Role of Riemannian geometry in double-bracket quantum imaginary-time evolution

TL;DR

This paper reframes imaginary-time evolution as a Riemannian gradient flow on the adjoint-unitary manifold, via Brockett's double-bracket flow, and introduces DB-QITE as a geometry-aware quantum algorithm that approximates ground-state preparation without measurement-based heuristics. The method uses group-commutator constructions (and a higher-order variant) to iteratively build unitaries that steer a state toward the ground state, with a clear link between energy decrease and the energy variance dictated by the manifold geometry. Numerical experiments on a 10-qubit Heisenberg model, implemented in the Qrisp framework, compare DB-QITE against GC and HOPF, showing GC's practicality due to similar convergence but significantly lower gate counts, while identifying saddle-point bottlenecks and the role of initializations. The work provides explicit gate-count assessments and outlines paths toward NISQ-era deployment, circuit optimization, and error-mitigation strategies to enable scalable quantum simulations of complex many-body systems.

Abstract

Double-bracket quantum imaginary-time evolution (DB-QITE) is a quantum algorithm which coherently implements steps in the Riemannian steepest-descent direction for the energy cost function. DB-QITE is derived from Brockett's double-bracket flow which exhibits saddle points where gradients vanish. In this work, we perform numerical simulations of DB-QITE and describe signatures of transitioning through the vicinity of such saddle points. We provide an explicit gate count analysis using quantum compilation programmed in Qrisp.

Figures (3)

• Figure 1: Qrisp implementation of DB-QITE. qarg is the QuantumVariable which is operated upon, U_0 is a state preparation function, exp_H is a function, which simulates the Hamiltonian in question. s is the array indicating the schedule and k is the recursion depth.
• Figure 2: DB-QITE for the 10-qubit Heisenberg model. a) When $B=0.5$ then $\ket{\text{Singlet}}$ has large ground state overlap $\mathcal{F}_0(0)=0.68$ and no overlap with the first excited state $\mathcal{F}_1(0)=0$, and similarly $\ket{\text{VQE}}$ has $\mathcal{F}_0(0)=0.88$ and $\mathcal{F}_1(0)=0$. DB-QITE rapidly converges to the ground state. b) For $B=1$ ground and first excited states change roles and then $\mathcal{F}_0(0)=0$, $\mathcal{F}_1(0)=0.68$ for $\ket{\text{Singlet}}$ and $\mathcal{F}_0(0)=0$, $\mathcal{F}_1(0)=0.88$ for $\ket{\text{VQE}}$ and DB-QITE to converges to $\ket{\lambda_1}$. c) Counts of gates $\{\mathrm{U3},\mathrm{CX}\}$ together with the circuit depth for panel a) for $\ket{\text{Singlet}}$ when using GC and HOPF formulas. d) GC and HOPF lead to similar fidelity convergence.
• Figure 3: ITE (a,b) and DB-QITE (c,d) for the 10-qubit Heisenberg model with $J=1,B=0.5$ starting from initial states $\ket{\Psi_0^{(j)}}$ with $j=1,2,4$ which are biased to transition from a high-energy state (approximately $\ket{\lambda_{10}}$) through the vicinity of a lower eigenstate $|\lambda_j\rangle$ before reaching the ground state. $|\lambda_j\rangle$ can be interpreted as a saddle point of the ITE because energy decrease can stall almost entirely. a) The ITE energy decrease proceeds in three phases. When $\tau$ is small, the energy $E(\tau)$ (shades of blue for $j=1,2,4$) decreases rapidly. As $\tau$ increases, the energy fluctuation $V(\tau)$ (green for $j=2$) drops to $0$ and we reach the saddle point at $\lambda_j$. b) The ITE energy becomes stagnant until $\ket{\Psi^{(j)}(\tau)}$ gains about $50\%$ fidelity with the ground state $\ket{\lambda_0}$. However, when the spectral gap is too small such as in the case of $\ket{\Psi^{(1)}(0)}$, leaving the saddle-point vicinity takes far too long time to reach the ground state. c) The energy expectations $E_k$ (shades of blue for $j=1,2,4$) with respect to the cumulative QITE duration for different initial states and energy fluctuations $V_k$ for $\ket{\Psi^{(2)}(\tau)}$ (green). Restricting circuit depths to current quantum hardware capacities, we can only explore the first region where the high-energy state phases out and before fully reaching the bottleneck from the underlying saddle point. Energy fluctuations $V_k$ are comparable in magnitude to those in the first peak of ITE. d) The fidelity to $\ket{\lambda_j}$, i.e. DB-QITE is approaching the bottleneck phase.