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Register a new userClassical algorithms for quantum mean values
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We consider the task of estimating the expectation value of an $n$-qubit tensor product observable $O_1otimes O_2otimes cdots otimes O_n$ in the output state of a shallow quantum circuit. This task is a cornerstone of variational quantum algorithms for optimization, machine learning, and the simulation of quantum many-body systems. Here we study its computational complexity for constant-depth quantum circuits and three types of single-qubit observables $O_j$ which are (a) close to the identity, (b) positive semidefinite, (c) arbitrary. It is shown that the mean value problem admits a classical approximation algorithm with runtime scaling as $mathrm{poly}(n)$ and $2^{tilde{O}(sqrt{n})}$ in cases (a,b) respectively. In case (c) we give a linear-time algorithm for geometrically local circuits on a two-dimensional grid. The mean value is approximated with a small relative error in case (a), while in cases (b,c) we satisfy a less demanding additive error bound. The algorithms are based on (respectively) Barvinoks polynomial interpolation method, a polynomial approximation for the OR function arising from quantum query complexity, and a Monte Carlo method combined with Matrix Product State techniques. We also prove a technical lemma characterizing a zero-free region for certain polynomials associated with a quantum circuit, which may be of independent interest.
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We give efficient quantum algorithms to estimate the partition function of (i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi 2D square lattice, and (iv) the Z_2 lattice gauge theory on a three-dimensional square lattice. Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation. The results are proven for a complex parameter regime of the models. The proofs are based on a mapping relating partition functions to quantum circuits introduced in [Van den Nest et al., Phys. Rev. A 80, 052334 (2009)] and extended here.
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.
Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorns algorithm for matrix scaling and Osbornes algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time $tilde O(sqrt{mn}/varepsilon^4)$ for scaling or balancing an $n times n$ matrix (given by an oracle) with $m$ non-zero entries to within $ell_1$-error $varepsilon$. Their classical analogs use time $tilde O(m/varepsilon^2)$, and every classical algorithm for scaling or balancing with small constant $varepsilon$ requires $Omega(m)$ queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of $n$, at the expense of a worse polynomial dependence on the obtained $ell_1$-error $varepsilon$. We emphasize that even for constant $varepsilon$ these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorns and Osbornes algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorns algorithm for entrywise-positive matrices to the $ell_1$-setting, leading to an $tilde O(n^{1.5}/varepsilon^3)$-time quantum algorithm for $varepsilon$-$ell_1$-scaling in this case. We also prove a lower bound, showing that our quantum algorithm for matrix scaling is essentially optimal for constant $varepsilon$: every quantum algorithm for matrix scaling that achieves a constant $ell_1$-error with respect to uniform marginals needs to make at least $Omega(sqrt{mn})$ queries.
We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to fs block sensitivity.
We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are * For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. * There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Theta(n*log n) queries. We give several extensions and applications of these results.
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