Quantum complexity of the Kronecker coefficients

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a QMA verifier, where QMA is the quantum analogue of NP. This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of quantum approximate counting problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in QMA, complementing a recent NP-hardness result of Ikenmeyer, Mulmuley and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error.Comment: Updated missing appendix for Kronecker distribution samplin

Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

We consider the problem of approximating the partition function of a classical Hamiltonian using simulated annealing. This requires the computation of a cooling schedule, and the ability to estimate the mean of the Gibbs distributions at the corresponding inverse temperatures. We propose classical and quantum algorithms for these two tasks, achieving two goals: (i) we simplify the seminal work of \v{S}tefankovi\v{c}, Vempala and Vigoda (\emph{J.~ACM}, 56(3), 2009), improving their running time and almost matching that of the current classical state of the art; (ii) we quantize our new simple algorithm, improving upon the best known algorithm for computing partition functions of many problems, due to Harrow and Wei (SODA 2020). A key ingredient of our method is the paired-product estimator of Huber (\emph{Ann.\ Appl.\ Probab.}, 25(2),~2015). The proposed quantum algorithm has two advantages over the classical algorithm: it has quadratically faster dependence on the spectral gap of the Markov chains as well as the precision, and it computes a shorter cooling schedule, which matches the length conjectured to be optimal by \v{S}tefankovi\v{c}, Vempala and Vigoda.Comment: Comments welcom

Supervised learning with quantum enhanced feature spaces

Machine learning and quantum computing are two technologies each with the potential for altering how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous for pattern recognition, with support vector machines (SVMs) being the most well-known method for classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element to computational speed-ups afforded by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference. Here, we propose and experimentally implement two novel methods on a superconducting processor. Both methods represent the feature space of a classification problem by a quantum state, taking advantage of the large dimensionality of quantum Hilbert space to obtain an enhanced solution. One method, the quantum variational classifier builds on [1,2] and operates through using a variational quantum circuit to classify a training set in direct analogy to conventional SVMs. In the second, a quantum kernel estimator, we estimate the kernel function and optimize the classifier directly. The two methods present a new class of tools for exploring the applications of noisy intermediate scale quantum computers [3] to machine learning.Comment: Fixed typos, added figures and discussion about quantum error mitigatio

Colour texture modelling.

The main goal of the thesis was to find mathematical models which can be used for texture modelling. The advantages of mathematical texture modelling are the huge compression ratio, unachievable by any other means and needed in both distributed and real-time graphical systems, and the possibility to model any required size texture. We focused on smooth colour textures and we present several methods which we used to synthesize artificial natural colour textures. Rough textures modelling is the possible generalization of our presented approach.. We developed three main models in the thesis - a novel multi-resolution Markov random field model, two and tree dimensional causal autoregressive random field models, respectively. The whole model of texture modelling is divided into four main steps - virtual model geometry inference, texture analysis, texture synthesis and texture mapping step.Available from STL Prague, CZ / NTK - National Technical LibrarySIGLECZCzech Republi

Amplitude Ratios and Neural Network Quantum States

Neural Network Quantum States (NQS) represent quantum wavefunctions by artificial neural networks. Here we study the wavefunction access provided by NQS defined in [Science, \textbf{355}, 6325, pp. 602-606 (2017)] and relate it to results from distribution testing. This leads to improved distribution testing algorithms for such NQS. It also motivates an independent definition of a wavefunction access model: the amplitude ratio access. We compare it to sample and sample and query access models, previously considered in the study of dequantization of quantum algorithms. First, we show that the amplitude ratio access is strictly stronger than sample access. Second, we argue that the amplitude ratio access is strictly weaker than sample and query access, but also show that it retains many of its simulation capabilities. Interestingly, we only show such separation under computational assumptions. Lastly, we use the connection to distribution testing algorithms to produce an NQS with just three nodes that does not encode a valid wavefunction and cannot be sampled from.Comment: 31 pages, 5 figs. Typos corrected. Comments encouraged and very welcome