Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding

2024-01-16

arXiv.org

AI Summary

Efficiently simulating exponentially sparse Hamiltonians on Noisy Intermediate-Scale Quantum (NISQ) devices remains challenging due to prohibitive gate complexity and hardware limitations on Hilbert space dimensionality. Method: This paper introduces a novel Hamiltonian embedding paradigm: encoding the target Hamiltonian into the low-energy subspace dynamics of a structured expanded system—such as binary trees or glued trees—thereby circumventing conventional black-box oracle queries and deep, high-complexity gate sequences. Contribution/Results: The approach exponentially enlarges the effectively controllable Hilbert space dimension without increasing quantum gate count, overcoming hardware-imposed dimensional bottlenecks for sparse Hamiltonian simulation. By integrating hardware-efficient, structured Hamiltonian design with optimized gate compilation, the scheme is compatible with trapped-ion and neutral-atom platforms. Experimental demonstrations include quantum walk, spatial search, and real-space Schrödinger equation simulation, all achieving substantial reductions in quantum resource overhead.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Quantum Machine Learning
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Many promising quantum applications depend on the efficient quantum simulation of an exponentially large sparse Hamiltonian, a task known as sparse Hamiltonian simulation, which is fundamentally important in quantum computation. Although several theoretically appealing quantum algorithms have been proposed for this task, they typically require a black-box query model of the sparse Hamiltonian, rendering them impractical for near-term implementation on quantum devices. In this paper, we propose a technique named Hamiltonian embedding. This technique simulates a desired sparse Hamiltonian by embedding it into the evolution of a larger and more structured quantum system, allowing for more efficient simulation through hardware-efficient operations. We conduct a systematic study of this new technique and demonstrate significant savings in computational resources for implementing prominent quantum applications. As a result, we can now experimentally realize quantum walks on complicated graphs (e.g., binary trees, glued-tree graphs), quantum spatial search, and the simulation of real-space Schr""odinger equations on current trapped-ion and neutral-atom platforms. Given the fundamental role of Hamiltonian evolution in the design of quantum algorithms, our technique markedly expands the horizon of implementable quantum advantages in the NISQ era.

Problem

• Simulating large sparse Hamiltonians efficiently on quantum hardware
• Overcoming impractical black-box query models for near-term devices
• Enabling experimental quantum applications with hardware-efficient operations

Innovation

• Embedding sparse Hamiltonian into larger structured system
• Enabling efficient simulation via hardware-efficient operations
• Expanding implementable quantum advantages on NISQ devices