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Research Briefing 
Efficient optimization of deep neural quantum states
An improved optimization algorithm enables the training of largescale neural quantum states in which the enormous number of neuron connections capture the intricate complexity of quantum manybody wavefunctions. This advance leads to unprecedented accuracy in paradigmatic quantum models, opening up new avenues for simulating and understanding complex quantum phenomena.

Article
 Open AccessUnravelling quantum dynamics using flow equations
The complexity of a manybody quantum state grows exponentially with system size, hindering numerical studies. A unitary flowbased method now enables accurate estimates of longterm properties of one and twodimensional quantum systems.
 S. J. Thomson
 & J. Eisert

Article
 Open AccessEmpowering deep neural quantum states through efficient optimization
An optimization algorithm reduces the cost of training largescale neural quantum states. This leads to accurate computations of the ground states of frustrated magnets and provides evidence of gapless quantumspinliquid phases.
 Ao Chen
 & Markus Heyl

Article 
Timeliness criticality in complex systems
Timeliness is a crucial property for the optimal functioning of sociotechnical systems where delays can propagate. Now it is shown that a stylized model of delay propagation on temporal networks shows a phase transition that can trigger avalanches.
 José Moran
 , Matthijs Romeijnders
 & JeanPhilippe Bouchaud


News & Views 
The importance of spatial heterogeneity in disease transmission
Spatial heterogeneity in disease transmission rates and in mixing patterns between regions makes predicting epidemic trajectories hard. Quantifying the mixing rates within and between spatial regions can improve predictions.
 Emily Paige Harvey
 & Dion R. J. O’Neale

Article 
Quantum spherical codes
Many recent experiments have stored quantum information in bosonic modes, such as photons in resonators or optical fibres. Now an adaptation of the classical spherical codes provides a framework for designing quantum error correcting codes for these platforms.
 Shubham P. Jain
 , Joseph T. Iosue
 & Victor V. Albert

News & Views 
Efficient learning of manybody systems
The Hamiltonian describing a quantum manybody system can be learned using measurements in thermal equilibrium. Now, a learning algorithm applicable to many natural systems has been found that requires exponentially fewer measurements than existing methods.
 Sitan Chen

Article 
Learning quantum Hamiltonians from hightemperature Gibbs states and realtime evolutions
Complexity of learning Hamiltonians from Gibbs states is an important issue for both manybody physics and machine learning. The optimal sample and time complexities of quantum Hamiltonian learning for high temperature has now been proven.
 Jeongwan Haah
 , Robin Kothari
 & Ewin Tang

World View 
Artificial intelligence needs a scientific methoddriven reset
AI needs to develop more solid assumptions, falsifiable hypotheses, and rigorous experimentation.
 Luís A. Nunes Amaral

World View 
Why even specialists struggle with black hole proofs
Mathematical proofs of black hole physics are becoming too complex even for specialists.
 Alejandro Penuela Diaz

Article 
Selforganized intracellular twisters
Cytoplasmic flows in the fruit fly oocyte can reorganize cellular components. These structured vortical flows arise through selforganizing dynamics of microtubules, molecular motors and cytoplasm.
 Sayantan Dutta
 , Reza Farhadifar
 & Michael J. Shelley

Article
 Open AccessFalse vacuum decay via bubble formation in ferromagnetic superfluids
The transition from a metastable state to the ground state in classical manybody systems is mediated by bubble nucleation. This transition has now been experimentally observed in a quantum setting using coupled atomic superfluids.
 A. Zenesini
 , A. Berti
 & G. Ferrari

News & Views 
Intrinsic simplicity of complex systems
Predicting the largescale behaviour of complex systems is challenging because of their underlying nonlinear dynamics. Theoretical evidence now verifies that many complex systems can be simplified and still provide an insightful description of the phenomena of interest.
 Jianxi Gao

Article 
The lowrank hypothesis of complex systems
Although using lowrank matrices is the goto approach to model the dynamics of complex systems, its validity remains formally unconfirmed. An analysis of random networks and realworld data now sheds light on this lowrank hypothesis and its implications.
 Vincent Thibeault
 , Antoine Allard
 & Patrick Desrosiers

Article 
Quantuminspired classical algorithms for molecular vibronic spectra
It has been suggested that Gaussian boson sampling may provide a quantum computational advantage for calculating the vibronic spectra of molecules. Now, an equally efficient classical algorithm has been identified.
 Changhun Oh
 , Youngrong Lim
 & Liang Jiang

News & Views 
Graph theory captures hardcore exclusion
Physical networks, composed of nodes and links that occupy a spatial volume, are hard to study with conventional techniques. A metagraph approach that elucidates the impact of physicality on network structure has now been introduced.
 Zoltán Toroczkai

Article 
Impact of physicality on network structure
Physical networks are systems composed of physical entities, which conventional graphbased approaches fail to capture. Theoretical work now introduces a metagraph technique to uncover the impact of physicality on the structure of networks.
 Márton Pósfai
 , Balázs Szegedy
 & AlbertLászló Barabási

Article 
Geometric description of clustering in directed networks
Network geometry is an emerging framework used to describe several topological and organizational features of complex networks. Now this approach has been extended to directed networks, which contain both symmetric and asymmetric interactions.
 Antoine Allard
 , M. Ángeles Serrano
 & Marián Boguñá

Article 
Laser ablation and fluid flows reveal the mechanism behind spindle and centrosome positioning
Cell division is governed by the positioning of a cytoskeletal structure called the spindle. Two methods, one based on laser ablation and the other on fluid flow assessments, are now shown to be useful tools for studying spindle positioning.
 HaiYin Wu
 , Gökberk Kabacaoğlu
 & Daniel J. Needleman


Article 
Interactive cryptographic proofs of quantumness using midcircuit measurements
Being able to perform qubit measurements within a quantum circuit and adapt to their outcome broadens the power of quantum computers. These midcircuit measurements have now been used to implement a cryptographic proof of nonclassical behaviour.
 Daiwei Zhu
 , Gregory D. KahanamokuMeyer
 & Christopher Monroe

Article 
The hardness of random quantum circuits
Quantum computers are believed to exponentially outperform classical computers at some tasks, but it is hard to make guarantees about the limits of classical computers. It has now been proven that classical computers cannot efficiently simulate most quantum circuits.
 Ramis Movassagh

Article
 Open AccessScaling and intermittency in turbulent flows of elastoviscoplastic fluids
Elastoviscoplastic fluids combine solid and liquidlike behaviour depending on applied stress. Simulations of elastoviscoplastic fluids at high Reynolds number now show that plasticity plays a key role in the turbulent flows seen in these systems, leading for example to intermittency.
 Mohamed S. Abdelgawad
 , Ianto Cannon
 & Marco E. Rosti

Article 
A quantum complexity lower bound from differential geometry
Quantum operations can be considered as points in a highdimensional space in which distance reflects the similarity of two operations. Applying differentialgeometric methods in this picture gives insights into the complexity of quantum systems.
 Adam R. Brown


Comment 
Depictions of fluid phenomena in art
An analysis of representations of fluid flows in classical paintings reveals scientific inaccuracies. Some of these misrepresentations might be caused by a limited understanding of fluid dynamics and others by deliberate artistic choices.
 Rouslan Krechetnikov

News & Views 
Sheep lead the way
Experiments with small flocks of sheep show intermittent collective motion events driven by random leaders that guide the group. A model reveals information pooling capabilities, suggesting a mechanism for swarm intelligence.
 Cristián Huepe

Research Briefing 
Understanding the formation of gas bubbles at liquid–liquid interfaces
The formation of bubbles at liquid–liquid interfaces is challenging to explain because gas pockets cannot be stabilized by cracks on solid impurities. Experiments show that a difference in the gas solubilities of two immiscible liquids provides a gas reservoir, which allows gas to accumulate at the interface, leading to bubble formation.

Perspective 
Citation inequity and gendered citation practices in contemporary physics
The undercitation of woman authors in physics is quantified and measures that could overcome this inequity are presented.
 Erin G. Teich
 , Jason Z. Kim
 & Dani S. Bassett

Comment 
The flow from simulation to reality
Fluid simulations today are remarkably realistic. In this Comment I discuss some of the most striking results from the past 20 years of computer graphics research that made this happen.
 Károly ZsolnaiFehér

News & Views 
A cool quantum simulator
Experiments with ultracold atoms can be used to create nearly ideal quantum simulations of theoretical models. A realization of a model of exotic magnetism has tested the limits of what can be studied numerically on a classical computer.
 Evgeny Kozik

Article 
Exact solutions for the wrinkle patterns of confined elastic shells
Wrinkling happens because of mechanical instabilities arising from length mismatches. A theory now describes wrinkling in confined elastic shells and is expected to be relevant for the controlled design of complex wrinkle patterns.
 Ian Tobasco
 , Yousra Timounay
 & Eleni Katifori

Article
 Open AccessTransition from subRayleigh anticrack to supershear crack propagation in snow avalanches
Avalanches can occur when a porous snow layer lies beneath a dense cohesive snow slab. Field experiments and simulations now reveal different crackpropagation regimes in slab avalanches, similar to rupture propagation following an earthquake.
 Bertil Trottet
 , Ron Simenhois
 & Johan Gaume

Article 
The temporal rich club phenomenon
Uncovering structures in temporal networks requires different tools than in their static counterparts. A metric now quantifies whether the nodes with a large number of connections also tend to stay simultaneously connected for longer times.
 Nicola Pedreschi
 , Demian Battaglia
 & Alain Barrat



Article 
Degreepreserving network growth
Network models rarely fix the number of connections of each node during evolution, despite this being needed in realworld applications. Addressing this need, a new approach can grow scalefree networks without preferential attachment.
 Shubha R. Kharel
 , Tamás R. Mezei
 & Zoltan Toroczkai

Article 
Measuring the capabilities of quantum computers
Evaluations of quantum computers across architectures need reliable benchmarks. A class of benchmarks that can directly reflect the structure of any algorithm shows that different quantum computers have considerable variations in performance.
 Timothy Proctor
 , Kenneth Rudinger
 & Robin BlumeKohout



Letter 
Clonal dominance in excitable cell networks
As tissues grow, a small fraction of cells can give rise to a large fraction of the tissue. A model borrowed from forest fires suggests that this can occur spontaneously in development as a collective property of the cell interaction network.
 Jasmin Imran Alsous
 , Jan Rozman
 & Stanislav Y. Shvartsman


Perspective 
The physics of higherorder interactions in complex systems
Network representations of complex systems are limited to pairwise interactions, but realworld systems often involve higherorder interactions. This Perspective looks at the new physics emerging from attempts to characterize these interactions.
 Federico Battiston
 , Enrico Amico
 & Giovanni Petri

Article 
A rigorous and robust quantum speedup in supervised machine learning
Many quantum machine learning algorithms have been proposed, but it is typically unknown whether they would outperform classical methods on practical devices. A specially constructed algorithm shows that a formal quantum advantage is possible.
 Yunchao Liu
 , Srinivasan Arunachalam
 & Kristan Temme

Letter 
Quantum advantage for computations with limited space
In general, it isn’t known when a quantum computer will have an advantage over a classical device. Now it’s proven that computers with limited working memory are more powerful if they are quantum.
 Dmitri Maslov
 , JinSung Kim
 & Sarah Sheldon

Article 
Learning models of quantum systems from experiments
Quantum systems make it challenging to determine candidate Hamiltonians from experimental data. An automated protocol is presented and its capabilities to infer the correct Hamiltonian are demonstrated in a nitrogenvacancy centre setup.
 Antonio A. Gentile
 , Brian Flynn
 & Anthony Laing

Measure for Measure 
One outstanding path from A to B
What does it mean for an individual to be ‘important’ or for a connection to be ‘outstanding’? The answer depends on context, as Sarah Shugars and Samuel V. Scarpino explain.
 Sarah Shugars
 & Samuel V. Scarpino

Article 
Topological limits to the parallel processing capability of network architectures
The ability to perform multiple tasks simultaneously is a key characteristic of parallel architectures. Using methods from statistical physics, this study provides analytical results that quantify the limitations of processing capacity for different types of tasks in neural networks.
 Giovanni Petri
 , Sebastian Musslick
 & Jonathan D. Cohen