2 Search Results for "Piddock, Stephen"

Document
DOI: 10.4230/LIPIcs.ITCS.2022.75
On Polynomially Many Queries to NP or QMA Oracles

Authors: Sevag Gharibian and Dorian Rudolph

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as P^NP and P^QMA, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes P^NP[log] and P^QMA[log], defined identically to P^NP and P^QMA, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a P^NP machine have a "query graph" which is a tree, then this computation can be simulated in P^NP[log]. In this work, we first show that for any verification class C ∈ {NP, MA, QCMA, QMA, QMA(2), NEXP, QMA_exp}, any P^C machine with a query graph of "separator number" s can be simulated using deterministic time exp(slog n) and slog n queries to a C-oracle. When s ∈ O(1) (which includes the case of O(1)-treewidth, and thus also of trees), this gives an upper bound of P^C[log], and when s ∈ O(log^k(n)), this yields bound QP^{C[log^{k+1}]} (QP meaning quasi-polynomial time). We next show how to combine Gottlob’s "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding P^C computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial p specified via an arithmetic circuit, if one can "weakly compress" p so that its optimal value requires m bits to represent, then P^NP can be decided with only m queries to an NP-oracle.
Cite as

Sevag Gharibian and Dorian Rudolph. On Polynomially Many Queries to NP or QMA Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 75:1-75:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gharibian_et_al:LIPIcs.ITCS.2022.75,
  author =	{Gharibian, Sevag and Rudolph, Dorian},
  title =	{{On Polynomially Many Queries to NP or QMA Oracles}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{75:1--75:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.75},
  URN =		{urn:nbn:de:0030-drops-156717},
  doi =		{10.4230/LIPIcs.ITCS.2022.75},
  annote =	{Keywords: admissible weighting function, oracle complexity class, quantum complexity theory, Quantum Merlin Arthur (QMA), simulation of local measurement}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.20
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians

Authors: Sevag Gharibian, Stephen Piddock, and Justin Yirka

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
The canonical hard problems for NP and its quantum analogue, Quantum Merlin-Arthur (QMA), are MAX-k-SAT and the k-local Hamiltonian problem (k-LH), the quantum generalization of MAX-k-SAT, respectively. In recent years, however, an arguably even more physically motivated problem than k-LH has been formalized - the problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM). Perhaps surprisingly, [Ambainis, CCC 2014] showed that APX-SIM is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^{QMA[log]}-complete, for P^{QMA[log]} the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^{QMA[log]}-complete even when restricted to physically motivated Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. Specifically, we first give a sequence of results which together yield P^{QMA[log]}-hardness for APX-SIM on well-motivated Hamiltonians such as the 2D Heisenberg model: - We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. Formally, P^{NP[log]}=P^{||NP}, P^{StoqMA[log]}=P^{||StoqMA}, and P^{QMA[log]}=P^{||QMA}. (The result for NP was previously shown using a different proof technique.) These equalities simplify the proofs of our subsequent results. - Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (à la [Cubitt, Montanaro, Piddock, 2017]) by studying a seemingly weaker problem, ∀-APX-SIM. As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^{||NP},P^{||StoqMA}, or P^{||QMA} depending on the Hamiltonians employed. - Leveraging the above, we show that APX-SIM is P^{QMA[log]}-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians. This implies APX-SIM is P^{QMA[log]}-complete even on physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^{QMA[log]}-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.
Cite as

Sevag Gharibian, Stephen Piddock, and Justin Yirka. Oracle Complexity Classes and Local Measurements on Physical Hamiltonians. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 20:1-20:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gharibian_et_al:LIPIcs.STACS.2020.20,
  author =	{Gharibian, Sevag and Piddock, Stephen and Yirka, Justin},
  title =	{{Oracle Complexity Classes and Local Measurements on Physical Hamiltonians}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{20:1--20:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.20},
  URN =		{urn:nbn:de:0030-drops-118818},
  doi =		{10.4230/LIPIcs.STACS.2020.20},
  annote =	{Keywords: Quantum Merlin Arthur (QMA), simulation of local measurement, local Hamiltonian, oracle complexity class, physical Hamiltonians}
}
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