As commonly understood, the noise spectroscopy problem—characterizing the statistical properties of a noise process affecting a quantum system by measuring its response—is mathematically ill-posed, in the sense that no unique noise process leads to a set of responses unless extra assumptions are taken into account. Ad-hoc solutions assume an implicit structure, which is often never determined. Thus, it is unclear when the method will succeed or whether one should trust the solution obtained. Here, we propose to treat the problem from the point of view of statistical estimation theory. We develop a Bayesian solution to the problem which allows one to easily incorporate assumptions which render the problem solvable. We compare several numerical techniques for noise spectroscopy and find the Bayesian approach to be superior in many respects.
Qubit noise spectroscopy (QNS) is a valuable tool for both the characterization of a qubit's environment and as a precursor to more effective qubit control to improve qubit fidelities. Existing approaches to QNS are what the classical spectrum estimation literature would call "non-parametric" approaches, in that a series of probe sequences are used to estimate noise power at a set of points or bands. In contrast, model-based approaches to spectrum estimation assume additional structure in the form of the spectrum and leverage this for improved statistical accuracy or other capabilities, such as superresolution. Here, we derive model-based QNS approaches using inspiration from classical signal processing, primarily though the recently developed Schrodinger wave autoregressive moving-average (SchWARMA) formalism for modeling correlated noise. We show, through both simulation and experimental data, how these model-based QNS approaches maintain the statistical and computational benefits of their classical counterparts, resulting in powerful new estimation approaches. Beyond the direct application of these approaches to QNS and quantum sensing, we anticipate that the flexibility of the underlying models will find utility in adaptive feedback control for quantum systems, in analogy with their role in classical adaptive signal processing and control.
This paper investigates the application of Bayesian inference for optimizing quantum measurement schemes. Traditional quantum measurement strategies often rely on heuristic approaches or specific system constraints, potentially leading to suboptimal outcomes. We propose a framework where the quantum measurement process is modeled as a Bayesian inference problem. This allows us to systematically explore the measurement landscape, incorporating prior knowledge and iteratively refining the measurement strategy based on observed outcomes. The core of this approach lies in formulating the measurement problem as a probabilistic model and utilizing Bayesian updating to determine the optimal measurement basis or parameters. We demonstrate, through theoretical analysis and conceptual design, that Bayesian optimization can lead to improved measurement accuracy and efficiency compared to conventional methods. The key benefits include reduced measurement errors, enhanced signal-to-noise ratios, and potentially the ability to extract more information from quantum systems. This work presents a novel methodology for quantum measurement, bridging the gap between classical Bayesian optimization and the unique properties of quantum mechanics.
This paper proposes a novel approach to Bayesian inference that exploits the inherent noise present in quantum systems. The core idea is to utilize entangled quantum systems to generate noise patterns exhibiting strong statistical correlations. These correlations, when measured, are then directly incorporated as prior probabilities within a Bayesian inference engine, leading to improved accuracy compared to traditional Bayesian methods relying solely on classical noise. The methodology moves beyond merely utilizing quantum randomness, actively harnessing the correlated information contained within quantum noise to reshape the inference process. We demonstrate, theoretically, the potential for significant improvements in model accuracy by strategically leveraging these correlations. The framework presented offers a promising avenue for enhancing Bayesian inference in complex scenarios, particularly where classical noise sources are limited in their ability to provide correlated information. The mathematical formalism presented outlines the key principles and equations governing this process.
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.
This paper explores a novel approach to Bayesian inference within quantum systems, specifically addressing the significant challenge posed by decoherence. Traditional Bayesian inference relies on maintaining quantum coherence, which is highly susceptible to environmental noise. We introduce a framework where decoherence is actively mitigated through dynamically controlled environmental interactions. The core concept involves utilizing precisely engineered external fields, coupled with real-time feedback loops, to counteract decoherence effects during the inference process. Unlike passive isolation or conventional error correction strategies, our method offers an active, responsive solution that maintains coherence for extended periods. We detail the theoretical underpinnings of this approach, focusing on the mathematical formulation of the inference process, incorporating decoherence models and the control strategies employed. The proposed system leverages quantum measurement to update Bayesian probabilities, with the control parameters designed to minimize the impact of decoherence on the measurement outcomes. This work lays the groundwork for performing accurate Bayesian inference on quantum systems, potentially unlocking new possibilities in quantum computing and information processing. The key mathematical concepts employed include the Schrödinger equation, density matrix formalism, and Bayesian probability theory. The system is designed to achieve a target accuracy in the Bayesian inference, even in the presence of significant decoherence.