StoqMA meets distribution testing

11/11/2020
βˆ™
by   Yupan Liu, et al.
βˆ™
0
βˆ™

π–²π—π—ˆπ—Šπ–¬π–  captures the computational hardness of approximating the ground energy of local Hamiltonians that do not suffer the so-called sign problem. We provide a novel connection between π–²π—π—ˆπ—Šπ–¬π–  and the distribution testing via reversible circuits. First, we prove that easy-witness π–²π—π—ˆπ—Šπ–¬π–  (viz. π–Ύπ–²π—π—ˆπ—Šπ–¬π– , a sub-class of π–²π—π—ˆπ—Šπ–¬π– ) is contained in 𝖬𝖠. Easy witness is a generalization of a subset state such that the associated set's membership can be efficiently verifiable, and all non-zero coordinates are not necessarily uniform. Second, by showing distinguishing reversible circuits with random ancillary bits is π–²π—π—ˆπ—Šπ–¬π– -complete (as a comparison, distinguishing quantum circuits is 𝖰𝖬𝖠-complete [JWB05]), we construct soundness error reduction of π–²π—π—ˆπ—Šπ–¬π– . This new π–²π—π—ˆπ—Šπ–¬π– -complete problem further signifies that π–²π—π—ˆπ—Šπ–¬π–  with perfect completeness (π–²π—π—ˆπ—Šπ–¬π– _1) is contained in π–Ύπ–²π—π—ˆπ—Šπ–¬π– , which leads us to an alternating proof for π–²π—π—ˆπ—Šπ–¬π– _1 βŠ†π–¬π–  previously proved in [BBT06, BT10]. Additionally, we show that both variants of π–²π—π—ˆπ—Šπ–¬π–  that without any random ancillary bit and with perfect soundness are contained in 𝖭𝖯. Our results make a step towards collapsing the hierarchy π–¬π– βŠ†π–²π—π—ˆπ—Šπ–¬π– βŠ†π–²π–‘π–― [BBT06], in which all classes are contained in 𝖠𝖬 and collapse to 𝖭𝖯 under derandomization assumptions.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

βˆ™ 11/21/2021

Simple circuit simulations of classical and quantum Turing machines

We construct reversible Boolean circuits efficiently simulating reversib...
βˆ™ 08/18/2018

The Distribution of Reversible Functions is Normal

The distribution of reversible programs tends to a limit as their size i...
βˆ™ 11/28/2017

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

It is a long-standing open problem whether quantum computing can be veri...
βˆ™ 02/02/2018

On The Complexity of the Cayley Semigroup Membership Problem

We investigate the complexity of deciding, given a multiplication table ...
βˆ™ 04/13/2022

Clifford Circuits can be Properly PAC Learned if and only if =

Given a dataset of input states, measurements, and probabilities, is it ...
βˆ™ 10/06/2020

StoqMA vs. MA: the power of error reduction

StoqMA characterizes the computational hardness of stoquastic local Hami...
βˆ™ 08/17/2021

Testable Designs of Toffoli Fredkin Reversible Circuits

Loss of every bit in traditional logic circuits involves dissipation of ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.