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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2022.9
URN: urn:nbn:de:0030-drops-165163
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16516/
Dontha, Suchetan ;
Tan, Shi Jie Samuel ;
Smith, Stephen ;
Choi, Sangheon ;
Coudron, Matthew
Approximating Output Probabilities of Shallow Quantum Circuits Which Are Geometrically-Local in Any Fixed Dimension
Abstract
We present a classical algorithm that, for any D-dimensional geometrically-local, quantum circuit C of polylogarithmic-depth, and any bit string x ∈ {0,1}ⁿ, can compute the quantity |<x|C|0^{⊗ n}>|² to within any inverse-polynomial additive error in quasi-polynomial time, for any fixed dimension D. This is an extension of the result [Nolan J. Coble and Matthew Coudron, 2021], which originally proved this result for D = 3. To see why this is interesting, note that, while the D = 1 case of this result follows from a standard use of Matrix Product States, known for decades, the D = 2 case required novel and interesting techniques introduced in [Sergy Bravyi et al., 2020]. Extending to the case D = 3 was even more laborious, and required further new techniques introduced in [Nolan J. Coble and Matthew Coudron, 2021]. Our work here shows that, while handling each new dimension has historically required a new insight, and fixed algorithmic primitive, based on known techniques for D ≤ 3, we can now handle any fixed dimension D > 3.
Our algorithm uses the Divide-and-Conquer framework of [Nolan J. Coble and Matthew Coudron, 2021] to approximate the desired quantity via several instantiations of the same problem type, each involving D-dimensional circuits on about half the number of qubits as the original. This division step is then applied recursively, until the width of the recursively decomposed circuits in the D^{th} dimension is so small that they can effectively be regarded as (D-1)-dimensional problems by absorbing the small width in the D^{th} dimension into the qudit structure at the cost of a moderate increase in runtime. The main technical challenge lies in ensuring that the more involved portions of the recursive circuit decomposition and error analysis from [Nolan J. Coble and Matthew Coudron, 2021] still hold in higher dimensions, which requires small modifications to the analysis in some places. Our work also includes some simplifications, corrections and clarifications of the use of block-encodings within the original classical algorithm in [Nolan J. Coble and Matthew Coudron, 2021].
BibTeX - Entry
@InProceedings{dontha_et_al:LIPIcs.TQC.2022.9,
author = {Dontha, Suchetan and Tan, Shi Jie Samuel and Smith, Stephen and Choi, Sangheon and Coudron, Matthew},
title = {{Approximating Output Probabilities of Shallow Quantum Circuits Which Are Geometrically-Local in Any Fixed Dimension}},
booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
pages = {9:1--9:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-237-2},
ISSN = {1868-8969},
year = {2022},
volume = {232},
editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16516},
URN = {urn:nbn:de:0030-drops-165163},
doi = {10.4230/LIPIcs.TQC.2022.9},
annote = {Keywords: Low-Depth Quantum Circuits, Matrix Product States, Block-Encoding}
}
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Keywords: |
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Low-Depth Quantum Circuits, Matrix Product States, Block-Encoding |
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Collection: |
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17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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04.07.2022 |
