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

Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time T for the simulation greatly affect algorithm runtime as expected. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time o(T), for some large classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time T. On the other hand, while there exist lower bounds of Ω(T) circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. As a result, physical systems with system size scaling with T can potentially do a fast-forwarding simulation. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism.
In this work, we give a negative result for the above open problem in various settings. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth o(T). In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians on n qubits that cannot be simulated via an oracle circuit of depth o(T/n^c), where the Hamiltonians act on n qubits, and c is a constant. Lastly, we generalize the above results and show that any simulators that are geometrically local Hamiltonians cannot do the simulation much faster than parallel quantum algorithms.

Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen. On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 33:1-33:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chia_et_al:LIPIcs.CCC.2023.33, author = {Chia, Nai-Hui and Chung, Kai-Min and Hsieh, Yao-Ching and Lin, Han-Hsuan and Lin, Yao-Ting and Shen, Yu-Ching}, title = {{On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {33:1--33:45}, 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.2023.33}, URN = {urn:nbn:de:0030-drops-183038}, doi = {10.4230/LIPIcs.CCC.2023.33}, annote = {Keywords: Hamiltonian simulation, Depth lower bound, Parallel query lower bound} }

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

In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory - its hardness is mysterious, and a better understanding of its hardness can have surprising implications to many fields in computer science.
We first define and investigate the basic complexity-theoretic properties of minimum quantum circuit size problems for three natural objects: Boolean functions, unitaries, and quantum states. We show that these problems are not trivially in NP but in QCMA (or have QCMA protocols). Next, we explore the relations between the three quantum MCSPs and their variants. We discover that some reductions that are not known for classical MCSP exist for quantum MCSPs for unitaries and states, e.g., search-to-decision reductions and self-reductions. Finally, we systematically generalize results known for classical MCSP to the quantum setting (including quantum cryptography, quantum learning theory, quantum circuit lower bounds, and quantum fine-grained complexity) and also find new connections to tomography and quantum gravity. Due to the fundamental differences between classical and quantum circuits, most of our results require extra care and reveal properties and phenomena unique to the quantum setting. Our findings could be of interest for future studies, and we post several open problems for further exploration along this direction.

Nai-Hui Chia, Chi-Ning Chou, Jiayu Zhang, and Ruizhe Zhang. Quantum Meets the Minimum Circuit Size Problem. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chia_et_al:LIPIcs.ITCS.2022.47, author = {Chia, Nai-Hui and Chou, Chi-Ning and Zhang, Jiayu and Zhang, Ruizhe}, title = {{Quantum Meets the Minimum Circuit Size Problem}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {47:1--47:16}, 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.2022.47}, URN = {urn:nbn:de:0030-drops-156433}, doi = {10.4230/LIPIcs.ITCS.2022.47}, annote = {Keywords: Quantum Computation, Quantum Complexity, Minimum Circuit Size Problem} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

We present two efficient classical analogues of the quantum matrix inversion algorithm [Harrow et al., 2009] for low-rank matrices. Inspired by recent work of Tang [Tang, 2019], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired "smooth" function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [András Gilyén et al., 2019], our results indicate that more low-rank quantum algorithms can be effectively "dequantised" into classical length-square sampling algorithms.

Nai-Hui Chia, András Gilyén, Han-Hsuan Lin, Seth Lloyd, Ewin Tang, and Chunhao Wang. Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chia_et_al:LIPIcs.ISAAC.2020.47, author = {Chia, Nai-Hui and Gily\'{e}n, Andr\'{a}s and Lin, Han-Hsuan and Lloyd, Seth and Tang, Ewin and Wang, Chunhao}, title = {{Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {47:1--47:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.47}, URN = {urn:nbn:de:0030-drops-133916}, doi = {10.4230/LIPIcs.ISAAC.2020.47}, annote = {Keywords: sublinear algorithms, quantum-inspired, regression, importance sampling, quantum machine learning} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with m constraint matrices, each of dimension n and rank r, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time O(m⋅poly(log n,r,1/ε)) given access to a sampling-based low-overhead data structure for the constraint matrices, where ε is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest:
- Weighted sampling: assuming sampling access to each individual constraint matrix A₁,…,A_τ, we propose a procedure that gives a good approximation of A = A₁+⋯+A_τ.
- Symmetric approximation: we propose a sampling procedure that gives the spectral decomposition of a low-rank Hermitian matrix A. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chia_et_al:LIPIcs.MFCS.2020.23, author = {Chia, Nai-Hui and Li, Tongyang and Lin, Han-Hsuan and Wang, Chunhao}, title = {{Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.23}, URN = {urn:nbn:de:0030-drops-126919}, doi = {10.4230/LIPIcs.MFCS.2020.23}, annote = {Keywords: Spectral decomposition, Semi-definite programming, Quantum-inspired algorithm, Sublinear algorithm} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a d-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(n log n) time in constant dimensions (i.e., when d = O(1)). This paper asks and answers the question of the problem’s quantum time complexity. Specifically, we give an Õ(n^(2/3)) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference.
In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover’s algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover’s algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an n^o(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

Scott Aaronson, Nai-Hui Chia, Han-Hsuan Lin, Chunhao Wang, and Ruizhe Zhang. On the Quantum Complexity of Closest Pair and Related Problems. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 16:1-16:43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aaronson_et_al:LIPIcs.CCC.2020.16, author = {Aaronson, Scott and Chia, Nai-Hui and Lin, Han-Hsuan and Wang, Chunhao and Zhang, Ruizhe}, title = {{On the Quantum Complexity of Closest Pair and Related Problems}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {16:1--16:43}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, 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.CCC.2020.16}, URN = {urn:nbn:de:0030-drops-125681}, doi = {10.4230/LIPIcs.CCC.2020.16}, annote = {Keywords: Closest pair, Quantum computing, Quantum fine grained reduction, Quantum strong exponential time hypothesis, Fine grained complexity} }

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**Published in:** LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)

We study the hardness of the dihedral hidden subgroup problem. It is known that lattice problems reduce to it, and that it reduces to random subset sum with density > 1 and also to quantum sampling subset sum solutions. We examine a decision version of the problem where the question asks whether the hidden subgroup is trivial or order two. The decision problem essentially asks if a given vector is in the span of all coset states. We approach this by first computing an explicit basis for the coset space and the perpendicular space. We then look at the consequences of having efficient unitaries that use this basis. We show that if a unitary maps the basis to the standard basis in any way, then that unitary can be used to solve random subset sum with constant density >1. We also show that if a unitary can exactly decide membership in the coset subspace, then the collision problem for subset sum can be solved for density >1 but approaching 1 as the problem size increases. This strengthens the previous hardness result that implementing the optimal POVM in a specific way is as hard as quantum sampling subset sum solutions.

Nai-Hui Chia and Sean Hallgren. How Hard Is Deciding Trivial Versus Nontrivial in the Dihedral Coset Problem?. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chia_et_al:LIPIcs.TQC.2016.6, author = {Chia, Nai-Hui and Hallgren, Sean}, title = {{How Hard Is Deciding Trivial Versus Nontrivial in the Dihedral Coset Problem?}}, booktitle = {11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)}, pages = {6:1--6:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-019-4}, ISSN = {1868-8969}, year = {2016}, volume = {61}, editor = {Broadbent, Anne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.6}, URN = {urn:nbn:de:0030-drops-66877}, doi = {10.4230/LIPIcs.TQC.2016.6}, annote = {Keywords: Quantum algorithms, hidden subgroup problem, random subset sum problem} }

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