Quantum Archive

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

A pure qubit state is a normalized vector in a two-dimensional complex vector space. It is not a hidden classical bit and it is not simply two classical values stored at once.

One vector in a chosen coordinate system for describing quantum states.

For a single qubit, the computational-basis states are |0> and |1>. A basis is a descriptive coordinate system, not a claim that the system secretly has one of those values.

The standard basis {|0>, |1>} used for most circuit descriptions and Z-basis measurement.

Other bases are possible. Changing the basis can change the outcome probabilities even when the physical state is unchanged.

A vector representation of a pure quantum state.

The same physical state can be represented in different bases. State vectors encode amplitudes and relative phase, not just likely measurement outcomes.

A geometric representation of single-qubit pure states up to global phase.

The sphere encodes relative amplitude and relative phase for one qubit. It is useful, but it does not generalize visually to many qubits in a simple way.

A complex coefficient whose squared magnitude gives a measurement probability.

Amplitudes carry magnitude and phase. Probabilities discard phase information, which is why states with identical probabilities can later behave differently.

The requirement that the total probability over a complete measurement basis is 1.

For |psi> = alpha|0> + beta|1>, normalization requires |alpha|^2 + |beta|^2 = 1.

A coherent linear combination of basis states.

Superposition is not ordinary ignorance about a pre-existing value. Relative phase within the combination can affect later interference.

The phase difference between components of a superposition.

Relative phase is physically meaningful because later gates can turn phase differences into measurable probability differences.

A common phase factor multiplying the entire state vector.

Vectors that differ only by e^{i gamma} represent the same pure state: unitary evolution preserves the common factor and every measurement gives the same outcome probabilities. It differs from relative phase.

Constructive or destructive combination of probability amplitudes.

Quantum algorithms use interference to amplify amplitudes of useful outcomes and suppress others. This does not mean every answer is simply tested in parallel.

A controlled operation causing information about a function to appear as phase on another register.

Phase kickback lets algorithms convert a function property into relative signs, which later Hadamards can turn into measurable probability differences.

A physical operation that produces a classical outcome according to a chosen basis.

Projective measurement also changes the post-measurement state. It is not merely revealing a stored classical value.

The orthonormal basis whose alternatives define the possible measurement outcomes.

A computational-basis measurement and an X-basis measurement ask different physical questions of the same state.

The rule that maps amplitudes to outcome probabilities by squared magnitude.

For measurement in a basis, each outcome probability is obtained from the magnitude squared of the component along that basis vector.

The state assigned after a measurement outcome has occurred.

After projective measurement, repeating the same measurement immediately gives the same outcome in the ideal model.

Unwanted interaction or imperfection that changes a quantum state or recorded outcome.

Common introductory models include bit flips, phase flips, depolarizing noise, amplitude damping, and measurement error. These are models, not full descriptions of every device.

Loss of coherence in a system's reduced state through uncontrolled environmental coupling.

Dephasing suppresses off-diagonal coherence in a chosen basis, while amplitude damping models energy relaxation and also changes populations. Decoherence is an open-system physical process, not merely missing knowledge.

A unitary linear operation in the ideal circuit model.

Ideal gates transform amplitudes unitarily and are reversible. Noisy open-system evolution, loss, reset, and measurement require more general quantum-channel descriptions.

A single-qubit gate that connects computational-basis states with |+> and |-> states.

H creates equal superpositions from |0> and |1>, and it can also resolve phase differences into deterministic outcomes.

A gate that exchanges |0> and |1> in the computational basis.

On a general superposition it acts linearly, moving each amplitude to the other basis component.

A multi-qubit gate whose action on a target depends on the control qubit.

CNOT flips the target when the control is |1>. On superpositions, linearity applies the rule to every basis component, which is why H followed by CNOT can create a Bell state.

The result that an arbitrary unknown quantum state cannot be copied perfectly by a physical operation.

A copying rule that works for |0> and |1> cannot also copy every superposition linearly. The theorem is a statement about unknown arbitrary states, not about preparing many known states.

A black-box operation used in algorithm analysis to encode a promised function or marked item.

Oracle algorithms make precise claims about query counts under a promise. They should not be presented as unlimited practical speedups without considering input loading, noise, and resources.

A multi-qubit state that cannot be factored into independent single-qubit states.

Entanglement produces correlations stronger than separable state descriptions, but it does not allow faster-than-light communication.

The operation used to combine individual system spaces into a joint state space.

For two qubits the ordered computational basis is often |00>, |01>, |10>, |11>. Keeping the basis order explicit prevents many circuit and CNOT mistakes.

One of four maximally entangled two-qubit states.

The state |Phi+> = (|00> + |11>)/sqrt(2) has perfectly correlated Z-basis outcomes, but neither qubit has its own independent pure state.

Encoding quantum information across multiple physical qubits so some errors can be detected and corrected.

Quantum error correction protects information through redundancy in a code space and syndrome measurements. It does not permit perfect copying of an unknown state.

A pattern of check outcomes used to identify an error class relative to a quantum code.

A syndrome guides recovery only together with a code and assumed error model. It need not uniquely identify every physical error, and errors outside the correctable set can be ambiguous or uncorrectable.

A design principle for computing reliably despite imperfect physical operations.

Fault-tolerant quantum computing requires error rates, code choices, overhead, and control systems to work together. It is the route toward long reliable algorithms, not a single gate.

A demonstrated or argued advantage of a quantum method over classical alternatives for a defined task.

A responsible advantage claim specifies the task, assumptions, resources, error model, and comparison baseline. It does not mean quantum computers are faster for every problem.