17 research outputs found
The Power of Natural Properties as Oracles
We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq P/poly. We also show that, assuming the existence of Indistinguishability Obfuscators (IO), SAT and MCSP are equivalent in the sense that one has a ZPP algorithm if and only the other one does. We interpret our results as providing some evidence that MCSP may be NP-hard under randomized polynomial-time reductions
Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection
We define rewinding operators that invert quantum measurements. Then, we
define complexity classes , , and as
sets of decision problems solvable by polynomial-size quantum circuits with a
polynomial number of rewinding operators, cloning operators, and adaptive
postselections, respectively. Our main result is that . As a
byproduct of this result, we show that any problem in can be
solved with only postselections of outputs whose probabilities are polynomially
close to one. Under the strongly believed assumption that , or the shortest independent vectors problem cannot be
efficiently solved with quantum computers, we also show that a single rewinding
operator is sufficient to achieve tasks that are intractable for quantum
computation. In addition, we consider rewindable Clifford and instantaneous
quantum polynomial time circuits.Comment: 29 pages, 3 figures, v2: Added Result 3 and improved Result
Exponential separations between classical and quantum learners
Despite significant effort, the quantum machine learning community has only
demonstrated quantum learning advantages for artificial cryptography-inspired
datasets when dealing with classical data. In this paper we address the
challenge of finding learning problems where quantum learning algorithms can
achieve a provable exponential speedup over classical learning algorithms. We
reflect on computational learning theory concepts related to this question and
discuss how subtle differences in definitions can result in significantly
different requirements and tasks for the learner to meet and solve. We examine
existing learning problems with provable quantum speedups and find that they
largely rely on the classical hardness of evaluating the function that
generates the data, rather than identifying it. To address this, we present two
new learning separations where the classical difficulty primarily lies in
identifying the function generating the data. Furthermore, we explore
computational hardness assumptions that can be leveraged to prove quantum
speedups in scenarios where data is quantum-generated, which implies likely
quantum advantages in a plethora of more natural settings (e.g., in condensed
matter and high energy physics). We also discuss the limitations of the
classical shadow paradigm in the context of learning separations, and how
physically-motivated settings such as characterizing phases of matter and
Hamiltonian learning fit in the computational learning framework.Comment: this article supersedes arXiv:2208.0633
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
StoqMA Meets Distribution Testing
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 and
distribution testing via reversible circuits. First, we prove that easy-witness
(viz. , a sub-class of )
is contained in . 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. This sub-class
contains with perfect completeness
(), which further signifies a simplified proof for
[BBT06, BT10]. Second, by showing
distinguishing reversible circuits with ancillary random bits is
-complete (as a comparison, distinguishing quantum circuits is
-complete [JWB05]), we construct soundness error reduction of
. Additionally, we show that both variants of
that without any ancillary random bit and with perfect
soundness are contained in . Our results make a step towards
collapsing the hierarchy [BBT06], in which all classes are contained in and
collapse to under derandomization assumptions.Comment: 24 pages. v2: mostly adds corrections and clarifications. v3: add a
connection between eStoqMA and Guided Stoquastic Hamiltonian Proble