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Approximating Output Probabilities of Shallow Quantum Circuits Which Are Geometrically-Local in Any Fixed Dimension

Authors Suchetan Dontha, Shi Jie Samuel Tan, Stephen Smith, Sangheon Choi, Matthew Coudron

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Author Details

Suchetan Dontha
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Shi Jie Samuel Tan
  • Department of Computer Science, Haverford College, PA, USA
Stephen Smith
  • Department of Mathematics, University of South Carolina, Columbia, SC, USA
Sangheon Choi
  • Department of Computer Science, Rose-Hulman Institute of Technology, Terre Haute, IN, USA
Matthew Coudron
  • Department of Computer Science, University of Maryland, College Park, MD, USA


MC thanks Sergey Bravyi for helpful discussions. We thank Gorjan Alagic and Nolan Coble for attending and contributing to some project group meetings. We thank the REU-CAAR program (NSF grant number 1852352) which funded SD, SJST, SC over the summer of 2021.

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Suchetan Dontha, Shi Jie Samuel Tan, Stephen Smith, Sangheon Choi, and Matthew Coudron. Approximating Output Probabilities of Shallow Quantum Circuits Which Are Geometrically-Local in Any Fixed Dimension. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 9:1-9:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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].

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Divide and conquer
  • Low-Depth Quantum Circuits
  • Matrix Product States
  • Block-Encoding


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