22 Search Results for "Gharibian, Sevag"

Document
DOI: 10.4230/LIPIcs.TQC.2024.10
Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture

Authors: Jordi Weggemans, Marten Folkertsma, and Chris Cade

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
We study "Merlinized" versions of the recently defined Guided Local Hamiltonian problem, which we call "Guidable Local Hamiltonian" problems. Unlike their guided counterparts, these problems do not have a guiding state provided as a part of the input, but merely come with the promise that one exists. We consider in particular two classes of guiding states: those that can be prepared efficiently by a quantum circuit; and those belonging to a class of quantum states we call classically evaluatable, for which it is possible to efficiently compute expectation values of local observables classically. We show that guidable local Hamiltonian problems for both classes of guiding states are QCMA-complete in the inverse-polynomial precision setting, but lie within NP (or NqP) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ansätze selected by classical heuristics are just as powerful as quantum Ansätze prepared by quantum heuristics, as long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a complexity class capturing quantum-classical probabilistically checkable proof systems and show that it is contained in BQP^NP[1] for constant proof queries; (ii) give a no-go result on "dequantizing" the known quantum reduction which maps a QPCP-verification circuit to a local Hamiltonian with constant promise gap; (iii) give several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iv) propose two conjectures that can be viewed as stronger versions of the NLTS theorem. Finally, we show that many of our results can be directly modified to obtain similar results for the class MA.
Cite as

Jordi Weggemans, Marten Folkertsma, and Chris Cade. Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{weggemans_et_al:LIPIcs.TQC.2024.10,
  author =	{Weggemans, Jordi and Folkertsma, Marten and Cade, Chris},
  title =	{{Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.10},
  URN =		{urn:nbn:de:0030-drops-206804},
  doi =		{10.4230/LIPIcs.TQC.2024.10},
  annote =	{Keywords: Quantum complexity theory, local Hamiltonian problem, quantum state ansatzes, QCMA, quantum PCP conjecture}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.7
Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds

Authors: Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
The Polynomial-Time Hierarchy (PH) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to "quantum advantage" analyses for near-term quantum computers. Quantumly, however, despite the fact that at least four definitions of quantum PH exist, it has been challenging to prove analogues for these of even basic facts from PH. This work studies three quantum-verifier based generalizations of PH, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings (QCPH) and quantum mixed states (QPH) as proofs, and one of which is new to this work, utilizing quantum pure states (QPHpure) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for QCPH. Then, for our new class QPHpure, we show one-sided error reduction QPHpure, as well as the first bounds relating these quantum variants of PH, namely QCPH ⊆ QPHpure ⊆ EXP^PP.
Cite as

Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph. Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agarwal_et_al:LIPIcs.MFCS.2024.7,
  author =	{Agarwal, Avantika and Gharibian, Sevag and Koppula, Venkata and Rudolph, Dorian},
  title =	{{Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.7},
  URN =		{urn:nbn:de:0030-drops-205632},
  doi =		{10.4230/LIPIcs.MFCS.2024.7},
  annote =	{Keywords: Quantum complexity, polynomial hierarchy}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.6
The Entangled Quantum Polynomial Hierarchy Collapses

Authors: Sabee Grewal and Justin Yirka

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We introduce the entangled quantum polynomial hierarchy, QEPH, as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove QEPH collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, QEPH = QRG(1), the class of problems having one-turn quantum refereed games, which is known to be contained in PSPACE. This is in contrast to the unentangled quantum polynomial hierarchy, QPH, which contains QMA(2). We also introduce DistributionQCPH, a generalization of the quantum-classical polynomial hierarchy QCPH where the provers send probability distributions over strings (instead of strings). We prove DistributionQCPH = QCPH, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that, without loss of generality, the provers can send uniform distributions over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., DistributionPH = PH. Finally, we show that PH and QCPH are contained in QPH, resolving an open question of Gharibian et al. (2022).
Cite as

Sabee Grewal and Justin Yirka. The Entangled Quantum Polynomial Hierarchy Collapses. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{grewal_et_al:LIPIcs.CCC.2024.6,
  author =	{Grewal, Sabee and Yirka, Justin},
  title =	{{The Entangled Quantum Polynomial Hierarchy Collapses}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{6:1--6:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.6},
  URN =		{urn:nbn:de:0030-drops-204028},
  doi =		{10.4230/LIPIcs.CCC.2024.6},
  annote =	{Keywords: Polynomial hierarchy, Entangled proofs, Correlated proofs, Minimax}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.70
BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting

Authors: Sevag Gharibian and Jonas Kamminga

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log n) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(log n) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(log n) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(log n) NP queries, then BPP^NP[o(n)] contains a 𝖯^NP-complete problem if the algorithm is classical and FBQP^NP[o(n)] contains an FP^NP-complete problem if the algorithm is quantum.
Cite as

Sevag Gharibian and Jonas Kamminga. BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 70:1-70:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gharibian_et_al:LIPIcs.ICALP.2024.70,
  author =	{Gharibian, Sevag and Kamminga, Jonas},
  title =	{{BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{70:1--70:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.70},
  URN =		{urn:nbn:de:0030-drops-202134},
  doi =		{10.4230/LIPIcs.ICALP.2024.70},
  annote =	{Keywords: Approximate Counting, Search to Decision Reduction, BQP, NP, Oracle Complexity Class}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.105
An Improved Quantum Max Cut Approximation via Maximum Matching

Authors: Eunou Lee and Ojas Parekh

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Finding a high (or low) energy state of a given quantum Hamiltonian is a potential area to gain a provable and practical quantum advantage. A line of recent studies focuses on Quantum Max Cut, where one is asked to find a high energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work, we present a classical approximation algorithm for Quantum Max Cut that achieves an approximation ratio of 0.595, outperforming the previous best algorithms of Lee [Eunou Lee, 2022] (0.562, generic input graph) and King [King, 2023] (0.582, triangle-free input graph). The algorithm is based on finding the maximum weighted matching of an input graph and outputs a product of at most 2-qubit states, which is simpler than the fully entangled output states of the previous best algorithms.
Cite as

Eunou Lee and Ojas Parekh. An Improved Quantum Max Cut Approximation via Maximum Matching. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 105:1-105:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lee_et_al:LIPIcs.ICALP.2024.105,
  author =	{Lee, Eunou and Parekh, Ojas},
  title =	{{An Improved Quantum Max Cut Approximation via Maximum Matching}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{105:1--105:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.105},
  URN =		{urn:nbn:de:0030-drops-202482},
  doi =		{10.4230/LIPIcs.ICALP.2024.105},
  annote =	{Keywords: approximation, optimization, local Hamiltonian, rounding, SDP, matching}
}
Document
DOI: 10.4230/LIPIcs.TQC.2023.14
Local Hamiltonians with No Low-Energy Stabilizer States

Authors: Nolan J. Coble, Matthew Coudron, Jon Nelson, and Seyed Sajjad Nezhadi

Published in: LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

Abstract
The recently-defined No Low-energy Sampleable States (NLSS) conjecture of Gharibian and Le Gall [Sevag Gharibian and François {Le Gall}, 2022] posits the existence of a family of local Hamiltonians where all states of low-enough constant energy do not have succinct representations allowing perfect sampling access. States that can be prepared using only Clifford gates (i.e. stabilizer states) are an example of sampleable states, so the NLSS conjecture implies the existence of local Hamiltonians whose low-energy space contains no stabilizer states. We describe families that exhibit this requisite property via a simple alteration to local Hamiltonians corresponding to CSS codes. Our method can also be applied to the recent NLTS Hamiltonians of Anshu, Breuckmann, and Nirkhe [Anshu et al., 2022], resulting in a family of local Hamiltonians whose low-energy space contains neither stabilizer states nor trivial states. We hope that our techniques will eventually be helpful for constructing Hamiltonians which simultaneously satisfy NLSS and NLTS.
Cite as

Nolan J. Coble, Matthew Coudron, Jon Nelson, and Seyed Sajjad Nezhadi. Local Hamiltonians with No Low-Energy Stabilizer States. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{coble_et_al:LIPIcs.TQC.2023.14,
  author =	{Coble, Nolan J. and Coudron, Matthew and Nelson, Jon and Nezhadi, Seyed Sajjad},
  title =	{{Local Hamiltonians with No Low-Energy Stabilizer States}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-283-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{266},
  editor =	{Fawzi, Omar and Walter, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.14},
  URN =		{urn:nbn:de:0030-drops-183243},
  doi =		{10.4230/LIPIcs.TQC.2023.14},
  annote =	{Keywords: Hamiltonian complexity, Stabilizer codes, Low-energy states}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.34
The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate

Authors: Lennart Bittel, Sevag Gharibian, and Martin Kliesch

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

Abstract
Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the depth of the variational ansatz used - the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant ε > 0, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor N^(1-ε), for N denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even "simpler" QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.
Cite as

Lennart Bittel, Sevag Gharibian, and Martin Kliesch. The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 34:1-34:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bittel_et_al:LIPIcs.CCC.2023.34,
  author =	{Bittel, Lennart and Gharibian, Sevag and Kliesch, Martin},
  title =	{{The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{34:1--34:24},
  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.2023.34},
  URN =		{urn:nbn:de:0030-drops-183045},
  doi =		{10.4230/LIPIcs.CCC.2023.34},
  annote =	{Keywords: Variational quantum algorithms (VQA), Quantum Approximate Optimization Algorithm (QAOA), circuit depth minimization, Quantum-Classical Merlin-Arthur (QCMA), hardness of approximation, hybrid quantum algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.32
Improved Hardness Results for the Guided Local Hamiltonian Problem

Authors: Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.
Cite as

Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans. Improved Hardness Results for the Guided Local Hamiltonian Problem. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cade_et_al:LIPIcs.ICALP.2023.32,
  author =	{Cade, Chris and Folkertsma, Marten and Gharibian, Sevag and Hayakawa, Ryu and Le Gall, Fran\c{c}ois and Morimae, Tomoyuki and Weggemans, Jordi},
  title =	{{Improved Hardness Results for the Guided Local Hamiltonian Problem}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.32},
  URN =		{urn:nbn:de:0030-drops-180840},
  doi =		{10.4230/LIPIcs.ICALP.2023.32},
  annote =	{Keywords: Quantum computing, Quantum advantage, Quantum Chemistry, Guided Local Hamiltonian Problem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.54
The Complexity of Translationally Invariant Problems Beyond Ground State Energies

Authors: James D. Watson, Johannes Bausch, and Sevag Gharibian

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
The physically motivated quantum generalisation of k-SAT, the k-Local Hamiltonian (k-LH) problem, is well-known to be QMA-complete ("quantum NP"-complete). What is surprising, however, is that while the former is easy on 1D Boolean formulae, the latter remains hard on 1D local Hamiltonians, even if all constraints are identical [Gottesman, Irani, FOCS 2009]. Such "translation-invariant" systems are much closer in structure to what one might see in Nature. Moving beyond k-LH, what is often more physically interesting is the computation of properties of the ground space (i.e. "solution space") itself. In this work, we focus on two such recent problems: Simulating local measurements on the ground space (APX-SIM, analogous to computing properties of optimal solutions to MAX-SAT formulae) [Ambainis, CCC 2014], and deciding if the low energy space has an energy barrier (GSCON, analogous to classical reconfiguration problems) [Gharibian, Sikora, ICALP 2015]. These problems are known to be P^{QMA[log]}- and QCMA-complete, respectively, in the general case. Yet, to date, it is not known whether they remain hard in such simple 1D translationally invariant systems. In this work, we show that the 1D translationally invariant versions of both APX-SIM and GSCON are intractable, namely are P^{QMA_{EXP}}- and QCMA^{EXP}-complete ("quantum P^{NEXP}" and "quantum NEXP"), respectively. Each of these results is attained by giving a respective generic "lifting theorem". For APX-SIM we give a framework for lifting any abstract local circuit-to-Hamiltonian mapping H satisfying mild assumptions to hardness of APX-SIM on the family of Hamiltonians produced by H, while preserving the structural properties of H (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we compress the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we show strong robustness, i.e. soundness even against adversaries acting on all but a single qudit in the system.
Cite as

James D. Watson, Johannes Bausch, and Sevag Gharibian. The Complexity of Translationally Invariant Problems Beyond Ground State Energies. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 54:1-54:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{watson_et_al:LIPIcs.STACS.2023.54,
  author =	{Watson, James D. and Bausch, Johannes and Gharibian, Sevag},
  title =	{{The Complexity of Translationally Invariant Problems Beyond Ground State Energies}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{54:1--54:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.54},
  URN =		{urn:nbn:de:0030-drops-177065},
  doi =		{10.4230/LIPIcs.STACS.2023.54},
  annote =	{Keywords: Complexity, Quantum Computing, Physics, Constraint Satisfaction, Combinatorial Reconfiguration, Many-Body Physics}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.53
Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement

Authors: Sevag Gharibian and Dorian Rudolph

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
A celebrated result in classical complexity theory is Savitch’s theorem, which states that non-deterministic polynomial-space computations (NPSPACE) can be simulated by deterministic poly-space computations (PSPACE). In this work, we initiate the study of a quantum analogue of NPSPACE, denoted Streaming-QCMASPACE (SQCMASPACE), in which an exponentially long classical proof is streamed to a poly-space quantum verifier. We first show that a quantum analogue of Savitch’s theorem is unlikely to hold, in that SQCMASPACE = NEXP. For completeness, we also introduce the companion class Streaming-QMASPACE (SQMASPACE) with an exponentially long streamed quantum proof, and show SQMASPACE = QMAEXP (the quantum analogue of NEXP). Our primary focus, however, is on the study of exponentially long streaming classical proofs, where we next show the following two main results. The first result shows that, in strong contrast to the classical setting, the solution space of a quantum constraint satisfaction problem (i.e. a local Hamiltonian) is always connected when exponentially long proofs are permitted. For this, we show how to simulate any Lipschitz continuous path on the unit hypersphere via a sequence of local unitary gates, at the expense of blowing up the circuit size. This shows that quantum error-correcting codes can be unable to detect one codeword erroneously evolving to another if the evolution happens sufficiently slowly, and answers an open question of [Gharibian, Sikora, ICALP 2015] regarding the Ground State Connectivity problem. Our second main result is that any SQCMASPACE computation can be embedded into "unentanglement", i.e. into a quantum constraint satisfaction problem with unentangled provers. Formally, we show how to embed SQCMASPACE into the Sparse Separable Hamiltonian problem of [Chailloux, Sattath, CCC 2012] (QMA(2)-complete for 1/poly promise gap), at the expense of scaling the promise gap with the streamed proof size. As a corollary, we obtain the first systematic construction for obtaining QMA(2)-type upper bounds on arbitrary multi-prover interactive proof systems, where the QMA(2) promise gap scales exponentially with the number of bits of communication in the interactive proof. Our construction uses a new technique for exploiting unentanglement to simulate quadratic Boolean functions, which in some sense allows history states to encode the future.
Cite as

Sevag Gharibian and Dorian Rudolph. Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 53:1-53:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gharibian_et_al:LIPIcs.ITCS.2023.53,
  author =	{Gharibian, Sevag and Rudolph, Dorian},
  title =	{{Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{53:1--53:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.53},
  URN =		{urn:nbn:de:0030-drops-175564},
  doi =		{10.4230/LIPIcs.ITCS.2023.53},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), QMA(2), ground state connectivity (GSCON), quantum error correction}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.54
The Importance of the Spectral Gap in Estimating Ground-State Energies

Authors: Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman

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

Abstract
The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the Local Hamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class QMA, a quantum generalization of the class NP. A major challenge in the field is to understand the complexity of the Local Hamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamiltonian in many-body physics is the spectral gap, which is the difference between the smallest two eigenvalues. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of the Local Hamiltonian problem is less well-understood. In this work, we make progress on this question by considering the precise regime, in which one estimates the ground-state energy to within inverse exponential precision. Computing ground-state energies precisely is a task that is important for quantum chemistry and quantum many-body physics. In the setting of inverse-exponential precision (promise gap), there is a surprising result that the complexity of Local Hamiltonian is magnified from QMA to PSPACE, the class of problems solvable in polynomial space (but possibly exponential time). We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high precision case only comes about when the spectral gap is exponentially small. As a consequence of the proof techniques developed to show our results, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection.
Cite as

Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman. The Importance of the Spectral Gap in Estimating Ground-State Energies. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 54:1-54:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{deshpande_et_al:LIPIcs.ITCS.2022.54,
  author =	{Deshpande, Abhinav and Gorshkov, Alexey V. and Fefferman, Bill},
  title =	{{The Importance of the Spectral Gap in Estimating Ground-State Energies}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{54:1--54:6},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.54},
  URN =		{urn:nbn:de:0030-drops-156501},
  doi =		{10.4230/LIPIcs.ITCS.2022.54},
  annote =	{Keywords: Local Hamiltonian problem, PSPACE, PP, QMA}
}
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.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/DagRep.11.5.76
Quantum Complexity: Theory and Application (Dagstuhl Seminar 21261)

Authors: Bill Fefferman, Sevag Gharibian, Norbert Schuch, and Barbara Terhal

Published in: Dagstuhl Reports, Volume 11, Issue 5 (2021)

Abstract
This report documents the program and outcomes of Dagstuhl Seminar 21261 "Quantum Complexity: Theory and Application". The seminar ran from June 27 to July 2 , 2021, and was held in a hybrid format (due to COVID travel restrictions). Of the 55 total participants from 14 countries, 17 participants were on-site, and 38 were remote. Recent advances in both theoretic and experimental aspects of quantum complexity theory were presented and discussed, ranging from new theoretical developments via a "Quantum Strong Exponential Time Hypothesis", to more experimentally oriented talks involving benchmarking of random circuits in quantum supremacy experiments. In addition, an open problem session and a discussion session regarding the current state of the field were included.
Cite as

Bill Fefferman, Sevag Gharibian, Norbert Schuch, and Barbara Terhal. Quantum Complexity: Theory and Application (Dagstuhl Seminar 21261). In Dagstuhl Reports, Volume 11, Issue 5, pp. 76-88, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Article{fefferman_et_al:DagRep.11.5.76,
  author =	{Fefferman, Bill and Gharibian, Sevag and Schuch, Norbert and Terhal, Barbara},
  title =	{{Quantum Complexity: Theory and Application (Dagstuhl Seminar 21261)}},
  pages =	{76--88},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2021},
  volume =	{11},
  number =	{5},
  editor =	{Fefferman, Bill and Gharibian, Sevag and Schuch, Norbert and Terhal, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.11.5.76},
  URN =		{urn:nbn:de:0030-drops-155715},
  doi =		{10.4230/DagRep.11.5.76},
  annote =	{Keywords: complexity theory, many-body systems, proof and verification systems, quantum computation, quantum supremacy}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.59
An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem

Authors: Sean Hallgren, Eunou Lee, and Ojas Parekh

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
We present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor.
Cite as

Sean Hallgren, Eunou Lee, and Ojas Parekh. An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hallgren_et_al:LIPIcs.APPROX/RANDOM.2020.59,
  author =	{Hallgren, Sean and Lee, Eunou and Parekh, Ojas},
  title =	{{An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{59:1--59:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.59},
  URN =		{urn:nbn:de:0030-drops-126629},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.59},
  annote =	{Keywords: approximation algorithm, quantum computing, local Hamiltonian, mean-field theory, randomized rounding}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.6
Towards Quantum One-Time Memories from Stateless Hardware

Authors: Anne Broadbent, Sevag Gharibian, and Hong-Sheng Zhou

Published in: LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Abstract
A central tenet of theoretical cryptography is the study of the minimal assumptions required to implement a given cryptographic primitive. One such primitive is the one-time memory (OTM), introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008], which is a classical functionality modeled after a non-interactive 1-out-of-2 oblivious transfer, and which is complete for one-time classical and quantum programs. It is known that secure OTMs do not exist in the standard model in both the classical and quantum settings. Here, we propose a scheme for using quantum information, together with the assumption of stateless (i.e., reusable) hardware tokens, to build statistically secure OTMs. Via the semidefinite programming-based quantum games framework of Gutoski and Watrous [STOC 2007], we prove security for a malicious receiver, against a linear number of adaptive queries to the token, in the quantum universal composability framework, but leave open the question of security against a polynomial amount of queries. Compared to alternative schemes derived from the literature on quantum money, our scheme is technologically simple since it is of the "prepare-and-measure" type. We also show our scheme is "tight" according to two scenarios.
Cite as

Anne Broadbent, Sevag Gharibian, and Hong-Sheng Zhou. Towards Quantum One-Time Memories from Stateless Hardware. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 6:1-6:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{broadbent_et_al:LIPIcs.TQC.2020.6,
  author =	{Broadbent, Anne and Gharibian, Sevag and Zhou, Hong-Sheng},
  title =	{{Towards Quantum One-Time Memories from Stateless Hardware}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{6:1--6:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Flammia, Steven T.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.6},
  URN =		{urn:nbn:de:0030-drops-120654},
  doi =		{10.4230/LIPIcs.TQC.2020.6},
  annote =	{Keywords: quantum cryptography, one-time memories, semi-definite programming}
}
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