253 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Group-theoretic error mitigation enabled by classical shadows and symmetries

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    Estimating expectation values is a key subroutine in many quantum algorithms. However, near-term implementations face two major challenges: a limited number of samples to learn a large collection of observables, and the accumulation of errors in devices without quantum error correction. To address these challenges simultaneously, we develop a quantum error-mitigation strategy which unifies the group-theoretic structure of classical-shadow tomography with symmetries in quantum systems of interest. We refer to our protocol as "symmetry-adjusted classical shadows," as it mitigates errors by adjusting estimators according to how known symmetries are corrupted under those errors. As a concrete example, we highlight global U(1)\mathrm{U}(1) symmetry, which manifests in fermions as particle number and in spins as total magnetization, and illustrate their unification with respective classical-shadow protocols. One of our main results establishes rigorous error and sampling bounds under readout errors obeying minimal assumptions. Furthermore, to probe mitigation capabilities against a more comprehensive class of gate-level errors, we perform numerical experiments with a noise model derived from existing quantum processors. Our analytical and numerical results reveal symmetry-adjusted classical shadows as a flexible and low-cost strategy to mitigate errors from noisy quantum experiments in the ubiquitous presence of symmetry.Comment: 45 pages, 13 figures. Typos corrected and references updated. Open-source code available at https://github.com/zhao-andrew/symmetry-adjusted-classical-shadow

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    An annotated graph model with differential degree heterogeneity for directed networks

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    Directed networks are conveniently represented as graphs in which ordered edges encode interactions between vertices. Despite their wide availability, there is a shortage of statistical models amenable for inference, specially when contextual information and degree heterogeneity are present. This paper presents an annotated graph model with parameters explicitly accounting for these features. To overcome the curse of dimensionality due to modelling degree heterogeneity, we introduce a sparsity assumption and propose a penalized likelihood approach with â„“1 -regularization for parameter estimation. We study the estimation and selection consistency of this approach under a sparse network assumption, and show that inference on the covariate parameter is straightforward, thus bypassing the need for the kind of debiasing commonly employed in â„“1 -penalized likelihood estimation. Simulation and data analysis corroborate our theoretical findings

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    Global Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming

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    The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. [SIAM J. Optim. 29(2), 1211-1239, (2019)]. However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods

    Gaussian resource theories and semidefinite programming hierarchies for quantum information

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    Determining which quantum tasks we can perform with currently available tools and devices is one of the most important goals of quantum information science today. To achieve this requires careful investigation of the capability of current quantum tools as well as development of classical protocols which can assist quantum tasks and amplify their abilities. In this thesis, we approach this problem through two different topics in quantum information theory: Gaussian resource theories and semidefinite programming hierarchies. In the first part of this thesis, we examine the possibility of implementing quantum information processing tasks in the Gaussian platform through the eyes of quantum resource theories. Gaussian states and operations are primary tools for the study of continuous-variable quantum information processing due to their easy accessibility and concise mathematical descriptions, although it has been discovered that they are subject to a number of limitations for advanced quantum information processing tasks. We explore the capability of the Gaussian platform further in the first part of this thesis. Firstly, we investigate whether introducing convex structure to the Gaussian framework can circumvent the known no-go theorem of Gaussian resource distillation. Surprisingly, we find that resource distillation becomes possible — albeit in a limited fashion — when convexity is introduced. Then, we consider the quantum resource theory of Gaussian thermal operations when catalysts are allowed, and examine the abilities of catalytic Gaussian thermal operations by characterising all possible state transformations under them. In the second part of this thesis, we address the problem of characterising quantum cor- relations via semidefinite programming hierarchies. In particular, we focus on characterising quantum correlations of fixed dimension, which is practically relevant to the field of semi- device-independent quantum information processing. Semidefinite programming is a special type of mathematical optimisation, and it is known that some important but difficult problems in quantum information theory admit semidefinite programming relaxations; these include the characterisation of general quantum correlations in the context of non-locality and the distinction of quantum separable states from entangled states. In this second part, we show how to construct a hierarchy of semidefinite programming relaxations for quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. For the proof, we make a connection to a variant of quantum separability problem and employ multipartite quantum de Finetti theorems with linear constraints.Open Acces

    Robust Active and Passive Beamforming for RIS-Assisted Full-Duplex Systems under Imperfect CSI

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    The sixth-generation (6G) wireless technology recognizes the potential of reconfigurable intelligent surfaces (RIS) as an effective technique for intelligently manipulating channel paths through reflection to serve desired users. Full-duplex (FD) systems, enabling simultaneous transmission and reception from a base station (BS), offer the theoretical advantage of doubled spectrum efficiency. However, the presence of strong self-interference (SI) in FD systems significantly degrades performance, which can be mitigated by leveraging the capabilities of RIS. Moreover, accurately obtaining channel state information (CSI) from RIS poses a critical challenge. Our objective is to maximize downlink (DL) user data rates while ensuring quality-of-service (QoS) for uplink (UL) users under imperfect CSI from reflected channels. To address this, we introduce the robust active BS and passive RIS beamforming (RAPB) scheme for RIS-FD, accounting for both SI and imperfect CSI. RAPB incorporates distributionally robust design, conditional value-at-risk (CVaR), and penalty convex-concave programming (PCCP) techniques. Additionally, RAPB extends to active and passive beamforming (APB) with perfect channel estimation. Simulation results demonstrate the UL/DL rate improvements achieved considering various levels of imperfect CSI. The proposed RAPB/APB schemes validate their effectiveness across different RIS deployment and RIS/BS configurations. Benefited from robust beamforming, RAPB outperforms existing methods in terms of non-robustness, deployment without RIS, conventional successive convex approximation, and half-duplex systems
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