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DOI: 10.4230/LIPIcs.TQC.2017.2
URN: urn:nbn:de:0030-drops-85776
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8577/
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Gharibian, Sevag ; Yirka, Justin

The Complexity of Simulating Local Measurements on Quantum Systems

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LIPIcs-TQC-2017-2.pdf (0.7 MB)


Abstract

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, Ambainis defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following results. The P^QMA[log]-completeness result of Ambainis requires O(log n)-local observ- ables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete. P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of Beigel, Hemachandra, and Wechsung to show P^QMA[log] \subseteq PP. This improves the containment QMA \subseteq PP from Kitaev and Watrous. A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, we identify a flaw in Ambainis' prior work regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on his prior work to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions.

BibTeX - Entry

@InProceedings{gharibian_et_al:LIPIcs:2018:8577,
  author =	{Sevag Gharibian and Justin Yirka},
  title =	{{The Complexity of Simulating Local Measurements on Quantum Systems}},
  booktitle =	{12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)},
  pages =	{2:1--2:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-034-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{73},
  editor =	{Mark M. Wilde},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8577},
  URN =		{urn:nbn:de:0030-drops-85776},
  doi =		{10.4230/LIPIcs.TQC.2017.2},
  annote =	{Keywords: Complexity theory, Quantum Merlin Arthur (QMA), local Hamiltonian, local measurement, spectral gap}
}

Keywords: Complexity theory, Quantum Merlin Arthur (QMA), local Hamiltonian, local measurement, spectral gap
Seminar: 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)
Issue Date: 2018
Date of publication: 26.02.2018


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