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.

In this paper we outline the extension of recently introduced the sub-system embedding subalgebras coupled cluster (SES-CC) formalism to the unitary CC formalism. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC formalism (DUCC) the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects -used to define the effective Hamiltonian -from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of complete active space representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction (FCI) type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing.

Identifying an accurate model for the dynamics of a quantum system is a vexing problem that underlies a range of problems in experimental physics and quantum information theory. Recently, a method called quantum Hamiltonian learning has been proposed by the present authors that uses quantum simulation as a resource for modeling an unknown quantum system. This approach can, under certain circumstances, allow such models to be efficiently identified. A major caveat of that work is the assumption of that all elements of the protocol are noise free. Here we show that quantum Hamiltonian learning can tolerate substantial amounts of depolarizing noise and show numerical evidence that it can tolerate noise drawn from other realistic models. We further provide evidence that the learning algorithm will find a model that is maximally close to the true model in cases where the hypothetical model lacks terms present in the true model. Finally, we also provide numerical evidence that the algorithm works for noncommuting models. This work illustrates that quantum Hamiltonian learning can be performed using realistic resources and suggests that even imperfect quantum resources may be valuable for characterizing quantum systems.

In recent years, Bayesian methods have been proposed as a solution to a wide range of issues in quantum state and process tomography. State-of-the-art Bayesian tomography solutions suffer from three problems: numerical intractability, a lack of informative prior distributions, and an inability to track time-dependent processes. Here, we address all three problems. First, we use modern statistical methods, as pioneered by Huszár and Houlsby (2012 Phys. Rev. A 85 052120) and by Ferrie (2014 New J. Phys.16 093035), to make Bayesian tomography numerically tractable. Our approach allows for practical computation of Bayesian point and region estimators for quantum states and channels. Second, we propose the first priors on quantum states and channels that allow for including useful experimental insight. Finally, we develop a method that allows tracking of time-dependent states and estimates the drift and diffusion processes affecting a state. We provide source code and animated visual examples for our methods.

Quantum information processing offers promising advances for a wide range of fields and applications, provided that we can efficiently assess the performance of the control applied in candidate systems. That is, we must be able to determine whether we have implemented a desired gate, and refine accordingly. Randomized benchmarking reduces the difficulty of this task by exploiting symmetries in quantum operations. Here, we bound the resources required for benchmarking and show that, with prior information, we can achieve several orders of magnitude better accuracy than in traditional approaches to benchmarking. Moreover, by building on state-of-the-art classical algorithms, we reach these accuracies with near-optimal resources. Our approach requires an order of magnitude less data to achieve the same accuracies and to provide online estimates of the errors in the reported fidelities. We also show that our approach is useful for physical devices by comparing to simulations.Quantum information processing devices offer great promise in a variety of different fields, including chemistry and material science, data analysis and machine learning [1][2][3][4], as well as cryptography [5]. Over the past few years, proposals have been advanced for quantum information processing past the classical scale, based on nodebased architectures [6,7]. In addition, rapid progress has been made towards experimental implementations that might allow for developing such devices [8,9]. An impediment in this effort, however, is presented by the difficulty of calibrating and diagnosing quantum devices.In particular, in the development of quantum information processing, an important experimental challenge is to efficiently characterize the quality with which we can control a quantum system. By characterizing the quality of a quantum gate that is implemented by a control pulse, we can then reason about the utility of that gate for quantum information processing tasks. For instance, we can estimate the feasibility of and the resources required to implement error correction using that control by comparing to proven and numerically estimated fault-tolerance thresholds [10,11]. Alternately, we can adjust our control sequences to account for differences between our control model and the actual system.In cases where only the quality of a quantum gate or set of gates is required, randomized benchmarking has proven to be a useful means of extracting this information with relatively little experimental effort [12]. This has been demonstrated in a variety of experimental settings [9,[13][14][15][16][17][18][19]. Randomized benchmarking has also been used to improve gate fidelities by characterizing cross-talk [20] or distortions [21]. Extracting fidelity information can often be useful in diagnosing performance and problems with a device in lieu of full characterization [22]. Moreover, randomized benchmarking has also been used to extract information about the completely positive and unital parts of linear maps [23].Here, using near-optimal data proces...