Estimating Local Observables via Cluster-Level Light-Cone Decomposition

TL;DR

This work tackles the challenge of simulating local quantum observables on large, modular quantum devices by introducing a cluster-level light-cone framework. It presents two complementary algorithms: a Causal Decoupling method that leverages geometric light-cone disconnections for polynomial sampling cost, and an Algebraic Decomposition method that reduces quantum hardware to minimal clusters at the expense of exponential light-cone-volume sampling. Theoretical results show that local observables can be probed with costs depending on circuit depth and connectivity rather than total system size, enabling scalable near-term quantum simulations on modular architectures. The framework offers practical trade-offs for VQE, correlation function studies, and QEC benchmarking, and highlights a pathway beyond circuit cutting toward locality-focused quantum-classical hybrids.

Abstract

Simulating large quantum circuits on hardware with limited qubit counts is often attempted through methods like circuit knitting, which typically incur sample costs that grow exponentially with the number of connections cut. In this work, we introduce a framework based on Cluster-level Light-cone analysis that leverages the natural locality of quantum workloads. We propose two complementary algorithms: the Causal Decoupling Algorithm, which exploits geometric disconnections in the light cone for sampling efficiency, and the Algebraic Decomposition Algorithm, which utilizes algebraic expansion to minimize hardware requirements. These methods allow simulation costs to depend on circuit depth and connectivity rather than system size. Together, our results generalize Lieb-Robinson-inspired locality to modular architectures and establish a quantitative framework for probing local physics on near-term quantum devices by decoupling the simulation cost from the global system size.

Figures (2)

• Figure 1: Illustration of the cluster‐level light‐cone decomposition for an $s$-local Pauli operator acting on a one‐dimensional chain of clusters under a shallow circuit $U$. The pale blue boxes mark the clusters $\mathcal{C}_\alpha$ where the Pauli term acts non-trivially. For each such cluster $C_j$, its causal light cone $L(C_j)$ under $U$ is shown in green. The union of overlapping light cones splits into connected supports $L_1$ and $L_2$, whose boundaries are outlined in red dashes. We simulate each subcircuit $U_{\mathrm{loc}}^{(\alpha,i)}$ by retaining only the gates within the corresponding support $L_i$, since gates outside cancel between $U$ and $U^\dagger$.
• Figure 2: Causal light cone and task decomposition in a cluster-level brick-wall circuit. Each vertical line represents a quantum cluster $C_i$. For a local observable $O_j$ measured on a target cluster $C_j$ (here $C_5$) at the final time step, the support of its Heisenberg evolution, $U^\dagger O_j U$, is confined to a causal light cone (bounded by the red dashed line). Only the inter-cluster gates within this cone (green blocks) contribute to the measurement outcome, while gates outside of it (gray blocks) can be disregarded. Crucially, each gate within the light cone is decomposed, which allows the complex global evolution to be transformed into a series of local sub-problems. Each sub-problem can then be solved independently on a small quantum computer encompassing only the clusters within that light cone.

Theorems & Definitions (21)

• Definition 1: Qubit Light Cone
• Definition 2: Qubit-Level Range on Lattice Systems
• Definition 3: Clustered Quantum System
• Definition 4: $D$-Dimensional Clustered Lattice
• Definition 5: All-to-All Connected Clustered System
• Definition 6: Cluster Light Cone
• Definition 7: Volume of the Light Cone Circuit
• Definition 8: Cluster-Level Range on Lattice Clustered Systems
• Definition 9: $m$-Sparse Operator
• Definition 10: Cluster-Level $s$-local Operator