Sampling from the Thermal Quantum Gibbs State and Evaluating Partition Functions with a Quantum Computer

We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound $D^{\alpha }$ on the thermalization time of a quantum system, where $D$ is the system’s Hilbert space dimension and $\alpha \le \frac{1}{2}$ is proportional to the Helmholtz free energy density. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system’s thermalization time and inversely proportional to the targeted accuracy squared.

Figure 1

Scaling exponent as a function of the inverse temperature for the one dimensional Ising model defined by the Hamiltonian $H(J,g)=\sum _{i}g{\sigma }_x^{(i)}+J{\sigma }_z^{(i)}{\sigma }_z^{(i+1)}$ for different values of $g/J$ ($g/J=1$ is critical). The absolute values of $g$ and $J$ are fixed by $\Vert H(g,J)\Vert =\Vert H(0,1)\Vert$. Like the partition function, $\alpha$ is symmetric under $J\leftrightarrow g$.