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On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation

Authors Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, Yu-Ching Shen

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Author Details

Nai-Hui Chia
  • Rice University, Houston, TX, USA
Kai-Min Chung
  • Academia Sinica, Taipei, Taiwan
Yao-Ching Hsieh
  • University of Washington, Seattle, WA, USA
Han-Hsuan Lin
  • National Tsing Hua University, Hsinchu, Taiwan
Yao-Ting Lin
  • University of California at Santa Barbara, CA, USA
Yu-Ching Shen
  • Academia Sinica, Taipei, Taiwan

Funding

NH. Chia is supported by NSF award FET-2243659 and Google Scholar Award. KM. Chung’s research is partially supported by NSTC QC project under Grant no. NSTC 111-2119-M-001-004- and the 2021 Academia Sinica Investigator Award (AS-IA-110-M02). HH. Lin is supported by NSTC QC project under Grant no. NSTC 111-2119-M-001-004- and MOST Grant no. 110-2222-E-007-002-MY3. YC. Hsieh and YC. Shen are supported by NSTC QC project under Grant no. NSTC 111-2119-M-001-004-. YT. Lin is partially supported by Executive Yuan Data Safety and Talent Cultivation Project (AS-KPQ-110-DSTCP). Part of the work was done when YC. Hsieh and YT. Lin were working at Academia Sinica.

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Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen. On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 33:1-33:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.33

Abstract

Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time T for the simulation greatly affect algorithm runtime as expected. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time o(T), for some large classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time T. On the other hand, while there exist lower bounds of Ω(T) circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. As a result, physical systems with system size scaling with T can potentially do a fast-forwarding simulation. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism. In this work, we give a negative result for the above open problem in various settings. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth o(T). In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians on n qubits that cannot be simulated via an oracle circuit of depth o(T/n^c), where the Hamiltonians act on n qubits, and c is a constant. Lastly, we generalize the above results and show that any simulators that are geometrically local Hamiltonians cannot do the simulation much faster than parallel quantum algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Hamiltonian simulation
  • Depth lower bound
  • Parallel query lower bound

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