Variational Quantum Subspace Construction via Symmetry-Preserving Cost Functions

TL;DR

This article proposes a variational strategy based on symmetry-preserving cost functions to iteratively construct a reduced subspace for the extraction of low-lying energy states and shows that, under certain conditions, this approach leads to a tridiagonal representation similar to that obtained with the Lanczos algorithm.

Abstract

Determining low-energy eigenstates in electronic many-body quantum systems is a key challenge in computational chemistry and condensed-matter physics. Hybrid quantum-classical approaches, such as the Variational Quantum Eigensolver and Quantum Subspace Methods, offer practical solutions but face limitations in circuit depth and measurement overhead. In this article, we propose a variational strategy based on symmetry-preserving cost functions to iteratively construct a reduced subspace for the extraction of low-lying energy states. We show that, under certain conditions, our approach leads to a tridiagonal representation similar to that obtained with the Lanczos algorithm. The iterative process allows control over the trade-off between circuit depth, the number of variational parameters, and the number of measurements required to achieve the desired accuracy, making it suitable for current quantum hardware. As a proof of concept, we test the proposed algorithms on H4 chain and ring, targeting both the ground-state energy and the charge gap.

Figures (12)

• Figure 1: (a) Schematic representation of the ground-state many-body density matrices ${\bf\Gamma}$, ${\bf \overline{\Gamma}}$, and $\boldsymbol{\widetilde{\Gamma}}$ in the three different representations, namely the many-body basis set, eigenvector, and Householder representation. The unitary transformations ${\bf P}$, ${\bf U}$, and ${\bf R}$ linking the three representations are displayed. (b) Schematic representation of the two-level decomposition of the ground state. We propose to optimize the gain energy $E_G$ coming from the interaction between the good-guess $|\Phi_0\rangle$ and a variational vector $|\widetilde{\Phi}_1({\bm \theta})\rangle$.
• Figure 2: Schematic depiction of the iterative algorithm (top panel). Only the states $\lbrace |\widetilde{\Phi}_1^{(n)}({\bm \theta}^{(n)})\rangle \rbrace$ in blue are optimized on a quantum computer to minimize the cost function $C({\bm \theta}^{(n)})$.
• Figure 3: Schematic representation of the HEA used in this work. The circuit inside the square brackets representing one layer $l$ is repeated $N_L$ times.
• Figure 4: Ground-state energy at the first iteration. Panels (a) and (b): ground-state energy (in Hartree) as a function of the interatomic distance $d_{\rm H-H}$ (in Angström) for different circuit depths corresponding to $N_L =1, 3$, or $5$ layers in the HEA using $E_G({\bm\theta})$ and $E_I({\bm\theta})$ cost functions, respectively. Results are compared with values obtained using the Hartree--Fock approximation and FCI method. Panels (c) and (d): absolute error $\Delta E = E(\bm\theta^*) - E^{\rm FCI}$ (in Hartree) as a function of the number of layers in the HEA for selected interatomic distances of $d_{\rm H-H} = 1.0, 2.0, 3.0$, and $4.0 {\mathrm \AA}$ using $E_G({\bm\theta})$ and $E_I({\bm\theta})$ cost functions, respectively.
• Figure 5: Convergence with respect to the circuit depth and optimization issues at first iteration. Panels (a) and (c): Difference between the cost function $C(\bm \theta^*)$ (in Hartree) with the optimal value $C^{\rm min}$, as a function of the number of layers in the HEA for selected interatomic distances of $d_{\rm H-H} = 1.0, 2.0, 3.0$, and $4.0 {\mathrm \AA}$, using $E_G({\bm\theta})$ and $E_I({\bm\theta})$ cost functions, respectively. Panels (b) and (d): Difference between the cost function $C(\bm \theta^*)$ (in Hartree) with the optimal value $C^{\rm min}$, as a function of the number of layers in the HEA with box-plot representing 25% (colored box), 50% (empty box) and 75% (whiskers) of optimization outcomes for the cost function when starting with different random variational parameters. Results are given as a function of the number of layers in the HEA for interatomic distances of $d_{\rm H-H} = 2.0 {\mathrm \AA}$, using $E_G({\bm\theta})$ (b) and $E_I({\bm\theta})$ (d) cost functions.