Riemannian quantum circuit optimization based on matrix product operators

TL;DR

This work advances quantum circuit optimization for Hamiltonian time evolution by marrying first-order Riemannian optimization on the complex Stiefel manifold with tensor-network representations. By representing the reference propagator $U_t$ as an MPO and optimizing gates simultaneously under unitarity constraints, it achieves substantial accuracy gains across spin and fermionic models, plus a molecular example (LiH). The approach avoids symmetry assumptions, scales to larger systems via MPO compression, and leverages tensor-network environments to efficiently compute cost and gradients. The results show up to four orders of magnitude improvement for 50-qubit spin chains and up to eight orders for LiH, underscoring the method’s potential for scalable quantum simulations and hardware-agnostic circuit compression.

Abstract

We significantly enhance the simulation accuracy of initial Trotter circuits for Hamiltonian simulation of quantum systems by integrating first-order Riemannian optimization with tensor network methods. Unlike previous approaches, our method imposes no symmetry assumptions, such as translational invariance, on the quantum systems. This technique is scalable to large systems through the use of a matrix product operator representation of the reference time evolution propagator. Our optimization routine is applied to various spin chains and fermionic systems described by the transverse-field Ising Hamiltonian, the Heisenberg Hamiltonian, and the spinful Fermi-Hubbard Hamiltonian. In these cases, our approach achieves a relative error improvement of up to four orders of magnitude for systems of 50 qubits, although our method is also applicable to larger systems. Furthermore, we demonstrate the versatility of our method by applying it to molecular systems, specifically lithium hydride, achieving an error improvement of up to eight orders of magnitude. This proof of concept highlights the potential of our approach for broader applications in quantum simulations.

Figures (14)

• Figure 1: Visualization of concepts on the Riemannian manifold $\mathcal{M}$. A point $y\in\mathcal{M}$ is "moved" in the direction of the vector $\vec{v}\in T_y\mathcal{M}$ via the retraction $R$. A vector $\vec{\xi}$ is moved in the direction of a vector $\vec{\eta}\in T_y\mathcal{M}$ to the tangent space $T_x \mathcal{M}$ in point $x\in\mathcal{M}$ using the vector transport $\tau$. Lastly, the Riemannian gradient $\grad_{\text{Rie}} f(x)$ can be obtained as the projected Euclidean gradient $\grad f(x)$ through the projector $P$.
• Figure 2: Illustration of how to merge a brickwall layer into an MPO in a left-to-right-sweep using the tensor network framework. Similarly, a right-to-left-sweep can be used, which is not explicitly illustrated here.
• Figure 3: How to obtain the reference MPO via a suitable fine higher-order Trotterization for system sizes of $N\geq 12$. The corresponding deep Trotter circuit is interpreted as a tensor network, and its brickwall layers are sequentially merged into an initial identity MPO.
• Figure 4: Diagrammatic sketch of $\partial_{G_i^\ell} \Tr(U_\text{ref}^\dagger W)$, where $G_i^\ell$ is the $i$-th gate in the $\ell$-th layer of the brickwall circuit. The partial derivative can be obtained by "cutting out" the considered gate $G_i^\ell$. For the efficient evaluation, a top environment MPO $E_\text{top}^{\ell}$ and a bottom environment MPO $E_\text{bottom}^\ell$ are computed. From there, the left and right environment, $E_\text{left}^{\ell,i}$ and $E_\text{right}^{\ell,i}$, are evaluated. Finally, contracting the resulting tensor network can compute the partial derivative.
• Figure 5: Results for the Ising model on a chain with $N=50$ sites, $J=1$, $g=0.75$, $h=0.6$, and $t=2$. (a) The error of the Riemannian optimized brickwall circuit compared to that of different Trotter circuits. (b) The error scaling in Trotter time step $\Delta t$ for the Riemannian optimized brickwall circuit compared to that of various Trotter circuits.