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For a Boolean function $f$, its standard oracle acts by

The Toffoli, or controlled-controlled-NOT, gate maps $|x_1{x}_{2}y⟩$ to $|x_1{x}_2,y\oplus x_1{x}_2⟩$.

The Born approximation replaces the unknown total field inside a scattering integral by the known incident field, thereby retaining only single scattering. In nonrelativistic quantum mechanics, first-order Lippmann-Schwinger iteration gives

$|\alpha ⟩=D(\alpha )|0⟩$ is normalized, satisfies $A|\alpha ⟩=\alpha |\alpha ⟩$, and evolves to a phase times $|\alpha e^{-i\omega t}⟩$.

For $U(x)=k{x}^2$ and $k>0$, the ansatz $\psi =C{e}^{-\alpha x^2}$ solves the stationary Schrödinger equation whenIt has $\Delta x=1/(2\sqrt{\alpha })$ and $\Delta p=\hslash \sqrt{\alpha }$, so it saturates the Heisenberg bound.

$\Delta A\Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$.

To distinguish two gates in one use without an ancilla, choose an input $|\psi ⟩$ and discriminate the output states $U_1|\psi ⟩$ and $U_2|\psi ⟩$.

Equiprobable pure states of overlap magnitude $c$ have optimal discrimination probability $\frac{1}{2}(1+\sqrt{1-c^2})$.

There are a unitary $U$ and normalized environment states $|e_i⟩$ satisfyingexactly whenNecessity follows from preservation of inner products. For sufficiency, choose $⟨e_0|e_1⟩$ to make the input and output inner products equal; equal two-vector Gram matrices define an isometry that extends to a unitary.

A cloning operation for a family of pure states would mapfor every state in that family. A single physical operation cannot clone every unknown quantum state.

An uncommunicated local trace-preserving operation cannot change the other subsystem's reduced state.

A finite-dimensional density operator is Hermitian, positive semidefinite, and has trace one. In dimension $N$ it has $N^2-1$ real parameters. Schrödinger evolution iswhen ${\rho }_H$ denotes the time-independent Heisenberg-picture state.

If a state has good amplitude $\sin \theta$, the product of the reflection in the bad axis and the reflection in the state line rotates its good-bad plane by $2\theta$. After $k$ iterations the good probability isso $O(1/\sin \theta )$ iterations raise a small good amplitude close to one.

A quantum measurement assigns probabilities to classical outcomes through positive operators that sum to the identity. A projective measurement uses mutually orthogonal projections.

For a particle of charge $-e$, the changes $A\mapsto A+\nabla f$ and $\varphi \mapsto \varphi -{\partial }_{t}f$ leave the minimally coupled Schrödinger equation invariant when $\psi \mapsto e^{-ief/\hslash }\psi$.

The Pauli $X$ eigenstates are $(|0⟩\pm |1⟩)/\sqrt{2}$ with eigenvalues $\pm 1$.

First-order degenerate perturbation theory diagonalizes the perturbing operator within each degenerate eigenspace of the unperturbed Hamiltonian.

Time-independent perturbation theory expands the eigenvalues and eigenstates of $H_0+V$ in powers of a small perturbation $V$.

Expanding a state in unperturbed energy eigenstates with their free phases converts the time-dependent Schrodinger equation into coupled equations for slowly varying amplitudes. To first order, the perturbation on the right-hand side acts on the initial unperturbed state.

Independent quantum systems combine by tensor product, with local operators acting as $A\otimes I$ or $I\otimes B$.