Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userApproximating the Determinant of Well-Conditioned Matrices by Shallow Circuits
72
0
0.0
(
0
)
Added by
Enric Boix Adser\\`a
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The determinant can be computed by classical circuits of depth $O(log^2 n)$, and therefore it can also be computed in classical space $O(log^2 n)$. Recent progress by Ta-Shma [Ta13] implies a method to approximate the determinant of Hermitian matrices with condition number $kappa$ in quantum space $O(log n + log kappa)$. However, it is not known how to perform the task in less than $O(log^2 n)$ space using classical resources only. In this work, we show that the condition number of a matrix implies an upper bound on the depth complexity (and therefore also on the space complexity) for this task: the determinant of Hermitian matrices with condition number $kappa$ can be approximated to inverse polynomial relative error with classical circuits of depth $tilde O(log n cdot log kappa)$, and in particular one can approximate the determinant for sufficiently well-conditioned matrices in depth $tilde{O}(log n)$. Our algorithm combines Barvinoks recent complex-analytic approach for approximating combinatorial counting problems [Bar16] with the Valiant-Berkowitz-Skyum-Rackoff depth-reduction theorem for low-degree arithmetic circuits [Val83].
rate research
Read More
We give lower bounds on the performance of two of the most popular sampling methods in practice, the Metropolis-adjusted Langevin algorithm (MALA) and multi-step Hamiltonian Monte Carlo (HMC) with a leapfrog integrator, when applied to well-conditioned distributions. Our main result is a nearly-tight lower bound of $widetilde{Omega}(kappa d)$ on the mixing time of MALA from an exponentially warm start, matching a line of algorithmic results up to logarithmic factors and answering an open question of Chewi et. al. We also show that a polynomial dependence on dimension is necessary for the relaxation time of HMC under any number of leapfrog steps, and bound the gains achievable by changing the step count. Our HMC analysis draws upon a novel connection between leapfrog integration and Chebyshev polynomials, which may be of independent interest.
Recently, Bravyi, Gosset, and K{o}nig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Relaxed Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem. As a step towards even stronger lower bounds, we present a search problem that we call the Parity Bending Problem, which is in QNC^0/qpoly (QNC^0 circuits that are allowed to start with a quantum state of their choice that is independent of the input), but is not even in AC^0[2] (the class AC^0 with unbounded fan-in XOR gates). All the quantum circuits in our paper are simple, and the main difficulty lies in proving the classical lower bounds. For this we employ a host of techniques, including a refinement of H{aa}stads switching lemmas for multi-output circuits that may be of independent interest, the Razborov-Smolensky AC^0[2] lower bound, Vaziranis XOR lemma, and lower bounds for non-local games.
We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netov{c}ny and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.
We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander. We consider the spectral definition of high-dimensional expansion used by Dinur and Kaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure the expansion in terms of a parameter $gamma$ which is the analogue of the second singular value for expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS 2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet $[q]$ is a high-dimensional expander with parameter $gamma$, then it is possible to approximate the maximum fraction of satisfiable constraints up to an additive error $epsilon$ using $q^{O(k)} cdot (k/epsilon)^{O(1)}$ levels of the sum-of-squares SDP hierarchy, provided $gamma leq epsilon^{O(1)} cdot (1/(kq))^{O(k)}$. Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs, which can be used to extend the results for approximating 2-CSPs on low threshold-rank graphs. We show that if an instance of MAX k-CSP has threshold rank $r$ for a threshold $tau = (epsilon/k)^{O(1)} cdot (1/q)^{O(k)}$, then it is possible to approximately solve the instance up to additive error $epsilon$, using $r cdot q^{O(k)} cdot (k/epsilon)^{O(1)}$ levels of the sum-of-squares hierarchy. As in the case of graphs, high-dimensional expanders (with sufficiently small $gamma$) have threshold rank 1 according to our definition.
We consider the problem of scattering $n$ robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that $n log n$ random bits are necessary to scatter $n$ robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in $o(log log n)$ rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal ($n log n$ random bits are used) and time optimal ($log log n$ rounds are used). This improves the time complexity of previous results in the same setting by a $log n$ factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes its time complexity gap with and without strong multiplicity detection (that is, $O(1)$ time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach it as needed otherwise).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
