Appendix E — Math Identities Cheat-Sheet

The identities you'll reach for constantly. Conventions match Appendix C: $\hbar = 1$ , big-endian qubits, rotation gates use the half-angle $e^{-i\theta\sigma/2}$ .

1. Pauli Matrices

I=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad Y=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\quad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.

Basic properties (let $\sigma_j \in \{X,Y,Z\}$ ):

Hermitian and unitary: $\sigma_j = \sigma_j^\dagger$ and $\sigma_j^2 = I$ .
Traceless: $\mathrm{Tr}(\sigma_j) = 0$ ; eigenvalues $\pm 1$ .
$\det(\sigma_j) = -1$ .

Products and (anti)commutators (with Levi-Civita $\varepsilon_{jkl}$ , $\{j,k,l\}$ a permutation of $\{x,y,z\}$ ):

XY = iZ,\quad YZ = iX,\quad ZX = iY \qquad(\text{cyclic});

[\sigma_j,\sigma_k] = 2i\,\varepsilon_{jkl}\,\sigma_l,\qquad \{\sigma_j,\sigma_k\} = 2\,\delta_{jk} I,\qquad \sigma_j\sigma_k = \delta_{jk} I + i\,\varepsilon_{jkl}\sigma_l.

Completeness: $\{I, X, Y, Z\}$ is an orthogonal basis for $2\times2$ Hermitian matrices under the Hilbert–Schmidt inner product $\langle A,B\rangle = \mathrm{Tr}(A^\dagger B)$ , with $\mathrm{Tr}(\sigma_j\sigma_k) = 2\delta_{jk}$ . Hence any single-qubit density matrix is

\rho = \tfrac12\big(I + \vec r\cdot\vec\sigma\big),\qquad \vec r = (\langle X\rangle,\langle Y\rangle,\langle Z\rangle),\ \ |\vec r|\le 1.

2. Common Gate Matrices

H=\frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix},\quad S=\begin{pmatrix}1&0\\0&i\end{pmatrix},\quad T=\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}.

Relations: $S = T^2$ , $Z = S^2$ , $H X H = Z$ , $H Z H = X$ , $H Y H = -Y$ , $S X S^\dagger = Y$ .

Two-qubit (big-endian, control = qubit 0):

\text{CNOT}=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix},\quad \text{CZ}=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{pmatrix},\quad \text{SWAP}=\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}.

Useful: $\text{CZ} = (I\otimes H)\,\text{CNOT}\,(I\otimes H)$ , and $\text{SWAP} = \text{CNOT}_{01}\,\text{CNOT}_{10}\,\text{CNOT}_{01}$ .

3. Rotation Gates (Exponentials of Paulis)

Since $\sigma_j^2 = I$ , the exponential closes into sines and cosines:

e^{-i\theta\,\hat n\cdot\vec\sigma/2} = \cos\!\tfrac{\theta}{2}\,I - i\sin\!\tfrac{\theta}{2}\,(\hat n\cdot\vec\sigma), \qquad |\hat n| = 1.

The single-axis rotations:

R_x(\theta)=\begin{pmatrix}\cos\frac\theta2 & -i\sin\frac\theta2\\ -i\sin\frac\theta2 & \cos\frac\theta2\end{pmatrix},\ R_y(\theta)=\begin{pmatrix}\cos\frac\theta2 & -\sin\frac\theta2\\ \sin\frac\theta2 & \cos\frac\theta2\end{pmatrix},\ R_z(\theta)=\begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}.

Identities:

$R_z(\pi) = e^{-i\pi/2}Z$ , $\ R_x(\pi) = -iX$ , $\ R_y(\pi) = -iY$ (note the global phases).
$H = e^{i\pi/2} R_y(\tfrac{\pi}{2}) R_z(\pi)$ (up to global phase); more simply $H$ is a $\pi$ rotation about the $\hat n = (\hat x + \hat z)/\sqrt2$ axis.

Euler (ZYZ) decomposition. Any single-qubit unitary $U$ can be written, for some $\alpha,\beta,\gamma,\delta\in\mathbb R$ ,

U = e^{i\alpha}\,R_z(\beta)\,R_y(\gamma)\,R_z(\delta).

This is why $\{R_y, R_z\}$ (plus phase) generate all of $SU(2)$ — see Term 2.2.

4. Operator Identities

Adjoint of a product: $(AB)^\dagger = B^\dagger A^\dagger$ ; $(A\otimes B)^\dagger = A^\dagger\otimes B^\dagger$ .
Trace is cyclic: $\mathrm{Tr}(ABC) = \mathrm{Tr}(CAB) = \mathrm{Tr}(BCA)$ (basis-independent).
Trace of a tensor product: $\mathrm{Tr}(A\otimes B) = \mathrm{Tr}(A)\,\mathrm{Tr}(B)$ .
Expectation: $\langle A\rangle = \mathrm{Tr}(\rho A)$ ; for pure $\rho=|\psi\rangle\langle\psi|$ , $=\langle\psi|A|\psi\rangle$ .
Spectral decomposition: a normal operator $A=\sum_i \lambda_i |v_i\rangle\langle v_i|$ with $\{|v_i\rangle\}$ orthonormal; then $f(A)=\sum_i f(\lambda_i)|v_i\rangle\langle v_i|$ for any function $f$ .
Function of a unitary's generator: if $H=\sum_i E_i|E_i\rangle\langle E_i|$ then $e^{-iHt}=\sum_i e^{-iE_i t}|E_i\rangle\langle E_i|$ .

Baker–Campbell–Hausdorff (BCH):

e^{A}e^{B} = e^{A+B+\frac12[A,B]+\frac1{12}[A,[A,B]]-\frac1{12}[B,[A,B]]+\cdots}.

If $[A,B]=0$ then $e^A e^B = e^{A+B}$ . This is exactly why Trotterization is needed when $[A,B]\neq 0$ (see Term 3.4):

e^{-i(A+B)t} = \lim_{n\to\infty}\Big(e^{-iAt/n}e^{-iBt/n}\Big)^{n},\qquad e^{-i(A+B)\delta} = e^{-iA\delta}e^{-iB\delta} + O(\delta^2).

Hadamard lemma (operator conjugation):

e^{A}Be^{-A} = B + [A,B] + \tfrac1{2!}[A,[A,B]] + \cdots.

5. Tensor-Product (Kronecker) Identities

Mixed-product: $(A\otimes B)(C\otimes D) = (AC)\otimes(BD)$ .
Action on product states: $(A\otimes B)(|a\rangle\otimes|b\rangle) = (A|a\rangle)\otimes(B|b\rangle)$ .
Adjoint / inverse / transpose distribute: $(A\otimes B)^{-1} = A^{-1}\otimes B^{-1}$ , etc.
Eigenvalues: if $A|u\rangle=a|u\rangle$ and $B|v\rangle=b|v\rangle$ , then $(A\otimes B)(|u\rangle\otimes|v\rangle) = ab\,(|u\rangle\otimes|v\rangle)$ .
Dimension: $A\,(m\times m)\otimes B\,(n\times n)$ is $mn\times mn$ .

6. Complex Numbers, Roots of Unity, and the QFT Kernel

Euler: $e^{i\theta} = \cos\theta + i\sin\theta$ ; $\ |e^{i\theta}|=1$ .
$N$ -th roots of unity: $\omega = e^{2\pi i/N}$ satisfies $\omega^N = 1$ and $\sum_{k=0}^{N-1}\omega^{k} = 0$ (for $N>1$ ).
Geometric sum: $\displaystyle\sum_{k=0}^{N-1} r^k = \frac{1-r^N}{1-r}$ for $r\neq1$ .
Orthogonality (DFT/QFT core): $\displaystyle \frac1N\sum_{x=0}^{N-1} e^{2\pi i x(j-k)/N} = \delta_{jk}.$
QFT action: $\displaystyle \text{QFT}\,|x\rangle = \frac{1}{\sqrt N}\sum_{y=0}^{N-1} e^{2\pi i xy/N}\,|y\rangle,\quad N=2^n.$

7. Probability & Statistics (for shots and estimation)

Born rule: $p_i = |\langle i|\psi\rangle|^2$ ; $\sum_i p_i = 1$ .
Sample-mean estimator of $\langle A\rangle$ from $M$ shots has standard error $\sigma/\sqrt M$ ; halving the error costs $4\times$ the shots. Plan QPU budgets accordingly.
Binomial outcome: estimating a probability $p$ from $M$ shots gives variance $p(1-p)/M$ .
Hoeffding bound: $\Pr[\,|\hat p - p| \ge \epsilon\,] \le 2e^{-2M\epsilon^2}$ — how many shots guarantee a target confidence.

8. Frequently Used Special Values

Quantity	Value
$\cos(\pi/8),\ \sin(\pi/8)$	$\approx 0.9239,\ 0.3827$
$\langle 0	H
Optimal Grover iterations (database size $N$ , one marked)	$\big\lfloor \frac{\pi}{4}\sqrt N \big\rfloor$
CHSH classical bound / quantum (Tsirelson) bound	$2$ / $2\sqrt2$
Purity range for $d$ -dim state	$\big[\tfrac1d,\,1\big]$
QFT gate count ( $n$ qubits)	$\Theta(n^2)$