An Improved Approximation Algorithm for Quantum Max-Cut

We give an approximation algorithm for Quantum Max-Cut which works by rounding an SDP relaxation to an entangled quantum state. The SDP is used to choose the parameters of a variational quantum circuit. The entangled state is then represented as the quantum circuit applied to a product state. It achieves an approximation ratio of 0.582 on triangle-free graphs. The previous best algorithms of Anshu, Gosset, Morenz, and Parekh, Thompson achieved approximation ratios of 0.531 and 0.533 respectively. In addition, we study the EPR Hamiltonian, which we argue is a natural intermediate problem which isolates some key quantum features of local Hamiltonian problems. For the EPR Hamiltonian, we give an approximation algorithm with approximation ratio $1 / \sqrt{2}$ on all graphs

Promise Clique Homology on weighted graphs is QMA1\text{QMA}_1-hard and contained in QMA\text{QMA}

We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space $X$ and an integer $k$, decide if $X$ contains a $k$-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics [Wit82]. We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every $k+1$-clique in the graph to be a $k$-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise, is $\text{QMA}_1$-hard and contained in $\text{QMA}$. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as spectral sequences. In particular, we exploit a connection between spectral sequences and Hodge theory [For94]. Spectral sequences will play a role analogous to perturbation theory for combinatorial Laplacians. In addition, we develop the simplicial surgery technique used in prior work [CK22]. Our result provides some suggestion that the quantum TDA algorithm [LGZ16] cannot be dequantized. More broadly, we hope that our results will open up new possibilities for quantum advantage in topological data analysis

Bipartite Measurements in Minkowski Space

We study the measurements which Alice and Bob can perform on a bipartite quantum system, where Alice and Bob are spacelike separated. For a measurement to be possible, it must be causal i.e. non-signalling. Within causal measurements, we define four notions of ‘localisability’. Each of the four classes of measurement restricts the actions of Alice and Bob in different ways, and we study their relative power. We end with a discussion of the difficulties posed by non-local measurements for the idea of wavefunction collapse

An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions

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Do Ask, Do Tell: High Levels of Acceptability by Patients of Routine Collection of Sexual Orientation and Gender Identity Data in Four Diverse American Community Health Centers

Background: The Institute of Medicine and The Joint Commission have recommended asking sexual orientation and gender identity (SOGI) questions in clinical settings and including such data in Electronic Health Records (EHRs). This is increasingly viewed as a critical step toward systematically documenting and addressing health disparities affecting lesbian, gay, bisexual, and transgender (LGBT) people. The U.S. government is currently considering whether to include SOGI data collection in the Stage 3 guidelines for the incentive program promoting meaningful use of EHR. However, some have questioned whether acceptable standard measures to collect SOGI data in clinical settings exist. Methods: In order to better understand how a diverse group of patients would respond if SOGI questions were asked in primary care settings, 301 randomly selected patients receiving primary care at four health centers across the U.S. were asked SOGI questions and then asked follow-up questions. This sample was mainly heterosexual, racially diverse, and geographically and regionally broad. Results: There was a strong consensus among patients surveyed about the importance of asking SOGI questions. Most of the LGBT respondents thought that the questions presented on the survey allowed them to accurately document their SOGI. Most respondents—heterosexual and LGBT—answered the questions, and said that they would answer such questions in the future. While there were some age-related differences, respondents of all ages overwhelmingly expressed support for asking SOGI questions and understood the importance of providers' knowing their patients' SOGI. Conclusions: Given current deliberations within national health care regulatory bodies and the government's increased attention to LGBT health disparities, the finding that patients can and will answer SOGI questions has important implications for public policy. This study provides evidence that integrating SOGI data collection into the meaningful use requirements is both acceptable to diverse samples of patients, including heterosexuals, and feasible

Airborne investigation of quasi-specular Ku-band radar scattering for satellite altimetry over snow-covered Arctic sea ice

Surface-based Ku-band radar altimetry investigations indicate the radar signal is typically backscattered from well above the snow-sea ice interface. However, this would induce a bias in satellite altimeter sea ice thickness retrievals not reflected by buoy validation. Our study presents a mechanism to potentially explain this paradox: probabilistic quasi-specular radar scattering from the snow-ice interface. We introduce the theory for this mechanism before identifying it in airborne Ku-band radar observations collected over landfast first year Arctic sea ice near Eureka, Canada, in spring 2016. Based on SAR data, this study area likely represents level first year sea ice across the Arctic. Radar backscatter from the snow and ice interfaces were estimated by co-aligning laser scanner and radar observations with in situ measurements. On average, 4-5 times more radar power was scattered from the snow-ice than the air-snow interface over first-year ice. However, return power varied by up to 20 dB between consecutive radar echoes, particularly from the snow-ice interface, depending on local slope and roughness. Measured laser-radar snow depths were more accurate when radar returns were specular, but there was no systematic bias between airborne and in situ snow depths. The probability and strength of quasi-specular returns depend on the measuring height above and slope distribution of sea ice, so these findings have implications for satellite altimetry snow depth and freeboard estimates. This mechanism could explain the apparent differences in Ku-band radar penetration into snow on sea ice when observed from the range of a surface-, airborne- or satellite-based sensor

On quantum backpropagation, information reuse, and cheating measurement collapse

The success of modern deep learning hinges on the ability to train neural networks at scale. Through clever reuse of intermediate information, backpropagation facilitates training through gradient computation at a total cost roughly proportional to running the function, rather than incurring an additional factor proportional to the number of parameters - which can now be in the trillions. Naively, one expects that quantum measurement collapse entirely rules out the reuse of quantum information as in backpropagation. But recent developments in shadow tomography, which assumes access to multiple copies of a quantum state, have challenged that notion. Here, we investigate whether parameterized quantum models can train as efficiently as classical neural networks. We show that achieving backpropagation scaling is impossible without access to multiple copies of a state. With this added ability, we introduce an algorithm with foundations in shadow tomography that matches backpropagation scaling in quantum resources while reducing classical auxiliary computational costs to open problems in shadow tomography. These results highlight the nuance of reusing quantum information for practical purposes and clarify the unique difficulties in training large quantum models, which could alter the course of quantum machine learning.Comment: 29 pages, 2 figure

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