Quantum Computing Basics

Flashcards covering fundamental concepts in quantum computing, including qubits, quantum states, superposition, and unitary operations.

Qubit

A unit of information in the quantum world, the quantum bit.

Classical Bit

A unit of information that can be either a zero or a one, mapped onto physical states like voltage levels or the presence/absence of a photon.

Quantum States of Zero and One

Physical realizations of information in the quantum world, such as an electron's spin orientation (up/down) or a photon's polarization.

Wave Functions

Mathematical descriptions used in quantum mechanics to represent the states of matter.

Ket Vector

A column vector of numbers that represents the state of a quantum system in a specific formulation of quantum physics, denoted as |ψ⟩ (e.g., |0⟩ or |1⟩).

Hilbert Space

A mathematical space where quantum states are represented as vectors, often infinite-dimensional, but finite-dimensional for most quantum computing applications.

Pure Quantum States

Orthogonal quantum states (like |0⟩ and |1⟩) meaning that if a system is in one state, it definitely cannot be in the other.

Superposition

A fundamental concept in quantum mechanics where a quantum system can exist in a combination of two or more pure states simultaneously, expressed as α|0⟩ + β|1⟩.

Amplitudes (Quantum)

Complex numbers (α and β) associated with each pure state in a superposition (α|0⟩ + β|1⟩), which determine the probability of measuring that state.

Measurement (Quantum)

An operation that causes a quantum superposition to 'collapse' into one of its pure states (|0⟩ or |1⟩) with a probability determined by the square of its amplitude.

Probability of Collapse

For a superposition α|0⟩ + β|1⟩, the probability of measuring |0⟩ is |α|^2 and the probability of measuring |1⟩ is |β|^2. The sum |α|^2 + |β|^2 must always equal 1.

Complex Numbers in Quantum Physics

Used to describe the amplitudes (α and β) of quantum states; they have both a real and an imaginary part, influencing both the magnitude and phase of a state.

Quantum Operation (Unitary Operation)

An operation that transforms a quantum system (qubit) from one superposition to another, always described by a unitary matrix.

Unitary Matrix (U)

A special 2x2 matrix of complex numbers that represents a quantum operation. It satisfies the condition U†U = UU† = I (identity matrix), where U† is its conjugate transpose.

Conjugate Transpose (U Dagger, U†)

For a complex matrix, it is the transpose of the matrix where all elements have been replaced by their complex conjugates.

Complex Conjugate

For a complex number (a + bi), its complex conjugate is (a - bi), where the sign of the imaginary part is negated.

Quantum Gates

Specific unitary operations applied to one or more qubits to perform computations, transforming their quantum states.

Euler's/De Moivre's Theorem

A mathematical theorem that relates complex exponentials to trigonometric functions (e.g., e^(iθ) = cosθ + i sinθ), crucial for understanding complex numbers in quantum mechanics.