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Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit’s generators. The barren plateau problem represents one of the major bottlenecks for parametrized quantum circuits algorithms. Here, the authors study the known sources of BP using the lens of Lie algebraic theory, finding an expression of the variance of the loss function depending on the dynamical Lie algebra of the circuit.

The expansion in the northeastern direction of the Qinghai-Tibet Plateau has had a significant impact on the topography and geomorphology of the Alxa Block. Among the numerous faults on the southern margin of the Alxa Block, the Beidashan Fault is the fault with the strongest tectonic activity. However, at present, there are few studies on the tectonic distribution characteristics of the Beidashan Fault and its coupling relationship with the surrounding basins. In this paper, by using the measured gravity and magnetotelluric sounding data and through the inversion of gravity and magnetotelluric sounding data, the structural distribution characteristics of the Beidashan Fault and its coupling relationship with the Yabulai Basin and the Chaoshui Basin are studied. The research finds that the Beidashan Fault on the southern margin of the Alxa Block is a normal fault in the shape of a “V”, located at the leading edge of the Beidashan. The Beidashan separates the Yablai Basin from the Chaoshui Basin and provides material sources for the two basins during the evolution process. The low-resistance anomaly area of Beidashan gradually decreases with the increase of depth and converges into a small area in the eastern section. Moreover, the vertical section shows that the low-resistance anomaly beneath Beidashan is divided into two layers at about 8 kilometers, forming an hourglass shape. It might be a volcanic channel that transported the deep, low-resistance material to the upper part. At the same time, both layers of materials have the phenomenon of expanding and moving northward, indicating that the compression of the Beidashan Mountain by the Qinghai-Tibet Plateau causes it to thrust northward, resulting in the Beidashan Fault having both thrust and strike-slip properties.

Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n . These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model. Variational quantum algorithms (VQAs) are a leading candidate for useful applications of near-term quantum computing, but limitations due to unavoidable noise have not been clearly characterized. Here, the authors prove that local Pauli noise can cause vanishing gradients rendering VQAs untrainable.

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V ( θ ) to minimize a cost function C . While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V ( θ ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V ( θ ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V ( θ ) is O ( log n ) . Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation. Parametrised quantum circuits are a promising hybrid classical-quantum approach, but rigorous results on their effective capabilities are rare. Here, the authors explore the feasibility of training depending on the type of cost functions, showing that local ones are less prone to the barren plateau problem.

Majorana fermion is a hypothetical particle that is its own antiparticle. We report transport measurements that suggest the existence of one-dimensional chiral Majorana fermion modes in the hybrid system of a quantum anomalous Hall insulator thin film coupled with a superconductor. As the external magnetic field is swept, half-integer quantized conductance plateaus are observed at the locations of magnetization reversals, giving a distinct signature of the Majorana fermion modes. This transport signature is reproducible over many magnetic field sweeps and appears at different temperatures. This finding may open up an avenue to control Majorana fermions for implementing robust topological quantum computing.

The fractional quantum anomalous Hall effect (FQAHE), the analogue of the fractional quantum Hall effect 1 at zero magnetic field, is predicted to exist in topological flat bands under spontaneous time-reversal-symmetry breaking 2 – 6 . The demonstration of FQAHE could lead to non-Abelian anyons that form the basis of topological quantum computation 7 – 9 . So far, FQAHE has been observed only in twisted MoTe 2 at a moiré filling factor v  > 1/2 (refs. 10 – 13 ). Graphene-based moiré superlattices are believed to host FQAHE with the potential advantage of superior material quality and higher electron mobility. Here we report the observation of integer and fractional QAH effects in a rhombohedral pentalayer graphene–hBN moiré superlattice. At zero magnetic field, we observed plateaus of quantized Hall resistance R x y = h v e 2 at v  = 1, 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moiré superlattice, respectively, accompanied by clear dips in the longitudinal resistance R xx . R xy equals 2 h e 2 at v  = 1/2 and varies linearly with v , similar to the composite Fermi liquid in the half-filled lowest Landau level at high magnetic fields 14 – 16 . By tuning the gate-displacement field D and v , we observed phase transitions from composite Fermi liquid and FQAH states to other correlated electron states. Our system provides an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field 7 – 9 , 17 – 19 , especially considering a lateral junction between FQAHE and superconducting regions in the same device 20 – 22 . Integer and fractional quantum anomalous Hall effects in a rhombohedral pentalayer graphene–hBN moiré superlattice are observed, providing an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field.

One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models—which are shallow, and exhibit no barren plateaus—have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms. Implementations of shallow quantum machine learning models are a promising application of near-term quantum computers, but rigorous results on their trainability are sparse. Here, the authors demonstrate settings where such models are untrainable.

Magic-angle twisted bilayer graphene exhibits intriguing quantum phase transitions triggered by enhanced electron–electron interactions when its flat bands are partially filled. However, the phases themselves and their connection to the putative non-trivial topology of the flat bands are largely unexplored. Here we report transport measurements revealing a succession of doping-induced Lifshitz transitions that are accompanied by van Hove singularities, which facilitate the emergence of correlation-induced gaps and topologically non-trivial subbands. In the presence of a magnetic field, well-quantized Hall plateaus at a filling of 1,2,3 carriers per moiré cell reveal the subband topology and signal the emergence of Chern insulators with Chern numbers, C  = 3,2,1, respectively. Surprisingly, for magnetic fields exceeding 5 T we observe a van Hove singularity at a filling of 3.5, suggesting the possibility of a fractional Chern insulator. This van Hove singularity is accompanied by a crossover from low-temperature metallic, to high-temperature insulating behaviour, characteristic of entropically driven Pomeranchuk-like transitions. A magneto-transport study of twisted bilayer graphene near the magic angle further reveals its rich physics.

Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied. Gradient-based hybrid quantum-classical algorithms are often initialised with random, unstructured guesses. Here, the authors show that this approach will fail in the long run, due to the exponentially-small probability of finding a large enough gradient along any direction.

The demography of human longevity is a contentious topic. On the basis of high-quality data from Italians aged 105 and older, Barbi et al. show that mortality is constant at extreme ages but at levels that decline somewhat across cohorts. Human death rates increase exponentially up to about age 80, then decelerate, and plateau after age 105. Science , this issue p. 1459 A study of centenarians in Italy suggests that human mortality is approximately constant in extreme old age. Theories about biological limits to life span and evolutionary shaping of human longevity depend on facts about mortality at extreme ages, but these facts have remained a matter of debate. Do hazard curves typically level out into high plateaus eventually, as seen in other species, or do exponential increases persist? In this study, we estimated hazard rates from data on all inhabitants of Italy aged 105 and older between 2009 and 2015 (born 1896–1910), a total of 3836 documented cases. We observed level hazard curves, which were essentially constant beyond age 105. Our estimates are free from artifacts of aggregation that limited earlier studies and provide the best evidence to date for the existence of extreme-age mortality plateaus in humans.