7 Search Results for "Aharonov, Dorit"

Document
DOI: 10.4230/LIPIcs.CCC.2023.23
Translationally Invariant Constraint Optimization Problems

Authors: Dorit Aharonov and Sandy Irani

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the computational complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class FP^NEXP. Gottesman and Irani (FOCS 2009) also studied classical constraint satisfaction problems using this strong notion of translational-invariance; they show that the problem of deciding whether the cost of the optimal assignment is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel’s famous result from 1988, showing that the function version of SAT is complete for the class FP^NP. An essential ingredient in the proof is a study of the computational complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of Ω(N^(1/4)), where the grid size is N × N. To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. This might be of independent interest. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for P^NEXP.
Cite as

Dorit Aharonov and Sandy Irani. Translationally Invariant Constraint Optimization Problems. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{aharonov_et_al:LIPIcs.CCC.2023.23,
  author =	{Aharonov, Dorit and Irani, Sandy},
  title =	{{Translationally Invariant Constraint Optimization Problems}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.23},
  URN =		{urn:nbn:de:0030-drops-182932},
  doi =		{10.4230/LIPIcs.CCC.2023.23},
  annote =	{Keywords: Constraint satisfaction, Tiling, Translational-invariance}
}
Document
DOI: 10.4230/LIPIcs.TQC.2021.4
StoqMA Meets Distribution Testing

Authors: Yupan Liu

Published in: LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

Abstract
StoqMA captures the computational hardness of approximating the ground energy of local Hamiltonians that do not suffer the so-called sign problem. We provide a novel connection between StoqMA and distribution testing via reversible circuits. First, we prove that easy-witness StoqMA (viz. eStoqMA, a sub-class of StoqMA) is contained in MA. Easy witness is a generalization of a subset state such that the associated set’s membership can be efficiently verifiable, and all non-zero coordinates are not necessarily uniform. This sub-class eStoqMA contains StoqMA with perfect completeness (StoqMA₁), which further signifies a simplified proof for StoqMA₁ ⊆ MA [Bravyi et al., 2006; Bravyi and Terhal, 2010]. Second, by showing distinguishing reversible circuits with ancillary random bits is StoqMA-complete (as a comparison, distinguishing quantum circuits is QMA-complete [Janzing et al., 2005]), we construct soundness error reduction of StoqMA. Additionally, we show that both variants of StoqMA that without any ancillary random bit and with perfect soundness are contained in NP. Our results make a step towards collapsing the hierarchy MA ⊆ StoqMA ⊆ SBP [Bravyi et al., 2006], in which all classes are contained in AM and collapse to NP under derandomization assumptions.
Cite as

Yupan Liu. StoqMA Meets Distribution Testing. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{liu:LIPIcs.TQC.2021.4,
  author =	{Liu, Yupan},
  title =	{{StoqMA Meets Distribution Testing}},
  booktitle =	{16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)},
  pages =	{4:1--4:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-198-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{197},
  editor =	{Hsieh, Min-Hsiu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.4},
  URN =		{urn:nbn:de:0030-drops-139995},
  doi =		{10.4230/LIPIcs.TQC.2021.4},
  annote =	{Keywords: StoqMA, distribution testing, error reduction, reversible circuits}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.36
Two Combinatorial MA-Complete Problems

Authors: Dorit Aharonov and Alex B. Grilo

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Despite the interest in the complexity class MA, the randomized analog of NP, there are just a few known natural (promise-)MA-complete problems. The first such problem was found by Bravyi and Terhal (SIAM Journal of Computing 2009); this result was then followed by Crosson, Bacon and Brown (PRE 2010) and then by Bravyi (Quantum Information and Computation 2015). Surprisingly, each of these problems is either from or inspired by quantum computation. This fact makes it hard for classical complexity theorists to study these problems, and prevents potential progress, e.g., on the important question of derandomizing MA. In this note we define two new natural combinatorial problems and we prove their MA-completeness. The first problem, that we call approximately-clean approximate-connected-component (ACAC), gets as input a succinctly described graph, some of whose vertices are marked. The problem is to decide whether there is a connected component whose vertices are all unmarked, or the graph is far from having this property. The second problem, called SetCSP, generalizes in a novel way the standard constraint satisfaction problem (CSP) into constraints involving sets of strings. Technically, our proof that SetCSP is MA-complete is a fleshing out of an observation made in (Aharonov and Grilo, FOCS 2019), where it was noted that a restricted case of Bravyi and Terhal’s MA complete problem (namely, the uniform case) is already MA complete; and, moreover, that this restricted case can be stated using classical, combinatorial language. The fact that the first, arguably more natural, problem of ACAC is MA-hard follows quite naturally from this proof as well; while containment of ACAC in MA is simple, based on the theory of random walks. We notice that this work, along with a translation of the main result of Aharonov and Grilo to the SetCSP problem, implies that finding a gap-amplification procedure for SetCSP (in the spirit of the gap-amplification procedure introduced in Dinur’s PCP proof) would imply MA=NP. In fact, the problem of finding gap-amplification for SetCSP is equivalent to the MA=NP problem. This provides an alternative new path towards the major problem of derandomizing MA. Deriving a similar statement regarding gap amplification of a natural restriction of $ACAC$ remains an open question.
Cite as

Dorit Aharonov and Alex B. Grilo. Two Combinatorial MA-Complete Problems. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{aharonov_et_al:LIPIcs.ITCS.2021.36,
  author =	{Aharonov, Dorit and Grilo, Alex B.},
  title =	{{Two Combinatorial MA-Complete Problems}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.36},
  URN =		{urn:nbn:de:0030-drops-135754},
  doi =		{10.4230/LIPIcs.ITCS.2021.36},
  annote =	{Keywords: Merlin-Arthur proof systems, Constraint sastifation problem, Random walks}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.49
Robust Quantum Entanglement at (Nearly) Room Temperature

Authors: Lior Eldar

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We formulate an average-case analog of the NLTS conjecture of Freedman and Hastings (QIC 2014) by asking whether there exist topologically ordered systems with corresponding local Hamiltonians for which the thermal Gibbs state for constant temperature cannot even be approximated by shallow quantum circuits. We then prove this conjecture for nearly optimal parameters: we construct a quantum error correcting code whose corresponding (log) local Hamiltonian has the following property: for nearly constant temperature (temperature decays as 1/log²log(n)) the thermal Gibbs state of that Hamiltonian cannot be approximated by any circuit of depth less than log(n), and it is highly entangled in a well-defined way. This implies that appropriately chosen local Hamiltonians can give rise to ground-state long-range entanglement which can survive without active error correction at temperatures which are nearly independent of the system size: thereby improving exponentially upon previously known bounds.
Cite as

Lior Eldar. Robust Quantum Entanglement at (Nearly) Room Temperature. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 49:1-49:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{eldar:LIPIcs.ITCS.2021.49,
  author =	{Eldar, Lior},
  title =	{{Robust Quantum Entanglement at (Nearly) Room Temperature}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{49:1--49:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.49},
  URN =		{urn:nbn:de:0030-drops-135886},
  doi =		{10.4230/LIPIcs.ITCS.2021.49},
  annote =	{Keywords: Quantum error-correcting codes, Quantum Entanglement, Quantum Locally-Testable Codes, Local Hamiltonians, quantum PCP, NLTS}
}
Document
Abstract
DOI: 10.4230/LIPIcs.ITCS.2020.63
Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality (Abstract)

Authors: Adam Bouland, Bill Fefferman, and Umesh Vazirani

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
The AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics, a fundamental goal of physics. It posits a duality between a gravitational theory in Anti de Sitter (AdS) space and a quantum mechanical conformal field theory (CFT), embodied in a map known as the AdS/CFT dictionary mapping states to states and operators to operators. This dictionary map is not well understood and has only been computed on special, structured instances. In this work we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity. Our argument has its origins in a fundamental paradox in the AdS/CFT correspondence known as the wormhole growth paradox. The paradox is that the CFT is believed to be "scrambling" - i.e. the expectation value of local operators equilibrates in polynomial time - whereas the gravity theory is not, because the interiors of certain black holes known as "wormholes" do not equilibrate and instead their volume grows at a linear rate for at least an exponential amount of time. So what could be the CFT dual to wormhole volume? Susskind’s proposed resolution was to equate the wormhole volume with the quantum circuit complexity of the CFT state. From a computer science perspective, circuit complexity seems like an unusual choice because it should be difficult to compute, in contrast to physical quantities such as wormhole volume. We show how to create pseudorandom quantum states in the CFT, thereby arguing that their quantum circuit complexity is not "feelable", in the sense that it cannot be approximated by any efficient experiment. This requires a specialized construction inspired by symmetric block ciphers such as DES and AES, since unfortunately existing constructions based on quantum-resistant one way functions cannot be used in the context of the wormhole growth paradox as only very restricted operations are allowed in the CFT. By contrast we argue that the wormhole volume is "feelable" in some general but non-physical sense. The duality between a "feelable" quantity and an "unfeelable" quantity implies that some aspect of this duality must have exponential complexity. More precisely, it implies that either the dictionary is exponentially complex, or else the quantum gravity theory is exponentially difficult to simulate on a quantum computer. While at first sight this might seem to justify the discomfort of complexity theorists with equating computational complexity with a physical quantity, a further examination of our arguments shows that any resolution of the wormhole growth paradox must equate wormhole volume to an "unfeelable" quantity, leading to the same conclusions. In other words this discomfort is an inevitable consequence of the paradox.
Cite as

Adam Bouland, Bill Fefferman, and Umesh Vazirani. Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality (Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 63:1-63:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bouland_et_al:LIPIcs.ITCS.2020.63,
  author =	{Bouland, Adam and Fefferman, Bill and Vazirani, Umesh},
  title =	{{Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{63:1--63:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.63},
  URN =		{urn:nbn:de:0030-drops-117486},
  doi =		{10.4230/LIPIcs.ITCS.2020.63},
  annote =	{Keywords: Quantum complexity theory, pseudorandomness, AdS/CFT correspondence}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.2
Hamiltonian Sparsification and Gap-Simulation

Authors: Dorit Aharonov and Leo Zhou

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
Analog quantum simulation - simulation of one Hamiltonian by another - is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian H~ whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian H (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where H~ is only required to simulate the groundstate(s) and spectral gap of H instead of its full spectrum, and we believe it is of independent interest. Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the "first questions" one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves as a good starting point for future research of these topics.
Cite as

Dorit Aharonov and Leo Zhou. Hamiltonian Sparsification and Gap-Simulation. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{aharonov_et_al:LIPIcs.ITCS.2019.2,
  author =	{Aharonov, Dorit and Zhou, Leo},
  title =	{{Hamiltonian Sparsification and Gap-Simulation}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.2},
  URN =		{urn:nbn:de:0030-drops-100956},
  doi =		{10.4230/LIPIcs.ITCS.2019.2},
  annote =	{Keywords: quantum simulation, quantum Hamiltonian complexity, sparsification, quantum PCP}
}
Document
DOI: 10.4230/LIPIcs.TQC.2018.2
On the Complexity of Two Dimensional Commuting Local Hamiltonians

Authors: Dorit Aharonov, Oded Kenneth, and Itamar Vigdorovich

Published in: LIPIcs, Volume 111, 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

Abstract
The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice, but this problem too remained open apart from some very specific cases [Aharonov and Eldar, 2011; Hastings, 2012; Schuch, 2011]. Here we consider a wide class of two dimensional CLH instances; these are k-local CLHs, for any constant k; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [Schuch, 2011] which regarded the special case of 4-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.
Cite as

Dorit Aharonov, Oded Kenneth, and Itamar Vigdorovich. On the Complexity of Two Dimensional Commuting Local Hamiltonians. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{aharonov_et_al:LIPIcs.TQC.2018.2,
  author =	{Aharonov, Dorit and Kenneth, Oded and Vigdorovich, Itamar},
  title =	{{On the Complexity of Two Dimensional Commuting Local Hamiltonians}},
  booktitle =	{13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-080-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{111},
  editor =	{Jeffery, Stacey},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.2},
  URN =		{urn:nbn:de:0030-drops-92498},
  doi =		{10.4230/LIPIcs.TQC.2018.2},
  annote =	{Keywords: local Hamiltonian complexity, commuting Hamiltonians, local Hamiltonian problem, trivial states, toric code, ground states, quantum NP, QMA, topological order, multiparticle entanglement, logical operators, ribbon}
}
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