The Complexity of Simulating Local Measurements on Quantum Systems

S. Gharibian, J. Yirka, in: M. Wilde (Ed.), 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018, pp. 2:1-2:17.

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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.
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Proceedings Title
12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)
Volume
73
Page
2:1-2:17
Conference
12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)
Conference Location
Paris, France
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Gharibian S, Yirka J. The Complexity of Simulating Local Measurements on Quantum Systems. In: Wilde M, ed. 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Vol 73. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik; 2018:2:1-2:17. doi:10.4230/LIPIcs.TQC.2017.2
Gharibian, S., & Yirka, J. (2018). The Complexity of Simulating Local Measurements on Quantum Systems. In M. Wilde (Ed.), 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017) (Vol. 73, pp. 2:1-2:17). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.TQC.2017.2
@inproceedings{Gharibian_Yirka_2018, place={Dagstuhl, Germany}, series={Leibniz International Proceedings in Informatics (LIPIcs)}, title={The Complexity of Simulating Local Measurements on Quantum Systems}, volume={73}, DOI={10.4230/LIPIcs.TQC.2017.2}, booktitle={12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, publisher={Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, author={Gharibian, Sevag and Yirka, Justin}, editor={Wilde, MarkEditor}, year={2018}, pages={2:1-2:17}, collection={Leibniz International Proceedings in Informatics (LIPIcs)} }
Gharibian, Sevag, and Justin Yirka. “The Complexity of Simulating Local Measurements on Quantum Systems.” In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), edited by Mark Wilde, 73:2:1-2:17. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. https://doi.org/10.4230/LIPIcs.TQC.2017.2.
S. Gharibian and J. Yirka, “The Complexity of Simulating Local Measurements on Quantum Systems,” in 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), Paris, France, 2018, vol. 73, pp. 2:1-2:17.
Gharibian, Sevag, and Justin Yirka. “The Complexity of Simulating Local Measurements on Quantum Systems.” 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), edited by Mark Wilde, vol. 73, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018, pp. 2:1-2:17, doi:10.4230/LIPIcs.TQC.2017.2.

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