Entanglement and spin squeezing in non-Hermitian phase transitions

We show that non-Hermitian dynamics generate substantial entanglement in many-body systems. We consider the non-Hermitian Lipkin-Meshkov-Glick model and show that its phase transition occurs with maximum multiparticle entanglement: there is full N-particle entanglement at the transition, in contrast to the Hermitian case. The non-Hermitian model also exhibits more spin squeezing than the Hermitian model, showing that non-Hermitian dynamics are useful for quantum metrology. Experimental implementations with trapped ions and cavity QED are discussed.Comment: 5 pages + appendi

Steady state entanglement of two superconducting qubits engineered by dissipation

We present a scheme for the dissipative preparation of an entangled steady state of two superconducting qubits in a circuit QED setup. Combining resonator photon loss, a dissipative process already present in the setup, with an effective two-photon microwave drive, we engineer an effective decay mechanism which prepares a maximally entangled state of the two qubits. This state is then maintained as the steady state of the driven, dissipative evolution. The performance of the dissipative state preparation protocol is studied analytically and verified numerically. In view of the experimental implementation of the presented scheme we investigate the effects of potential experimental imperfections and show that our scheme is robust to small deviations in the parameters. We find that high fidelities with the target state can be achieved both with state-of-the-art 3D, as well as with the more commonly used 2D transmons. The promising results of our study thus open a route for the demonstration of an entangled steady state in circuit QED.Comment: 12 pages, 5 figures; close to published versio

Effective operator formalism for open quantum systems

We present an effective operator formalism for open quantum systems. Employing perturbation theory and adiabatic elimination of excited states for a weakly driven system, we derive an effective master equation which reduces the evolution to the ground-state dynamics. The effective evolution involves a single effective Hamiltonian and one effective Lindblad operator for each naturally occurring decay process. Simple expressions are derived for the effective operators which can be directly applied to reach effective equations of motion for the ground states. We compare our method with the hitherto existing concepts for effective interactions and present physical examples for the application of our formalism, including dissipative state preparation by engineered decay processes.Comment: 11 pages, 6 figure

Photon Scattering from a System of Multi-Level Quantum Emitters. I. Formalism

We introduce a formalism to solve the problem of photon scattering from a system of multi-level quantum emitters. Our approach provides a direct solution of the scattering dynamics. As such the formalism gives the scattered fields amplitudes in the limit of a weak incident intensity. Our formalism is equipped to treat both multi-emitter and multi-level emitter systems, and is applicable to a plethora of photon scattering problems including conditional state preparation by photo-detection. In this paper, we develop the general formalism for an arbitrary geometry. In the following paper (part II), we reduce the general photon scattering formalism to a form that is applicable to $1$-dimensional waveguides, and show its applicability by considering explicit examples with various emitter configurations.Comment: This is first part of a two part series of papers. It has 11 pages, double column, and one figur

Characterization of Coherent Errors in Noisy Quantum Devices

Characterization of quantum devices generates insights into their sources of disturbances. State-of-the-art characterization protocols often focus on incoherent noise and eliminate coherent errors when using Pauli or Clifford twirling techniques. This approach biases the structure of the effective noise and adds a circuit and sampling overhead. We motivate the extension of an incoherent local Pauli noise model to coherent errors and present a practical characterization protocol for an arbitrary gate layer. We demonstrate our protocol on a superconducting hardware platform and identify the leading coherent errors. To verify the characterized noise structure, we mitigate its coherent and incoherent components using a gate-level coherent noise mitigation scheme in conjunction with probabilistic error cancellation. The proposed characterization procedure opens up possibilities for device calibration, hardware development, and improvement of error mitigation and correction techniques.Comment: 12 pages, 8 figure

Dissipative self-interference and robustness of continuous error-correction to miscalibration

We derive an effective equation of motion within the steady-state subspace of a large family of Markovian open systems (i.e., Lindbladians) due to perturbations of their Hamiltonians and system-bath couplings. Under mild and realistic conditions, competing dissipative processes destructively interfere without the need for fine-tuning and produce no dissipation within the steady-state subspace. In quantum error-correction, these effects imply that continuously error-correcting Lindbladians are robust to calibration errors, including miscalibrations consisting of operators undetectable by the code. A similar interference is present in more general systems if one implements a particular Hamiltonian drive, resulting in a coherent cancellation of dissipation. On the opposite extreme, we provide a simple implementation of universal Lindbladian simulation

Deep learning-based quantum algorithms for solving nonlinear partial differential equations

Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the potential for enhancing a classical deep learning-based method for solving high-dimensional nonlinear partial differential equations with suitable quantum subroutines. First, with near-term noisy intermediate-scale quantum computers in mind, we construct architectures employing variational quantum circuits and classical neural networks in conjunction. While the hybrid architectures show equal or worse performance than their fully classical counterparts in simulations, they may still be of use in very high-dimensional cases or if the problem is of a quantum mechanical nature. Next, we identify the bottlenecks imposed by Monte Carlo sampling and the training of the neural networks. We find that quantum-accelerated Monte Carlo methods offer the potential to speed up the estimation of the loss function. In addition, we identify and analyse the trade-offs when using quantum-accelerated Monte Carlo methods to estimate the gradients with different methods, including a recently developed backpropagation-free forward gradient method. Finally, we discuss the usage of a suitable quantum algorithm for accelerating the training of feed-forward neural networks. Hence, this work provides different avenues with the potential for polynomial speedups for deep learning-based methods for nonlinear partial differential equations.Comment: 48 pages, 17 figure

Provable advantages of kernel-based quantum learners and quantum preprocessing based on Grover's algorithm

There is an ongoing effort to find quantum speedups for learning problems. Recently, [Y. Liu et al., Nat. Phys. $\textbf{17}$, 1013--1017 (2021)] have proven an exponential speedup for quantum support vector machines by leveraging the speedup of Shor's algorithm. We expand upon this result and identify a speedup utilizing Grover's algorithm in the kernel of a support vector machine. To show the practicality of the kernel structure we apply it to a problem related to pattern matching, providing a practical yet provable advantage. Moreover, we show that combining quantum computation in a preprocessing step with classical methods for classification further improves classifier performance.Comment: 14 pages, 5 figure

Dissipative self-interference and robustness of continuous error-correction to miscalibration

We derive an effective equation of motion within the steady-state subspace of a large family of Markovian open systems (i.e., Lindbladians) due to perturbations of their Hamiltonians and system-bath couplings. Under mild and realistic conditions, competing dissipative processes destructively interfere without the need for fine-tuning and produce no dissipation within the steady-state subspace. In quantum error-correction, these effects imply that continuously error-correcting Lindbladians are robust to calibration errors, including miscalibrations consisting of operators undetectable by the code. A similar interference is present in more general systems if one implements a particular Hamiltonian drive, resulting in a coherent cancellation of dissipation. On the opposite extreme, we provide a simple implementation of universal Lindbladian simulation