Abstract

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

Highlights

  • Sampling output probability distributions of sub-universal quantum computing models is known to be impossible under certain classical complexity conjectures

  • We say that a quantum probability distribution {pz}z is classically sampled in time T within a multiplicative error if there exists a classical T -time probabilistic algorithm that outputs z with probability qz such that |pz − qz| ≤ pz for all z

  • We have shown additive-error fine-grained quantum supremacy based on several conjectures

Read more

Summary

Introduction

Sampling output probability distributions of sub-universal quantum computing models is known to be impossible under certain classical complexity conjectures. The standard proof technique of additive-error quantum supremacy [2, 4], namely, the combination of Markov’s inequality, Stockmeyer’s theorem, and the anti-concentration lemma, cannot be directly used for fine-grained quantum supremacy, because Stockmeyer’s theorem is a result for polynomial-time probabilistic computing. The conjecture on which additive-error fine-grained quantum supremacy of the IQP model is based is stated as follows. Theorem 1 If Conjecture 1 is true, there exists an N -qubit IQP circuit whose output probability distribution cannot be classically sampled in O(2aN )-time within a certain constant additive error.

IQP plus log-depth Boolean circuit
Apply Z on the last qubit to obtain
Clifford plus T
Conjectures
Other models
A pairwise independent hash family and the leftover hash lemma
Algorithm Ak
Fine-grained Stockmeyer’s theorem

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.