## Abstract

Analog quantum simulation – simulation of one Hamiltonian by another – is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian H whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian H (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where H is only required to simulate the groundstate(s) and spectral gap of H instead of its full spectrum, and we believe it is of independent interest. Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the “first questions” one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves as a good starting point for future research of these topics.

Original language | American English |
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Title of host publication | 10th Innovations in Theoretical Computer Science, ITCS 2019 |

Editors | Avrim Blum |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770958 |

DOIs | |

State | Published - 1 Jan 2019 |

Event | 10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States Duration: 10 Jan 2019 → 12 Jan 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 124 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 10th Innovations in Theoretical Computer Science, ITCS 2019 |
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Country/Territory | United States |

City | San Diego |

Period | 10/01/19 → 12/01/19 |

### Bibliographical note

Publisher Copyright:© Dorit Aharonov and Leo Zhou.

## Keywords

- Quantum Hamiltonian complexity
- Quantum PCP
- Quantum simulation
- Sparsification