Quantum Computing   •   Diploma

Quantum-Mechanical Foundations

Introduction

Quantum computing is based on a physical theory that differs from classical mechanics at very small scales. Electrons, photons, atoms, and other quantum systems do not always possess definite classical properties before measurement. Instead, their possible states are described by mathematical objects called state vectors. These vectors encode the probabilities and phase relationships needed to predict experimental outcomes.

This chapter develops the quantum-mechanical ideas required for introductory quantum computing. The presentation uses linear algebra but avoids advanced physics. The central message is that a quantum computer does not simply manipulate unknown classical bits. It manipulates probability amplitudes, whose magnitudes determine measurement probabilities and whose phases allow interference.

Learning Objectives

By the end of this chapter, you should be able to:

  1. Describe a qubit using a state vector and explain normalization.
  2. Explain superposition, relative phase, and interference.
  3. Distinguish measurement probabilities from probability amplitudes.
  4. Explain collapse as an update associated with measurement.
  5. Relate observables to Hermitian operators and eigenvalues.
  6. State the uncertainty principle in conceptual and mathematical form.
  7. Explain why unknown quantum states cannot be perfectly cloned.
  8. Apply these ideas to simple quantum-circuit and communication scenarios.

1. From Classical Bits to Qubits

A classical bit has one of two logical values: 0 or 1. A qubit is a physical quantum system with two selected basis states, conventionally written as |0⟩ and |1⟩. Examples include two energy levels of an atom, two polarization states of a photon, or two current directions in a superconducting circuit.

A qubit can be in a state that is neither simply 0 nor simply 1. Its general pure state is

|ψ⟩ = α|0⟩ + β|1⟩

where α and β are generally complex numbers called probability amplitudes. The state must be normalized:

|α|² + |β|² = 1

Here |α|² and |β|² are real probabilities. The symbol |z| denotes the magnitude of a complex number z. Normalization means that the probabilities of all mutually exclusive measurement outcomes add to one.

In column-vector notation, the computational basis is

|0⟩ = [1, 0]ᵀ, |1⟩ = [0, 1]ᵀ

and therefore

|ψ⟩ = [α, β]ᵀ

The superscript T means transpose. The vector is not itself a list of classical probabilities: α and β can be negative or complex, whereas probabilities cannot. This distinction is essential because amplitudes can interfere.

Global and Relative Phase

A complex amplitude can be written in polar form as α = reiφ, where r is non-negative, φ is a phase angle, and i = √(-1). Multiplying an entire state by the same phase does not change any measurement probability:

eiγ|ψ⟩

represents the same physical state as |ψ⟩ for any real γ. This is called global phase and is unobservable in an isolated measurement.

By contrast, the phase difference between terms matters. For example,

|+⟩ = (|0⟩ + |1⟩)/√2

and

|−⟩ = (|0⟩ − |1⟩)/√2

both produce 0 and 1 with equal probabilities when measured in the computational basis. However, they behave differently under later operations because their relative phases differ.

2. Superposition

A superposition is a linear combination of basis states. It should not be interpreted as an ordinary mixture in which the qubit is secretly either 0 or 1 and we merely lack information. A classical mixture represents uncertainty about a definite alternative. A coherent quantum superposition contains phase information that can produce interference.

For the state

|ψ⟩ = (√3/2)|0⟩ + (1/2)|1⟩

measurement in the computational basis gives

P(0) = |√3/2|² = 3/4

and

P(1) = |1/2|² = 1/4

The amplitudes must be squared in magnitude because amplitudes, not probabilities, combine linearly.

A useful geometric representation is the Bloch sphere. Every pure qubit state can be written, apart from a global phase, as

|ψ⟩ = cos(θ/2)|0⟩ + eiφ sin(θ/2)|1⟩

where 0 ≤ θ ≤ π and 0 ≤ φ < 2π. The angles specify a point on a unit sphere. The north and south poles correspond to |0⟩ and |1⟩. The equator contains equal-magnitude superpositions, including |+⟩ and |−⟩.

The Bloch sphere is a visualization, not a physical claim that the qubit is literally located on a small sphere. It is especially useful for understanding single-qubit rotations and phase changes.

3. Interference and Quantum Operations

Quantum evolution of an isolated system is reversible and represented by a unitary operator U. It transforms a state according to

|ψ'⟩ = U|ψ⟩

A unitary operator satisfies

U†U = UU† = I

where U† is the conjugate transpose and I is the identity operator. This condition preserves inner products, state normalization, and reversibility.

The Hadamard gate is a fundamental example:

H = (1/√2)[[1, 1], [1, −1]]

Its action on the computational basis is

H|0⟩ = |+⟩

and

H|1⟩ = |−⟩

Applying H again returns the original state because H² = I. In particular,

H|+⟩ = |0⟩

The amplitudes associated with different computational paths add before probabilities are calculated. If paths contribute amplitudes with the same phase, they reinforce one another; if they have opposite phases, they cancel. This is constructive or destructive interference.

For example, beginning with |0⟩, applying H creates |+⟩. Applying H once more gives |0⟩, not a random result, because the two amplitude contributions to |1⟩ cancel exactly. Quantum algorithms are designed to make incorrect computational paths interfere destructively and useful paths interfere constructively.

A common misconception is that superposition alone provides computational speedup. It does not. Speedup depends on a complete algorithm that prepares states, controls phases, creates interference, and finally extracts useful information through measurement.

4. Measurement and Collapse

Measurement connects a quantum description with a classical recorded result. For a computational-basis measurement of

|ψ⟩ = α|0⟩ + β|1⟩

the Born rule states that the outcome 0 occurs with probability |α|² and outcome 1 with probability |β|².

After an ideal measurement gives 0, the post-measurement state is |0⟩. After it gives 1, the state is |1⟩. This transition is commonly called collapse of the state vector. Collapse is not a unitary evolution; it is a probabilistic state update associated with the measurement process.

For a projective measurement with projectors P₀ and P₁, the probability of outcome k is

P(k) = ⟨ψ|Pk|ψ⟩

and, when P(k) ≠ 0, the normalized post-measurement state is

|ψk⟩ = Pk|ψ⟩ / √P(k)

For computational-basis measurement,

P₀ = |0⟩⟨0|, P₁ = |1⟩⟨1|

The formula assumes an ideal projective measurement and a normalized pure state. Real devices may require more general measurement models because detectors are noisy and measurements can be incomplete.

Measurement is basis-dependent. The state |+⟩ is definite in the basis {|+⟩, |−⟩}, but measuring it in the computational basis gives 0 or 1 with equal probability. A measurement does not merely reveal a pre-existing value independent of the chosen basis.

Repeated measurement of the same observable immediately after an ideal measurement gives the same result, provided no intervening operation or environmental interaction changes the state. This is often called measurement repeatability.

5. Observables and Eigenstates

A measurable physical quantity is represented by an observable, mathematically a Hermitian operator A. Hermitian means

A† = A

This property guarantees real eigenvalues, which are appropriate for measurement results.

An eigenvector |a⟩ of A satisfies

A|a⟩ = a|a⟩

where a is the corresponding eigenvalue. If a system is in an eigenstate |a⟩, measuring A returns a with certainty.

If a state is expanded in an orthonormal eigenbasis,

|ψ⟩ = ∑j cj|aj⟩

then the probability of obtaining eigenvalue a_j is |cj|², assuming distinct eigenvalues and an ideal projective measurement. The expected or average value of repeated measurements is

⟨A⟩ = ⟨ψ|A|ψ⟩

This is an average over an ensemble of identically prepared systems, not a claim that one individual measurement must equal the average.

For a qubit, the Pauli operators are important observables:

X = [[0, 1], [1, 0]], Y = [[0, −i], [i, 0]], Z = [[1, 0], [0, −1]]

The Z observable distinguishes |0⟩ and |1⟩. The X observable distinguishes |+⟩ and |−⟩. The Y observable uses states with relative phases of ±i. These are different measurement bases for the same qubit.

6. Uncertainty and Non-Commuting Observables

In classical physics, quantities such as position and momentum can ideally be known simultaneously. Quantum mechanics places a limit on the precision with which certain pairs of observables can be jointly specified.

For observables A and B, define their standard deviations in state |ψ⟩ by

∆A = √(⟨A²⟩ − ⟨A⟩²), ∆B = √(⟨B²⟩ − ⟨B⟩²)

The general Robertson uncertainty relation is

∆A∆B ≥ (1/2)|⟨[A,B]⟩|

where the commutator is

[A,B] = AB − BA

and ⟨[A,B]⟩ is its expectation value. The relation assumes a normalized state and observables for which the relevant variances exist.

For position x and momentum p, the canonical commutation relation leads to

∆x∆p ≥ ℏ/2

where ℏ is the reduced Planck constant. This is not merely a limitation of imperfect instruments; it is a property of quantum states and the structure of the observables.

In quantum computing, the Pauli X and Z observables do not commute:

[X,Z] = −2iY

Consequently, a state that is definite in the Z basis is not definite in the X basis. A qubit prepared as |0⟩ has definite Z measurement outcome but gives equal probabilities for X outcomes. This basis incompatibility is central to quantum cryptography and quantum information.

Uncertainty should not be confused with ignorance. A mixed state may reflect incomplete knowledge of preparation, whereas uncertainty relations also apply to a fully specified pure state.

7. The No-Cloning Principle

Classical information can be copied without changing the original bit. Quantum information behaves differently. The no-cloning theorem states that there is no physical operation that can perfectly copy every unknown quantum state.

Suppose a hypothetical copying operation C uses a blank state |0⟩ and claims to copy any state |ψ⟩:

C(|ψ⟩|0⟩) = |ψ⟩|ψ⟩

Assume it also copies two distinct states |ψ⟩ and |φ⟩. Unitary operations preserve inner products. Therefore, the inner product between the input states must equal the inner product between the output states:

⟨ψ|φ⟩ = (⟨ψ|φ⟩)²

Let s = ⟨ψ|φ⟩. Then s = s², which is possible only for s = 0 or s = 1. Thus perfect copying is possible for identical states and for mutually orthogonal states, but not for arbitrary nonorthogonal unknown states.

A controlled-NOT gate can copy computational-basis information in a restricted sense. For example,

CNOT(|0⟩|0⟩) = |0⟩|0⟩

and

CNOT(|1⟩|0⟩) = |1⟩|1⟩

However, applying it to an arbitrary superposition produces

CNOT((α|0⟩ + β|1⟩)|0⟩) = α|00⟩ + β|11⟩

which is generally an entangled state, not (α|0⟩ + β|1⟩)². The theorem does not prohibit copying classical labels, preparing a known state repeatedly, or transmitting classical descriptions of quantum states. It prohibits a universal perfect copier for unknown quantum states.

8. Worked Example: Phase Revealed by Interference

Consider the two states |+⟩ and |−⟩. Both give equal computational-basis probabilities:

P(0) = P(1) = 1/2

Therefore, a single computational-basis measurement cannot distinguish them reliably. Apply a Hadamard gate.

For |+⟩:

H|+⟩ = |0⟩

For |−⟩:

H|−⟩ = |1⟩

Thus the phase difference, initially hidden from a direct computational-basis measurement, is converted into a definite measurement result by interference. This example illustrates a central design principle of quantum algorithms: operations can transform phase information into observable probability differences.

9. Applied Case: Quantum Key Distribution

The BB84 quantum key-distribution protocol illustrates superposition, measurement, basis dependence, and no-cloning. This is an illustrative application rather than a complete security proof.

Background

Two parties, traditionally called Alice and Bob, want to establish a shared secret key while detecting an eavesdropper, Eve. Alice encodes random bits using one of two incompatible bases: the computational basis {|0⟩, |1⟩} and the diagonal basis {|+⟩, |−⟩}.

Problem

Eve may intercept and measure the transmitted qubits. If quantum states could be copied perfectly, Eve could copy each signal, retain one copy, and forward the other without introducing detectable disturbance. The no-cloning principle prevents this strategy for unknown nonorthogonal states.

Analysis and Approach

Bob independently chooses a random measurement basis for each received qubit. After transmission, Alice and Bob publicly compare bases, but not the encoded bit values. They retain only positions where their bases agree; this is called sifting.

If Eve measures a qubit in the wrong basis, her measurement changes the state relative to the basis Alice used. When Bob later measures in Alice’s original basis, errors can appear. Alice and Bob estimate the error rate by comparing a sample of their sifted bits. A sufficiently high error rate indicates noise or possible interception, so the raw key is discarded or processed using an appropriate security protocol.

Outcome

The protocol does not guarantee secrecy merely because quantum states are used. Practical security also requires authenticated classical communication, carefully characterized devices, error correction, privacy amplification, and protection against implementation attacks. Under suitable assumptions, however, incompatible measurements and no-cloning make undetected interception fundamentally different from classical copying.

Learning Points

  1. Measurement basis affects the information obtained.
  2. Nonorthogonal states cannot be perfectly distinguished in one measurement.
  3. Unknown quantum states cannot be universally cloned.
  4. A quantum protocol requires classical post-processing and engineering controls.
  5. Security claims depend on explicit assumptions about devices, channels, and adversaries.

10. Practical Limitations and Best Practices

Quantum states are fragile. Interaction with the environment can destroy phase relationships, a process called decoherence. Device imperfections can cause gate errors, readout errors, leakage outside the intended two-level system, and unwanted coupling between qubits.

For introductory circuit development, the following practices are valuable:

  • Track whether a quantity is an amplitude, probability, or expectation value.
  • Normalize state vectors before interpreting their probabilities.
  • Treat global phase as physically irrelevant but preserve relative phase.
  • Specify the measurement basis rather than saying simply that a state is measured.
  • Remember that measurement is generally irreversible and is not an ordinary gate.
  • Use repeated trials to estimate probabilities and expectation values.
  • Separate ideal mathematical behavior from hardware noise and calibration error.
  • Do not infer that a simulator's exact output will occur on every real-device run.
  • State the assumptions behind security or performance claims.

Quantum software tools can simulate state vectors and measurement distributions, but simulation does not remove the conceptual need to understand the underlying physics. For n qubits, a general pure state has 2ⁿ complex amplitudes, so classical simulation becomes costly as n increases. This is one reason quantum computing is studied as a distinct computational model, although it does not imply that every quantum task is faster than every classical task.

Contemporary Perspective

Modern quantum processors use several physical platforms, including superconducting circuits, trapped ions, neutral atoms, photonic systems, and semiconductor spin devices. Their engineering details differ, but all must control quantum states, implement approximately unitary operations, measure selected observables, and manage noise.

Quantum error correction does not violate no-cloning. It distributes logical information across an entangled code space and measures error-related information without directly measuring the encoded logical state. The objective is to detect and correct certain errors while preserving the quantum information. Fault-tolerant computing requires carefully designed codes, operations, thresholds, and hardware conditions.

Quantum machine learning and quantum simulation also rely on the same foundations. Algorithms may encode data in amplitudes or angles, manipulate interference, and measure observables. Claims about practical advantage should be evaluated using a defined problem, resource model, noise assumptions, and a fair classical comparison.

Chapter Summary

A qubit is represented by a normalized state vector |ψ⟩ = α|0⟩ + β|1⟩, where squared amplitude magnitudes give measurement probabilities. Superposition is a coherent combination of basis states, not an ordinary classical mixture. Relative phase is physically meaningful, while global phase is not.

Unitary operations preserve normalization and reversibility. Interference occurs because amplitudes add before probabilities are calculated, enabling quantum algorithms to amplify useful outcomes and suppress others. Measurement is basis-dependent and produces probabilistic outcomes according to the Born rule. An ideal projective measurement updates, or collapses, the state into the subspace associated with the observed result.

Observables are Hermitian operators. Their eigenvalues are possible measurement results, and their eigenvectors are states with definite values. Non-commuting observables obey uncertainty relations, so a state can be definite in one measurement basis while intrinsically uncertain in another. Finally, the no-cloning theorem prevents perfect copying of arbitrary unknown quantum states, supporting applications such as quantum key distribution.

Key Terms

State vector: A mathematical vector representing the state of a quantum system.

Qubit: A two-level quantum system used as the basic unit of quantum information.

Basis state: A state belonging to a chosen orthonormal set used to describe measurements.

Superposition: A coherent linear combination of possible basis states.

Probability amplitude: A generally complex coefficient whose squared magnitude gives a measurement probability.

Normalization: The condition that the total probability of all outcomes equals one.

Global phase: A common phase factor multiplying an entire state; it has no observable effect by itself.

Relative phase: The phase difference between components of a superposition.

Interference: The addition of amplitudes that can reinforce or cancel outcomes.

Unitary operator: A reversible linear operator satisfying U†U = I.

Measurement: A physical process that produces a classical outcome from a quantum system.

Collapse: The state update associated with an ideal measurement outcome.

Observable: A Hermitian operator representing a measurable quantity.

Eigenvalue: A possible measurement result of an observable.

Eigenstate: A state that produces a definite result for a particular observable.

Expectation value: The statistical average of repeated measurements, written ⟨A⟩ = ⟨ψ|A|ψ⟩.

Commutator: The operator [A,B] = AB − BA, indicating whether two observables commute.

Uncertainty principle: A limit on the simultaneous precision of certain non-commuting observables.

Decoherence: Loss of quantum coherence caused by interaction with the environment.

No-cloning theorem: The principle that no universal physical operation can perfectly copy an arbitrary unknown quantum state.

Review Questions

Short Answer Questions

  1. What condition must the amplitudes of a normalized qubit satisfy?
  2. Why is relative phase observable while global phase is not?
  3. What does the Born rule predict?
  4. What mathematical property must an observable satisfy?
  5. State the no-cloning principle in your own words.

Long Answer Questions

  1. Explain the difference between a quantum superposition and a classical probabilistic mixture.
  2. Describe how the Hadamard gate demonstrates interference and phase sensitivity.
  3. Explain projective measurement, including probabilities and post-measurement states.
  4. Discuss the uncertainty relation and its connection with non-commuting observables.
  5. Explain why a CNOT gate does not provide a universal quantum cloning machine.

Analytical / Application Questions

  1. A qubit is prepared as |ψ⟩ = (√3/2)|0⟩ + (1/2)|1⟩. Calculate the probabilities of the two computational-basis outcomes and verify normalization.
  2. Compare the results of measuring |+⟩ and |−⟩ directly in the computational basis and after applying a Hadamard gate. What information becomes observable?
  3. In the BB84 scenario, explain why an eavesdropper measuring in a randomly chosen basis can introduce errors detectable by the legitimate parties.

Multiple Choice Questions with Answers

Question 1

For a qubit state |ψ⟩ = α|0⟩ + β|1⟩, which condition expresses normalization?

A. α + β = 1

B. |α|² + |β|² = 1

C. |α| + |β| = 0

D. αβ = 1

Correct Answer: B. |α|² + |β|² = 1

Question 2

What is the probability of obtaining 1 when measuring (√3/2)|0⟩ + (1/2)|1⟩ in the computational basis?

A. 1/4

B. 1/2

C. 3/4

D. 1

Correct Answer: A. 1/4

Question 3

Which statement about global phase is correct?

A. It changes all measurement probabilities.

B. It is always equal to zero.

C. It has no observable effect when applied to the entire state.

D. It converts a qubit into a classical bit.

Correct Answer: C. It has no observable effect when applied to the entire state.

Question 4

What is the principal effect of applying the Hadamard gate twice to a qubit?

A. It produces a random state.

B. It measures the qubit.

C. It destroys normalization.

D. It returns the original state.

Correct Answer: D. It returns the original state.

Question 5

An observable in quantum mechanics is represented mathematically by a:

A. Hermitian operator

B. Classical probability table only

C. Non-invertible bit string

D. Random complex number

Correct Answer: A. Hermitian operator

Question 6

For an ideal projective measurement, what determines the probability of an outcome associated with projector P_k?

A. ⟨ψ|Pk|ψ⟩

B. ⟨ψ|ψ⟩ + 1

C. The global phase only

D. The number of qubits in the computer

Correct Answer: A. ⟨ψ|Pk|ψ⟩

Question 7

The uncertainty relation is especially relevant when two observables:

A. Have identical eigenvalues

B. Are represented by real matrices only

C. Do not commute

D. Are measured on classical bits

Correct Answer: C. Do not commute

Question 8

What does the no-cloning theorem prohibit?

A. Copying a known classical bit

B. Preparing a known quantum state repeatedly

C. Copying an arbitrary unknown quantum state perfectly

D. Using two identical quantum devices

Correct Answer: C. Copying an arbitrary unknown quantum state perfectly

Question 9

Why do |+⟩ and |−⟩ produce different definite results after a Hadamard gate?

A. They have different relative phases that interference converts into outcomes.

B. They have different normalization factors.

C. One of them is not a valid quantum state.

D. The Hadamard gate performs two independent measurements.

Correct Answer: A. They have different relative phases that interference converts into outcomes.

Question 10

What is decoherence?

A. The creation of a global phase

B. Loss of quantum coherence through environmental interaction

C. Perfect copying of a qubit

D. The normalization of an observable

Correct Answer: B. Loss of quantum coherence through environmental interaction
Introduction to Quantum Computing
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