---
_id: '8161'
abstract:
- lang: eng
text: The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing
separations in computational complexity theory (modulo standard conjectures such
as PH does not collapse). Here, we study whether two quantum generalizations of
PH can similarly prove separations in the quantum setting. The first generalization,
QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the
former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem.
For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid
Method for efficiently solving semidefinite programs. These results yield two
implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled
proofs, a complexity class whose characterization has proven difficult. First,
if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to
make classical and quantum proofs "equivalent"), then QMA(2) is in the Counting
Hierarchy (specifically, in P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e.,
alternating quantifiers do not help in the presence of "unentanglement"), QMA(2)
is strictly contained in NEXP.
author:
- first_name: Sevag
full_name: Gharibian, Sevag
id: '71541'
last_name: Gharibian
orcid: 0000-0002-9992-3379
- first_name: Miklos
full_name: Santha, Miklos
last_name: Santha
- first_name: Jamie
full_name: Sikora, Jamie
last_name: Sikora
- first_name: Aarthi
full_name: Sundaram, Aarthi
last_name: Sundaram
- first_name: Justin
full_name: Yirka, Justin
last_name: Yirka
citation:
ama: 'Gharibian S, Santha M, Sikora J, Sundaram A, Yirka J. Quantum Generalizations
of the Polynomial Hierarchy with Applications to QMA(2). In: Potapov I, Spirakis
P, Worrell J, eds. 43rd International Symposium on Mathematical Foundations
of Computer Science (MFCS 2018). Vol 117. Leibniz International Proceedings
in Informatics (LIPIcs). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik; 2018:58:1-58:16.
doi:10.4230/LIPIcs.MFCS.2018.58'
apa: Gharibian, S., Santha, M., Sikora, J., Sundaram, A., & Yirka, J. (2018).
Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2).
In I. Potapov, P. Spirakis, & J. Worrell (Eds.), 43rd International Symposium
on Mathematical Foundations of Computer Science (MFCS 2018) (Vol. 117, p.
58:1-58:16). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2018.58
bibtex: '@inproceedings{Gharibian_Santha_Sikora_Sundaram_Yirka_2018, place={Dagstuhl,
Germany}, series={Leibniz International Proceedings in Informatics (LIPIcs)},
title={Quantum Generalizations of the Polynomial Hierarchy with Applications to
QMA(2)}, volume={117}, DOI={10.4230/LIPIcs.MFCS.2018.58},
booktitle={43rd International Symposium on Mathematical Foundations of Computer
Science (MFCS 2018)}, publisher={Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik},
author={Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi
and Yirka, Justin}, editor={Potapov, Igor and Spirakis, Paul and Worrell, James},
year={2018}, pages={58:1-58:16}, collection={Leibniz International Proceedings
in Informatics (LIPIcs)} }'
chicago: 'Gharibian, Sevag, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin
Yirka. “Quantum Generalizations of the Polynomial Hierarchy with Applications
to QMA(2).” In 43rd International Symposium on Mathematical Foundations of
Computer Science (MFCS 2018), edited by Igor Potapov, Paul Spirakis, and James
Worrell, 117:58:1-58:16. Leibniz International Proceedings in Informatics (LIPIcs).
Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. https://doi.org/10.4230/LIPIcs.MFCS.2018.58.'
ieee: 'S. Gharibian, M. Santha, J. Sikora, A. Sundaram, and J. Yirka, “Quantum Generalizations
of the Polynomial Hierarchy with Applications to QMA(2),” in 43rd International
Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Liverpool,
UK, 2018, vol. 117, p. 58:1-58:16, doi: 10.4230/LIPIcs.MFCS.2018.58.'
mla: Gharibian, Sevag, et al. “Quantum Generalizations of the Polynomial Hierarchy
with Applications to QMA(2).” 43rd International Symposium on Mathematical
Foundations of Computer Science (MFCS 2018), edited by Igor Potapov et al.,
vol. 117, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018, p. 58:1-58:16,
doi:10.4230/LIPIcs.MFCS.2018.58.
short: 'S. Gharibian, M. Santha, J. Sikora, A. Sundaram, J. Yirka, in: I. Potapov,
P. Spirakis, J. Worrell (Eds.), 43rd International Symposium on Mathematical Foundations
of Computer Science (MFCS 2018), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik,
Dagstuhl, Germany, 2018, p. 58:1-58:16.'
conference:
location: Liverpool, UK
name: 43rd International Symposium on Mathematical Foundations of Computer Science
(MFCS 2018)
date_created: 2019-03-01T11:29:44Z
date_updated: 2023-02-28T11:01:03Z
department:
- _id: '623'
- _id: '7'
doi: 10.4230/LIPIcs.MFCS.2018.58
editor:
- first_name: Igor
full_name: Potapov, Igor
last_name: Potapov
- first_name: Paul
full_name: Spirakis, Paul
last_name: Spirakis
- first_name: James
full_name: Worrell, James
last_name: Worrell
external_id:
arxiv:
- '1805.11139'
intvolume: ' 117'
keyword:
- Complexity Theory
- Quantum Computing
- Polynomial Hierarchy
- Semidefinite Programming
- QMA(2)
- Quantum Complexity
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=9640
oa: '1'
page: 58:1-58:16
place: Dagstuhl, Germany
publication: 43rd International Symposium on Mathematical Foundations of Computer
Science (MFCS 2018)
publication_identifier:
unknown:
- 978-3-95977-086-6
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
series_title: Leibniz International Proceedings in Informatics (LIPIcs)
status: public
title: Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)
type: conference
user_id: '71541'
volume: 117
year: '2018'
...
---
_id: '8160'
abstract:
- lang: eng
text: 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.
author:
- first_name: Sevag
full_name: Gharibian, Sevag
id: '71541'
last_name: Gharibian
orcid: 0000-0002-9992-3379
- first_name: Justin
full_name: Yirka, Justin
last_name: Yirka
citation:
ama: '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). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik; 2018:2:1-2:17.
doi:10.4230/LIPIcs.TQC.2017.2'
apa: 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, p. 2:1-2:17).
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.TQC.2017.2
bibtex: '@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, Mark},
year={2018}, pages={2:1-2:17}, collection={Leibniz International Proceedings in
Informatics (LIPIcs)} }'
chicago: '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.'
ieee: '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, p.
2:1-2:17, doi: 10.4230/LIPIcs.TQC.2017.2.'
mla: 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, p. 2:1-2:17, doi:10.4230/LIPIcs.TQC.2017.2.
short: '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, p. 2:1-2:17.'
conference:
location: Paris, France
name: 12th Conference on the Theory of Quantum Computation, Communication and Cryptography
(TQC 2017)
date_created: 2019-03-01T11:25:27Z
date_updated: 2023-02-28T11:00:48Z
department:
- _id: '623'
- _id: '7'
doi: 10.4230/LIPIcs.TQC.2017.2
editor:
- first_name: Mark
full_name: Wilde, Mark
last_name: Wilde
external_id:
arxiv:
- '1606.05626'
intvolume: ' 73'
keyword:
- Complexity theory
- Quantum Merlin Arthur (QMA)
- local Hamiltonian
- local measurement
- spectral gap
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=8577
oa: '1'
page: 2:1-2:17
place: Dagstuhl, Germany
publication: 12th Conference on the Theory of Quantum Computation, Communication and
Cryptography (TQC 2017)
publication_identifier:
unknown:
- 978-3-95977-034-7
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
series_title: Leibniz International Proceedings in Informatics (LIPIcs)
status: public
title: The Complexity of Simulating Local Measurements on Quantum Systems
type: conference
user_id: '71541'
volume: 73
year: '2018'
...