TEPID-ADAPT: Adaptive variational method for simultaneous preparation of low-temperature Gibbs and low-lying eigenstates

TL;DR

The paper tackles efficient preparation of Gibbs states at low temperature on quantum hardware by introducing TEPID-ADAPT, a partially adaptive variational algorithm that uses a truncated eigenspectrum and a parametrized diagonal reference density $\rho_m$ to represent the target Gibbs state as $\rho_G \approx \frac{1}{Z_m}\sum_{k=1}^m e^{-\beta E_k} |\psi_k\rangle\langle \psi_k|$. The method minimizes the free energy $F=\langle H\rangle-\beta^{-1}S$ with an entropy $S=-\sum_{k=1}^m \mu_k \log\mu_k$ that is easily computed because $\rho_m$ is diagonal, and then uses an adaptive unitary $V_A$ to rotate into the truncated eigenbasis. It provides the ability to obtain low-lying eigenstates and to lower the temperature to $\beta>\beta_0$ without re-optimizing, and it can prepare thermofield double (TFD) states without variational optimization on a doubled system. An ancilla-free variant further reduces resource requirements by interpreting the mixed state as a classical ensemble and evaluating observables as ensemble averages.

Abstract

Preparing Gibbs states, which describe systems in equilibrium at finite temperature, is of great importance, particularly at low temperatures. In this work, we propose a new method -- TEPID-ADAPT -- that prepares the thermal Gibbs state of a given Hamiltonian at low temperatures using a variational method that is partially adaptive and uses a purification with a minimal number of ancillary qubits. We also present an alternative implementation without ancillary qubits. A key technical innovation here is to use a mixed-state ansatz where the entropy can be efficiently calculated, with no computational overhead. Our algorithm uses a truncated, parametrized eigenspectrum of the Hamiltonian. Beyond preparing Gibbs states, this approach also straightforwardly gives us access to the truncated low-energy eigenspectrum of the Hamiltonian, making it also a method that prepares excited states simultaneously. As a result of this, we are also able to prepare thermal states at any lower temperature of the same Hamiltonian without further optimization.

Figures (21)

• Figure 1: A diagrammatic workflow for TEPID-ADAPT. The white shapes indicate the inputs-- the Hamiltonian $H$, the inverse temperature $\beta_0$, an $m$-dimensional computational subspace, and an operator pool for ADAPT. The black circuit, labeled $U_m(\vec{\mu})$ is the static part of the ansatz, which prepares $\rho_m$ on the system register. The pink circuit parametrized by $\vec{\theta}$ is the adaptively generated portion that maps the computational subspace to the truncated eigenspace of the Hamiltonian. The two solid green arrows (from left-right) respectively indicate the convergence of the VQE subroutine and the pool gradients vanishing. The primary output of TEPID-ADAPT is the target Gibbs state at $\rho_G(\beta_0)$. We can also readily prepare the low-lying eigenstates of the Hamiltonian in the truncated subspace $\{\ket{\psi_k}\}_{k=1}^m$, the Gibbs states at lower temperatures $\rho_G(\beta>\beta_0)$, and the TFD state at inverse temperatures $\beta\geq\beta_0$.
• Figure 2: General block diagram for the variational ansatz of TEPID-ADAPT. $U_m(\mu)$ prepares $\rho_m$ on the system register (top), as indicated by the red line. $V_A(\vec{\theta})$ is an adaptively generated unitary on the system register that approximately evolves $\rho_m$ to the target Gibbs state.
• Figure 3: The circuit for preparing a low-temperature TFD state using TEPID-ADAPT without any further variational optimization. The parameters in the unitaries are the converged ones from preparing the low-temperature Gibbs state. $P({\{\ket{c_k}}\})$ is a permutation matrix that maps the first $m$ computational basis states on the ancillary registers to the chosen elements $\{\ket{c_k}\}_{k=1}^m$.
• Figure 4: The infidelities (left axis, circle markers) and relative errors (right axis, triangle markers) of the low-lying eigenenergies, and of the free energy of the $\beta_0=3.0$ Gibbs state (truncated and full) of the XXZ model with $N_s=6$ and $J_z=-1.5$. The numbered state indices refer to the eigenstates, $\rho_G$ to the Gibbs state, and $\rho^{(m)}_G$ is the truncated $m$-rank Gibbs state. The inset shows the low-lying eigenspectrum.
• Figure 5: The relative free energy error as a function of $\beta$ for the XXZ model with $N_s=6$ and $J_z=-1.5$. For $\beta\leq\beta_0$, we prepare the Gibbs state using TEPID-ADAPT, as indicated by the markers. For $\beta>\beta_0$, the same converged adaptive unitary is used with no parameter re-optimization, as indicated by the dashed lines.