Energy and centrality dependence of particle multiplicity in heavy ion collisions from sNN\sqrt{s_{_{NN}}} = 20 to 2760 GeV

The centrality dependence of midrapidity charged-particle multiplicities at a nucleon-nucleon center-of-mass energy of 2.76 TeV from CMS are compared to PHOBOS data at 200 and 19.6 GeV. The results are first fitted with a two-component model which parameterizes the separate contributions of nucleon participants and nucleon-nucleon collisions. A more direct comparison involves ratios of multiplicity densities per participant pair between the different collision energies. The results support and extend earlier indications that the influences of centrality and collision energy on midrapidity charged-particle multiplicities are to a large degree independent.Comment: 5 pages, 2 figures, 1 table, Replaced with published version, v3 has fixed typ

Symmetry-protected dissipative preparation of matrix product states

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare the Affleck-Kennedy-Lieb-Tasaki (AKLT) states, which features symmetry-protected topological order and non-trivial edge excitations. We show that the use of symmetry allows for robust experimental implementation without fine-tuned control parameters. Numerical simulations show that the preparation time scales polynomially in system size $n$. Furthermore, we demonstrate that this scaling can be improved to $\mathcal{O}(\log^2n)$ by using parallel preparation of AKLT segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into Rydberg state via Electromagnetically Induced Transparency is proposed, and generalizations to a broader class of matrix product states are discussed

Error suppression in Hamiltonian based quantum computation using energy penalties

We consider the use of quantum error detecting codes, together with energy penalties against leaving the codespace, as a method for suppressing environmentally induced errors in Hamiltonian based quantum computation. This method was introduced in [1] in the context of quantum adiabatic computation, but we consider it more generally. Specifically, we consider a computational Hamiltonian, which has been encoded using the logical qubits of a single-qubit error detecting code, coupled to an environment of qubits by interaction terms that act one-locally on the system. Energy penalty terms are added that penalize states outside of the codespace. We prove that in the limit of infinitely large penalties, one-local errors are completely suppressed, and we derive some bounds for the finite penalty case. Our proof technique involves exact integration of the Schrodinger equation, making no use of master equations or their assumptions. We perform long time numerical simulations on a small (one logical qubit) computational system coupled to an environment and the results suggest that the energy penalty method achieves even greater protection than our bounds indicate.Comment: 26 pages, 7 figure

Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models

The Quantum Approximate Optimization Algorithm (QAOA) is a general purpose quantum algorithm designed for combinatorial optimization. We analyze its expected performance and prove concentration properties at any constant level (number of layers) on ensembles of random combinatorial optimization problems in the infinite size limit. These ensembles include mixed spin models and Max-$q$-XORSAT on sparse random hypergraphs. To enable our analysis, we prove a generalization of the multinomial theorem which is a technical result of independent interest. We then show that the performance of the QAOA at constant levels for the pure $q$-spin model matches asymptotically the ones for Max-$q$-XORSAT on random sparse Erd\H{o}s-R\'{e}nyi hypergraphs and every large-girth regular hypergraph. Through this correspondence, we establish that the average-case value produced by the QAOA at constant levels is bounded away from optimality for pure $q$-spin models when $q\ge 4$ is even. This limitation gives a hardness of approximation result for quantum algorithms in a new regime where the whole graph is seen.Comment: 12+46 page

The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers $p$. While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of $n$ spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can be tailored to efficiently find an approximate solution for a typical instance of the SK model to within $(1-\epsilon)$ times the ground state energy. We hope to match its performance with the QAOA. Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the $2p$ QAOA parameters, in the infinite size limit that can be evaluated on a computer with $O(16^p)$ complexity. We evaluate the formula up to $p=12$, and find that the QAOA at $p=11$ outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as $n\to\infty$, measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.Comment: 32 pages, 2 figures, 2 tables; improved presentation of figures and iterative formul

Bayesian Inference using the Proximal Mapping: Uncertainty Quantification under Varying Dimensionality

In statistical applications, it is common to encounter parameters supported on a varying or unknown dimensional space. Examples include the fused lasso regression, the matrix recovery under an unknown low rank, etc. Despite the ease of obtaining a point estimate via the optimization, it is much more challenging to quantify their uncertainty -- in the Bayesian framework, a major difficulty is that if assigning the prior associated with a $p$-dimensional measure, then there is zero posterior probability on any lower-dimensional subset with dimension $d<p$; to avoid this caveat, one needs to choose another dimension-selection prior on $d$, which often involves a highly combinatorial problem. To significantly reduce the modeling burden, we propose a new generative process for the prior: starting from a continuous random variable such as multivariate Gaussian, we transform it into a varying-dimensional space using the proximal mapping. This leads to a large class of new Bayesian models that can directly exploit the popular frequentist regularizations and their algorithms, such as the nuclear norm penalty and the alternating direction method of multipliers, while providing a principled and probabilistic uncertainty estimation. We show that this framework is well justified in the geometric measure theory, and enjoys a convenient posterior computation via the standard Hamiltonian Monte Carlo. We demonstrate its use in the analysis of the dynamic flow network data.Comment: 26 pages, 4 figure