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Complexity Bounds for Hamiltonian Simulation in Unitary Representations

Naihuan Jing, Molena Nguyen

Abstract

For any unitary representation $ρ$ on a finite-dimensional Hilbert space \(V\) with differential \(dρ: \mathfrak{g} \to \mathfrak{u}(V)\) for the Lie algebra $\mathfrak g$, we consider the Hamiltonian evolution \[ U_X(t) \coloneqq ρ(\exp(tX)) = e^{t\,dρ(X)}, \qquad t\in\mathbb{R}. \] For any complexification $ X_\mathbb{C} = X_0 + \sum\limits_{α\inΔ} x_αE_α$ associated with the root system $Δ$, we introduce the numerical invariants %\emph{root activity} and \emph{root curvature} functionals \begin{align*} \mathcal{A}_p(X) &\coloneqq \Bigl(\sum_{α\inΔ} |x_α|^p \,\|dρ(E_α)\|_{\mathrm{op}}^p\Bigr)^{1/p}, \quad 1\le p<\infty\\ \mathcal{C}(X) &\coloneqq \Bigl(\sum_{α\inΔ} |α(X_0)|^2\,|x_α|^2 \,\|dρ(E_α)\|_{\mathrm{op}}^2\Bigr)^{1/2}, \end{align*} where \(\|\cdot\|_{\mathrm{op}}\) is the operator norm on \(\mathrm{End}(V)\). We first describe how the Hamiltonian \(dρ(X)\) is distributed along the directions of root spaces $\mathfrak{g}_α$. Our main result shows that for each fixed \(X\in\mathfrak{g}\) there exists a constant \(C_X>0\) such that \[ \bigl\| e^{t(dρ(X_0)+dρ(X_{\mathrm{root}}))} - e^{\frac{t}{2}dρ(X_0)} e^{t dρ(X_{\mathrm{root}})} e^{\frac{t}{2}dρ(X_0)} \bigr\|_{\mathrm{op}} \le C_X\,t^{3}\,\bigl(\mathcal{C}(X)+\mathcal{A}_1(X_{\mathrm{root}})\bigr) \] for all sufficiently small \(|t|\). We also introduce a root-gate circuit model and test this on spin$-$chain Hamiltonians on \((\mathbb{C}^2)^{\otimes n}\subset\mathfrak{su}(2^n)\), where root spaces are spanned by matrix units, \(\mathcal{A}_p\), and \(\mathcal{C}\), which gives sharper complexity bounds and dimension$-$free representation$-$theoretic invariants.

Complexity Bounds for Hamiltonian Simulation in Unitary Representations

Abstract

For any unitary representation on a finite-dimensional Hilbert space with differential \(dρ: \mathfrak{g} \to \mathfrak{u}(V)\) for the Lie algebra , we consider the Hamiltonian evolution For any complexification associated with the root system , we introduce the numerical invariants %\emph{root activity} and \emph{root curvature} functionals \begin{align*} \mathcal{A}_p(X) &\coloneqq \Bigl(\sum_{α\inΔ} |x_α|^p \,\|dρ(E_α)\|_{\mathrm{op}}^p\Bigr)^{1/p}, \quad 1\le p<\infty\\ \mathcal{C}(X) &\coloneqq \Bigl(\sum_{α\inΔ} |α(X_0)|^2\,|x_α|^2 \,\|dρ(E_α)\|_{\mathrm{op}}^2\Bigr)^{1/2}, \end{align*} where is the operator norm on \(\mathrm{End}(V)\). We first describe how the Hamiltonian \(dρ(X)\) is distributed along the directions of root spaces . Our main result shows that for each fixed there exists a constant such that for all sufficiently small . We also introduce a root-gate circuit model and test this on spinchain Hamiltonians on \((\mathbb{C}^2)^{\otimes n}\subset\mathfrak{su}(2^n)\), where root spaces are spanned by matrix units, , and , which gives sharper complexity bounds and dimensionfree representationtheoretic invariants.
Paper Structure (36 sections, 13 theorems, 170 equations)

This paper contains 36 sections, 13 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Representation-theoretic setting and notation
  3. Overview
  4. Novelty and significance
  5. Outline of the paper
  6. Preliminaries
  7. Compact semisimple Lie algebras and the Killing form
  8. Maximal tori, roots, and root-space decompositions
  9. Weights, highest weights, and Hamiltonian action
  10. Hamiltonian evolution in unitary representations
  11. Decompositions and product formulas
  12. Root Profiles and Representation-Theoretic Functionals
  13. Torus--root decomposition of a Hamiltonian generator
  14. Root activity functionals
  15. Root curvature and commutator strength
  16. ...and 21 more sections

Key Result

Theorem 1

Let $X \in \mathfrak{g}$ and decompose $X = X_0 + X_{\mathrm{root}}$ as above. Then there exist constants $t_0(X) > 0$ and $C(X) > 0$, depending on $\mathfrak{g}$, $\rho$, and $X$, such that for all real $t$ with $|t| \le t_0(X)$,

Theorems & Definitions (58)

  • Definition 1.1: Hamiltonian simulation problem
  • Theorem : cf. Theorem \ref{['thm:curvature-error']}
  • Example 1.2: Curvature-free toral Hamiltonians
  • Example 2.1: $\mathfrak{su}(n)$ as a compact semisimple algebra
  • Definition 2.2: Roots
  • Example 2.3: Roots of $\mathfrak{su}(3)$
  • Definition 2.4: Weights and weight spaces
  • Example 2.5: Weights of the standard representation of $\mathrm{SU}(3)$
  • Definition 2.6: Gate set associated with a representation
  • Example 2.7: Single-spin Hamiltonian evolution in $\mathrm{SU}(2)$
  • ...and 48 more