Adaptive time Compressed QITE (ACQ) and its geometrical interpretation

TL;DR

The paper develops Adaptive Compressed QITE (ACQ), a geometry-inspired method to efficiently approximate Quantum Imaginary Time Evolution on quantum hardware. By interpreting ITE as a gradient flow on CP^N and leveraging geodesic structure, the authors introduce line-search based adaptive time steps and a one-parameter unitary compression to reduce circuit depth and classical overhead. They derive Newton-style bounds for adaptive steps and demonstrate that ACQ achieves comparable ground-state fidelities to standard QITE with fewer unitary applications, exemplified in the TFIM. This approach offers a practical pathway toward resource-efficient ground-state preparation on near-term quantum devices, with a clear link between geometric trajectories and algorithmic efficiency.

Abstract

Preparing the ground state of a given Hamiltonian is a computational task of interest in many fields, such as material science, chemistry and even some optimization problems, to name a few. Efficiently preparing ground states for large, strongly correlated systems is a challenging task for both classical and quantum hardware. Drawing from classical optimization methods, e.g. dynamical optimization techniques, one may deduce the spectral decomposition in manner that avoids direct spectral decomposition and is amenable to Trotterization methods. An instance of the latter is ground state preparation by Imaginary Time Evolution (ITE), understood in physical terms as a natural cooling process. Its quantum version QITE (Quantum Imaginary Time Evolution) aims at implementing ITE in a quantum computer. In this paper we introduce a novel QITE algorithm, which leverages underlying geometric properties for algorithm-runtime and circuit depth reduction. This will materialize in the form of an iterative Line Search approach for minimization of energy as well as a Newton's method approach for the deduction of the optimal time-steps for each iteration of QITE. The depth-reduction will be carried out via approximating the resulting unitary operator estimated from the QITE algorithm with unitary operator which is an element of a one-parameter group; making expressible as a single unitary in a quantum circuit. Furthermore, we perform a numerical study to stablish the scaling of fidelities with the different truncation parameters and give gate counts estimates for each.

Figures (4)

• Figure 1: Trajectory distance defined in Eq. \ref{['eq:traj_distance']} between ITE and the geodesic connecting the initial state and the ground state of the TFIM with values $J=0.5$ and $h=1$.
• Figure 2: Sketch representation of the geodesic trajectory of $\ket{\gamma_2(\tau)}$ and the trajectory obtained by a step-wise evolution $\ket{\gamma_1(\tau)}$ such as the one from QITE. The black dashed lines represent the shortest distance from points of $\ket{\gamma_1(\tau)}$ to the geodesic. Note that step-wise evolutions may not reach exactly the ground state $\ket{E_0}$.
• Figure 3: Sketch of the trajectories taken by different methods. Geodesic/unitary trajectories are represented by straight lines. The ITE trajectory given by the gradient descent equation is tightly reproduced by discrete time steps of QITE. Adaptive QITE extends the unitary evolution of a QITE step until the energy of the evolved state starts increasing, at which point a new QITE step is computed and propagated.
• Figure 4: Comparison of QITE and ACQ for the TFIM with a ground state in the disordered phase ($J=0.5$ and $h=1$). In the two left plots, we plot the energy and fidelity evolution for a system of $N=12$ qubits, respectively. Blue lines represent the evolution generated by regular QITE, while the red lines represent the evolution generated by ACQ. The black horizontal line in the left panel indicates the exact energy of the ground state. The black crosses represent the points where the QITE routine of approximating ITE by unitaries is performed. On the rightmost panel we plot the maximum fidelities reached by each algorithm for an increasing system size. We plot the fidelities reached for different truncation values of $D$, in dashed lines $D=2$ and in solid lines $D=4$, with the same colors as before for QITE and ACQ.

Theorems & Definitions (13)

• Definition 1: Fubini-Study distance
• Theorem 1: Geodesics in $\mathbb{CP}^{N}$
• Lemma 1
• proof
• Remark 1
• Definition 2
• Theorem 2
• Definition 3
• Lemma 2
• Lemma 3: Equivalence of ITE and commutator flow for projector Hamiltonians arxivgluzagrover