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Apply a Hadamard gate to the first qubit, then a controlled-NOT gate from the first to the second. This maps a computational-basis state $|ij⟩$ to the Bell state with phase bit $i$ and parity bit $j$. Its inverse uses the same gates in reverse order because each gate is self-inverse but they do not commute. Applying the forward sequence twice is not generally a decoding operation.

The $m$-qubit bit-flip repetition code stores one logical qubit in $|0_L⟩=|0{⟩}^{\otimes m}$ and $|1_L⟩=|1{⟩}^{\otimes m}$. Its stabilizer generators are $Z_j{Z}_{j+1}$ for $j=1,\dots ,m-1$, and its logical operators are $\overline{X}=X_1\cdots X_m$ and $\overline{Z}=Z_1$. Interchanging the Pauli X gate and Pauli Z gate gives the phase-flip repetition code. For odd $m$, majority-vote decoding of a repetition code corrects up to $(m-1)/2$ physical Pauli X gate errors, assuming ideal operations and a noise family containing only these bit flips.

For $m=3$, prepare two quantum ancillas in $|00⟩$ and apply controlled-NOT gates from the input qubit to each of them. By linearity, encoding sendsThis does not produce $(\alpha |0⟩+\beta |1⟩{)}^{\otimes 3}$, which would contain cross terms. For example, encoding $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ produces $(|000⟩+|111⟩)/\sqrt{2}$, whereas three independent copies would have eight equal computational basis amplitudes. The encoding creates quantum entanglement when both coefficients are nonzero and respects the no-cloning theorem.

The three-qubit bit-flip syndrome table corrects the family $\{I,X_1,X_2,X_3\}$ and its entire linear span, as specified by the Knill--Laflamme condition. It cannot correct arbitrary errors on one physical qubit: the phase flipcommutes with both checks and acts as logical $\overline{Z}$. In particular, it takes the encoded $|+⟩$ to the orthogonal encoded $|-⟩$ with error syndrome $00$. Thus its full quantum code distance is one, despite correcting one error of the restricted Pauli X gate type. The Shor code, Steane code and five-qubit error correcting code provide protection against arbitrary one-qubit errors.

The group of unitary Clifford gates, defined by normalization of the Pauli group. For a single qubit, $H$ and $S$ generate its usual finite gate group, and modulo global phase its action permutes the six Pauli eigenstates. On multiple qubits, one adds an entangling controlled-NOT gate. This quantum-computing use of the name is distinct from groups attached to a general Clifford algebra.

Encode $u|0⟩+v|1⟩$ as $u|000⟩+v|111⟩$. Four controlled-NOT gates copy the displayed two data parities into initially zero quantum ancillas. No logical amplitude is measured. The error syndromes $00,01,10,11$ correspond respectively to no bit flip and bit flips on data qubits $1,2,3$. Conditional Toffoli gates, with temporary Pauli X gates implementing zero-valued controls, undo the indicated error. Inverse encoding yields the original logical state and a separate syndrome state. A phase flip has zero bit-flip syndrome and remains as a logical phase error; this three-qubit code is not a general single-qubit-error code.

The controlled-Z gate is $CP(\pi )=\mathrm{diag}(1,1,1,-1)$. Conjugating the target of a controlled-NOT gate by Hadamard gates turns controlled-NOT into controlled-Z.

For odd $n=2k+1$, a symmetric quantum teleportation resource can encode an unknown qubit into $E(\alpha |0⟩+\beta |1⟩)=\alpha |D_k^n⟩+\beta |D_{k+1}^n⟩$. Begin with $|D_{k+1}^{n+1}⟩$, apply a controlled-NOT gate from the input to the sender's resource qubit, apply a Pauli X gate to that resource qubit and a Hadamard gate to the input, and measure those two qubits. The four unnormalized remaining vectors are $E|\psi ⟩/2$, $EX|\psi ⟩/2$, $EZ|\psi ⟩/2$ and $EXZ|\psi ⟩/2$. Each outcome has probability $1/4$. Since $X^{\otimes n}$ interchanges $|D_k^n⟩$ and $|D_{k+1}^n⟩$, and $Z^{\otimes n}$ acts as logical $Z$ up to the common phase $(-1{)}^k$, local Pauli corrections make the output independent of the result. The one-qubit reduction of Dicke-state superpositions givesFor $n>1$ this distributes imperfect copies, as required by the no-cloning theorem. For $n=1$ it reduces to exact quantum teleportation.

In most-significant-bit-first order, let $x=x_1\cdots x_n$ and $0.x_j\cdots x_n={\sum }_{l=j}^n{x}_l{2}^{-(l-j+1)}$. On a computational-basis state, process wire $j$ with a Hadamard gate, then controlled phase gates $R_{l-j+1}$ controlled by unprocessed wire $l$ for each $l>j$. Wire $j$ becomes $(|0⟩+e^{2\pi i0.x_j\cdots x_n}|1⟩)/\sqrt{2}$. Reversing the final wire order yields the positive-exponent quantum Fourier transform. There are $n$ Hadamard gates and $n(n-1)/2$ controlled phase gates. Physical reversal costs $\lfloor n/2\rfloor$ swap operators, each expressible as three controlled-NOT gates, themselves implemented with Hadamard gates and a Controlled-Z gate. Thus the entire construction uses $O(n^2)$ gates from the specified family.

A unitary gate on two quantum systems is entangling if it maps at least one pure state that is a product state to an entangled state. The controlled-NOT gate qualifies because it maps $|+⟩|0⟩$ to a Bell state. A gate that merely exchanges the two systems need not be a local unitary operation, but cannot create quantum entanglement from a product state.

For $n\ge 2$, apply a Hadamard gate to one qubit, then controlled-NOTs from it to all others. Replacing each controlled-NOT gate by a target Hadamard, a Controlled-Z gate, and another target Hadamard uses $1+3(n-1)$ allowed gates. Each one-qubit reduced state is $I/2$, proving entanglement across each qubit-versus-rest bipartition.

For a quantum cloning device with pure state output $U|\varphi ,0⟩$, the global cloning fidelity is its squared quantum fidelity with the desired joint two-copy pure state. It is the probability that a joint projector test for two ideal copies accepts the actual output. It differs from a one-clone marginal fidelity: correlations between the output qubits affect the joint overlap. For a controlled-NOT gate with input $|+⟩|0⟩$, the output is a Bell state and its global cloning fidelity with $|++⟩$ is $1/2$; for input $|0⟩|0⟩$ it is one.

The inverse of a quantum circuit applies the adjoints of its gates in reverse temporal order. The Hadamard gate, Pauli Z gate, and controlled-NOT gate are each self-inverse.

Apply a controlled-NOT gate from data $a|0⟩+b|1⟩$ to the resource $|A⟩$, then measure the resource in the standard basis. Outcome zero leaves data $T|\psi ⟩$; outcome one leaves $e^{i\pi /4}{T}^{†}|\psi ⟩$. Both outcomes have probability $1/2$. Correct outcome one with $S$, using $S{T}^{†}=T$, so the data has undergone the T gate irrespective of the measurement result. The resource is consumed, and its Bloch vector lies outside the stabilizer octahedron.

A quantum computing apparatus needs a scalable quantum register of controllable qubits, reproducible initialization, a universal quantum gate set, sufficiently long quantum coherence, and selective readout. A qubit is a two-dimensional state space; several qubits occupy a tensor product space and can develop entanglement. Quantum state preparation must establish a known low-entropy input and supply fresh ancillas during quantum error correction. A universal quantum gate set may use arbitrary single-qubit rotations with an entangling two-qubit gate such as controlled-NOT gate. Quantum coherence and control fidelity must suffice for the intended circuit, and measurement must recover chosen output probabilities without uncontrolled disturbance of other qubits. These are the five DiVincenzo criteria.^[ref]

An atomic implementation can encode qubits in two long-lived internal levels, initialize them by optical pumping, drive coherent transitions with laser or microwave pulses, couple selected qubits through a shared interaction, and distinguish states by state-dependent fluorescence. For trapped ions, a shared vibrational mode can mediate entangling gates, so motional control and cooling are part of the computational apparatus. The motion is a resource only while uncontrolled heating does not erase the required quantum coherence.

The computational resource is coherent control of amplitudes and entanglement, with interference making useful outcomes likely. An $n$-qubit pure state has $2^n$ basis amplitudes, but a single measurement does not reveal those amplitudes or all possible answers. Algorithms exploit structure rather than simply reading out parallel classical computations. The Shor algorithm gives factoring with polynomial resource scaling in the input bit length, compared with the best known classical algorithms' superpolynomial scaling; an exponential classical lower bound has not been proved.^[ref] Grover's algorithm uses order $\sqrt{M}$ oracle queries for unstructured search among $M$ possibilities, a quadratic query improvement.^[ref] Quantum simulation is a further motivation because interacting quantum systems can require exponentially many classical amplitudes for a direct generic representation.

The main challenges are quantum decoherence, fluctuating control fields, imperfect gates, leakage out of the qubit subspace, readout errors, crosstalk, and the difficulty of scaling connectivity and calibration. Faster control can introduce extra errors, while stronger isolation can make preparation and readout harder. Quantum error correction encodes logical information redundantly in several physical qubits and extracts error syndromes without measuring the unknown logical state itself. Fault-tolerant quantum computation prevents the correction apparatus and gates from spreading a few faults into uncorrectable logical errors. It requires error rates below an architecture- and noise-dependent threshold and substantial resources; there is no universal numerical threshold for all implementations. Advantages are task-dependent, while reliable scaling requires both coherent hardware and error control.

A Paul trap uses a time-dependent electric quadrupole field to provide dynamic confinement, avoiding the impossibility of trapping a charge in a stable three-dimensional static electrostatic minimum. Its dynamics follows a Mathieu equation: stable drive parameters produce slow Paul trap secular motion with superposed fast Paul trap micromotion. If ${\mathbf{E}}_{\mathrm{rf}}$ denotes the peak oscillating field, the effective energy for small Paul trap micromotion isThe radiofrequency pseudopotential of a Paul trap is an averaged mechanical energy, not a static electrostatic potential. Doppler cooling and, where needed, resolved sideband cooling reduce Paul trap secular motion; the Lamb-Dicke regime suppresses higher motional sidebands.^[ref]

Two long-lived internal states encode a qubit, for example hyperfine states, while an optical metastable transition provides another option. Optical pumping initializes it. Resonant or Raman transition pulses make coherent single-qubit rotations, and state-dependent fluorescence distinguishes a bright state from a dark shelved state. Shared quantized normal modes let one ion's internal state influence another, providing entangling gates in the proposal of J. I. Cirac and P. Zoller; the motional bus must be returned to its intended state rather than left carrying unwanted information.^[ref]

A concrete early experiment demonstrated a controlled-NOT gate between the internal hyperfine and motional degrees of freedom of a single trapped ion. This was an internal–motion two-qubit gate, not yet a gate between two separate ions.^[ref]

Challenges include electric-field-noise heating of the motional bus, excess Paul trap micromotion from stray fields, magnetic and laser phase noise, off-resonant spontaneous scattering, imperfect preparation and readout, and unwanted excitation of spectator modes. Scaling to many ions adds mode crowding, individual-addressing crosstalk and transport requirements. Cooling and quantum error correction must remove entropy without erasing stored qubits. Long storage coherence is necessary but not sufficient: gates and measurements must also be accurate and reproducible. The original gate experiment explicitly identified laser, field, trap-drive and motional decoherence limitations.^[ref]

A bipartite unitary operator is an entangling gate if it maps at least one product state to an entangled state. It need not entangle every product input. For example, the controlled-NOT gate entangles $|+⟩|0⟩$ but leaves $|00⟩$ as a product. A gate that preserves every product input is not entangling, even if it exchanges the two subsystems.

Both the controlled-NOT gate and Pauli X gate permute computational basis vectors. Any finite composition remains a permutation matrix, and so sends $|0\cdots 0⟩$ to another computational basis vector. It cannot prepare a superposition such as $|+⟩\otimes |0\cdots 0⟩$.

This also rules out approximation: the squared overlap of that target with any computational basis vector is at most $1/2$, so even after optimizing global phase the pure-state norm distance is at least $\sqrt{2-\sqrt{2}}>0$. Hence $\{CNOT,X\}$ is not universal. Basis permutations can create entanglement on some superposed inputs, but this gate set cannot create the required superposition from the specified basis input.

Compute the parity into the second wire, phase that wire with the T gate, then uncompute with the same controlled-NOT gate. The answer isFor computational bits $x,y$, the successive states are $|x,y⟩\mapsto |x,x\oplus y⟩\mapsto e^{i\pi (x\oplus y)/4}|x,x\oplus y⟩\mapsto e^{i\pi (x\oplus y)/4}|x,y⟩$. The parity $x\oplus y$ is one precisely on the two odd-parity inputs. Thus, in the ordered computational basis $00,01,10,11$, the matrix is exactly $\mathrm{diag}(1,e^{i\pi /4},e^{i\pi /4},1)$, with no omitted overall phase. This is a two-qubit parity phase gate.

Here $H^2=I$. Both remaining gates fix the all-zero computational vector: the T gate fixes $|0⟩$, and the controlled-NOT gate fixes $|00⟩$. Therefore every finite circuit, on any number of initially zero wires, leaves $|0\cdots 0⟩$ unchanged.

It cannot even approximate $|+⟩\otimes |0\cdots 0⟩$, whose squared overlap with that unchanged vector is $1/2$. Thus the gate set is not universal. The needed quantum superposition cannot be created from this input by basis permutations and diagonal phases alone.

The controlled-NOT gate is its own inverse, so ${\mathrm{CNOT}}^2=I$. Every remaining nontrivial gate acts on one wire. By the tensor product rule, any circuit on a product input stays a product state; in particular, starting at $|00⟩$ it cannot produce a Bell state.

This also prevents arbitrary approximation, not merely exact preparation. For normalized single-qubit vectors $|u⟩,|v⟩$,by the Cauchy-Schwarz inequality. Hence the quantum fidelity with $|{\Phi }^{+}⟩$ is bounded away from one for every reachable state. This gate set is not universal. The obstruction is preservation of product states, a special case of the existing product-state obstruction to SWAP universality.

Prepare Bob in $|0{⟩}_B$ and encode the bit using Alice's preparation $|m{⟩}_A$. The controlled-NOT gate givesBob measures in the computational basis and reads $m$ with probability one. Alice's final measurement is unnecessary. The interaction transmits a classical bit; it does not clone an arbitrary unknown qubit.