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

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.

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.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} }

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

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.

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.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} }

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

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.

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.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} }

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

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.

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.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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