37 research outputs found
Analyzing XOR-Forrelation Through Stochastic Calculus
In this note we present a simplified analysis of the quantum and classical complexity of the k-XOR Forrelation problem (introduced in the paper of Girish, Raz and Zhan [Uma Girish et al., 2020]) by a stochastic interpretation of the Forrelation distribution
The Computational Complexity of Linear Optics
We give new evidence that quantum computers -- moreover, rudimentary quantum
computers built entirely out of linear-optical elements -- cannot be
efficiently simulated by classical computers. In particular, we define a model
of computation in which identical photons are generated, sent through a
linear-optical network, then nonadaptively measured to count the number of
photons in each mode. This model is not known or believed to be universal for
quantum computation, and indeed, we discuss the prospects for realizing the
model using current technology. On the other hand, we prove that the model is
able to solve sampling problems and search problems that are classically
intractable under plausible assumptions. Our first result says that, if there
exists a polynomial-time classical algorithm that samples from the same
probability distribution as a linear-optical network, then P^#P=BPP^NP, and
hence the polynomial hierarchy collapses to the third level. Unfortunately,
this result assumes an extremely accurate simulation. Our main result suggests
that even an approximate or noisy classical simulation would already imply a
collapse of the polynomial hierarchy. For this, we need two unproven
conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is
#P-hard to approximate the permanent of a matrix A of independent N(0,1)
Gaussian entries, with high probability over A; and the "Permanent
Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with
high probability over A. We present evidence for these conjectures, both of
which seem interesting even apart from our application. This paper does not
assume knowledge of quantum optics. Indeed, part of its goal is to develop the
beautiful theory of noninteracting bosons underlying our model, and its
connection to the permanent function, in a self-contained way accessible to
theoretical computer scientists.Comment: 94 pages, 4 figure
Fractional Pseudorandom Generators from Any Fourier Level
We prove new results on the polarizing random walk framework introduced in
recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit
Fourier tail bounds for classes of Boolean functions to construct pseudorandom
generators (PRGs). We show that given a bound on the -th level of the
Fourier spectrum, one can construct a PRG with a seed length whose quality
scales with . This interpolates previous works, which either require Fourier
bounds on all levels [CHHL19], or have polynomial dependence on the error
parameter in the seed length [CHLT10], and thus answers an open question in
[CHLT19]. As an example, we show that for polynomial error, Fourier bounds on
the first levels is sufficient to recover the seed length in
[CHHL19], which requires bounds on the entire tail.
We obtain our results by an alternate analysis of fractional PRGs using
Taylor's theorem and bounding the degree- Lagrange remainder term using
multilinearity and random restrictions. Interestingly, our analysis relies only
on the \emph{level-k unsigned Fourier sum}, which is potentially a much smaller
quantity than the notion in previous works. By generalizing a connection
established in [CHH+20], we give a new reduction from constructing PRGs to
proving correlation bounds. Finally, using these improvements we show how to
obtain a PRG for polynomials with seed length close to the
state-of-the-art construction due to Viola [Vio09], which was not known to be
possible using this framework
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde