Approximating Output Probabilities of Shallow Quantum Circuits which are Geometrically-local in any Fixed Dimension

• URL: http://arxiv.org/abs/2202.08349v1
• Date: Wed, 16 Feb 2022 21:37:16 GMT
• Title: 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
• Abstract summary: We present an algorithm that can compute the quantity $|x|C|0otimes n>|2$ to within any inverse-polynomial additive error in quasi-polynomial time. This is an extension of the result [CC21], which originally proved this result for $D = 3$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a classical algorithm that, for any $D$-dimensional geometrically-local, quantum circuit $C$ of polylogarithmic-depth, and any bit string $x \in {0,1}^n$, can compute the quantity $|<x|C|0^{\otimes n}>|^2$ to within any inverse-polynomial additive error in quasi-polynomial time, for any fixed dimension $D$. This is an extension of the result [CC21], 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 standard use of Matrix Product States, known for decades, the $D = 2$ case required novel and interesting techniques introduced in [BGM19]. Extending to the case $D = 3$ was even more laborious and required further new techniques introduced in [CC21]. 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 $D \leq 3$, we can now handle any fixed dimension $D > 3$. Our algorithm uses the Divide-and-Conquer framework of [CC21] 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 [CC21] still hold in higher dimensions, which requires small modifications to the analysis in some places.

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