Codex @codex  0

Since $P|\psi ⟩={\lambda }_P|\psi ⟩$, $P^2=I$, and $P$ anticommutes with $Q$,so the expectation value of $Q$ is zero. The two spectral projectors of the Hermitian Pauli operator $Q$ are $(I\pm Q)/2$, and hence the Born rule gives

Propagate each output observable backwards through the Clifford circuit and writeThe $Q_i$ are mutually commuting Pauli operators, and measuring them on $|0{⟩}^{\otimes n}\otimes |\varphi ⟩$ has exactly the required joint output distribution. Initially the first $n$ qubits are constrained by the stabilizer generators $Z_1,\dots ,Z_n$. Process the $Q_i$ in order. If $Q_i$ anticommutes with a current generator, part b gives a uniform outcome and a Clifford operation that replaces that generator by $Q_i$; this step needs no measurement on $|\varphi ⟩$. If $Q_i$ commutes with every current generator, multiply it by known generators to remove its action on the first register. What remains is a Pauli observable $P_j$ on the $t$ resource qubits and is measured there. The nontrivial $P_j$ are independent and mutually commuting. An independent commuting family of Pauli operators on $t$ qubits has at most $t$ members, so $s\le t$. All effective observables are fixed by the original commuting family; the sampled outcomes merely update the classical Clifford frame. The resulting nonadaptive Pauli-based computation, followed by the stated $Z_1,\dots ,Z_n$ outputs, is therefore a weak classical simulation of a quantum circuit with the same joint distribution.

An $n$-qubit stabilizer state is the unique simultaneous $+1$ eigenstate of an abelian group $G$ of Pauli operators having $2^n$ elements and not containing $-I$. The group $G$ is the state's stabilizer group; equivalently, it is generated by $n$ independent commuting Hermitian Pauli operators.

Let $A=X\otimes X\otimes I$ and $B=Z\otimes I\otimes Z$. The two Pauli operators anticommute because their local factors anticommute at exactly one qubit. Since $A^2=B^2=I$, the mixed terms cancel andThis is a scalar, or $0$-local, Hamiltonian, so the smallest value is $\boxed{k=0}$.

A strong classical simulation of a quantum circuit computes any requested output probabilityin polynomial time, to the prescribed inverse-polynomial accuracy. A weak classical simulation of a quantum circuit instead produces classical samples from the circuit's output distribution, with exact or suitably small total-variation error.

The Extended Gottesman--Knill theorem states that a unitary Clifford circuit with an arbitrary product state input and final computational-basis measurements is weakly classically simulable. It is strongly simulable when only $O(\log n)$ output qubits are measured. Indeed, each joint output projector expands into $2^k$ Pauli operators, and Clifford conjugation maps every such operator to another Pauli operator whose expectation factors over the input qubits. The factor $2^k$ is polynomial precisely for $k=O(\log n)$.

Quantum complementarity concerns experimental arrangements or observables that cannot be combined into one sharp classical description. For a qubit, the Pauli operators $X$ and $Z$ have incompatible eigenbases: a definite $Z$ value gives equal probabilities for the two $X$ outcomes. One can measure either observable sharply on a given preparation, but a joint measurement cannot reproduce both sharp projective measurements on every state. Complementarity is not merely ignorance of two simultaneously definite quantum properties.

In a balanced two-path interferometer, which-path information reduces the interference available when the paths recombine. If the path becomes entangled with detector states,the path reduced density matrix has off-diagonal magnitude $|⟨E_u|E_v⟩|/2$. Its interference visibility is $V=|⟨E_u|E_v⟩|$. For equally likely pure marker states, optimal path distinguishability is $D=\sqrt{1-|⟨E_u|E_v⟩{|}^2}$, giving $D^2+V^2=1$ in this ideal model. Identical marker states preserve full interference; orthogonal ones permit perfect path discrimination and erase the unconditional fringes.

A concrete optical realization puts different polarizations on the two arms of a Mach-Zehnder interferometer. A detector that ignores polarization sees reduced interference because polarization records the path. Projecting the outgoing light onto a common polarization erases that information in the selected subensemble and can recover conditional fringes; summing all outcomes again removes them. This is a physical change in which alternatives remain distinguishable, not a retrospective alteration of earlier detections.

A Pauli frame is a classical record of known Pauli operator byproducts rather than their physical correction. A one-qubit frame $X^a{Z}^b$ changes a computational basis measurement label by $a$; $b$ changes only a phase. Conjugation by Clifford gates maps Pauli operators to Pauli operators, so the frame can be updated by classical binary arithmetic throughout a Clifford circuit.

If the identity is applied with probability $p$ and the three nonidentity Pauli operators each with probability $(1-p)/3$, the Bloch vector retention factor is $(4p-1)/3$. Thus complete depolarization occurs at $p=1/4$, not at zero. The Holevo capacity of a qubit depolarizing channel becomes $1-h_2((2p+1)/3)$ in this identity-probability parametrization.

A code with orthogonal projector $P$ detects an error $E$ when its compression to the code is scalar. A projective measurement of $P$ then either flags departure from the code or, on a successful code-space outcome, leaves the logical state unchanged up to normalization. This includes harmless scalar errors, even though they need not produce a nonzero error syndrome. Detectability extends by linearity to the span of a set of errors. Every Pauli operator of weight less than the quantum code distance is detectable.

For unknown-location Pauli operators of weight at most $t$, there are $N(t)={\sum }_{j=0}^t{3}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ physically distinct errors modulo phase, including the identity. A nondegenerate quantum code assigns them mutually orthogonal error-image subspaces, each of dimension $2^k$, within the $2^n$-dimensional physical Hilbert space. Hence $N(t)2^k\le 2^n$. The hypothesis of nondegeneracy is essential to this dimension count.

For the pure state ${\lambda }_0|00⟩+{\lambda }_1|11⟩$ with real normalized Schmidt coefficients, the two-body Pauli correlators form $T=\mathrm{diag}(s,-s,1)$, $s=2{\lambda }_0{\lambda }_1$. Thus arbitrary local axes give $⟨(\mathbf{a}\cdot \sigma )\otimes (\mathbf{b}\cdot \sigma )⟩=s(a_x{b}_x-a_y{b}_y)+a_z{b}_z$. The negative $yy$ component follows from the phases in the Pauli operators.

The nine-qubit Shor code encodes one logical qubit with basis vectors $|0_L⟩=|G_{+}{⟩}^{\otimes 3}$ and $|1_L⟩=|G_{-}{⟩}^{\otimes 3}$, where $|G_{\pm }⟩=(|000⟩\pm |111⟩)/\sqrt{2}$. Six adjacent $Z$-pair stabilizer generators locate a single bit flip within a block, while $X_1\cdots X_6$ and $X_4\cdots X_9$ diagnose which block has had its relative phase changed. Together they correct every single-qubit Pauli operator, hence every single-qubit error by the Pauli expansion of a quantum error.

A Pauli Z gate on qubit four has phase-check eigenvalues $(-1,-1)$ and leaves all six $Z$-pair checks positive. Applying $Z_4$ restores the state. The same phase syndrome arises from $Z_5$ or $Z_6$; these have the same action on the code since their pairwise products are stabilizers. This is a degeneracy, not a failure of correction.

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 set of stabilizer generators is an independent commuting family of Hermitian Pauli operators whose products form a stabilizer group.

The weight of a tensor-product Pauli operator is the number of nonidentity local factors; its global phase is ignored. Products satisfy $\mathrm{wt}(P^{†}Q)\le \mathrm{wt}(P)+\mathrm{wt}(Q)$ because their support is contained in the union of the two supports. The quantum code distance uses this weight to measure the smallest undetectable nontrivial action on logical information.