On the Feasibility of Exact Unitary Transformations for Many-body Hamiltonians

TL;DR

This work identifies a general, unifying criterion for exact unitary transformations of many-body Hamiltonians: if the adjoint action $\mathrm{ad}_{i\hat{G}}$ acts in a finite-dimensional space, Cayley–Hamilton theory ensures a finite polynomial that annihilates the adjoint map, turning the otherwise infinite Baker–Campbell–Hausdorff expansion into a manageable finite sum. It unifies two main routes—finite-dimensional Lie-algebra representations and finite-spectrum generators—plus an eigen-subspace projector perspective, and shows how block structure and additional algebraic relations can drastically reduce BCH depth. The authors apply the framework to orbital rotations ($\mathfrak{u}(N)$ modules), the Heisenberg algebra, Pauli-product generators, and UCC-type generators, illustrating exact transformations and deriving explicit, compact polynomial forms. They further introduce fermionic reflections to construct a new class of number-conserving, finite-spectrum generators, enabling exact transformations with potentially reduced cost and preserved symmetries. The work points to futures directions in designing symmetry-respecting, low-spectrum generators and automating fragment identification, with practical impact on quantum simulation and related classical-quantum algorithms.

Abstract

Exact unitary transformations play a central role in the analysis and simulation of many-body quantum systems, yet the conditions under which they can be carried out exactly and efficiently remain incompletely understood. We show that exact transformations arise whenever the adjoint action of a unitary's generator defines a linear map within a finite-dimensional operator space. In this regime, there exists a finite-degree polynomial that annihilates the adjoint map, rendering the Baker-Campbell-Hausdorff (BCH) expansion finite. We identify the role of Lie algebras and their modules in producing finite BCH expansions in all known cases. This perspective brings together previously disparate examples of exact transformations under a single unifying principle and clarifies how algebraic relations between generators and transformed operators determine the polynomial degree of the transformation. We illustrate this framework for previously known cases of efficient unitary transformations including unitary coupled-cluster and Pauli product generators. Using this framework, we propose a new class of fermionic generators that can be used for efficient transformations. The result establishes sufficient algebraic conditions for when exact unitary transformations are possible and provides new strategies for reducing their computational cost in quantum simulation and constructing feasible unitary transformations.

Figures (1)

• Figure 1: Two patterns of nonvanishing projected blocks (depicted by $*$) of $\hat{H}_\alpha=\hat{T}_\alpha+\hat{T}_\alpha^\dagger$ in the eigenspaces $\{\mathcal{V}_-,\mathcal{V}_0,\mathcal{V}_+\}$ of $\hat{G}=(-i)(\hat{T}_G-\hat{T}_G^\dagger)$, depending on whether $\hat{P}_G \hat{H}_\alpha \hat{P}_G$ vanishes, where $\hat{P}_G := \hat{P}_+ + \hat{P}_-$. When the operators $\hat{T}_G := \tilde{T}_G \tilde{L}_G$ and $\hat{T}_\alpha := \tilde{T}_\alpha \tilde{L}_\alpha$ take the forms specified, additional projected blocks vanish. A superscript $(n)$ denotes a product of occupation operators, $\{\hat{a}_p^\dagger \hat{a}_p\}_{p=1}^N$.

Theorems & Definitions (8)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof