Complexity of Digital Quantum Simulation in the Low-Energy Subspace: Applications and a Lower Bound

TL;DR

This work establishes a rigorous framework for digital quantum simulation restricted to the low-energy subspace, showing that the simulation error can be governed by an effective low-energy norm and that resource costs (Trotter numbers and gates) can be dramatically reduced relative to full-space simulations. It develops and analyzes randomized product-formulas (qDRIFT, random permutation) and symmetry-protected schemes, plus extensions to power-law Hamiltonians, with concrete, quantified bounds on step complexity and gate counts under low-energy assumptions. A key contribution is the robustness result against imperfect state preparation due to thermalization, ensuring practical relevance for near-term devices. The paper also proves a lower bound indicating inherent queries-to-time limits in the low-energy setting, highlighting both the potential gains and fundamental constraints of low-energy quantum simulation, and it provides extensive appendices with auxiliary lemmas to support the main results.

Abstract

Digital quantum simulation has broad applications in approximating unitary evolution of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations due to thermalization. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller Trotter numbers. Such improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.

Figures (4)

• Figure 1: Quantum simulation in full Hilbert space vs. low-energy subspace: Trotter error induced by product formulas for a $\mathbf{2\times 6}$ homogeneous Heisenberg spin-$\mathbf{1/2}$ model. The solid lines denote simulation errors in the full Hilbert space while the dashed lines denote errors in the low-energy subspace. The plot distinguishes different Trotter numbers $r$ by the color of the lines.
• Figure 2: Full Hilbert space vs. low-energy subspace: simulation error induced by the qDRIFT and random permutation algorithms for localized Heisenberg spin-$\mathbf{1/2}$ models. The solid lines denote errors in the full Hilbert space while the dashed lines denote errors in the low-energy subspace. We distinguish the total evolution time $t$ or the order of the product formula $p$ by the color of the lines. We also compare the numerical and the theoretical error scalings. (a) We plot the worst-case and the low-energy simulation errors of the qDRIFT algorithm as a function of step number $r$. (b) The same as the upper left subfigure, but for the random permutation approach at $p=2$ and $p=4$. (c) We plot the worst-case and the low-energy simulation errors of the qDRIFT algorithm as a function of system size $n$. (d) The same as the lower left subfigure, but for the random permutation approach at $p=2$ and $p=4$.
• Figure 3: Full Hilbert space vs. low-energy subspace: Error induced by the symmetry protection approach for Heisenberg spin-$\mathbf{1/2}$ models. The solid lines denote errors in the full Hilbert space while the dashed lines denote errors in the low-energy subspace. We distinguish the experiments with different schemes by the color of the lines. We also compare the numerical and the theoretical error scalings. (a) We plot the worst-case and the low-energy simulation errors as a function of step number $r$. (b) We plot the worst-case and the low-energy simulation errors as a function of system size $n$.
• Figure 4: Full Hilbert space vs. low-energy subspace: Trotter error induced by the product formulas for a $\mathbf{3\times 3}$ power-law model. The solid lines denote errors in the full Hilbert space while the dashed lines denote errors in the low-energy subspace, and the plot distinguishes interaction power $\alpha$ by the color of the lines.

Theorems & Definitions (20)

• Definition 1
• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Theorem 5: Informal, see \ref{['thm:lower']} for the formal version
• Lemma 1: Theorem 2.1 of arad2016connecting
• Lemma 2: Eq. (111) of csahinouglu2021hamiltonian
• Lemma 3: Lemma 7 of tran2021faster