Quantum Computing Glossary

A step-by-step procedure for calculations or problem-solving. Quantum algorithms are specifically designed to run on quantum computers and may offer computational advantages over classical algorithms for certain problems.

See also: Grover's Algorithm, Shor's Algorithm

In quantum computing, amplitudes are complex numbers associated with each basis state in a quantum superposition. The square of the amplitude's magnitude gives the probability of measuring that particular state.

A quantum technique that increases the probability of measuring desired states while decreasing the probability of measuring undesired states. This technique is central to Grover's search algorithm.

An auxiliary or "helper" qubit used in quantum circuits to assist with computations but not part of the input or output register. Ancilla qubits are commonly used in error correction and complex quantum operations.

A set of states that forms a reference frame for describing quantum systems. In quantum computing, the computational basis states are typically |0⟩ and |1⟩, analogous to classical 0 and 1 bits.

Maximally entangled quantum states of two qubits. The four Bell states form an orthonormal basis and are essential for many quantum protocols including quantum teleportation and superdense coding.

Common examples include:

• |Φ⁺⟩ = (|00⟩ + |11⟩)/√2
• |Φ^-⟩ = (|00⟩ - |11⟩)/√2
• |Ψ⁺⟩ = (|01⟩ + |10⟩)/√2
• |Ψ^-⟩ = (|01⟩ - |10⟩)/√2

The first quantum key distribution protocol, named after its inventors Bennett and Brassard and the year of publication (1984). BB84 uses quantum properties to establish a secure encryption key between two parties, with the ability to detect eavesdropping attempts.

Try it: BB84 Quantum Key Distribution Simulation

A geometric representation of a single qubit's pure state as a point on a unit sphere. The poles typically represent the |0⟩ and |1⟩ basis states, while points on the equator represent equal superpositions with different phases.

The Bloch sphere provides an intuitive visualization for:

• Qubit rotations (quantum gates)
• Superposition states
• Phase information
• Measurement probabilities

A sequence of quantum gates, measurements, and other operations that process quantum information. Quantum circuits are the quantum computing equivalent of classical logical circuits and form the basis of quantum algorithms.

A two-qubit quantum gate that flips the state of a target qubit if and only if the control qubit is in state |1⟩. The CNOT gate is essential for creating entanglement between qubits and is a fundamental component of most quantum algorithms.

Matrix representation:

[1 0 0 0]
[0 1 0 0]
[0 0 0 1]
[0 0 1 0]

The property of a quantum system to maintain its quantum state over time. Quantum coherence enables phenomena like superposition and interference to persist and is essential for quantum computation. Loss of coherence due to environmental interactions is called decoherence.

The standard basis used in quantum computing, consisting of states |0⟩ and |1⟩ for a single qubit, and their tensor products for multi-qubit systems (|00⟩, |01⟩, |10⟩, |11⟩, etc.). Measurements in quantum computing are typically performed in this basis.

The process by which quantum systems lose their quantum properties (particularly superposition and entanglement) due to interaction with the environment. Decoherence is one of the main challenges in building practical quantum computers.

A mathematical representation of a quantum state that can describe both pure and mixed states. Density matrices are particularly useful for describing systems that are entangled with other systems or subject to noise and decoherence.

A quantum algorithm that determines whether a function is constant (returns all 0s or all 1s) or balanced (returns 0s for half the inputs and 1s for the other half) with just one evaluation. This algorithm demonstrates quantum advantage over classical algorithms, which would require multiple evaluations.

Try it: Deutsch-Jozsa Algorithm Simulation

A quantum phenomenon where the quantum states of two or more particles become correlated in such a way that the quantum state of each particle cannot be described independently of the others, regardless of the distance separating them. Einstein famously referred to this as "spooky action at a distance."

Entanglement is a key resource in quantum computing that enables:

• Quantum teleportation
• Superdense coding
• Quantum key distribution
• Quantum computational speedups

Techniques to protect quantum information from errors caused by decoherence, gate imperfections, and other sources of noise. Quantum error correction typically encodes logical qubits using multiple physical qubits to detect and correct errors.

Try it: Shor's Code Error Correction Simulation

In quantum mechanics, eigenvalues represent the possible values that can be obtained when measuring a quantum observable. Quantum algorithms like quantum phase estimation extract eigenvalues of unitary operators.

In quantum mechanics, eigenvectors (or eigenstates) are states that remain unchanged in direction when a specific operator is applied to them. When measured, a quantum system in an eigenstate of an observable will yield the corresponding eigenvalue with certainty.

A fundamental quantum operation that transforms quantum states. Analogous to classical logic gates, quantum gates are represented by unitary matrices that act on qubits. Common quantum gates include:

• Hadamard (H): Creates superposition
• Pauli-X: Quantum equivalent of the NOT gate
• Pauli-Y and Pauli-Z: Phase operations
• CNOT: Two-qubit gate for entanglement
• Toffoli: Three-qubit gate for universal computation

A quantum algorithm for unstructured search that provides a quadratic speedup over classical algorithms. Grover's algorithm can find a specific item in an unsorted database of N items in approximately √N steps, compared to the classical O(N) requirement.

The algorithm works through:

• Creating a superposition of all possible states
• Applying an oracle function that marks the target state
• Using amplitude amplification to increase the probability of the target state
• Measuring to obtain the target state with high probability

Try it: Grover's Algorithm Simulation

A single-qubit quantum gate that creates superpositions by transforming basis states to equal superpositions:

• |0⟩ → (|0⟩ + |1⟩)/√2
• |1⟩ → (|0⟩ - |1⟩)/√2

Matrix representation:

H = 1/√2 [1  1]
         [1 -1]

The Hadamard gate is essential for creating superpositions and is used in virtually all quantum algorithms.

An operator representing the total energy of a quantum system. In quantum simulation, Hamiltonians describe the physics of the system being simulated. VQE and other quantum algorithms are designed to find the ground state (lowest energy eigenstate) of Hamiltonians.

A quantum phenomenon where probability amplitudes of quantum states combine constructively or destructively, similar to wave interference. Quantum algorithms leverage interference to enhance probabilities of desired outcomes while suppressing undesired ones.

The process of extracting classical information from a quantum system. Measurement causes a quantum state to collapse from a superposition to a specific basis state, with probabilities determined by the amplitudes in the superposition.

Key properties of quantum measurement:

• Probabilistic outcomes based on quantum state amplitudes
• Collapse of the quantum state upon measurement
• Irreversible process (information loss)
• Can be performed in different bases

A combinatorial optimization problem where the goal is to divide the vertices of a graph into two sets such that the number of edges between the sets is maximized. MaxCut is a common benchmark problem for QAOA and other quantum optimization algorithms.

Try it: QAOA MaxCut Solver Simulation

A term referring to current-generation quantum computers that have limited numbers of qubits (50-1000) and are subject to significant noise and decoherence. NISQ devices are not capable of full error correction but can run algorithms designed to work with noisy qubits.

A fundamental theorem in quantum mechanics stating that it is impossible to create an exact copy of an unknown quantum state. This property underlies the security of quantum key distribution protocols like BB84.

Unwanted interactions between a quantum system and its environment that cause errors in quantum states and operations. Quantum noise leads to decoherence and is a major challenge in building practical quantum computers.

A black-box function in quantum algorithms that encodes the problem to be solved. Oracles are used in algorithms like Grover's search and Deutsch-Jozsa to provide information about the problem without explicitly revealing the solution.

A set of single-qubit quantum gates that are fundamental to quantum computation. There are three Pauli gates:

• Pauli-X: Bit-flip operation (analogous to classical NOT)
• Pauli-Y: Combined bit and phase flip
• Pauli-Z: Phase-flip operation

The complex argument of a quantum amplitude. Phases are not directly observable but affect measurement outcomes through interference. Quantum gates like the phase gate and Z gate manipulate the phase of quantum states.

A quantum algorithm that estimates the eigenvalues of a unitary operator. It is a fundamental subroutine in many quantum algorithms, including Shor's algorithm for factoring integers.

Try it: Quantum Phase Estimation Simulation

A hybrid quantum-classical algorithm designed to solve combinatorial optimization problems. QAOA alternates between quantum evolution under problem and mixing Hamiltonians, with the parameters optimized classically to approximate the optimal solution.

Try it: QAOA Simulation

A quantum analogue of the discrete Fourier transform that transforms quantum states from the computational basis to the Fourier basis. The QFT is a fundamental building block of many quantum algorithms, including Shor's algorithm and quantum phase estimation.

Try it: Quantum Fourier Transform Simulation

The fundamental unit of quantum information, analogous to a classical bit. Unlike classical bits that can only be 0 or 1, qubits can exist in superpositions of states represented as α|0⟩ + β|1⟩, where α and β are complex numbers with |α|² + |β|² = 1.

Qubits can be physically implemented using various quantum systems:

• Superconducting circuits
• Trapped ions
• Photons
• Semiconductor quantum dots
• Neutral atoms

A method that uses quantum mechanics to establish a secure encryption key between two parties. QKD leverages the no-cloning theorem and the fact that measurement disturbs quantum states to detect eavesdropping attempts. BB84 is the most well-known QKD protocol.

Try it: BB84 Quantum Key Distribution Simulation

The demonstration that a programmable quantum device can solve a problem that no classical computer can solve in a feasible amount of time. Google claimed to achieve quantum supremacy in 2019 with a 53-qubit processor performing a specific sampling task.

A device that uses quantum mechanical phenomena to generate truly random numbers. QRNGs leverage the inherent randomness of quantum measurements to produce unpredictable output that cannot be determined in advance, even in principle.

Try it: Quantum Random Number Generator Simulation

Quantum gates that rotate the state of a qubit around different axes of the Bloch sphere. Common rotation gates include:

• RX(θ): Rotation around the X-axis by angle θ
• RY(θ): Rotation around the Y-axis by angle θ
• RZ(θ): Rotation around the Z-axis by angle θ

Rotation gates are essential for creating arbitrary single-qubit states and are fundamental building blocks of quantum algorithms.

A quantum algorithm for integer factorization that runs in polynomial time, making it exponentially faster than the best-known classical algorithms. Shor's algorithm poses a significant threat to RSA and other cryptosystems based on the hardness of factoring.

A quantum error-correcting code that can correct both bit-flip and phase-flip errors. Shor's code encodes one logical qubit into nine physical qubits to protect against arbitrary single-qubit errors.

Try it: Shor's Code Simulation

A fundamental quantum phenomenon where a quantum system exists in multiple states simultaneously. In quantum computing, a qubit in superposition is represented as α|0⟩ + β|1⟩, where α and β are complex amplitudes.

Superposition enables quantum computers to process multiple possibilities in parallel, potentially leading to computational speedups for specific problems.

A quantum protocol that transfers the exact state of a qubit from one location to another using previously shared entanglement and classical communication. Quantum teleportation does not violate the no-cloning theorem or allow faster-than-light communication.

The protocol involves:

• Sharing an entangled pair between sender and receiver
• Performing a Bell measurement on the input state and sender's entangled qubit
• Communicating the measurement results (classical bits) to the receiver
• Applying appropriate corrections based on the measurement results

Try it: Quantum Teleportation Simulation

A three-qubit quantum gate that performs a controlled-controlled-NOT operation. The Toffoli gate flips the target qubit if and only if both control qubits are in state |1⟩. It is a universal gate for classical reversible computation and an important component in quantum error correction.

A property of quantum operations that preserve the norm (total probability) of quantum states. All quantum gates are represented by unitary matrices U, where U†U = I (the identity matrix). Unitarity ensures that quantum operations are reversible and preserve quantum information.

A hybrid quantum-classical algorithm designed to find the ground state energy of a quantum system. VQE uses a parameterized quantum circuit (ansatz) to prepare trial states, with the parameters optimized by a classical optimizer to minimize the energy.

VQE has applications in:

• Quantum chemistry
• Material science
• Optimization problems
• Many-body physics

Try it: Variational Quantum Eigensolver Simulation

A mathematical description of a quantum state. In quantum computing, the wavefunction of a system of n qubits is represented as a complex vector in a 2^n-dimensional Hilbert space, describing the amplitudes of each possible computational basis state.