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Documents authored by Arad, Itai

Document
DOI: 10.4230/LIPIcs.ITCS.2017.46
Rigorous Rg Algorithms and Area Laws for Low Energy Eigenstates In 1D

Authors: Itai Arad, Zeph Landau, Umesh V. Vazirani, and Thomas Vidick

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance by Landau et al. gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unresolved, including whether there exist rigorous efficient algorithms when the ground space is degenerate (and poly(n) dimensional), or for the poly(n) lowest energy states for 1D systems, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm for finding low energy states for 1D systems, based on a rigorously justified renormalization group (RG)-type transformation. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n^{O(\log n)} algorithm for the poly(n) lowest energy states for 1D systems (under a mild density condition). We note that for these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is O(nM(n)), where M(n) is the time required to multiply two n by n matrices.
Cite as

Itai Arad, Zeph Landau, Umesh V. Vazirani, and Thomas Vidick. Rigorous Rg Algorithms and Area Laws for Low Energy Eigenstates In 1D. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arad_et_al:LIPIcs.ITCS.2017.46,
  author =	{Arad, Itai and Landau, Zeph and Vazirani, Umesh V. and Vidick, Thomas},
  title =	{{Rigorous Rg Algorithms and Area Laws for Low Energy Eigenstates In 1D}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.46},
  URN =		{urn:nbn:de:0030-drops-81920},
  doi =		{10.4230/LIPIcs.ITCS.2017.46},
  annote =	{Keywords: Hamiltonian complexity, area law, gapped ground states, algorithm}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.15
Linear Time Algorithm for Quantum 2SAT

Authors: Itai Arad, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Q_{ij} on a system of n qubits, and the task is to decide whether the Hamiltonian H = sum Q_{ij} has a 0-eigenvalue, or it is larger than 1/n^c for some c = O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin 1/2, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is O(n^4). In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.
Cite as

Itai Arad, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. Linear Time Algorithm for Quantum 2SAT. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.ICALP.2016.15,
  author =	{Arad, Itai and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu},
  title =	{{Linear Time Algorithm for Quantum 2SAT}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.15},
  URN =		{urn:nbn:de:0030-drops-62795},
  doi =		{10.4230/LIPIcs.ICALP.2016.15},
  annote =	{Keywords: Quantum SAT, Davis-Putnam Procedure, Linear Time Algorithm}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.12
On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems

Authors: Itai Arad, Adam Bouland, Daniel Grier, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC'13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.
Cite as

Itai Arad, Adam Bouland, Daniel Grier, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.MFCS.2016.12,
  author =	{Arad, Itai and Bouland, Adam and Grier, Daniel and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu},
  title =	{{On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{12:1--12:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.12},
  URN =		{urn:nbn:de:0030-drops-64284},
  doi =		{10.4230/LIPIcs.MFCS.2016.12},
  annote =	{Keywords: computational complexity, satisfiability problems, trial and error, quantum computing, learning theory}
}
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