11 Search Results for "Liu, Yupan"

Document
DOI: 10.4230/LIPIcs.STACS.2026.66
Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds

Authors: Yupan Liu

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We investigate the computational hardness of estimating the quantum α-Rényi entropy S^𝚁_α(ρ) = (ln Tr(ρ^α))/(1-α) and the quantum q-Tsallis entropy S^𝚃_q(ρ) = (1-Tr(ρ^q))/(q-1), both converging to the von Neumann entropy as the order approaches 1. The promise problems Quantum α-Rényi Entropy Approximation (RényiQEA_α) and Quantum q-Tsallis Entropy Approximation (TsallisQEA_q) ask whether S^𝚁_α(ρ) or S^𝚃_q(ρ), respectively, is at least τ_Y or at most τ_N, where τ_Y - τ_N is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order 1) and some cases of the quantum q-Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real orders, the rank-2 variants Rank2RényiQEA_α and Rank2TsallisQEA_q are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), our results imply: - For all real order α > 0 and 0 < q ≤ 1, LowRankRényiQEA_α and LowRankTsallisQEA_q are BQP-complete, where both are restricted versions of RényiQEA_α and TsallisQEA_q with ρ of polynomial rank. - For all real order q > 1, TsallisQEA_q is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the α-Rényi or q-Tsallis binary entropies of different orders, where the reductions differ substantially from previous approaches, and the inequalities are also of independent interest.
Cite as

Yupan Liu. Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 66:1-66:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{liu:LIPIcs.STACS.2026.66,
  author =	{Liu, Yupan},
  title =	{{Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{66:1--66:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.66},
  URN =		{urn:nbn:de:0030-drops-255550},
  doi =		{10.4230/LIPIcs.STACS.2026.66},
  annote =	{Keywords: computational hardness, quantum state testing, quantum R\'{e}nyi entropy, quantum Tsallis entropy, von Neumann entropy}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.4
Optimal Quantum Algorithm for Estimating Fidelity to a Pure State

Authors: Wang Fang and Qisheng Wang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We present an optimal quantum algorithm for fidelity estimation between two quantum states when one of them is pure. In particular, the (square root) fidelity of a mixed state to a pure state can be estimated to within additive error ε by using Θ(1/ε) queries to their state-preparation circuits, achieving a quadratic speedup over the folklore O(1/ε²). Our approach is technically simple, and can moreover estimate the quantity √{tr(ρσ²)} that is not common in the literature. To the best of our knowledge, this is the first query-optimal approach to fidelity estimation involving mixed states.
Cite as

Wang Fang and Qisheng Wang. Optimal Quantum Algorithm for Estimating Fidelity to a Pure State. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fang_et_al:LIPIcs.ESA.2025.4,
  author =	{Fang, Wang and Wang, Qisheng},
  title =	{{Optimal Quantum Algorithm for Estimating Fidelity to a Pure State}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{4:1--4:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.4},
  URN =		{urn:nbn:de:0030-drops-244727},
  doi =		{10.4230/LIPIcs.ESA.2025.4},
  annote =	{Keywords: Quantum computing, fidelity estimation, quantum algorithms, quantum query complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.73
Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians

Authors: François Le Gall

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We construct classical algorithms computing an approximation of the ground state energy of an arbitrary k-local Hamiltonian acting on n qubits. We first consider the setting where a good "guiding state" is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in poly (1/χ,n) time and poly(n) space, where χ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up to a polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in 2^O(n) time and poly(n) space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.
Cite as

François Le Gall. Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 73:1-73:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{legall:LIPIcs.ESA.2025.73,
  author =	{Le Gall, Fran\c{c}ois},
  title =	{{Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{73:1--73:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.73},
  URN =		{urn:nbn:de:0030-drops-245419},
  doi =		{10.4230/LIPIcs.ESA.2025.73},
  annote =	{Keywords: approximation algorithms, quantum computing, dequantization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.106
On Estimating the Quantum 𝓁_α Distance

Authors: Yupan Liu and Qisheng Wang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We study the computational complexity of estimating the quantum 𝓁_α distance T_α(ρ₀,ρ₁), defined via the Schatten α-norm ‖A‖_α := tr(|A|^α)^{1/α}, given poly(n)-size state-preparation circuits of n-qubit quantum states ρ₀ and ρ₁. This quantity serves as a lower bound on the trace distance for α > 1. For any constant α > 1, we develop an efficient rank-independent quantum estimator for T_α(ρ₀,ρ₁) with time complexity poly(n), achieving an exponential speedup over the prior best results of exp(n) due to Wang, Guan, Liu, Zhang, and Ying (IEEE Trans. Inf. Theory 2024). Our improvement leverages efficiently computable uniform polynomial approximations of signed positive power functions within quantum singular value transformation, thereby eliminating the dependence on the rank of the states. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten α-norm (QSD_α), which involves deciding whether T_α(ρ₀,ρ₁) is at least 2/5 or at most 1/5. This dichotomy arises between the cases of constant α > 1 and α = 1: - For any 1+Ω(1) ≤ α ≤ O(1), QSD_α is BQP-complete. - For any 1 ≤ α ≤ 1+1/n, QSD_α is QSZK-complete, implying that no efficient quantum estimator for T_α(ρ₀,ρ₁) exists unless BQP = QSZK. The hardness results follow from reductions based on new rank-dependent inequalities for the quantum 𝓁_α distance with 1 ≤ α ≤ ∞, which are of independent interest.
Cite as

Yupan Liu and Qisheng Wang. On Estimating the Quantum 𝓁_α Distance. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 106:1-106:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{liu_et_al:LIPIcs.ESA.2025.106,
  author =	{Liu, Yupan and Wang, Qisheng},
  title =	{{On Estimating the Quantum 𝓁\underline\alpha Distance}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{106:1--106:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.106},
  URN =		{urn:nbn:de:0030-drops-245758},
  doi =		{10.4230/LIPIcs.ESA.2025.106},
  annote =	{Keywords: quantum algorithms, quantum state testing, trace distance, Schatten norm}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.17
Space-Bounded Quantum Interactive Proof Systems

Authors: François Le Gall, Yupan Liu, Harumichi Nishimura, and Qisheng Wang

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP: - QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. - QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.
Cite as

François Le Gall, Yupan Liu, Harumichi Nishimura, and Qisheng Wang. Space-Bounded Quantum Interactive Proof Systems. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{legall_et_al:LIPIcs.CCC.2025.17,
  author =	{Le Gall, Fran\c{c}ois and Liu, Yupan and Nishimura, Harumichi and Wang, Qisheng},
  title =	{{Space-Bounded Quantum Interactive Proof Systems}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{17:1--17:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.17},
  URN =		{urn:nbn:de:0030-drops-237115},
  doi =		{10.4230/LIPIcs.CCC.2025.17},
  annote =	{Keywords: Intermediate measurements, Quantum interactive proofs, Space-bounded quantum computation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.18
Directed st-Connectivity with Few Paths Is in Quantum Logspace

Authors: Simon Apers and Roman Edenhofer

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We present a BQSPACE(O(log n))-procedure to count st-paths on directed graphs for which we are promised that there are at most polynomially many paths starting in s and polynomially many paths ending in t. For comparison, the best known classical upper bound in this case just to decide st-connectivity is DSPACE(O(log² n/ log log n)). The result establishes a new relationship between BQL and unambiguity and fewness subclasses of NL. Further, we also show how to recognize directed graphs with at most polynomially many paths between any two nodes in BQSPACE(O(log n)). This yields the first natural candidate for a language separating BQL from 𝖫 and BPL. Until now, all candidates potentially separating these classes were inherently promise problems.
Cite as

Simon Apers and Roman Edenhofer. Directed st-Connectivity with Few Paths Is in Quantum Logspace. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{apers_et_al:LIPIcs.CCC.2025.18,
  author =	{Apers, Simon and Edenhofer, Roman},
  title =	{{Directed st-Connectivity with Few Paths Is in Quantum Logspace}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.18},
  URN =		{urn:nbn:de:0030-drops-237128},
  doi =		{10.4230/LIPIcs.CCC.2025.18},
  annote =	{Keywords: Quantum computation, Space-bounded complexity classes, Graph connectivity, Unambiguous computation, Random walks}
}
Document
DOI: 10.4230/LIPIcs.ECOOP.2025.6
Detecting Functionality-Specific Vulnerabilities via Retrieving Individual Functionality-Equivalent APIs in Open-Source Repositories

Authors: Tianyu Chen, Zeyu Wang, Lin Li, Ding Li, Zongyang Li, Xiaoning Chang, Pan Bian, Guangtai Liang, Qianxiang Wang, and Tao Xie

Published in: LIPIcs, Volume 333, 39th European Conference on Object-Oriented Programming (ECOOP 2025)

Abstract
Functionality-specific vulnerabilities, which mainly occur in Application Programming Interfaces (APIs) with specific functionalities, are crucial for software developers to detect and avoid. When detecting individual functionality-specific vulnerabilities, the existing two categories of approaches are ineffective because they consider only the API bodies and are unable to handle diverse implementations of functionality-equivalent APIs. To effectively detect functionality-specific vulnerabilities, we propose APISS, the first approach to utilize API doc strings and signatures instead of API bodies. APISS first retrieves functionality-equivalent APIs for APIs with existing vulnerabilities and then migrates Proof-of-Concepts (PoCs) of the existing vulnerabilities for newly detected vulnerable APIs. To retrieve functionality-equivalent APIs, we leverage a Large Language Model for API embedding to improve the accuracy and address the effectiveness and scalability issues suffered by the existing approaches. To migrate PoCs of the existing vulnerabilities for newly detected vulnerable APIs, we design a semi-automatic schema to substantially reduce manual costs. We conduct a comprehensive evaluation to empirically compare APISS with four state-of-the-art approaches of detecting vulnerabilities and two state-of-the-art approaches of retrieving functionality-equivalent APIs. The evaluation subjects include 180 widely used Java repositories using 10 existing vulnerabilities, along with their PoCs. The results show that APISS effectively retrieves functionality-equivalent APIs, achieving a Top-1 Accuracy of 0.81 while the best of the baselines under comparison achieves only 0.55. APISS is highly efficient: the manual costs are within 10 minutes per vulnerability and the end-to-end runtime overhead of testing one candidate API is less than 2 hours. APISS detects 179 new vulnerabilities and receives 60 new CVE IDs, bringing high value to security practice.
Cite as

Tianyu Chen, Zeyu Wang, Lin Li, Ding Li, Zongyang Li, Xiaoning Chang, Pan Bian, Guangtai Liang, Qianxiang Wang, and Tao Xie. Detecting Functionality-Specific Vulnerabilities via Retrieving Individual Functionality-Equivalent APIs in Open-Source Repositories. In 39th European Conference on Object-Oriented Programming (ECOOP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 333, pp. 6:1-6:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.ECOOP.2025.6,
  author =	{Chen, Tianyu and Wang, Zeyu and Li, Lin and Li, Ding and Li, Zongyang and Chang, Xiaoning and Bian, Pan and Liang, Guangtai and Wang, Qianxiang and Xie, Tao},
  title =	{{Detecting Functionality-Specific Vulnerabilities via Retrieving Individual Functionality-Equivalent APIs in Open-Source Repositories}},
  booktitle =	{39th European Conference on Object-Oriented Programming (ECOOP 2025)},
  pages =	{6:1--6:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-373-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{333},
  editor =	{Aldrich, Jonathan and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2025.6},
  URN =		{urn:nbn:de:0030-drops-232999},
  doi =		{10.4230/LIPIcs.ECOOP.2025.6},
  annote =	{Keywords: Application Security, Vulnerability Detection, Large Language Model}
}
Document
Vision
DOI: 10.4230/TGDK.1.1.8
Machine Learning and Knowledge Graphs: Existing Gaps and Future Research Challenges

Authors: Claudia d'Amato, Louis Mahon, Pierre Monnin, and Giorgos Stamou

Published in: TGDK, Volume 1, Issue 1 (2023): Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge, Volume 1, Issue 1

Abstract
The graph model is nowadays largely adopted to model a wide range of knowledge and data, spanning from social networks to knowledge graphs (KGs), representing a successful paradigm of how symbolic and transparent AI can scale on the World Wide Web. However, due to their unprecedented volume, they are generally tackled by Machine Learning (ML) and mostly numeric based methods such as graph embedding models (KGE) and deep neural networks (DNNs). The latter methods have been proved lately very efficient, leading the current AI spring. In this vision paper, we introduce some of the main existing methods for combining KGs and ML, divided into two categories: those using ML to improve KGs, and those using KGs to improve results on ML tasks. From this introduction, we highlight research gaps and perspectives that we deem promising and currently under-explored for the involved research communities, spanning from KG support for LLM prompting, integration of KG semantics in ML models to symbol-based methods, interpretability of ML models, and the need for improved benchmark datasets. In our opinion, such perspectives are stepping stones in an ultimate view of KGs as central assets for neuro-symbolic and explainable AI.
Cite as

Claudia d'Amato, Louis Mahon, Pierre Monnin, and Giorgos Stamou. Machine Learning and Knowledge Graphs: Existing Gaps and Future Research Challenges. In Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge (TGDK), Volume 1, Issue 1, pp. 8:1-8:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{damato_et_al:TGDK.1.1.8,
  author =	{d'Amato, Claudia and Mahon, Louis and Monnin, Pierre and Stamou, Giorgos},
  title =	{{Machine Learning and Knowledge Graphs: Existing Gaps and Future Research Challenges}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{8:1--8:35},
  year =	{2023},
  volume =	{1},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.1.1.8},
  URN =		{urn:nbn:de:0030-drops-194824},
  doi =		{10.4230/TGDK.1.1.8},
  annote =	{Keywords: Graph-based Learning, Knowledge Graph Embeddings, Large Language Models, Explainable AI, Knowledge Graph Completion \& Curation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.22
A Distribution Testing Oracle Separating QMA and QCMA

Authors: Anand Natarajan and Chinmay Nirkhe

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

Abstract
It is a long-standing open question in quantum complexity theory whether the definition of non-deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson and Kuperberg, 2007; Bill Fefferman and Shelby Kimmel, 2018] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.
Cite as

Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 22:1-22:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{natarajan_et_al:LIPIcs.CCC.2023.22,
  author =	{Natarajan, Anand and Nirkhe, Chinmay},
  title =	{{A Distribution Testing Oracle Separating QMA and QCMA}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{22:1--22:27},
  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.2023.22},
  URN =		{urn:nbn:de:0030-drops-182928},
  doi =		{10.4230/LIPIcs.CCC.2023.22},
  annote =	{Keywords: quantum non-determinism, complexity theory}
}
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.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.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}
}
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