Geometric and holonomic quantum computation utilizes intrinsic geometric properties of quantum-mechanical state spaces to realize quantum logic gates. Since both geometric phases and quantum holonomies are global quantities depending only on the evolution paths of quantum systems, quantum gates based on them possess built-in resilience to certain kinds of errors. This review provides an introduction to the topic as well as gives an overview of the theoretical and experimental progress for constructing geometric and holonomic quantum gates and how to combine them with other error-resistant techniques. © 2023 Elsevier B.V. All rights reserved.

Lithium niobate (LNO) is a well established material for surface acoustic wave (SAW) devices including resonators, delay lines and filters. Recently, multi-layer substrates based on LNO thin films have become commercially available. Here, we present a systematic low-temperature study of the performance of SAW devices fabricated on LNO-on-insulator and LNO-on-Silicon substrates and compare them to bulk LNO devices. Our study aims at assessing the performance of these substrates for quantum acoustics, i.e. the integration with superconducting circuits operating in the quantum regime. To this end, we design SAW resonators with a target frequency of 5 GHz and perform experiments at millikelvin temperatures and microwave power levels corresponding to single photons or phonons. The devices are investigated regarding their internal quality factors as a function of the excitation power and temperature, which allows us to characterize and quantify losses and identify the dominating loss mechanism. For the measured devices, fitting the experimental data shows that the quality factors are limited by the coupling of the resonator to a bath of two-level-systems. Our results suggest that SAW devices on thin film LNO on silicon have comparable performance to devices on bulk LNO and are viable for use in SAW-based quantum acoustic devices.

Applications for noisy intermediate-scale quantum computing devices rely on the efficient entanglement of many qubits to reach a potential quantum advantage. Although entanglement is typically generated using two-qubit gates, direct control of strong multiqubit interactions can improve the efficiency of the process. Here, we investigate a system of three superconducting transmon-type qubits coupled via a single flux-tunable coupler. Tuning the frequency of the coupler by adiabatic flux pulses enables us to control the conditional energy shifts between the qubits and directly realize multiqubit interactions. To accurately adjust the resulting controlled relative phases, we describe a gate protocol involving refocusing pulses and adjustable interaction times. This enables the implementation of the full family of pairwise controlled -phase and controlled-controlled-phase gates. Numerical simulations result in fidelities around 99% and gate times below 300 ns using currently achievable system parameters and decoherence rates.

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.

The utility of classical neural networks as universal approximators suggests that their quantum analogues could play an important role in quantum generalizations of machine-learning methods. Inspired by the proposal in Torrontegui and Garcia-Ripoll [Europhys. Lett. 125, 30004 (2019)], we demonstrate a superconducting qubit implementation of a controlled gate, which generalizes the action of a classical perceptron as the basic building block of a quantum neural network. In a two-qubit setup we show full control over the steepness of the perceptron activation function, the input weight and the bias by tuning the gate length, the coupling between the qubits, and the frequency of the applied drive, respectively. In its general form, the gate realizes a multiqubit entangling operation in a single step, whose decomposition into single-and two-qubit gates would require a number of gates that is exponential in the number of qubits. Its demonstrated direct implementation as perceptron in quantum hardware may therefore lead to more powerful quantum neural networks when combined with suitable additional standard gates.

For the efficient implementation of quantum algorithms, practical ways to generate many-body entanglement are a basic requirement. Specifically, coupling multiple qubit pairs at once can be advantageous and may provide multiqubit operations useful in the construction of hardware-tailored algorithms. Here we extend the theory of fractional state transfer and harness the simultaneous coupling of qubits on a chain to engineer a set of nonlocal parity-dependent quantum operations suitable for a wide range of applications. The resulting effective long-range couplings directly implement a parametrizable Trotter-step for Jordan-Wigner fermions, and they can be used for simulations of quantum dynamics, efficient state generation in variational quantum eigensolvers, parity measurements for error-correction schemes, and the generation of efficient multiqubit gates. Moreover, we present numerical simulations of the gate operation in a superconducting quantum circuit architecture, which show a high gate fidelity for realistic experimental parameters.

Quantum algorithms often benefit from the ability to execute multi-qubit (> 2) gates. To date, such multi-qubit gates are typically decomposed into single- and two-qubit gates, particularly in superconducting qubit architectures. The ability to perform multi-qubit operations in a single step could vastly improve the fidelity and execution time of many algorithms. Here, we propose a single shot method for executing an i-Toffoli gate, a three-qubit gate with two control and one target qubit, using currently existing superconducting hardware. We show numerical evidence for a process fidelity over 99.5% and a gate time of 450 ns for superconducting qubits interacting via tunable couplers. Our method can straight forwardly be extended to implement gates with more than two control qubits at similar fidelities.

"In circuit-based quantum computing the available gate set typically consists of single-qubit gates acting on each individual qubit and at least one entangling gate between pairs of qubits. In certain physical architectures, however, some qubits may be ""hidden"" and lacking direct addressability through dedicated control and readout lines, for instance, because of limited on-chip routing capabilities, or because the number of control lines becomes a limiting factor for many-qubit systems. In this case, no single-qubit operations can be applied to the hidden qubits and their state cannot be measured directly. Instead, they may be controlled and read out only via single-qubit operations on connected ""control"" qubits and a suitable set of two-qubit gates. We first discuss the impact of such restricted control capabilities on the performance of specific qubit coupling networks. We then experimentally demonstrate full control and measurement capabilities in a superconducting two-qubit device with local single-qubit control and iSWAP and controlledphase two-qubit interactions enabled by a tunable coupler. We further introduce an iterative tune-up process required to completely characterize the gate set used for quantum process tomography and evaluate the resulting gate fidelities."