Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity, which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors. We also show that the unitarity provides a lower bound on the optimal achievable gate infidelity under a given noisy process.

In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.

In recent years quantum simulation has made great strides culminating in experiments that operate in a regime that existing supercomputers cannot easily simulate. Although this raises the possibility that special purpose analog quantum simulators may be able to perform computational tasks that existing computers cannot, it also introduces a major challenge: certifying that the quantum simulator is in fact simulating the correct quantum dynamics. We provide an algorithm that, under relatively weak assumptions, can be used to efficiently infer the Hamiltonian of a large but untrusted quantum simulator using a trusted quantum simulator. We illustrate the power of this approach by showing numerically that it can inexpensively learn the Hamiltonians for large frustrated Ising models, demonstrating that quantum resources can make certifying analog quantum simulators tractable.Quantum information processing promises to dramatically advance physics and chemistry by providing efficient simulators for the Schrödinger or Dirac equations [1][2][3]. This is important because conventional methods are inefficient, scaling exponentially in the number of interacting subsystems. Consequently, quantum simulations beyond a few tens of interacting particles are generally believed to be beyond the limitations of conventional supercomputers. This inability to simulate large quantum systems means that important questions in condensed matter, such as the shape of the phase diagram for the Fermi-Hubbard model, remain open. Analog quantum simulation raises the possibility that special purpose analog devices may be able to address such problems using current or near-future hardware [4][5][6]. A major objection to this avenue of inquiry is that analog simulators are not necessarily trustworthy [7,8] and certification of them is not known to be efficient. Without such certification, an analog simulator can at best only provide hints about the answer to a given computational question. A resolution to this problem is therefore essential if analog quantum simulators are to compete on an even footing with classical supercomputers.An important first step towards a resolution is provided in [9], where it is shown that quantum systems with local time-independent Hamiltonians can be efficiently characterized given ensemble readout. However, the method is not generally applicable, can be expensive and is not known to be either error robust or stable in cases where single shot measurements are used. A number of machine learning and statistical inference methods [10][11][12][13][14][15][16][17] have been recently introduced to address similar problems in metrology or Hamiltonian learning. In the context of Hamiltonian learning, such ideas have are known to be error-robust and lead to substantial reductions in the cost of high-precision Hamiltonian inference [15], albeit at the price of sacrificing the efficient scaling exhibited by [9].We overcome these challenges by providing a robust method that can be used to characterize unknown Hamiltonians by uni...

We introduce a new method called rejection filtering that we use to perform adaptive Bayesian phase estimation. Our approach has several advantages: it is classically efficient, easy to implement, achieves Heisenberg limited scaling, resists depolarizing noise, tracks time-dependent eigenstates, recovers from failures, and can be run on a field programmable gate array. It also outperforms existing iterative phase estimation algorithms such as Kitaev's method.