Glossary

Quantum, word by word

67 terms, each said plainly — with the usual misleading metaphor corrected where there is one, and a link to the simulation where you can watch it happen.

67 terms

Amplitudealso: probability amplitudeOwn page

The complex number attached to each possible outcome of a quantum state; its squared magnitude is that outcome’s probability.

Amplitudes are not probabilities. They can be negative or complex, which means two paths to the same outcome can cancel each other out — something probabilities can never do. That cancellation is interference, and it is where quantum algorithms get their power.

P(x) = |αₓ|², Σₓ |αₓ|² = 1

What Is a Qubit?Superposition & Measurement

Related: Born rule, Interference, Phase

Ancilla qubitalso: ancilla, helper qubit, scratch qubitOwn page

An extra qubit used as scratch space during a computation, usually returned to |0⟩ afterwards.

Ancillas hold intermediate results, mark states for an oracle, or collect syndromes in error correction. Leaving one entangled with the main register when you are done silently ruins interference, which is why “uncomputing” it matters.

Borrow a qubit and give it back Quantum Error Correction

Related: Uncomputation, Syndrome

Barren plateau Own page

A region of a variational circuit’s parameter space where the gradient vanishes exponentially with the number of qubits, so training stalls.

Deep, randomly initialised circuits tend to land on these plateaus, and no optimiser can find a slope that is exponentially small. It is one of the main reasons variational quantum machine learning has proved harder than it first looked.

Barren Plateau Explorer VQE and the Barren Plateau QML Reality Check

Bell inequalityalso: bell test, chsh inequality, chshOwn page

A limit on correlations that any theory of pre-set, local “hidden” answers must obey — and that quantum mechanics violates.

The CHSH version bounds a combination of correlations at 2 for any local hidden-variable theory; entangled qubits reach 2√2. Experiments closing the major loopholes earned the 2022 Nobel Prize. The violation rules out local hidden scripts; it does not allow signalling.

|S| ≤ 2 (local), |S| ≤ 2√2 (quantum)

The Bell Test

Bell statealso: bell pair, epr pair, phi plusOwn page

One of four maximally entangled two-qubit states; the simplest and most-used form of entanglement.

Made with a Hadamard followed by a CNOT. Measure either qubit and you get a random bit — but the other qubit’s result is then perfectly correlated with it. Neither qubit has a definite state of its own.

|Φ⁺⟩ = (|00⟩ + |11⟩)/√2

Make a Bell pair Entanglement Bell State (Φ+) in the simulator

Bloch sphere Own page

A sphere on which every pure state of a single qubit is one point, and every single-qubit gate is a rotation.

|0⟩ sits at the north pole, |1⟩ at the south, and equal superpositions around the equator, where the longitude is the relative phase. Note that opposite points on the sphere are orthogonal states — the angle on the sphere is twice the angle in the state vector.

|ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩

The Bloch Sphere What Is a Qubit?

Born rule Own page

The rule that the probability of a measurement outcome is the squared magnitude of its amplitude.

It is the only bridge between the mathematics of amplitudes and what an experiment records. Every histogram you get from a simulator or a real device is sampled according to it.

P(x) = |⟨x|ψ⟩|²

Superposition & Measurement Superposition & Measurement

Related: Amplitude, Measurement

Bra-ket notationalso: dirac notation, ket, braOwn page

Dirac’s notation for quantum states: a ket |ψ⟩ is a column vector, a bra ⟨ψ| its conjugate transpose.

⟨φ|ψ⟩ is an inner product (a number) and |ψ⟩⟨φ| an outer product (a matrix). Once it clicks, most quantum computing papers become readable.

|0⟩ = (1, 0)ᵀ, |1⟩ = (0, 1)ᵀ

What Is a Qubit?

Related: Qubit, Amplitude

Clifford gatesalso: Clifford group, Gottesman–Knill theorem, Clifford circuitOwn page

The gates — generated by H, S and CNOT — that turn every Pauli operator into another Pauli operator.

Because they only shuffle Paulis, a circuit of Clifford gates can be simulated efficiently on a classical computer however many qubits and however much entanglement it has (the Gottesman–Knill theorem). So entanglement alone is not what makes quantum computing hard to simulate: it takes a non-Clifford gate such as T.

Stabilizer Simulator Measurement-Induced Entanglement Transition Try the S gate

CNOT gatealso: controlled not, cx, controlled xOwn page

A two-qubit gate that flips the target qubit if and only if the control qubit is |1⟩.

Applied to a control in superposition it produces entanglement, which makes it the workhorse two-qubit gate. On hardware it is typically an order of magnitude noisier than a single-qubit gate, so the number of CNOTs is the cost metric that matters.

Broken CNOT Try the CX gate Gates & Circuits

Code distancealso: [[n,k,d]], distance, code parametersOwn page

The smallest number of qubits an error must touch to change the encoded information without being detected.

In the notation [[n, k, d]], n physical qubits encode k logical qubits at distance d. A distance-d code detects any error on up to d − 1 qubits and corrects any on up to ⌊(d − 1)/2⌋. It is not the number of errors the code survives: a distance-3 code corrects one.

corrects t = ⌊(d − 1)/2⌋ errors; Steane code: [[7, 1, 3]]

Stabilizer Code Explorer Fault-Tolerant Resource Estimator The Surface Code

Coherence time (T1, T2)also: t1, t2, relaxation time, dephasing timeOwn page

How long a physical qubit keeps its quantum information: T1 for energy relaxation, T2 for loss of phase.

T1 is the time for |1⟩ to decay to |0⟩; T2 is the time for a superposition to lose its phase relationship, and T2 ≤ 2T1. Divide the coherence time by the gate time and you get a rough budget for how many operations fit before the state degrades.

Decoherence Calibration Agent

Related: Decoherence, Noise

Coupling mapalso: connectivity, topology, heavy hexOwn page

The graph of which physical qubits on a device can interact directly with a two-qubit gate.

Most chips connect each qubit only to a few neighbours. A circuit that needs a gate between distant qubits must first move the states together with SWAPs, which a compiler inserts during routing — at a real cost in noise.

Route around the coupling map Routing Race

Related: Transpilation, SWAP gate

Decoherence Own page

The loss of quantum behaviour as a system becomes entangled with its uncontrolled environment.

Nothing mysterious “collapses”: information about the qubit leaks into its surroundings, and from the qubit’s point of view a superposition turns into an ordinary coin toss. It is the central engineering obstacle to building quantum computers.

Decoherence Mixed States and the Density Matrix

Density matrixalso: density operator, ρOwn page

A matrix description of a quantum state that can also represent uncertainty about which state you have.

Pure states give ρ = |ψ⟩⟨ψ|; mixtures give a weighted sum. The off-diagonal entries (“coherences”) are what decoherence destroys. It is the natural language for noise, and for one part of an entangled system.

ρ = Σᵢ pᵢ |ψᵢ⟩⟨ψᵢ|, Tr ρ = 1

Mixed States and the Density Matrix Tomography and No-Cloning

Related: Mixed state, Decoherence

Entanglement Own page

A correlation between quantum systems so strong that the whole has a definite state while the parts do not.

Measuring one entangled qubit tells you about the other, however far away — but the result you get is random, so no message can be sent this way. What makes it quantum is that the correlations hold in every measurement basis at once, which is what Bell tests detect.

Entanglement Entanglement The Bell Test

Entanglement entropyalso: von Neumann entropy, area law, volume lawOwn page

A number, in bits, for how entangled one part of a pure quantum state is with the rest.

It is the entropy of the part on its own: zero when the two parts are unentangled, and at most the size of the smaller part. When it grows with the size of the region the state follows a volume law; when it grows only with the boundary, an area law. It measures entanglement only for pure states — for a mixed state it also counts ordinary classical uncertainty.

S(A) = −Tr(ρ_A log₂ ρ_A)

Stabilizer Simulator Measurement-Induced Entanglement Transition Entanglement

Error mitigationalso: zne, zero noise extrapolation, readout mitigation, trexOwn page

Techniques that reduce the bias noise introduces into results, without the qubit overhead of full error correction.

Zero-noise extrapolation runs a circuit at deliberately amplified noise levels and extrapolates back to zero; readout mitigation inverts the measured confusion matrix. They cost extra shots rather than extra qubits, and they correct averages, not individual runs.

Noise Sandbox and Mitigation Run on real hardware

Error thresholdalso: threshold theoremOwn page

The physical error rate below which adding more qubits to an error-correcting code makes logical errors rarer, not more common.

Above threshold, a bigger code only collects more errors. Below it, logical error rate falls exponentially with code distance. Crossing it is the milestone that makes large-scale quantum computing an engineering problem rather than a hope.

Decoder Lab The Surface Code

Related: Fault tolerance, Surface code

Expectation valuealso: observable, ⟨O⟩Own page

The average result you would get measuring an observable on many copies of the same state.

Variational algorithms minimise an expectation value — usually an energy. On hardware it is estimated from finitely many shots, so it always comes with statistical error bars.

⟨O⟩ = ⟨ψ|O|ψ⟩

NISQ Algorithms and VQE Noise Sandbox and Mitigation

Fault tolerancealso: fault-tolerant quantum computing, ftqcOwn page

Running a quantum computation reliably even though every physical component is faulty, by encoding in error-correcting codes.

The threshold theorem says that if physical error rates are below a threshold, logical error rates can be made arbitrarily small by adding more qubits. Useful algorithms like Shor’s need fault tolerance; today’s devices are only just demonstrating logical qubits.

Quantum Error Correction The Surface Code

GHZ statealso: greenberger horne zeilinger, cat stateOwn page

An entangled state of three or more qubits that are all 0 or all 1, in superposition.

Lose any one qubit and the entanglement of the rest is destroyed, which makes GHZ states fragile — and a common benchmark of how well a device preserves multi-qubit coherence.

(|000⟩ + |111⟩)/√2

GHZ for three GHZ State (3 qubits) in the simulator

Related: Entanglement, Bell state

Grover's algorithmalso: grover search, amplitude amplificationOwn page

A quantum algorithm that finds a marked item among N in about √N queries, where classical search needs about N.

It works by repeatedly rotating the state towards the marked item. It is a rotation, not a convergence: run too many iterations and you rotate past the answer. The quadratic speedup is provably optimal for unstructured search.

≈ (π/4)√N iterations

Grover's Search Grover's Search One full turn of Grover

Related: Oracle, Interference

Hadamard gatealso: h gate, hadamardOwn page

The single-qubit gate that turns |0⟩ into an equal superposition of |0⟩ and |1⟩, and back again.

Applying H twice returns the original state — the two paths to |1⟩ cancel. That small fact is interference in its simplest form, and the reason H is the first gate of almost every algorithm.

H|0⟩ = (|0⟩+|1⟩)/√2, H² = I

Fair coin Try the H gate Undo the Hadamard

Related: Superposition, Interference

Hamiltonian Own page

The operator for a system’s total energy; it determines how the state evolves in time.

Its eigenvalues are the allowed energies and its lowest eigenvalue is the ground-state energy that VQE hunts for. Time evolution under a Hamiltonian H for time t is the unitary e^{−iHt/ħ}.

iħ ∂|ψ⟩/∂t = H|ψ⟩

The Quantum Harmonic Oscillator VQE and the Barren Plateau

Interference Own page

Amplitudes for the same outcome adding up (constructive) or cancelling (destructive).

This, not “trying every answer at once”, is the source of quantum speedups: an algorithm is choreographed so that wrong answers cancel and right answers reinforce before you measure.

Interference: Where the Power Comes From The Double-Slit Experiment The Mach-Zehnder Interferometer

Related: Amplitude, Phase, Superposition

Logical qubit Own page

A qubit encoded redundantly across many physical qubits so that errors can be detected and corrected.

A surface-code logical qubit uses on the order of 2d² physical qubits for code distance d. Estimates for breaking RSA-2048 run to around a million physical qubits, which is why qubit counts alone say little about capability.

Quantum Error Correction The Surface Code Decoder Lab

Measurementalso: collapse, readoutOwn page

Reading out a qubit, which returns a classical bit sampled by the Born rule and leaves the qubit in the state matching it.

A measurement in the computational basis only asks “0 or 1?”, and in answering it discards the phase information. Measuring in a different basis means rotating first — which is why the same state can look random in one basis and certain in another.

Superposition & Measurement The Stern-Gerlach Experiment

Related: Born rule, Superposition

Measurement-induced phase transitionalso: MIPT, entanglement transition, hybrid circuitOwn page

A sharp change in how entangled a quantum circuit’s state is, as the rate of mid-circuit measurements crosses a critical value.

Random gates build entanglement and measurements remove it. Below the critical measurement rate entanglement grows with system size; above it, it does not. The transition is invisible in the average state — it shows up only when each run is examined together with its own measurement record, which is what makes it hard to observe on hardware.

Measurement-Induced Entanglement Transition Stabilizer Simulator

Mixed state Own page

A state that is a statistical mixture of pure states, describing classical uncertainty on top of quantum uncertainty.

Mixed States and the Density Matrix

NISQalso: noisy intermediate-scale quantumOwn page

John Preskill’s name for today’s era: devices with tens to a few thousand noisy qubits and no full error correction.

NISQ algorithms — VQE, QAOA and variational classifiers — use short circuits and lean on classical optimisation to cope with noise. Whether they deliver a practical advantage before fault tolerance arrives is an open question.

NISQ Algorithms and VQE Run on real hardware

No-cloning theorem Own page

There is no process that makes an independent copy of an arbitrary unknown quantum state.

It follows directly from the linearity of quantum mechanics. It is why quantum error correction must be cleverer than keeping backups, and why an eavesdropper on quantum key distribution cannot copy the qubits and pass them on undisturbed.

Tomography and No-Cloning Quantum Key Distribution

No-communication theoremalso: no signalling, faster than lightOwn page

Entanglement cannot be used to send information: nothing done to one half changes the statistics seen at the other.

Correlations only show up when the two parties compare results over an ordinary channel. This is why teleportation needs two classical bits per qubit, and why entanglement does not beat the speed of light.

Entanglement Quantum Teleportation

Noisealso: quantum noise, gate error, depolarizingOwn page

Any unwanted process that disturbs a quantum computation: imperfect gates, decoherence, crosstalk, readout errors.

Noise is characterised by error rates per gate and per measurement. Two-qubit gates and readout typically dominate. Every result from real hardware carries its signature, which is why comparing against a noise model is part of reading one.

Noise Sandbox and Mitigation Decoherence

OpenQASMalso: qasm, openqasm 3Own page

A text language for describing quantum circuits, used to move circuits between tools and onto hardware.

Qiskit, Braket and most other stacks read and write it. The simulator here imports and exports OpenQASM 3, including exact π fractions for rotation angles.

Circuit simulator

Related: Transpilation

Oraclealso: black box, marking oracle, phase oracleOwn page

A subroutine that encodes the problem, often by flipping the phase of the states that are solutions.

Query complexity counts calls to the oracle. In real applications the oracle must be built out of gates, and its cost can swamp the speedup — a detail textbook treatments often skip.

Mark the state Deutsch–Jozsa: The First Quantum Speedup Grover's Search

Pauli gates (X, Y, Z)also: pauli x, pauli y, pauli z, bit flip, phase flipOwn page

The three basic single-qubit operations: X flips the bit, Z flips the phase, Y does both.

On the Bloch sphere each is a half-turn about its axis. Any single-qubit error can be written as a combination of them, which is why error-correcting codes only need to catch X and Z errors.

X|0⟩ = |1⟩, Z|1⟩ = −|1⟩, Y = iXZ

Flip it The Bloch Sphere The invisible sign

Phasealso: relative phase, global phaseOwn page

The angle of a complex amplitude; relative phases between outcomes decide how they interfere.

A global phase on the whole state is unobservable. A relative phase is invisible to a direct measurement but changes everything once the paths are recombined — the “invisible sign” behind most algorithms.

(|0⟩ + e^{iφ}|1⟩)/√2

The invisible sign The Bloch Sphere Quantum Phase Estimation

Phase kickback Own page

When a controlled operation acts on a target in an eigenstate, the eigenvalue’s phase appears on the control qubit instead.

It sounds backwards and it is the trick behind Deutsch–Jozsa, Bernstein–Vazirani, phase estimation and Shor. The target is unchanged; the information moves to the control.

Kick the phase back Deutsch–Jozsa: The First Quantum Speedup

Post-quantum cryptographyalso: pqc, quantum-safe, ml-kem, kyber, dilithiumOwn page

Classical cryptography designed to resist attack by quantum computers, running on ordinary hardware.

NIST standardised ML-KEM (from Kyber), ML-DSA (from Dilithium) and SLH-DSA (from SPHINCS+) in 2024. Unlike QKD it needs no special hardware, which is why it is the main migration path. Data encrypted today can be harvested and decrypted later, so the migration is urgent before the machines exist.

Post-Quantum Cryptography

QAOAalso: quantum approximate optimization algorithmOwn page

A variational algorithm for combinatorial optimisation that alternates a problem Hamiltonian with a mixing Hamiltonian.

Depth p controls the quality: as p grows it can in principle reach the optimum. Whether low-depth QAOA beats the best classical heuristics on practical problems is not established.

QAOA and Quantum Optimization

Quantum advantagealso: quantum supremacy, speedupOwn page

A quantum computer solving a task faster or better than any feasible classical method.

Demonstrations so far are on sampling tasks chosen for the purpose, and several have been narrowed by improved classical simulation. A useful advantage on a practical problem remains the field’s goal. Always ask: against which classical baseline?

QML Reality Check The HHL Algorithm

Related: NISQ, Fault tolerance

Quantum error correctionalso: qecOwn page

Encoding quantum information redundantly so errors can be detected and fixed without measuring — and destroying — the data.

Instead of reading qubits, you measure parity checks (stabilisers) that reveal where an error occurred but nothing about the encoded state. A classical decoder then decides the most likely correction.

Quantum Error Correction The Surface Code Decoder Duel: Lookup vs Learned

Quantum Fourier transformalso: qftOwn page

The quantum version of the discrete Fourier transform, acting on amplitudes with only O(n²) gates for n qubits.

You cannot read out all the Fourier coefficients — measuring gives one sample — so the QFT is useful only inside algorithms that arrange for the answer to be concentrated, such as period finding and phase estimation.

Quantum Fourier Transform The two-qubit QFT, missing its last step

Quantum key distributionalso: qkd, bb84, e91Own page

Protocols such as BB84 that let two parties agree a secret key, with any eavesdropping revealed by the errors it causes.

Security rests on measurement disturbance and no-cloning rather than a hard maths problem. In practice it needs dedicated optical links, is limited in distance, and still requires an authenticated classical channel.

Quantum Key Distribution Quantum Cryptography

Quantum phase estimationalso: qpeOwn page

An algorithm that reads out the eigenvalue phase of a unitary into a register of qubits.

It combines controlled unitaries, phase kickback and an inverse QFT. It is the core of Shor’s algorithm and of fault-tolerant quantum chemistry.

Quantum Phase Estimation Read the phase bit

Quantum teleportation Own page

Transferring an unknown qubit state using a shared Bell pair plus two classical bits.

The original is destroyed in the process (no cloning), and nothing moves faster than light — the receiver cannot reconstruct the state until the two classical bits arrive.

Quantum Teleportation Finish the teleporter Quantum Teleportation in the simulator

Quantum tunnellingalso: tunneling, barrier penetrationOwn page

A particle crossing an energy barrier it classically could not climb over.

The wavefunction decays exponentially inside the barrier rather than stopping, so a thin enough barrier lets some of it through. Flash memory, scanning tunnelling microscopes and alpha decay all depend on it.

Quantum Tunneling Quantum Tunneling Alpha Decay: Tunneling with a Half-life

Qubitalso: quantum bitOwn page

The basic unit of quantum information: a two-level system whose state is a normalised combination of |0⟩ and |1⟩.

A qubit is better pictured as a direction than a switch. It holds a continuum of possible states but yields just one classical bit when measured — so n qubits do not store 2ⁿ readable numbers.

|ψ⟩ = α|0⟩ + β|1⟩, |α|² + |β|² = 1

What Is a Qubit?The Bloch Sphere

Schrödinger equation Own page

The equation that governs how a quantum state changes in time.

Its time-independent form gives the allowed energies of bound systems — the particle in a box, the harmonic oscillator, the hydrogen atom. Its evolution is unitary and reversible; only measurement is not.

iħ ∂ψ/∂t = −(ħ²/2m)∇²ψ + Vψ

Particle in a Box Wave Packets and Scattering Quantum Tunneling

Related: Hamiltonian, Wavefunction

Shor's algorithm Own page

A quantum algorithm that factors integers and computes discrete logarithms in polynomial time.

It reduces factoring to finding the period of modular exponentiation, and finds the period with a QFT. It breaks RSA and elliptic-curve cryptography — but only on a large fault-tolerant machine that does not yet exist.

Shor's Algorithm Shor's Algorithm: Period Finding

Shots Own page

The number of times a circuit is run and measured to build up a histogram of outcomes.

Each shot yields a single bitstring, so probabilities are estimated statistically: the error shrinks like 1/√shots. Four times the shots halves the error bar.

Circuit simulator Run on real hardware

Related: Born rule, Expectation value

Spin Own page

An intrinsic angular momentum of particles; for an electron, a two-level system that is a natural qubit.

Measuring spin along any axis gives only ±ħ/2, as Stern and Gerlach found in 1922. Nothing is literally spinning — the name is historical.

The Stern-Gerlach Experiment Spin Precession

Related: Qubit, Measurement

Stabilizer formalismalso: stabilizer state, stabilizer code, stabilizer groupOwn page

A way to describe a quantum state by the Pauli operators that leave it unchanged, instead of by its amplitudes.

An n-qubit stabilizer state is pinned down by n commuting Pauli operators — n short strings rather than 2ⁿ numbers. It is not a simplification for beginners: nearly every quantum error-correcting code is defined this way, with fewer than n stabilizers so that a protected space is left over.

S|ψ⟩ = |ψ⟩ for every S in the stabilizer group; Bell pair: ⟨XX, ZZ⟩

Stabilizer Simulator Stabilizer Code Explorer Quantum Error Correction

Superposition Own page

A quantum state that is a weighted combination of basis states, with complex amplitudes.

It is not “being in both states at once” in the everyday sense, and not ignorance of a hidden answer. It is a single, definite state — one that simply is not one of the basis states you chose to measure in.

|ψ⟩ = Σₓ αₓ|x⟩

Superposition & Measurement Superposition & Measurement Fair coin

Surface code Own page

A quantum error-correcting code laid out on a 2D grid, needing only nearest-neighbour interactions.

Its high threshold (around 1% per operation) and hardware-friendly layout make it the leading candidate for fault-tolerant machines. In 2024 Google demonstrated logical error falling as the code grew — operation below threshold.

The Surface Code Decoder Lab Stabilizer Code Explorer Run This Paper

SWAP gate Own page

A two-qubit gate that exchanges the states of two qubits; it costs three CNOTs.

Compilers insert SWAPs to bring distant qubits together on hardware with limited connectivity, so a sloppy qubit layout can triple a circuit’s two-qubit gate count.

Routing Race Try the SWAP gate

Syndromealso: error syndrome, stabilizer measurementOwn page

The pattern of parity-check results that tells an error-correcting code where an error probably happened.

Syndrome measurements reveal the error without revealing the encoded data. Turning a syndrome into a correction is the decoder’s job, and it must keep up in real time.

The Surface Code Decoder Duel: Lookup vs Learned Decoder Lab Stabilizer Code Explorer

Transpilationalso: compilation, transpilerOwn page

Rewriting a circuit into the native gates and connectivity of a specific device, while optimising it.

It decomposes gates, chooses which physical qubit plays each logical one, inserts SWAPs, and cancels redundancy. The same circuit can transpile to very different gate counts on different devices, which is why device choice matters.

Routing Race Run on real hardware

Uncertainty principlealso: heisenbergOwn page

A limit on how sharply a state can define two incompatible quantities at once, such as position and momentum.

It is a property of the state, not a clumsiness of the measuring device: a narrow wave packet in position is necessarily a broad one in momentum, exactly as a short pulse needs many frequencies.

Δx · Δp ≥ ħ/2

Uncertainty as a Fourier Trade-off Wave Packets and Scattering

Related: Wavefunction

Uncomputation Own page

Running part of a circuit in reverse to return scratch qubits to |0⟩ and disentangle them from the result.

Leftover garbage entangled with your answer acts like a measurement of it and kills interference. Reversibility is what makes clean-up possible.

Borrow a qubit and give it back

Related: Ancilla qubit, Unitary

Unitaryalso: unitary operator, unitary matrixOwn page

A reversible, probability-preserving transformation; every quantum gate is a unitary matrix.

Unitarity means UU† = I, so every gate can be undone. That is why quantum circuits are reversible and why measurement, which is not, is the odd one out.

U†U = UU† = I

Gates & Circuits Undo the Hadamard

Related: Hamiltonian, Uncomputation

Universal gate setalso: clifford+t, native gatesOwn page

A small set of gates from which any quantum computation can be approximated to arbitrary accuracy.

A common choice is {H, S, T, CNOT}. Clifford gates alone (H, S, CNOT) can be simulated efficiently on a classical computer; adding the T gate is what makes the set universal — and what makes fault-tolerant computation expensive.

Gates & Circuits Circuit simulator

Variational circuitalso: parameterized circuit, ansatz, pqcOwn page

A circuit with tunable rotation angles that a classical optimiser adjusts to minimise a cost.

The chosen circuit shape is called the ansatz. Too simple and it cannot express the answer; too deep and noise and barren plateaus take over. Choosing it well is most of the art.

VQE and the Barren Plateau Tilt the coin Algorithm Evolution

Variational quantum eigensolveralso: vqeOwn page

A hybrid algorithm that estimates a Hamiltonian’s ground-state energy by optimising a variational circuit.

The quantum computer prepares a trial state and measures its energy; a classical optimiser updates the parameters. By the variational principle, the measured energy can never be below the true ground state — noise and shot error aside.

E(θ) = ⟨ψ(θ)|H|ψ(θ)⟩ ≥ E₀

NISQ Algorithms and VQE VQE and the Barren Plateau

Wave–particle duality Own page

Quantum objects show wave-like interference when unobserved and arrive as localised particles when detected.

Send electrons one at a time through two slits: each lands as a single dot, yet the dots build up an interference pattern. Find out which slit each went through and the pattern disappears.

The Double-Slit Experiment Davisson–Germer: Electron Diffraction The Quantum Eraser

Related: Interference, Wavefunction

Wavefunctionalso: ψ, psiOwn page

The function ψ(x) giving the amplitude for a particle to be found at each position.

|ψ(x)|² is a probability density. It evolves smoothly by the Schrödinger equation, spreads, reflects and interferes like a wave — but is detected as a single particle at a single point.

Wave Packets and Scattering Wavefunctions in Three Dimensions The Double-Slit Experiment