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A single confined electron's two spin states form a qubit when initialization, coherent rotations and measurement are controlled. Local driving rotates one spin; gate-controlled exchange can entangle neighboring dot spins. Spin-to-charge conversion permits electrical readout. Observing charge blockade alone does not demonstrate coherent qubit operation.

For a qubit, use the dimensionless Pauli matrices as observables $A={\sigma }_x$, $B={\sigma }_z$. Their Pauli matrix multiplication law givesThese observables have eigenvalues $\pm 1$; the physical spin components are $S_j=(\hslash /2){\sigma }_j$, where $\hslash$ is the reduced Planck constant, so their standard deviations acquire a factor $\hslash /2$.

In the computational basis, prepare $|+y⟩=(|0⟩+i|1⟩)/\sqrt{2}$, an eigenstate of ${\sigma }_y$ with eigenvalue $+1$. Direct multiplication givesBoth Pauli measurements return $+1$ and $-1$ with equal probabilities, and the Robertson uncertainty principle is saturated: $1\cdot 1=1$.

Instead prepare $|+z⟩=|0⟩$. Now the expectation values are $⟨{\sigma }_x⟩=⟨{\sigma }_y⟩=0$, $⟨{\sigma }_z⟩=1$, so $\Delta {\sigma }_x=1$, $\Delta {\sigma }_z=0$. The Pauli measurement of ${\sigma }_z$ is certain, and both sides of the bound vanish. The commutator is still nonzero as an operator: noncommuting observables do not guarantee a positive lower bound independent of the quantum state. This example is useful for distinguishing the algebra of the observables from the expectation values in a particular preparation.

For two spin-one-half qubits, the SWAP gate acts as one on the triplet sector and minus one on the singlet sector. The displayed unitary squares to SWAP. An isotropic exchange interaction with integrated angle $\pi /2$ implements it up to a common phase. There are other square-root branches; a pulse decomposition must use the declared branch consistently.

A stabilizer code is the common $+1$ eigenspace of a stabilizer group $S$, an abelian subgroup of the Pauli group excluding $-I$. Choose $r$ independent stabilizer generators $g_1,\dots ,g_r$ on $n$ physical qubits. They are commuting Hermitian operators, each squaring to $I$. The stabilizer subspace and its orthogonal projector areThe stabilizer-projector formula gives $\dim \mathcal{C}=2^{n-r}$, so $k=n-r$ logical qubits remain. Independent here means that no nonempty product of the chosen stabilizer generators is $I$. Excluding $-I$ is necessary: a vector cannot simultaneously be fixed by $I$ and $-I$ unless it is zero.

For the three-qubit bit-flip repetition code, take $g_1=Z_1{Z}_2$ and $g_2=Z_2{Z}_3$, with $Z_j$ the Pauli Z gate on site $j$ and identities elsewhere. The stabilizer group is $S=\{I,Z_1{Z}_2,Z_2{Z}_3,Z_1{Z}_3\}$. Both stabilizer generators have eigenvalue $+1$ exactly on the span of $|000⟩$ and $|111⟩$, giving $2^{3-2}=2$ dimensions for one logical qubit. A projective measurement of these checks distinguishes four orthogonal two-dimensional sectors rather than eight individual computational basis states.

A Pauli error that anticommutes with a stabilizer generator changes that check's sign, producing an error syndrome. A Pauli operator commuting with every check can still act on the logical qubits; the centralizer of a stabilizer group describes these possibilities. Thus the number of checks determines the encoded dimension, while the quantum code distance determines protection against arbitrary local errors. The five-qubit error correcting code, for instance, has four stabilizer generators and quantum code distance three; the three-qubit bit-flip repetition code has two checks but quantum code distance one because a single phase flip is unflagged. ^[ref].

A stabilizer state on $n$ qubits is the unique simultaneous positive eigenstate of a stabilizer group with $n$ independent generators.

The stabilizer subspace associated with a stabilizer group $\mathcal{S}$ on $n$ qubits is its simultaneous $+1$ eigenspace:Assume that $\mathcal{S}$ is a commuting subgroup of the Pauli group and excludes $-I$. Its elements are then Hermitian operators with eigenvalues $\pm 1$. If $r$ generators $g_1,\dots ,g_r$ are independent modulo scalar phases, thenThe exclusion of $-I$ is essential: the equation $-I|\psi ⟩=|\psi ⟩$ has only the zero solution.

A direct proof uses the orthogonal projectorAll nonidentity Pauli operators have zero trace, so $\mathrm{Tr}{P}_{\mathcal{S}}=2^{n-r}$. The trace of an orthogonal projector equals the dimension of its image. ^[ref]

For example, on two qubits, the single generator $Z\otimes Z$ fixes the two-dimensional space spanned by $|00⟩$ and $|11⟩$. Adding $X\otimes X$ leaves the one-dimensional space spanned by the Bell state $(|00⟩+|11⟩)/\sqrt{2}$. This dimension count determines how many logical qubits a stabilizer code can encode.

The Steane code is a $[[7,1,3]]$ CSS code: seven physical qubits encode one logical qubit, and the distance of a stabilizer code is three. It can correct an arbitrary error on one physical qubit when the ideal syndrome measurements and recovery are implemented correctly.

One choice of binary check rows is $1111000$, $1100110$ and $1010101$. For each row, put Pauli matrices $X$ on its four marked positions to obtain an X-type generator, and use $Z$ on the same positions for a Z-type generator. Any two rows overlap in an even number of positions, so all six stabilizer generators commute. Their common positive eigenspace has dimension $2^{7-6}=2$.

A single-qubit Z error reverses the X-check outcomes indicated by its column, identifying its location; X errors are diagnosed by the Z checks. The recovery learns the error syndrome without reading the unknown logical amplitudes.

Superdense coding uses a shared Bell state to communicate two classical bits by transmitting one qubit. The sender selects one of four orthogonal Bell states with a local Pauli operator, and the receiver performs a Bell-basis measurement. For an ideal shared Bell pair, the transmitted qubit alone has a maximally mixed state for every message and reveals no information about the chosen encoding.

A surface code is a family of topological stabilizer codes on a two-dimensional cellulation. In a standard square-lattice convention, qubits lie on edges, an X-type check multiplies X on edges meeting a vertex, and a Z-type check multiplies Z around a face.

Vertex and face checks commute because their overlaps contain zero or two edges. A local Pauli operator error changes nearby check signs, producing an error syndrome. Chains of errors have syndromes at their endpoints; a chain connecting appropriate boundaries or winding around a nontrivial cycle can instead implement a logical operation. Locality makes repeated syndrome extraction useful, but a sufficiently long undetected chain can still corrupt the encoded state. The associated surface-code anyon model describes the electric and magnetic check excitations and their mutual braiding.

Join $n$ spheres pole to pole, with $m$ longitude edges on each sphere and one qubit per edge. For this surface code, omit the Pauli X gates star checks at the two end poles. Include the Pauli Z gates checks on each two-edge face and the Pauli X gates star checks at the $n-1$ interior junctions. The boundary choice is essential for encoding one logical qubit.

Label an edge by $(r,j)$, with sphere $r$ and longitude $j$. Independent face checks are $Z_{r,j}{Z}_{r,j+1}$, $j=1,\dots ,m-1$, and the junction checks are $B_r{B}_{r+1}$ with $B_r={\prod }_{j=1}^m{X}_{r,j}$. There are $n(m-1)+(n-1)=nm-1$ independent stabilizer generators on $nm$ qubits, so the code space has dimension two. These are exactly the checks of a concatenated phase-and-bit-flip repetition code.

A pole-to-pole $Z$ path has weight $n$, and an equatorial dual $X$ cut has weight $m$. They cross once and have zero anticommutator, giving logical operations and quantum code distance $\min (m,n)$. If every pole star is instead included, the star rank is $n$ and the code dimension falls to one; the closed chain then carries no logical qubit.

The SWAP gate exchanges the states of two qubits, including any correlations they have with other systems. Its action on the computational basis fixes the operation on every quantum superposition by linearity.

The SWAP gate maps $|x⟩|y⟩$ to $|y⟩|x⟩$.

In particular, $\mathrm{SWAP}(|\psi ⟩\otimes |\varphi ⟩)=|\varphi ⟩\otimes |\psi ⟩$. This transfers rather than copies the two inputs, so it does not violate the no-cloning theorem. The Three-CNOT decomposition of the SWAP gate implements the same operation using alternating control directions. ^[ref]

The swap test estimates the overlap of two pure states without requiring their classical descriptions. Prepare an auxiliary qubit in $|0⟩$, apply a Hadamard gate, a Fredkin gate controlled by that auxiliary qubit, and another Hadamard gate. Then measure the auxiliary qubit in the computational basis.

For target states $|a⟩,|b⟩$, the auxiliary-zero branch has unnormalized target vector $(|a⟩|b⟩+|b⟩|a⟩)/2$. Its squared norm is thereforeIdentical pure states give $p_0=1$, while pure states with zero inner product give $p_0=1/2$. A single zero outcome does not certify equality; repeated trials estimate the probability. The calculation is a direct projection onto the symmetric target subspace using the SWAP gate.

Consider $q$ identical length-$n$ arms joined to one hub, with XY exchange interactions of strength $J_0/\sqrt{q}$ from the hub to each first site and strength $J_k$ between sites $k,k+1$ on each arm. Let $|s_0⟩$ be the hub excitation and $|s_k⟩=q^{-1/2}{\sum }_{a=1}^q|a,k⟩$. This orthonormal symmetric single-excitation subspace is invariant, and its restricted Hamiltonian operator iswith endpoint terms omitted. At the first link the $q$ hub contributions add to $J_0$; all other links act identically on the arms. Thus the linear isometry $F|k⟩=|s_k⟩$ intertwines $H$ with the path Hamiltonian operator $H_T={\sum }_{k=0}^{n-1}{J}_k(|k⟩⟨k+1|+|k+1⟩⟨k|)$: $HF=F{H}_T$. Expanding the matrix exponential gives $e^{-iHt}F=F{e}^{-i{H}_{T}t}$. Perfect quantum state transfer from zero to $n$ on the path therefore produces an equal excitation superposition at the $q$ arm tips. Since all other qubits are zero, the tip state is a pure W state, up to the transfer's common phase.

If two pure quantum states overlap by $(1/\sqrt{2}{)}^{1/n}$, their $n$-copy states have overlap $1/\sqrt{2}$. Each block spans an effective qubit, with orthonormal basis $e_0={\Psi }_1$ and $e_1=\sqrt{2}{\Psi }_2-{\Psi }_1$. Applying the two-block PBR exclusion measurement for zero and plus uses $2n$ independently prepared copies. A common single-copy ontic overlap of positive mass remains positive under this finite tensor power.

Choose the four unit Bloch vectors at the vertices of a regular tetrahedron, so ${\sum }_i{\mathbf{n}}_i=0$ and ${\sum }_i{\mathbf{n}}_i{\mathbf{n}}_i^T=4I/3$. The displayed four positive rank-one effects sum to $I$. On input $(I+\mathbf{r}\cdot \mathbf{\sigma })/2$, their probabilities are $p_i=(1+\mathbf{r}\cdot {\mathbf{n}}_i)/4$, giving the reconstruction $\mathbf{r}=3{\sum }_i{p}_i{\mathbf{n}}_i$. This realizes the smallest fixed qubit informationally complete POVM.

If two senders share a GHZ state with a receiver, one sender can encode two bits with $X^x{Z}^z$ and the other one bit with $X^b$. Sending both qubits to the receiver produces one of eight orthogonal GHZ-basis states, so a joint measurement recovers all three bits.

Measure the commuting stabilizer generators $g_1=Z_1{Z}_2$ and $g_2=Z_2{Z}_3$ in that order. Use error syndrome bit $0$ for eigenvalue $+1$ and bit $1$ for eigenvalue $-1$. For $X_j$ the Pauli X gate on physical qubit $j$, every row holds for arbitrary normalized $\alpha ,\beta$. This ordered parity-check convention follows ^[ref]:

Each pair of computational basis words within a row has the same two parities. Consequently the projective measurement reveals the row without distinguishing its two amplitudes. For example, $|{\psi }_L⟩=(\sqrt{3}/2)|000⟩+(i/2)|111⟩$ becomes $(\sqrt{3}/2)|010⟩+(i/2)|101⟩$ under $X_2$. Both checks give $-1$ with probability one, and applying $X_2$ restores both coefficients exactly, including the relative phase $i$.

One implementation uses two quantum ancillas initialized to $|00⟩$. Two controlled-NOT gates from data qubits 1 and 2 to the first quantum ancilla write their parity; two more from data qubits 2 and 3 to the second quantum ancilla write the other parity. Measuring only the quantum ancillas in the computational basis gives $(s_1,s_2)$. Measuring the individual data qubits instead would reveal the encoded computational basis value and destroy the unknown quantum superposition. The existing coherent syndrome extraction for the three-qubit repetition code uses the alternative ordered checks $(Z_2{Z}_3,Z_1{Z}_3)$, so its bit labels differ from this table.

The recovery also handles coherent errors. Put $E_0=I,E_1=X_1,E_2=X_2,E_3=X_3$. Their images of the code are mutually orthogonal, giving $P{E}_a^{†}{E}_{b}P={\delta }_{ab}P$, the Knill--Laflamme condition for this family. For any linear combination $E={\sum }_{a=0}^3{c}_a{E}_a$, its component in sector $s(E_a)$ is $c_a{E}_a|{\psi }_L⟩$. Measuring that error syndrome has probability $|c_a{|}^2/{\sum }_b|c_b{|}^2$ on the normalized erroneous state, and recovery $E_a$ returns $|{\psi }_L⟩$. More generally one recovery quantum channel corrects every noise quantum channel whose Kraus operators belong to this span. The assumption concerns the whole jointly correctable family; it does not extend to the span of arbitrary single-site Pauli errors. ^[ref].

A fully computed coherent example is the unitary operator $U=\exp (-i\theta X_2)=\cos \theta I-i\sin \theta X_2$. At $\theta =\pi /6$ it givesThe error syndromes $00$ and $11$ occur with probabilities $3/4$ and $1/4$, independently of $\alpha ,\beta$. Doing nothing for $00$ and applying $X_2$ for $11$ restores the same logical qubit in both branches; the scalar $-i$ in the second branch is an irrelevant global phase. This is an example of linearity of coherent quantum error correction, rather than an assumption that the original error was a classical random bit flip.

A path graph state on three qubits simulates two successive gates $U(\theta )=H\mathrm{diag}(1,e^{-i\theta })=J(-\theta )$. Measure vertex one in the equatorial qubit measurement basis $|v_r(\alpha )⟩=(|0⟩+(-1{)}^r{e}^{i\alpha }|1⟩)/\sqrt{2}$, then vertex two at angle $(-1{)}^r\beta$, obtaining bits $r,s$. Two one-bit teleportations and $U(\theta )X=e^{-i\theta }ZU(-\theta )$ leave vertex three in $X^s{Z}^{r}U(\beta )U(\alpha )|+⟩$, up to global phase. A computational output bit $z$ is corrected to $z\oplus s$; the $Z$ factor affects only phase. Adaptive sign choice is required for general angles, although the final bit correction is purely classical.

An experimental physicist who coauthored a programmable neutral-atom quantum computer demonstration. Focused light beams addressed individual qubits in a two-dimensional array, while interactions between Rydberg atoms enabled entangling operations. The experiment combined quantum logic gates into quantum algorithms, testing whether a collection of controlled chemical atoms could execute complete circuits.^[ref]

The Toffoli gate is a reversible three-qubit quantum gate with two controls and one target. It flips the target only when both controls are $1$.

The Toffoli gate maps $|x_1{x}_{2}y⟩$ to $|x_1{x}_2,y\oplus x_1{x}_2⟩$.

For a target initially in $|0⟩$, the output target records the classical AND $x_1{x}_2$ while both input labels remain available. Applying the Toffoli gate again erases that computed target reversibly. The same rule acts linearly on quantum superpositions, so it can be used inside a coherent quantum circuit. ^[ref]