138 research outputs found
Immunity and Simplicity for Exact Counting and Other Counting Classes
Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some
relativized world, PSPACE (in fact, ParityP) contains a set that is immune to
the polynomial hierarchy (PH). In this paper, we study and settle the question
of (relativized) separations with immunity for PH and the counting classes PP,
C_{=}P, and ParityP in all possible pairwise combinations. Our main result is
that there is an oracle A relative to which C_{=}P contains a set that is
immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A}
and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green
[IPL 37, 1991], we also show that, in suitable relativizations, NP contains a
C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the
existence of a C_{=}P^{B}-simple set for some oracle B, which extends results
of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the
first example of a simple set in a class not known to be contained in PH. Our
proof technique requires a circuit lower bound for ``exact counting'' that is
derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
Complexity-Theoretic Foundations of Quantum Supremacy Experiments
In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible.
First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by by the Quantum AI group at Google. We show that there\u27s a natural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure\u27s outputs do pass. Compared to previous work - for example, on BosonSampling and IQP - the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled.
Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Savitch\u27s Theorem, for simulating a general quantum circuit with n qubits and m gates in polynomial space and m^O(n) time. We then discuss why this and other known algorithms fail to refute our assumption.
Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem - of the form "if approximate quantum sampling is classically easy, then the polynomial hierarchy collapses" - must be non-relativizing. This sharply contrasts with the situation for exact sampling.
Fourth, refuting a conjecture by Aaronson and Ambainis, we show that the Fourier Sampling problem achieves a constant versus linear separation between quantum and randomized query complexities.
Fifth, in search of a "happy medium" between black-box and non-black-box arguments, we study quantum supremacy relative to oracles in P/poly. Previous work implies that, if one-way functions exist, then quantum supremacy is possible relative to such oracles. We show, conversely, that some computational assumption is needed: if SampBPP=SampBQP and NP is in BPP, then quantum supremacy is impossible relative to oracles with small circuits
A Counterexample to the Generalized Linial-Nisan Conjecture
In earlier work, we gave an oracle separating the relational versions of BQP
and the polynomial hierarchy, and showed that an oracle separating the decision
versions would follow from what we called the Generalized Linial-Nisan (GLN)
Conjecture: that "almost k-wise independent" distributions are
indistinguishable from the uniform distribution by constant-depth circuits. The
original Linial-Nisan Conjecture was recently proved by Braverman; we offered a
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by showing that the GLN Conjecture is false, at least for circuits of depth 3
and higher. As a byproduct, our counterexample also implies that Pi2P is not
contained in P^NP relative to a random oracle with probability 1. It has been
conjectured since the 1980s that PH is infinite relative to a random oracle,
but the highest levels of PH previously proved separate were NP and coNP.
Finally, our counterexample implies that the famous results of Linial, Mansour,
and Nisan, on the structure of AC0 functions, cannot be improved in several
interesting respects.Comment: 17 page
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