An Optimal Framework for Constructing Lie-Algebra Generator Pools: Application to Variational Quantum Eigensolvers for Chemistry

TL;DR

The paper tackles the costly problem of assembling Minimal Complete Pools (MCPs) for Lie Algebras in quantum algorithms, where prior methods relied on exponential greedy searches. It introduces a polynomial-scaling, optimal framework that maps Pauli strings to a binary vector space $\mathbb{F}_2^{2N}$ and uses the adjacency matrix $\mathbf{\Gamma}_{\mathcal{A}}$ to certify completeness via a congruence to a canonical MCP or via a rank criterion, for example $\text{rank}(\mathbf{\Gamma}_{\mathcal{A}})=\text{rank}(\mathbf{\Gamma}_{\mathcal{A}'})$. This enables the construction of symmetry-preserving MCPs that reduce gradient-evaluation overhead and scale to larger molecular systems, empowering MB-ADAPT-VQE and NI-DUCC-VQE to achieve faster convergence and handle stronger correlation. Beyond chemistry, the framework offers broad applicability to areas such as quantum error correction, quantum control, and quantum machine learning, wherever compact Pauli bases and complete Lie-algebras are advantageous.

Abstract

Lie Algebras are powerful mathematical structures used in physics to describe sets of operators and associated combinations. A central task is to identify a minimal set of generators from which the algebra can be constructed. The classical search for such generators has so far relied on greedy construction steps applied to an exponentially growing number of candidate operators, making it rapidly computationally intractable. We propose a general, polynomial-scaling and optimal strategy, based on Lie-Algebraic basic properties, to overcome this bottleneck. It allows for the efficient construction of these generators, also known as Minimal Complete Pools (MCPs), for a target Lie Algebra. As an immediate application, efficiently constructing user-defined MCPs that respect fermionic algebra is crucial in the context of adaptive Variational Quantum Eigensolver for quantum chemistry. Thus, we introduce MB-ADAPT-VQE, which incorporates optimally constructed MCPs into batched ADAPT-VQE to reduce quantum resources and improve convergence under strong correlation. These MCPs also unlock fixed-ansatz methods based on a Lie-algebraic structure such as the gradient-free NI-DUCC-VQE, enabling simulations surpassing prior MCP limits. The presented mathematical framework is general and applicable well beyond chemistry in fields including quantum error correction, quantum control, quantum machine learning, and more universally wherever compact Pauli basis are required.

Figures (5)

• Figure 1: We analyze the anti-commutation graph corresponding to the set of Pauli strings ${\mathcal{A} = \{Y_1Y_3, X_1Y_2, Z_1Z_3, Z_1Z_2X_3Y_4, X_4\}}$, based on the work of fuclass. As they demonstrate, such graphs can be simplified via vertex contraction. For example, contracting $P_3$ with $P_4$ produces a new string, $P_1' = Z_2Y_3Y_4$, which simplifies the graph by reducing its connectivity. Our analysis will focus on the adjacency matrices of these graphs, hereafter denoted as $\mathbf{\Gamma}_\mathcal{A}$.
• Figure 2: Convergence plots for ADAPT-VQE and MB-ADAPT-VQE methods across various molecules: LiH ($R=1.3\text{\AA}$, 12 qubits), the hydrogen chain H$_6$ ($R=3.0 \text{\AA}$, 12 qubits), H$_8$ ($R=0.8 \text{\AA}$, 16 qubits), and H$_2$O (26 qubits, geometry defined in Supplementary Material). The ADAPT-VQE method uses a QEB pool of size $\mathcal{O}(N^4)$. The composite MB-ADAPT-VQE method employs batched-ADAPT-VQE with efficient construction of MCPs of size $\mathcal{O}(N)$ for various values of $k$. Plots display energy errors versus CNOT count, function evaluations, and parameter count for different molecules.
• Figure 3: Convergence plots for NI-DUCC-VQE method across various molecules using different pool operators: the hydrogen chain H$_6$ ($R=3.0 \text{\AA}$, 12 qubits), H$_8$ ($R=0.8 \text{\AA}$, 16 qubits), and H$_2$O (26 qubits, geometry defined in Supplementary Material). Plots display energy errors versus function evaluations.
• Figure 4: Convergence plots for ADAPT-VQE and MB-ADAPT-VQE methods across H$_6$ ($R=3.0 \text{\AA}$, 12 qubits), and H$_8$ ($R=0.8 \text{\AA}$, 16 qubits) using different pool operators. Plots display energy errors versus number of iterations.
• Figure 5: Schematic of necessary steps towards a user-defined operator pool

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3
• Remark 4
• Definition 5
• Definition 6
• Definition 7
• Remark 8
• Proposition 9
• Definition 10