Single-qubit thermal (Gibbs) state

For a single-qubit Hamiltonian $H = -\frac{\Delta}{2}Z$ with eigenvalues $E_0=-\Delta/2$ (for $|0\rangle$) and $E_1=+\Delta/2$ (for $|1\rangle$), the Gibbs state at inverse temperature $\beta = 1/k_BT$ is

$\rho(\beta) = \frac{e^{-\beta H}}{Z} = p_0 |0\rangle\langle0| + p_1 |1\rangle\langle1|, \qquad p_0 = \frac{1}{1+e^{-\beta\Delta}},\quad p_1 = \frac{e^{-\beta\Delta}}{1+e^{-\beta\Delta}}.$

Purification construction. Introduce an ancilla and prepare $|\phi\rangle_{AS} = \cos(\theta/2)|0\rangle_A|0\rangle_S + \sin(\theta/2)|1\rangle_A|1\rangle_S$ by applying $R_y(\theta)$ to the ancilla and then $\mathrm{CNOT}_{A\to S}$. Tracing out the ancilla gives the system the reduced density matrix $\mathrm{diag}(\cos^2(\theta/2), \sin^2(\theta/2))$. Choosing

$\theta = 2\arctan\!\big(e^{-\beta\Delta/2}\big)$

makes $\cos^2(\theta/2) = p_0$ and $\sin^2(\theta/2) = p_1$ exactly — this is a standard identity since $\tan(\theta/2) = e^{-\beta\Delta/2}$ implies $\cos^2(\theta/2) = 1/(1+\tan^2(\theta/2)) = 1/(1+e^{-\beta\Delta}) = p_0$.

Limits. As $\beta \to 0$ (infinite temperature), $\theta \to \pi/2$ and $\rho \to I/2$, the maximally mixed state. As $\beta \to \infty$ (zero temperature), $\theta \to 0$ and $\rho \to |0\rangle\langle0|$, the ground state — recovering ordinary ground-state preparation as the $T=0$ limit of Gibbs-state preparation.

Beyond one qubit. For interacting many-body Hamiltonians this single-ancilla trick does not generalize directly (the purification's Schmidt rank must match the Hilbert space dimension in general), and practical quantum Gibbs-sampling proposals instead use quantum Metropolis sampling, quantum imaginary-time evolution (QITE), or variational thermofield-double constructions — each with its own resource and convergence tradeoffs.