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Quantum machine learning is a rapidly growing field at the intersection of quantum technology and artificial intelligence. This review provides a two-fold overview of several key approaches that can offer advancements in both the development of quantum technologies and the power of artificial intelligence. Among these approaches are quantum-enhanced algorithms, which apply quantum software engineering to classical information processing to improve keystone machine learning solutions. In this context, we explore the capability of hybrid quantum-classical neural networks to improve model generalization and increase accuracy while reducing computational resources. We also illustrate how machine learning can be used both to mitigate the effects of errors on presently available noisy intermediate-scale quantum devices, and to understand quantum advantage via an automatic study of quantum walk processes on graphs. In addition, we review how quantum hardware can be enhanced by applying machine learning to fundamental and applied physics problems as well as quantum tomography and photonics. We aim to demonstrate how concepts in physics can be translated into practical engineering of machine learning solutions using quantum software.

The efficient optimization of variational quantum algorithms (VQAs) is critical for their successful application in quantum computing. The Quantum Natural Gradient (QNG) method, which leverages the geometry of quantum state space, has demonstrated improved convergence compared to standard gradient descent (Stokes et al. in Quantum 4:269, 2020 ). In this work, we introduce the Modified Conjugate Quantum Natural Gradient (CQNG), an optimization algorithm that integrates QNG with principles from the nonlinear conjugate-gradient method. Unlike QNG, which employs a fixed learning rate, CQNG dynamically adjusts its hyperparameters at each step, enhancing both efficiency and flexibility. Numerical simulations show that CQNG achieves faster convergence and reduces quantum-resource requirements compared to QNG across various optimization scenarios, even when strict conjugacy conditions are not fully satisfied—hence the term “Modified Conjugate.” These results highlight CQNG as a promising optimization technique for improving the performance of VQAs.

A nice and interesting property of any pure tensor product state is that each such state has distillable entangled states at an arbitrarily small distance ϵ in its neighborhood. We say that such nearby states are ϵ -entangled, and we call the tensor product state in that case, a “boundary separable state,” as there is entanglement at any distance from this “boundary.” Here we find a huge class of separable states that also share the property mentioned above—they all have ϵ -entangled states at any small distance in their neighborhood. Furthermore, the entanglement they have is proved to be distillable. We then extend this result to the discordant/classical cut and show that all classical states (correlated and uncorrelated) have discordant states at distance ϵ , and provide a constructive method for finding ϵ -discordant states.

This volume contains ten lectures presented in the series ULB Lectures in Nonlinear Optics at the Universite Libre de Bruxelles during the period October 28 to November 4, 1991. A large part of the first six lectures is taken from material prepared for a book of somewhat larger scope which will be published,by Springer under the title Quantum Statistical Methods in Quantum Optics. The principal reason for the early publication of the present volume concerns the material contained in the last four lectures. Here I have put together, in a more or less systematic way, some ideas about the use of stochastic wavefunctions in the theory of open quantum optical systems. These ideas were developed with the help of two of my students, Murray Wolinsky and Liguang Tian, over a period of approximately two years. They are built on a foundation laid down in a paper written with Surendra Singh, Reeta Vyas, and Perry Rice on waiting-time distributions and wavefunction collapse in resonance fluorescence [Phys. Rev. A, 39, 1200 (1989)]. The ULB lecture notes contain my first serious atte~pt to give a complete account of the ideas and their potential applications. I am grateful to Professor Paul Mandel who, through his invitation to give the lectures, stimulated me to organize something useful out of work that may, otherwise, have waited considerably longer to be brought together.

Quantum computing promises to offer substantial speed-ups over its classical counterpart for certain problems. However, the greatest impediment to realizing its full potential is noise that is inherent to these systems. The widely accepted solution to this challenge is the implementation of fault-tolerant quantum circuits, which is out of reach for current processors. Here we report experiments on a noisy 127-qubit processor and demonstrate the measurement of accurate expectation values for circuit volumes at a scale beyond brute-force classical computation. We argue that this represents evidence for the utility of quantum computing in a pre-fault-tolerant era. These experimental results are enabled by advances in the coherence and calibration of a superconducting processor at this scale and the ability to characterize 1 and controllably manipulate noise across such a large device. We establish the accuracy of the measured expectation values by comparing them with the output of exactly verifiable circuits. In the regime of strong entanglement, the quantum computer provides correct results for which leading classical approximations such as pure-state-based 1D (matrix product states, MPS) and 2D (isometric tensor network states, isoTNS) tensor network methods 2 , 3 break down. These experiments demonstrate a foundational tool for the realization of near-term quantum applications 4 , 5 . Experiments on a noisy 127-qubit superconducting quantum processor report the accurate measurement of expectation values beyond the reach of current brute-force classical computation, demonstrating evidence for the utility of quantum computing before fault tolerance.

Future quantum computers capable of solving relevant problems will require a large number of qubits that can be operated reliably 1 . However, the requirements of having a large qubit count and operating with high fidelity are typically conflicting. Spins in semiconductor quantum dots show long-term promise 2 , 3 but demonstrations so far use between one and four qubits and typically optimize the fidelity of either single- or two-qubit operations, or initialization and readout 4 – 11 . Here, we increase the number of qubits and simultaneously achieve respectable fidelities for universal operation, state preparation and measurement. We design, fabricate and operate a six-qubit processor with a focus on careful Hamiltonian engineering, on a high level of abstraction to program the quantum circuits, and on efficient background calibration, all of which are essential to achieve high fidelities on this extended system. State preparation combines initialization by measurement and real-time feedback with quantum-non-demolition measurements. These advances will enable testing of increasingly meaningful quantum protocols and constitute a major stepping stone towards large-scale quantum computers. The universal control of six qubits in a 28 Si/SiGe quantum dot array is demonstrated, achieving Rabi oscillations for each qubit with visibilities of 93.5–98.0%, implying high readout and initialization fidelities.

The dynamics of quantum information in strongly interacting systems, known as quantum information scrambling, has recently become a common thread in our understanding of black holes, transport in exotic non-Fermi liquids, and many-body analogs of quantum chaos. To date, verified experimental implementations of scrambling have focused on systems composed of two-level qubits. Higher-dimensional quantum systems, however, may exhibit different scrambling modalities and are predicted to saturate conjectured speed limits on the rate of quantum information scrambling. We take the first steps toward accessing such phenomena, by realizing a quantum processor based on superconducting qutrits (three-level quantum systems). We demonstrate the implementation of universal two-qutrit scrambling operations and embed them in a five-qutrit quantum teleportation protocol. Measured teleportation fidelitiesFavg=0.568±0.001confirm the presence of scrambling even in the presence of experimental imperfections and decoherence. Our teleportation protocol, which connects to recent proposals for studying traversable wormholes in the laboratory, demonstrates how quantum technology that encodes information in higher-dimensional systems can exploit a larger and more connected state space to achieve the resource efficient encoding of complex quantum circuits.

A two-qubit quantum processor in a silicon device is demonstrated, which can perform the Deutsch–Josza algorithm and the Grover search algorithm. Taken for a spin in silicon The development of platforms for spin-based quantum computing continues apace. The individual components of such a system have been the subject of much investigation, and they have been assembled to implement specific quantum-computational algorithms. Thomas Watson and colleagues have now taken such component integration and control to the next level. Using two single-electron-spin qubits in a silicon-based double quantum dot, they realize a system that can be simply programmed to perform different quantum algorithms on demand. Now that it is possible to achieve measurement and control fidelities for individual quantum bits (qubits) above the threshold for fault tolerance, attention is moving towards the difficult task of scaling up the number of physical qubits to the large numbers that are needed for fault-tolerant quantum computing 1 , 2 . In this context, quantum-dot-based spin qubits could have substantial advantages over other types of qubit owing to their potential for all-electrical operation and ability to be integrated at high density onto an industrial platform 3 , 4 , 5 . Initialization, readout and single- and two-qubit gates have been demonstrated in various quantum-dot-based qubit representations 6 , 7 , 8 , 9 . However, as seen with small-scale demonstrations of quantum computers using other types of qubit 10 , 11 , 12 , 13 , combining these elements leads to challenges related to qubit crosstalk, state leakage, calibration and control hardware. Here we overcome these challenges by using carefully designed control techniques to demonstrate a programmable two-qubit quantum processor in a silicon device that can perform the Deutsch–Josza algorithm and the Grover search algorithm—canonical examples of quantum algorithms that outperform their classical analogues. We characterize the entanglement in our processor by using quantum-state tomography of Bell states, measuring state fidelities of 85–89 per cent and concurrences of 73–82 per cent. These results pave the way for larger-scale quantum computers that use spins confined to quantum dots.

The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling. Based on OTOCs, three different concepts—quantum chaos, holographic duality, and information scrambling—are found to be intimately related to each other. Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators. Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and nonintegrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.

Neutral-atom arrays have recently emerged as a promising platform for quantum information processing. One important remaining roadblock for the large-scale application of these systems is the ability to perform error-corrected quantum operations. To entangle the qubits in these systems, atoms are typically excited to Rydberg states, which could decay or give rise to various correlated errors that cannot be addressed directly through traditional methods of fault-tolerant quantum computation. In this work, we provide the first complete characterization of these sources of error in a neutral-atom quantum computer and propose hardware-efficient, fault-tolerant quantum computation schemes that mitigate them. Notably, we develop a novel and distinctly efficient method to address the most important errors associated with the decay of atomic qubits to states outside of the computational subspace. These advances allow us to significantly reduce the resource cost for fault-tolerant quantum computation compared to existing, general-purpose schemes. Our protocols can be implemented in the near term using state-of-the-art neutral-atom platforms with qubits encoded in both alkali and alkaline-earth atoms.