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arXiv:quant-ph/0508139 · Communications in Mathematical Physics

Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

Dominic William Berry, Graeme Ahokas, Richard Cleve, Barry C. Sanders

Published Dec 13, 2006• 692 citations
arXivHTMLPDF
DOI
10.1007/s00220-006-0150-x

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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 Schrödinger 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.