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DATE | 2021-06-25 |

TIME | 10:00 |

PLACE | Online |

FIELD | Quantum Information Science |

SPEAKER | Dr. Chi-Fang Chen - Department of Physics, Caltech, Pasadena, CA, USA |

TITLE | Random Product formula concentrates |

ABSTRACT | https://arxiv.org/abs/2008.11751
video link: https://www.youtube.com/watch?v=Qi1HLdYLcB0 Q&A section with Dr. Chi-Fang Chen: https://teams.microsoft.com/dl/launcher/launcher.html?url=%2F_%23%2Fl%2Fmeetup-join%2F19%3Abe36335ed90247778824ca18754a5af0%40thread.tacv2%2F1624258070992%3Fcontext%3D%257b%2522Tid%2522%253a%25221608350e-92d5-431a-8ea2-bd2f76141620%2522%252c%2522Oid%2522%253a%2522bbb10f0d-d6d4-464d-938b-2836778d560e%2522%257d%26anon%3Dtrue&type=meetup-join&deeplinkId=195158af-29c7-47b0-aa6d-fe9dee24a86f&directDl=true&msLaunch=true&enableMobilePage=true&suppressPrompt=true Quantum simulation has wide applications in quantum chemistry and physics. Recently, scientists have begun exploring the use of randomized methods for accelerating quantum simulation. Among them, a simple and powerful technique, called qDRIFT, is known to generate random product formulas for which the average quantum channel approximates the ideal evolution. Remarkably, it has a gate count not explicitly depending on the number of term in the Hamiltonian, which contrasts with Suzuki formulas. This work aims to understand the origin of this speed-up by comprehensively analyzing a single realization of the random product formula produced by qDRIFT. The main results prove that a typical realization of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error. The gate complexity is already independent of the number of terms in the Hamiltonian, but it depends on the system size and the sum of the interaction strengths in the Hamiltonian. Remarkably, the same random evolution starting from an arbitrary, but fixed, input state yields a much shorter circuit suitable for that input state. In contrast, in deterministic settings, such an improvement usually requires initial state knowledge. The proofs depend on concentration inequalities for vector and matrix martingales, and the framework is applicable to other randomized product formulas. Our bounds are saturated by certain commuting Hamiltonians. |