Quantum tomography is a class of characterization methods frequently used in current experiments, but its standard protocols suffer from unreliability originated from preknowledge assumptions. Self-consistent quantum tomography is an approach to avoid the problem, which treats every quantum operation in a characterization experiment as unknown objects to be characterized. As compensation for the beneficence, it leads to a problem that its characterization results cannot be determined uniquely only from experimental data due to the existence of experimentally undetectable gauge degrees of freedom, and we need to introduce a criterion to fix the gauge. Here, we propose to use a regularization technique to fix the gauge. First, we derive a sufficient condition on a characterization experiment to obtain all information of objects to be characterized except for the gauge. Second, we propose a self-consistent data-processing method with regularization and physicality constraints. A careless use of regularization can lead to non-negligible bias on the characterization result. As a solution for the concern, we propose a concrete way to tune the strength of the regularization, and mathematically prove that the method provides characterization results that converge to the gauge-equivalence class of the quantum operations of interest at the limit of data going to infinity. The asymptotic convergence guarantees the reliability of the method. We also derive the asymptotic convergence rate, which would be optimal. These theoretical results hold for any finite-dimensional quantum systems. Finally, as its first numerical implementation, we show numerical results on one-qubit system, which confirm the theoretical results and prove that the method proposed is practical.
Quantum tomography is a class of characterization methods frequently used in current experiments, but its standard protocols suffer from unreliability originated from preknowledge assumptions. Self-consistent quantum tomography is an approach to avoid the problem, which treats every quantum operation in a characterization experiment as unknown objects to be characterized. As compensation for the beneficence, it leads to a problem that its characterization results cannot be determined uniquely only from experimental data due to the existence of experimentally undetectable gauge degrees of freedom, and we need to introduce a criterion to fix the gauge. Here, we propose to use a regularization technique to fix the gauge. First, we derive a sufficient condition on a characterization experiment to obtain all information of objects to be characterized except for the gauge. Second, we propose a self-consistent data-processing method with regularization and physicality constraints. A careless use of regularization can lead to non-negligible bias on the characterization result. As a solution for the concern, we propose a concrete way to tune the strength of the regularization, and mathematically prove that the method provides characterization results that converge to the gauge-equivalence class of the quantum operations of interest at the limit of data going to infinity. The asymptotic convergence guarantees the reliability of the method. We also derive the asymptotic convergence rate, which would be optimal. These theoretical results hold for any finite-dimensional quantum systems. Finally, as its first numerical implementation, we show numerical results on one-qubit system, which confirm the theoretical results and prove that the method proposed is practical.
Improving accuracy of quantum operations is an indispensable task for realizing a practical quantum computer. A reliable method for characterizing quantum operations is a useful tool towards further improvements of the accuracy. Quantum tomography and randomized benchmarking protocols are current standard methods for evaluating accuracies of quantum operations, but their estimates lack reliability because of high sensitivity to state preparation and measurement (SPAM) errors or they use multiple approximations. Here we propose a self-consistent quantum tomographic method. First, we derive a sufficient condition on experiments for characterizing quantum operations except for the gauge degrees of freedom. Second, we propose a new self-consistent estimator. We mathematically prove that, when the sufficient condition is satisfied, the estimator provides estimates that are always physical and converge to the gauge-equivalence class of the quantum operations of interest at the limit of data size going to infinity. We also derive the rate of the asymptotic convergence. These results theoretically guarantee that the method makes it possible to reliably characterize accurate quantum operations.
In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that ℓ2-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.
As increasingly impressive quantum information processors are realized in laboratories around the world, robust and reliable characterization of these devices is now more urgent than ever. These diagnostics can take many forms, but one of the most popular categories is tomography , where an underlying parameterized model is proposed for a device and inferred by experiments. Here, we introduce and implement efficient operational tomography, which uses experimental observables as these model parameters. This addresses a problem of ambiguity in representation that arises in current tomographic approaches (the gauge problem ). Solving the gauge problem enables us to efficiently implement operational tomography in a Bayesian framework computationally, and hence gives us a natural way to include prior information and discuss uncertainty in fit parameters. We demonstrate this new tomography in a variety of different experimentally-relevant scenarios, including standard process tomography, Ramsey interferometry, randomized benchmarking, and gate set tomography.
Characterization of quantum processes is a preliminary step necessary in the development of quantum technology. The conventional method uses standard quantum process tomography, which requires d2 input states and d4 quantum measurements for a d-dimensional Hilbert space. These experimental requirements are compounded by the complexity of processing the collected data, which can take several orders of magnitude longer than the experiment itself. In this paper we propose an alternative self-guided algorithm for quantum process tomography, tuned for the task of finding an unknown unitary process. Our algorithm is a fully automated and adaptive process characterization technique. The advantages of our algorithm are the following: it has an inherent robustness to both statistical and technical noise; it requires less space and time since there is no postprocessing of the data; it requires only a single input state and measurement; and it provides on-the-fly diagnostic information while the experiment is running. Numerical results show our algorithm achieves the same 1/n scaling as standard quantum process tomography when n uses of the unknown process are used. We also present experimental results wherein the algorithm and its advantages are realized for the task of finding an element of SU(2).
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state and, therefore, have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions-a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.
We introduce a fast and accurate heuristic for adaptive tomography that addresses many of the limitations of prior methods.Previous approaches were either too computationally intensive or tailored to handle special cases such as single qubits or pure states.By contrast, our approach combines the efficiency of online optimization with generally applicable and well-motivated data-processing techniques.We numerically demonstrate these advantages in several scenarios including mixed states, higher-dimensional systems, and restricted measurements.Quantum information processing (QIP) promises advantages in a wide range of different contexts, including machine learning [2-4], chemistry simulation [5][6][7], and number theory [8,9].As such, the experimental effort to build useful QIP devices has exploded in recent years.In the course of this effort, quantum tomography is a valuable tool for diagnosing and debugging small quantum devices, and has subsequently seen a variety of different advances.In particular, Bayesian approaches to tomography which are especially well suited to utilizing prior information and adapting to changing experimental conditions have developed significantly in recent years [10-13], presenting a useful experimental tool [14][15][16].In this paper we demonstrate the efficiency and accuracy of an adaptive tomography protocol that we call PAQT: practical adaptive quantum tomography.PAQT intelligently selects new measurements based on the outcomes of previous ones [10,[17][18][19][20][21].Adaptivity has been experimentally demonstrated [14,15,[22][23][24][25], but is not currently standard practice.Though adaptivity increases accuracy, the computational costs incurred outweigh that of simply repeating standard measurements many times.The PAQT approach employs a simple heuristic that can be efficiently computed between measurements, even with embedded hardware [26][27][28][29].The algorithm we propose is therefore compatible with modern experimental design and avoids an important limitation of previous approaches.We base our algorithm off of self-guided quantum tomography (SGQT), which treats adaptive tomography as a direct optimization problem rather than a new optimization problem between each measurement [30].Though this affords an efficient and easy to implement adaptive heuristic, SGQT is not without its limitations.It requires assuming that the target state is pure, and it does not return rich region estimates for a state.What PAQT achieves is to effectively combine SGQT with conventional and easily-implemented tomographic estimators, such as the Bayesian particle filter or least-squares fit (LSF) estimators.Under this approach, an experimentalist can collect data using SGQT (even if its assumptions are not met), and then post-process this data using particle filtering or LSF.The benefit of PAQT is two-fold.(1) From the point of view of traditional tomography, it gives an adaptive tomography protocol requiring only modest computational resources, as the bulk of the computational cost is offloaded to post-processing.(2) From the point of view of simulation-based optimization tomography (such as SGQT), it effectively augments the output with region estimation providing a statistically robust quantification of uncertainty.Thus, while we do not explicitly demonstrate that the improved scaling of Ferrie [30] remains in the more general case considered here, PAQT does provide a practical and efficient procedure for performing adaptive quantum tomography with rigorous statistical principles.