Quantum information processing and computation requires high accuracy qubit configuration readout. In many practical schemes, the initial qubit configuration has to be inferred from readout that is a time-dependent weak measurement record. However, a combination of the measurement scheme and intrinsic dynamics can end up scrambling the initial state and lose information irretrievably. Here, we study the information physics of quantum trajectories based on weak measurements in order to address the optimal achievable performance in qubit configuration readout for two realistic models of single qubit readout: (i) Model I is informationally complete, but without intrinsic dynamics; (ii) Model II is informationally incomplete weak measurements with intrinsic dynamics. We first use mutual information to characterize how much intrinsic information about the initial state is encoded in the measurement record. Using a fixed discrete time-step formulation, we compute the mutual information while varying the measurement strength, duration of measurement record, and the relative strength of intrinsic dynamics in our measurement schemes. We also exploit the emergence of continuum scaling and the Stochastic Master Equation in the weak measurement limit. We develop an asymptotic expansion in the measurement efficiency parameter to calculate mutual information, which captures qualitative and quantitative features of the numerical data. The bounds on information extraction are manifested as plateaux in mutual information, our analysis obtains these bounds and also optimal duration of measurement required to saturate them. Our results should be useful both for quantum device operation and optimization and also, possibly, for improving the performance of recent machine learning approaches for qubit and multiqubit configuration readout in current Noisy Intermediate-Scale Quantum (NISQ) experiment regimes.
Quantum information processing and computation requires high accuracy qubit configuration readout. In many practical schemes, the initial qubit configuration has to be inferred from readout that is a time-dependent weak measurement record. However, a combination of the measurement scheme and intrinsic dynamics can end up scrambling the initial state and lose information irretrievably. Here, we study the information physics of quantum trajectories based on weak measurements in order to address the optimal achievable performance in qubit configuration readout for two realistic models of single qubit readout: (i) Model I is informationally complete, but without intrinsic dynamics; (ii) Model II is informationally incomplete weak measurements with intrinsic dynamics. We first use mutual information to characterize how much intrinsic information about the initial state is encoded in the measurement record. Using a fixed discrete time-step formulation, we compute the mutual information while varying the measurement strength, duration of measurement record, and the relative strength of intrinsic dynamics in our measurement schemes. We also exploit the emergence of continuum scaling and the Stochastic Master Equation in the weak measurement limit. We develop an asymptotic expansion in the measurement efficiency parameter to calculate mutual information, which captures qualitative and quantitative features of the numerical data. The bounds on information extraction are manifested as plateaux in mutual information, our analysis obtains these bounds and also optimal duration of measurement required to saturate them. Our results should be useful both for quantum device operation and optimization and also, possibly, for improving the performance of recent machine learning approaches for qubit and multiqubit configuration readout in current Noisy Intermediate-Scale Quantum (NISQ) experiment regimes.
In the context of quantum information, major efforts have been made to maximize the mutual information by measuring single copies of signal states. In general, one execution of optimal projective measurement extracts all the accessible mutual information. However, in some scenarios, weak measurements are preferred because of kinds of specific requirements, e.g., to distribute secret keys to multi-observers. In this study, we propose a method to construct optimal weak measurements for multi-party quantum communications. Utilizing the method in [Physical Review Letters 120, 160501 (2018)] to classify the mutual information, the theoretical study shows that by successively performing this optimal weak measurement, all accessible information can be obtained by multiple observers. This conclusion is experimentally verified by a cascaded measurement apparatus that can perform six successive weak measurements on heralded single photons. The experimental results clearly indicate that almost all accessible mutual information is extracted by this sequence of optimal weak measurements; meanwhile, none of the information is destroyed or residual. Thus, this optimal weak measurement is an efficient and reliable tool for performing quantum communication tasks. The consistence between the experimental and theoretical results verifies that the classifying method in [Phys. Rev. Lett.120, 160501 (2018)] can be applied to characterize realistic quantum measurements.
The field of quantum technologies has experienced explosive growth in the past decade. As hardware capabilities have advanced, quantum computing has exhibited a scaling trend analogous to Moore's law, often referred to as Rose's law, where the number of coherently controlled qubits roughly doubles every two years. Associated with such rapid growth is an equally growing need for benchmarking tools and error handling. While advancements in quantum hardware and software have mainly concentrated on enhancing gate fidelity and qubit connectivity, the progress in addressing readout errors has been relatively limited during the same period. This thesis tries to close this gap by advancing and developing new tools for information extraction in quantum systems. We do this in two different ways: In Part II, we focus on mitigating the effect of general readout errors in quantum systems. We developed a comprehensive protocol for readout-error-mitigated state tomography (REMST) and implemented it on a superconducting qubit system. The REMST protocol was designed to mitigate not only classical readout errors but also more challenging coherent and correlated readout errors. The protocol uses only a deterministic application of single-qubit readout, making it one of the most experimentally friendly approaches available. We developed a scalable version of the REMST protocol and showed numerically that it is suitable for systems with more than 100 qubits, matching the size of current state-of-the-art systems. The REMST protocol was applied to superconducting qubit experiments, where we saw an improvement in reconstruction accuracy by a factor of up to 30. In Part III, we move our focus to optimal measurement strategies. Adaptive measurement strategies promise a quadratic improvement in the statistical fluctuation of pure and close-to-pure state reconstruction by dynamically optimizing measurement settings during sampling. We investigated whether these adaptive measurement strategies were viable in realistic experimental settings. We demonstrated both analytically and in numerical simulations that going from ideal readout to noisy readout disproportionally deteriorates the quality of the adaptive measurement strategy, to a point where no asymptotic benefit exists for adaptive measurements. These results not only deepen our understanding of readout processes in superconducting qubits but also provide concrete implementations of more robust quantum measurement and tomography protocols, offering clearer insight into the interplay between readout noise and information extraction.
We study trade-off relations in information extraction from quantum systems subject to null-result weak measurements, where the absence of a detected photon continuously updates the system state. We present a detailed analysis of qubit and qutrit systems and investigate a general framework for a multilevel quantum system. We develop a dynamical characterization of null-result weak measurements that quantifies the information extracted over time, revealing the amount of the obtained information and also the rate of the information accumulation. The characterizations are obtained by examining the time-dependent evolution of the information theoretic quantities. More specifically, we consider Shannon entropy, mutual information, fidelity, and relative entropy to characterize the weak measurement dynamics. Our results provide an information theoretic analysis of the weak measurement process and highlight the dynamical nature of information extraction and reversibility in the weak measurement processes.
The conventional formulation of quantum sensing is based on the assumption that the probe is reset to its initial state after each measurement. In a very distinct approach, one can also pursue a sequential measurement scheme in which time-consuming resetting is avoided. In this situation, every measurement outcome effectively comes from a different probe, yet is correlated with other data samples. Finding a proper description for the precision of sequential measurement sensing is very challenging as it requires the analysis of long sequences with exponentially large outcomes. Here, we develop a recursive formula and an efficient Monte Carlo approach to calculate the Fisher information, as a figure of merit for sensing precision, for arbitrary lengths of sequential measurements. Our results show that the value of the Fisher information initially increases nonlinearly with the number of measurements and then asymptotically saturates to a linear scaling. This transition, which fundamentally constrains the extractable information about the parameter of interest, is directly linked to the finite memory of the probe when it undergoes multiple sequential measurements. Based on these findings, we establish a figure of merit to determine the optimal measurement sequence length and exemplify our results in three different physical systems.
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter.
I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement.
The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome
from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the…
We address the peculiarities of the quantum measurement process in the course of a continuous weak linear measurement (CWLM). As a tool, we implement an efficient numerical simulation scheme that allows us to generate single quantum trajectories of the measured system state as well as the recorded detector signal, and study statistics of these trajectories with and without post-selection. In this scheme, a linear detector is modelled with a qubit that is weakly coupled to the quantum system measured and is subject to projective measurement and re-initialization after a time interval at each simulation step. We explain the conditions under which the scheme provides an accurate description of CWLM. We restrict ourselves to a qubit non-demolition measurement. The qubit is initially in an equal-weight superposition of two quantum states. In the course of time, the detector signal is accumulated and the superposition is destroyed. The times required to resolve the quantum states and to destroy the superposition are of the same order. We prove numerically a rather counter intuitive fact: the average detector output conditioned on the final state does not depend on time. It looks like from the very beginning, the qubit knows in which state it is. We study statistics of decision times where the decision time is defined as time required for the density matrix along a certain trajectory to reach a threshold where it is close to one of the resulting states. This statistics is useful to estimate how fast a decisive CWLM can be. Basing on this, we devise and study a simple feedback scheme that attempts to keep the qubit in the equal-weight superposition. The detector readings are used to decide in which state the qubit is and which correction rotation to apply to bring it back to the superposition. We show how to optimize the feedback parameters and move towards more efficient feedback schemes.