The fundamental unit of quantum information is the qubit (quantum bit). Unlike classical bits that can only be 0 or 1, qubits can exist in superpositions of states, enabling quantum computers to process information in fundamentally different ways. Understanding qubits and quantum states is essential for grasping how quantum computing works.

For background, see Quantum Computing - Overview. For related concepts, see Quantum Computing - Entanglement and Quantum Computing - Quantum Gates and Circuits.

Classical Bits vs. Quantum Bits

Classical Bits

A classical bit is the basic unit of classical information:

• Can be in one of two states: 0 or 1
• States are mutually exclusive
• Operations are deterministic
• Information can be copied (no restrictions)
• Measurement always reveals the current state

Quantum Bits (Qubits)

A qubit is the quantum analog of a classical bit:

• Can be in a superposition of 0 and 1 simultaneously
• States are described by probability amplitudes
• Operations are probabilistic
• Information cannot be copied (no-cloning theorem)
• Measurement collapses the superposition to a definite state

Quantum State Representation

Single Qubit States

A qubit's state is represented as a vector in a two-dimensional complex vector space. The most general state of a single qubit is:

[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle]

where:

• (|\alpha|^2 + |\beta|^2 = 1) (normalization condition)
• (|0\rangle) and (|1\rangle) are the computational basis states
• (\alpha) and (\beta) are complex numbers called probability amplitudes
• (|\alpha|^2) is the probability of measuring 0
• (|\beta|^2) is the probability of measuring 1

Basis States

The computational basis consists of two orthogonal states:

[|0\rangle = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \ 1 \end{pmatrix}]

These correspond to the classical bit states 0 and 1, but a qubit can exist in any superposition of these states.

Examples of Qubit States

Equal Superposition: [|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)]

This state has equal probability (50%) of measuring 0 or 1. It's created by applying a Hadamard gate to (|0\rangle).

Opposite Superposition: [|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)]

This also has equal probability of measuring 0 or 1, but with opposite phase.

Arbitrary State: [|\psi\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle]

This represents any possible qubit state, parameterized by angles (\theta) and (\phi).

The Bloch Sphere

The Bloch sphere provides a geometric representation of a qubit's state. Every possible qubit state corresponds to a point on the surface of a unit sphere in three-dimensional space.

Bloch Sphere Coordinates

A qubit state can be written as: [|\psi\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle]

where:

• (\theta \in [0, \pi]) is the polar angle
• (\phi \in [0, 2\pi]) is the azimuthal angle

Key Points on the Bloch Sphere:

• North Pole ((\theta = 0)): (|0\rangle) state
• South Pole ((\theta = \pi)): (|1\rangle) state
• Equator ((\theta = \pi/2)): Equal superposition states
  • (\phi = 0): (|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle))
  • (\phi = \pi): (|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle))
  • (\phi = \pi/2): (\frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle))
  • (\phi = 3\pi/2): (\frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle))

The Bloch sphere is invaluable for visualizing qubit operations and understanding how quantum gates transform qubit states.

Superposition

Superposition is the ability of a quantum system to exist in multiple states simultaneously. For a qubit, this means it can be in a combination of (|0\rangle) and (|1\rangle) at the same time.

Understanding Superposition

When a qubit is in superposition:

• It's not "between" 0 and 1
• It's not "both" 0 and 1 in a classical sense
• It's in a quantum state that is a linear combination of (|0\rangle) and (|1\rangle)
• Measurement will reveal either 0 or 1 with probabilities determined by the amplitudes

Why Superposition Matters

Superposition enables quantum parallelism—a quantum computer can process multiple possibilities simultaneously. For example, with (n) qubits in superposition, you can represent (2^n) classical states at once. This exponential scaling is a key source of quantum advantage.

However, extracting useful information from this parallelism requires careful algorithm design, as measurement collapses the superposition to a single classical state.

Multiple Qubits

Two-Qubit States

A two-qubit system lives in a four-dimensional space. The computational basis consists of: [|00\rangle, |01\rangle, |10\rangle, |11\rangle]

A general two-qubit state is: [|\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle]

with normalization: (|\alpha_{00}|^2 + |\alpha_{01}|^2 + |\alpha_{10}|^2 + |\alpha_{11}|^2 = 1)

Tensor Product

The state of multiple independent qubits is given by the tensor product: [|\psi_1\rangle \otimes |\psi_2\rangle = (\alpha_1|0\rangle + \beta_1|1\rangle) \otimes (\alpha_2|0\rangle + \beta_2|1\rangle)] [= \alpha_1\alpha_2|00\rangle + \alpha_1\beta_2|01\rangle + \beta_1\alpha_2|10\rangle + \beta_1\beta_2|11\rangle]

(n)-Qubit Systems

For (n) qubits, the state space has dimension (2^n). A general (n)-qubit state is: [|\psi\rangle = \sum_{x=0}^{2^n-1} \alpha_x |x\rangle]

where (|x\rangle) represents the binary representation of (x) using (n) qubits.

This exponential growth in state space is what enables quantum computers to potentially solve certain problems exponentially faster than classical computers.

Measurement

Projective Measurement

When you measure a qubit in the computational basis ({|0\rangle, |1\rangle}), the measurement is projective:

• The qubit collapses to (|0\rangle) with probability (|\alpha|^2)
• The qubit collapses to (|1\rangle) with probability (|\beta|^2)
• After measurement, the qubit is in a definite classical state

Measurement in Other Bases

You can measure a qubit in any orthonormal basis. For example, measuring in the ({|+\rangle, |-\rangle}) basis:

• Probability of (|+\rangle): (|\langle+|\psi\rangle|^2)
• Probability of (|-\rangle): (|\langle-|\psi\rangle|^2)

Measurement and Information

Measurement is destructive—it collapses the quantum state. Once measured, the superposition is lost. This is why quantum algorithms must be carefully designed to extract useful information before measurement destroys the quantum state.

For more on measurement and its implications, see Quantum Computing - Measurement and Decoherence.

Phase and Interference

Global Phase

Multiplying a quantum state by a global phase factor (e^{i\phi}) doesn't change measurement probabilities: [|\psi'\rangle = e^{i\phi}|\psi\rangle]

Since (|e^{i\phi}\alpha|^2 = |\alpha|^2), the probabilities are unchanged. Global phases are unobservable.

Relative Phase

However, relative phases between terms in a superposition are crucial. They determine interference effects: [|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + e^{i\phi}|1\rangle)]

Different values of (\phi) give different interference patterns, which quantum algorithms exploit to amplify correct answers and cancel wrong ones.

Quantum vs. Classical Information

Information Capacity

• Classical bit: Stores 1 bit of information (0 or 1)
• Qubit: Stores infinite information in principle (continuous parameters), but measurement extracts only 1 bit

No-Cloning Theorem

Unlike classical bits, qubits cannot be copied. The no-cloning theorem states that there's no quantum operation that can create an identical copy of an arbitrary unknown quantum state.

This has important implications:

• Quantum error correction must work differently than classical error correction
• Quantum cryptography can provide security guarantees
• Quantum teleportation is possible (but destroys the original)

Physical Realizations

Qubits can be implemented using various physical systems. Each has different properties:

• Superconducting qubits: Charge or flux states in superconducting circuits
• Trapped ions: Internal energy levels of ions
• Photons: Polarization or path states
• Neutral atoms: Internal atomic states
• Quantum dots: Electron spin states

For detailed information on physical implementations, see Series 2: Hardware in the Quantum Computing Series Index.

Applications of Qubit Properties

The unique properties of qubits enable:

1. Quantum Parallelism: Processing multiple states simultaneously
2. Quantum Interference: Amplifying correct answers, canceling wrong ones
3. Quantum Entanglement: Creating correlations impossible classically
4. Quantum Tunneling: Escaping local minima in optimization

These properties are harnessed by quantum algorithms to achieve speedups over classical algorithms for certain problems.

Current Challenges

Working with qubits presents significant challenges:

1. Decoherence: Quantum states are fragile and easily disrupted
2. Measurement: Destroys superposition, limiting information extraction
3. Error Rates: Current qubits have high error rates
4. Scalability: Maintaining coherence becomes harder with more qubits

For solutions to these challenges, see Quantum Computing - Quantum Error Correction.

Conclusion

Qubits are the foundation of quantum computing. Their ability to exist in superpositions, combined with entanglement and interference, enables quantum algorithms that can solve certain problems exponentially faster than classical computers.

Understanding qubits and quantum states is essential for:

• Designing quantum algorithms
• Understanding quantum hardware
• Appreciating the power and limitations of quantum computing

The exponential growth of state space with qubit count is both a blessing (enables parallelism) and a curse (makes simulation difficult), but it's what makes quantum computing fundamentally different from classical computing.

Exploring Further

• Quantum Computing - Overview - Introduction to quantum computing
• Quantum Computing - Entanglement - How qubits become entangled and why it matters
• Quantum Computing - Quantum Gates and Circuits - How to manipulate qubit states
• Quantum Computing - Measurement and Decoherence - What happens when we measure qubits
• Quantum Computing Series Index - Complete guide to all quantum computing articles
• Science - Main science directory

Learning Resources

Online Courses

Beginner-Friendly:

• IBM Quantum Learning - Qubits and Quantum States - Interactive lessons on qubits: learning.quantum.ibm.com
• Qiskit Textbook - Quantum States and Qubits - Detailed chapter with examples: qiskit.org/textbook/ch-states/introduction.html

Intermediate to Advanced:

• Quantum Mechanics Courses (edX, Coursera) - Foundational quantum mechanics covering state representation and measurement

Video Resources

YouTube Channels:

• 3Blue1Brown - Mathematical visualizations of quantum mechanics and qubits (highly recommended)
• Qiskit YouTube Channel - Tutorials on qubits and quantum states: youtube.com/@qiskit

Educational Videos:

• Quantum Mechanics 101 by Dr. Ed Deveney - Introduction to quantum mechanics fundamentals: youtube.com/watch?v=ZjElRhZUoGs

Hands-On Platforms

• IBM Quantum Experience - Create and visualize qubit states: quantum.ibm.com
• Qiskit - Python framework for working with qubits: qiskit.org
• Cirq (Google) - Create and manipulate qubit states: quantumai.google/cirq