3 Search Results for "Mehta, Jenish C."

Document
DOI: 10.4230/LIPIcs.ITCS.2026.25
Commuting Local Hamiltonians Beyond 2D

Authors: John Bostanci and Yeongwoo Hwang

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Commuting local Hamiltonians provide a testing ground for studying many of the most interesting open questions in quantum information theory, including the quantum PCP conjecture and the nature of entanglement. However, unlike the general local Hamiltonian problem, the exact complexity of the commuting local Hamiltonian problem (CLH) remains unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit classical verifiers. Despite intense work, proofs placing CLH in NP rely heavily on an underlying 2D lattice structure, or a very constrained local dimension and locality. In this work, we present a new technique to analyze the complexity of various families of commuting local Hamiltonians: guided reductions. Intuitively, these are a generalization of typical reduction where the prover provides a guide so that the verifier can construct a simpler Hamiltonian. The core of our reduction is a new rounding technique based on a combination of Jordan’s Lemma for pairs of projectors and the Structure Lemma for C^* algebras. Our rounding technique is much more flexible than previous work and allows us to remove constraints on local dimension in exchange for a rank-1 assumption. Using our rounding technique, we prove the following two results: 1) 2D-CLH for rank-1 instances are contained in NP, independent of the qudit dimension. It is notable that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality of the Hamiltonian terms. 2) 3D-CLH for rank-1 instances are in NP. To our knowledge this is the first time a family of {3D} commuting local Hamiltonians has been contained in NP. Our results apply to Hamiltonians with large qudit degree and remain non-trivial despite the quantum Lovász Local Lemma. [Andris Ambainis et al., 2012]
Cite as

John Bostanci and Yeongwoo Hwang. Commuting Local Hamiltonians Beyond 2D. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bostanci_et_al:LIPIcs.ITCS.2026.25,
  author =	{Bostanci, John and Hwang, Yeongwoo},
  title =	{{Commuting Local Hamiltonians Beyond 2D}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.25},
  URN =		{urn:nbn:de:0030-drops-253129},
  doi =		{10.4230/LIPIcs.ITCS.2026.25},
  annote =	{Keywords: Quantum complexity, commuting Hamiltonians, complexity theory, C* algebras}
}
Document
DOI: 10.4230/LIPIcs.TQC.2025.6
Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes

Authors: Ricardo Rivera Cardoso, Alex Meiburg, and Daniel Nagaj

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract
Previously, all known variants of the Quantum Satisfiability (QSAT) problem - consisting of determining whether a k-local (k-body) Hamiltonian is frustration-free - could be classified as being either in 𝖯; or complete for NP, MA, or QMA₁. Here, we present new qubit variants of this problem that are complete for BQP₁, coRP, QCMA, PI(coRP,NP), PI(BQP₁,NP), PI(BQP₁,MA), SoPU(coRP,NP), SoPU(BQP₁,NP), and SoPU(BQP₁,MA). Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer’s dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial BQP₁-complete problem. We first construct QSAT problems on qudits that are complete for BQP₁, coRP, and QCMA. These are made by restricting the finite set of Hamiltonians to consist of elements similar to H_{init}, H_{prop}, and H_{out}, seen in the circuit-to-Hamiltonian transformation. Usually, these are used to demonstrate hardness of QSAT and Local Hamiltonian problems, and so our proofs of hardness are simple. The difficulty lies in ensuring that all Hamiltonians generated with these three elements can be decided in their respective classes. For this, we build our Hamiltonian terms with high-dimensional data and clock qudits, ternary logic, and either monogamy of entanglement or specific clock encodings. We then show how to express these problems in terms of qubits, by proving that any QCSP can be reduced to a qubit problem while maintaining the same complexity - something not believed possible classically. The remaining six problems are obtained by considering "sums" and "products" of some of the QSAT problems mentioned here. Before this work, the QSAT problems generated in this way resulted in complete problems for PI and SoPU classes that were trivially equal to NP, MA, or QMA₁. We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for coRP, BQP₁, and QCMA, we note that those constructions are flawed. Here, we rework them, provide correct proofs, and obtain improvements on the required qudit dimensionality.
Cite as

Ricardo Rivera Cardoso, Alex Meiburg, and Daniel Nagaj. Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{riveracardoso_et_al:LIPIcs.TQC.2025.6,
  author =	{Rivera Cardoso, Ricardo and Meiburg, Alex and Nagaj, Daniel},
  title =	{{Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{6:1--6:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.6},
  URN =		{urn:nbn:de:0030-drops-240557},
  doi =		{10.4230/LIPIcs.TQC.2025.6},
  annote =	{Keywords: Quantum complexity theory, quantum satisfiability, circuit-to-Hamiltonian, pairwise union of classes, pairwise intersection of classes}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.70
Tree Tribes and Lower Bounds for Switching Lemmas

Authors: Jenish C. Mehta

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
Let f be a Boolean function on n variables, rho a random p-restriction that independently keeps each variable unset (or free) with probability p and otherwise uniformly sets it to 0 or 1, and DT_{depth}(f) denote the depth of the smallest depth decision tree for f. Let R_d(f|rho) be the resilience of f to rho for depth d, defined as R_d(f|rho)=Pr_{rho < - rho}[DT_{depth}(f|rho)>= d]. If d >> pn, all functions have resilience close to 0 since less than d variables would remain unset with high probability. For d << pn, most functions f on n variables have resilience close to 1, and some functions, like AND and OR, have resilience close to 0. Håstad's Switching Lemma states that for t-DNFs, the resilience R_d(f|rho) is upper bounded by (5pt)^d, and from known upper bounds on the size of constant depth circuits computing the parity function, it follows that there exist t-DNFs whose resilience is close to the bound obtained by Håstad. However, the exact bounds for such maximally resilient DNFs or their structure is unclear, and moreover, the argument is non-constructive. In this work, we give an explicit construction of functions called Tree Tribes parameterized by an integer t and denoted Xi_t (on n variables), such that R_d(Xi_t|rho)<=(4p2^t)^d, and more importantly, the resilience is also lower bounded by the same quantity up to constants, R_d(Xi_t|rho)>=(c_0 p2^t)^d, for 0 <= p <= c_p 2^-t and 0 <= d <= c_d * (log n)/(2^t * t log t) (where c_0,c_p,c_d are universal constants). As a result, for sufficiently large n and small d, this gives a hierarchy of functions with strictly increasing resilience, and covers the entire region between the two extremes where functions have resilience (close to) 0 or 1.
Cite as

Jenish C. Mehta. Tree Tribes and Lower Bounds for Switching Lemmas. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 70:1-70:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{mehta:LIPIcs.MFCS.2018.70,
  author =	{Mehta, Jenish C.},
  title =	{{Tree Tribes and Lower Bounds for Switching Lemmas}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{70:1--70:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.70},
  URN =		{urn:nbn:de:0030-drops-96524},
  doi =		{10.4230/LIPIcs.MFCS.2018.70},
  annote =	{Keywords: Tree Tribes, Resilience, Switching lemmas, lower bounds, decision tree}
}
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