Root finding with threshold circuits

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur

DNF Sparsification and a Faster Deterministic Counting Algorithm

Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f). It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be $\epsilon$-approximated by a width $w$ DNF with at most $(w\log(1/\epsilon))^{O(w)}$ terms. We combine our sparsification result with the work of Luby and Velikovic to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic $n^{\tilde{O}(\log \log(n))}$ time algorithm that computes an additive $\epsilon$ approximation to the fraction of satisfying assignments of f for \epsilon = 1/\poly(\log n). The previous best result due to Luby and Velickovic from nearly two decades ago had a run-time of $n^{\exp(O(\sqrt{\log \log n}))}$.Comment: To appear in the IEEE Conference on Computational Complexity, 201

Anticoncentration theorems for schemes showing a quantum speedup

One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.Comment: 12+2 pages, added applications sectio

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist? We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of $[[N,k,d,\varepsilon]]$ approximate QLDPC codes that encode $k = \widetilde{\Omega}(N)$ logical qubits into $N$ physical qubits with distance $d = \widetilde{\Omega}(N)$ and approximation infidelity $\varepsilon = \mathcal{O}(1/\textrm{polylog}(N))$. The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in $\mathcal{O}(\textrm{polylog} N)$ projectors. We prove the existence of an efficient encoding map, and we show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth. Finally, we show that the spectral gap of the code Hamiltonian is $\widetilde{\Omega}(N^{-3.09})$ by analyzing a spacetime circuit-to-Hamiltonian construction for a bitonic sorting network architecture that is spatially local in $\textrm{polylog}(N)$ dimensions.Comment: 51 pages, 13 figure