144 research outputs found
Classical and quantum partition bound and detector inefficiency
We study randomized and quantum efficiency lower bounds in communication
complexity. These arise from the study of zero-communication protocols in which
players are allowed to abort. Our scenario is inspired by the physics setup of
Bell experiments, where two players share a predefined entangled state but are
not allowed to communicate. Each is given a measurement as input, which they
perform on their share of the system. The outcomes of the measurements should
follow a distribution predicted by quantum mechanics; however, in practice, the
detectors may fail to produce an output in some of the runs. The efficiency of
the experiment is the probability that the experiment succeeds (neither of the
detectors fails).
When the players share a quantum state, this gives rise to a new bound on
quantum communication complexity (eff*) that subsumes the factorization norm.
When players share randomness instead of a quantum state, the efficiency bound
(eff), coincides with the partition bound of Jain and Klauck. This is one of
the strongest lower bounds known for randomized communication complexity, which
subsumes all the known combinatorial and algebraic methods including the
rectangle (corruption) bound, the factorization norm, and discrepancy.
The lower bound is formulated as a convex optimization problem. In practice,
the dual form is more feasible to use, and we show that it amounts to
constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for
eff*). We give an example of a quantum distribution where the violation can be
exponentially bigger than the previously studied class of normalized Bell
inequalities.
For one-way communication, we show that the quantum one-way partition bound
is tight for classical communication with shared entanglement up to arbitrarily
small error.Comment: 21 pages, extended versio
The positive semidefinite Grothendieck problem with rank constraint
Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of
size m x m, the positive semidefinite Grothendieck problem with
rank-n-constraint (SDP_n) is
maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m
\in S^{n-1}.
In this paper we design a polynomial time approximation algorithm for SDP_n
achieving an approximation ratio of
\gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 -
\Theta(1/n).
We show that under the assumption of the unique games conjecture the achieved
approximation ratio is optimal: There is no polynomial time algorithm which
approximates SDP_n with a ratio greater than \gamma(n). We improve the
approximation ratio of the best known polynomial time algorithm for SDP_1 from
2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter
approximation ratio for SDP_n when A is the Laplacian matrix of a graph with
nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on
Automata, Languages and Programming, 12 page
Quantum query complexity of state conversion
State conversion generalizes query complexity to the problem of converting
between two input-dependent quantum states by making queries to the input. We
characterize the complexity of this problem by introducing a natural
information-theoretic norm that extends the Schur product operator norm. The
complexity of converting between two systems of states is given by the distance
between them, as measured by this norm.
In the special case of function evaluation, the norm is closely related to
the general adversary bound, a semi-definite program that lower-bounds the
number of input queries needed by a quantum algorithm to evaluate a function.
We thus obtain that the general adversary bound characterizes the quantum query
complexity of any function whatsoever. This generalizes and simplifies the
proof of the same result in the case of boolean input and output. Also in the
case of function evaluation, we show that our norm satisfies a remarkable
composition property, implying that the quantum query complexity of the
composition of two functions is at most the product of the query complexities
of the functions, up to a constant. Finally, our result implies that discrete
and continuous-time query models are equivalent in the bounded-error setting,
even for the general state-conversion problem.Comment: 19 pages, 2 figures; heavily revised with new results and simpler
proof
On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model
We study the communication complexity of linear algebraic problems over
finite fields in the multi-player message passing model, proving a number of
tight lower bounds. Specifically, for a matrix which is distributed among a
number of players, we consider the problem of determining its rank, of
computing entries in its inverse, and of solving linear equations. We also
consider related problems such as computing the generalized inner product of
vectors held on different servers. We give a general framework for reducing
these multi-player problems to their two-player counterparts, showing that the
randomized -player communication complexity of these problems is at least
times the randomized two-player communication complexity. Provided the
problem has a certain amount of algebraic symmetry, which we formally define,
we can show the hardest input distribution is a symmetric distribution, and
therefore apply a recent multi-player lower bound technique of Phillips et al.
Further, we give new two-player lower bounds for a number of these problems. In
particular, our optimal lower bound for the two-player version of the matrix
rank problem resolves an open question of Sun and Wang.
A common feature of our lower bounds is that they apply even to the special
"threshold promise" versions of these problems, wherein the underlying
quantity, e.g., rank, is promised to be one of just two values, one on each
side of some critical threshold. These kinds of promise problems are
commonplace in the literature on data streaming as sources of hardness for
reductions giving space lower bounds
Lifting for Constant-Depth Circuits and Applications to MCSP
Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas.
We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC?-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest.
We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC? circuit lower bounds
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
The Partition Bound for Classical Communication Complexity and Query Complexity
We describe new lower bounds for randomized communication complexity and
query complexity which we call the partition bounds. They are expressed as the
optimum value of linear programs. For communication complexity we show that the
partition bound is stronger than both the rectangle/corruption bound and the
\gamma_2/generalized discrepancy bounds. In the model of query complexity we
show that the partition bound is stronger than the approximate polynomial
degree and classical adversary bounds. We also exhibit an example where the
partition bound is quadratically larger than polynomial degree and classical
adversary bounds.Comment: 28 pages, ver. 2, added conten
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