Although single and two-qubit gates are sufficient for universal quantum computation, single-shot three-qubit gates greatly simplify quantum error correction schemes and algorithms. We design fast, high-fidelity three-qubit entangling gates based on microwave pulses for transmon qubits coupled through a superconducting resonator. We show that when interqubit frequency differences are comparable to single-qubit anharmonicities, errors occur primarily through a single unwanted transition. This feature enables the design of fast three-qubit gates based on simple analytical pulse shapes that are engineered to minimize such errors. We show that a three-qubit ccz gate can be performed in 260 ns with fidelities exceeding 99.38%, or 99.99% with numerical optimization.Quantum information processing is one of the most exciting and rapidly growing fields of modern science, in large part due to quantum algorithms, which promise exponential speedup in solving important problems. Quantum two-level systems (qubits) are the fundamental carriers of quantum information, and among the most promising of these are qubits based on superconducting circuits [1,2]. Such an architecture is attractive because of the mature circuit fabrication technology and the ability to couple qubits together via resonators to implement logic gates [3].For universal quantum computing, a certain set of high-fidelity logic gates suffices to implement any algorithm; this set is comprised of single-qubit gates along with one maximally entangling two-qubit gate. Threequbit gates, which play a prominent role in algorithms, can be decomposed in terms of a sequence of single-and two-qubit gates [4]. In the case of a maximally entangling three-qubit control-control-z (ccz) gate (described below), one must perform seven single-qubit gates and six entangling two-qubit gates, as shown in Fig. 1. The large number of gates needed makes it natural to explore whether a direct, single-shot three-qubit gate is preferable in terms of speed and fidelity.There are two ways to implement logic gates in superconducting qubit systems. One is via tuning of the energy levels, which brings states into and out of resonance. The other is via oscillating microwave fields that induce transitions between energy levels, which offers the advantage of less susceptibility to charge noise. Both approaches face the challenge of spectral crowding in these systems, especially in the context of superconducting transmon qubits where each qubit is a weakly anharmonic oscillator out of which the two lowest levels are selected to encode information [5]. In the case of many qubits coupled together, there is a large number of closely spaced transitions that need to be avoided during quantum gate operations. Generically, a way to avoid unwanted transitions is to consider the time-energy uncertainty principle, which tells us to make the operations slow. In realistic systems, however, this is not an option, as we need to perform operations at a time scale that is much faster than decay and decoherence. ...

The study of the impact of noise on quantum circuits is especially relevant to guide the progress of Noisy Intermediate-Scale Quantum (NISQ) computing. In this paper, we address the pulse-level simulation of noisy quantum circuits with the Quantum Toolbox in Python (QuTiP). We introduce new tools in qutip-qip, QuTiP's quantum information processing package. These tools simulate quantum circuits at the pulse level, leveraging QuTiP's quantum dynamics solvers and control optimization features. We show how quantum circuits can be compiled on simulated processors, with control pulses acting on a target Hamiltonian that describes the unitary evolution of the physical qubits. Various types of noise can be introduced based on the physical model, e.g., by simulating the Lindblad density-matrix dynamics or Monte Carlo quantum trajectories. In particular, the user can define environment-induced decoherence at the processor level and include noise simulation at the level of control pulses. We illustrate how the Deutsch-Jozsa algorithm is compiled and executed on a superconducting-qubit-based processor, on a spin-chain-based processor and using control optimization algorithms. We also show how to easily reproduce experimental results on cross-talk noise in an ion-based processor, and how a Ramsey experiment can be modeled with Lindblad dynamics. Finally, we illustrate how to integrate these features with other software frameworks.

The "hierarchical equations of motion" (HEOM) method is a powerful numerical approach to solve the dynamics and steady-state of a quantum system coupled to a non-Markovian and nonperturbative environment. Originally developed in the context of physical chemistry, it has also been extended and applied to problems in solid-state physics, optics, single-molecule electronics, and biological physics. Here we present a numerical library in Python, integrated with the powerful QuTiP platform, which implements the HEOM for both bosonic and fermionic environments. We demonstrate its utility with a series of examples. For the bosonic case, we present examples for fitting arbitrary spectral densities, modelling a Fenna-Matthews-Olsen photosynthetic complex, and simulating dynamical decoupling of a spin from its environment. For the fermionic case, we present an integrable single-impurity example, used as a benchmark of the code, and a more complex example of an impurity strongly coupled to a single vibronic mode, with applications in singlemolecule electronics.

Quantum simulators, machines that can replicate the dynamics of quantum systems, are being built as useful devices and are seen as a stepping stone to universal quantum computers. A key difference between the two is that computers have the ability to perform the logic gates that make up algorithms. We propose a method for learning how to construct these gates efficiently by using the simulator to perform optimal control on itself. This bypasses two major problems of purely classical approaches to the control problem: the need to have an accurate model of the system, and a classical computer more powerful than the quantum one to carry out the required simulations. Strong evidence that the scheme scales polynomially in the number of qubits, for systems of up to 9 qubits with Ising interactions, is presented from numerical simulations carried out in different topologies. This suggests that this in situ approach is a practical way of upgrading quantum simulators to computers.Recent and ongoing work on building large quantum systems is leading to simulators that are able to model physical phenomena, allowing questions about the underlying science to be answered [1][2][3]. These machines contain a register of quantum particles, typically two level quantum systems (qubits) storing quantum information. The presence of interactions between these leads to dynamics that, by varying control parameters in the system Hamiltonian, can replicate the quantum behaviour of systems of interest. This, however, is less general than a quantum computer which is able is to perform a universal set of logic gates on the qubits [4].Provided some control parameters can be varied in time, it is in principle possible to do an arbitrary gate on a quantum many-body system such as a quantum simulator [5][6][7]. Finding the right time-dependency however relies almost exclusively on numerical methods, especially when physical constraints on the control fields are taken into account [8,9]. These methods require a very precise knowledge of the parameters of a system, a daunting task for a machine with a huge number of degrees of freedom. Furthermore, they are intractable on a classical computer if the quantum simulator we want to solve the problem for is large enough to do something beyond the capabilities of classical computers. These two difficulties provide a major obstacle in using quantum simulators to perform arbitrary computation.We circumvent these problems by showing how well-known existing numerical methods can be translated to run in situ on the quantum simulator itself, as illustrated in Fig.1. A mix of an-

The question of open-loop control in the Gaussian regime may be cast by asking which Gaussian unitary transformations are reachable by turning on and off a given set of quadratic Hamiltonians. For compact groups, including finite dimensional unitary groups, the well known Lie algebra rank criterion provides a sufficient and necessary condition for the reachable set to cover the whole group. Because of the non-compact nature of the symplectic group, which corresponds to Gaussian unitary transformations, this criterion turns out to be still necessary but not sufficient for Gaussian systems. If the control Hamiltonians are unstable, in a sense made rigorous in the main text, the peculiar situation may arise where the rank criterion is satisfied and yet not all symplectic transformations are reachable. Here, we address this situation for one degree of freedom and study the properties of the reachable set under unstable control Hamiltonians. First, we provide a partial analytical characterisation of the reachable set and prove that no orthogonal ('energy-preserving' or 'passive' in the literature) symplectic operations may be reached with such controls. Then, we apply numerical optimal control algorithms to demonstrate a complete characterisation of the set in specific cases.