---
_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: '8171'
abstract:
- lang: eng
text: "The polynomial hierarchy plays a central role in classical complexity theory.
Here, we define\r\na quantum generalization of the polynomial hierarchy, and initiate
its study. We show that\r\nnot only are there natural complete problems for the
second level of this quantum hierarchy, but that these problems are in fact hard
to approximate. Using the same techniques, we\r\nalso obtain hardness of approximation
for the class QCMA. Our approach is based on the\r\nuse of dispersers, and is
inspired by the classical results of Umans regarding hardness of approximation
for the second level of the classical polynomial hierarchy [Umans, FOCS 1999].\r\nThe
problems for which we prove hardness of approximation for include, among others,
a\r\nquantum version of the Succinct Set Cover problem, and a variant of the local
Hamiltonian\r\nproblem with hybrid classical-quantum ground states."
article_type: original
author:
- first_name: Sevag
full_name: Gharibian, Sevag
id: '71541'
last_name: Gharibian
orcid: 0000-0002-9992-3379
- first_name: Julia
full_name: Kempe, Julia
last_name: Kempe
citation:
ama: Gharibian S, Kempe J. Hardness of approximation for quantum problems. Quantum
Information & Computation. 2014;14(5-6):517-540.
apa: Gharibian, S., & Kempe, J. (2014). Hardness of approximation for quantum
problems. Quantum Information & Computation, 14(5–6), 517–540.
bibtex: '@article{Gharibian_Kempe_2014, title={Hardness of approximation for quantum
problems}, volume={14}, number={5–6}, journal={Quantum Information & Computation},
author={Gharibian, Sevag and Kempe, Julia}, year={2014}, pages={517–540} }'
chicago: 'Gharibian, Sevag, and Julia Kempe. “Hardness of Approximation for Quantum
Problems.” Quantum Information & Computation 14, no. 5–6 (2014): 517–40.'
ieee: S. Gharibian and J. Kempe, “Hardness of approximation for quantum problems,”
Quantum Information & Computation, vol. 14, no. 5–6, pp. 517–540, 2014.
mla: Gharibian, Sevag, and Julia Kempe. “Hardness of Approximation for Quantum Problems.”
Quantum Information & Computation, vol. 14, no. 5–6, 2014, pp. 517–40.
short: S. Gharibian, J. Kempe, Quantum Information & Computation 14 (2014) 517–540.
date_created: 2019-03-01T11:56:55Z
date_updated: 2023-02-28T11:02:47Z
department:
- _id: '623'
- _id: '7'
extern: '1'
external_id:
arxiv:
- '1209.1055'
intvolume: ' 14'
issue: 5-6
keyword:
- Hardness of approximation
- polynomial time hierarchy
- succinct set cover
- quantum complexity
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1209.1055
oa: '1'
page: 517-540
publication: Quantum Information & Computation
publication_status: published
status: public
title: Hardness of approximation for quantum problems
type: journal_article
user_id: '71541'
volume: 14
year: '2014'
...