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

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

Beteiligte: Dontha, Suchetan [Verfasser:in]; Tan, Shi Jie Samuel [Verfasser:in]; Smith, Stephen [Verfasser:in]; Choi, Sangheon [Verfasser:in]; Coudron, Matthew [Verfasser:in]

Erschienen: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022

Sprache: Englisch

DOI: https://doi.org/10.4230/LIPIcs.TQC.2022.9

Schlagwörter: Block-Encoding ; Low-Depth Quantum Circuits ; Matrix Product States

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Beschreibung: 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 ...

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