Quantum state tomography strategy via weak value is valid for arbitrary dimensional systems, and can be easily realized. However, the efficiency of quantum state tomography strategy via weak value is lower than that of the conventional quantum state tomography (QST), especially for high-dimensional systems. To improve the efficiency of the scheme, we propose a two-step adaptive quantum state tomography via weak value for pure states (AQSTW). In AQSTW, by using appropriate measurement strength, the efficiency of AQSTW is significantly larger than that of the conventional QST, especially for high-dimensional systems. And AQSTW can be easily implemented. These results demonstrate AQSTW may be useful in the measurements of the unknown quantum states.
Quantum state tomography strategy via weak value is valid for arbitrary dimensional systems, and can be easily realized. However, the efficiency of quantum state tomography strategy via weak value is lower than that of the conventional quantum state tomography (QST), especially for high-dimensional systems. To improve the efficiency of the scheme, we propose a two-step adaptive quantum state tomography via weak value for pure states (AQSTW). In AQSTW, by using appropriate measurement strength, the efficiency of AQSTW is significantly larger than that of the conventional QST, especially for high-dimensional systems. And AQSTW can be easily implemented. These results demonstrate AQSTW may be useful in the measurements of the unknown quantum states.
Compared with the conventional quantum state tomography (QST), the direct state tomography (DST) using weak value is easily manipulated in experiments. However, the efficiency of the DST is lower than that of the conventional QST, especially for high-dimensional systems. For a pure state, the DST is revised to improve the efficiency. In the revised DST, the real or imaginary parts of the weak values can be obtained by measuring only one system observable. We constructed a hybrid DST by combining the original DST and the revised DST. By using the appropriate measurement strength, the efficiency of the hybrid DST is significantly larger than that of the conventional QST, especially for high-dimensional systems. The state reconstruction strategy investigated in this paper may be useful in actual experiments.
Adaptive tomography has been widely investigated to obtain faster state tomography process for quantum computation. Infidelity [1 − F(ρ,σ)] on nearly pure states in quantum information process scales generally as O(1/N), which requires a large number of statistical ensembles in comparison to the infidelity scaling on mixed states of O(1/N). Recently, optimization of a measurement basis in a photonic qubit system, whose state tomography uses projective measurements, reported improved infidelity scaled as O(1/N). However, this dramatic improvement cannot be applied to a weak-value based measurement system, which can be attributed to the fact that one cannot distinguish two quantum states with perfect measurement reliability. We introduce new optimal measurement basis to achieve a fast adaptive quantum state tomography and minimum magnitude of infidelity in this weak measurement system. This novel protocol allows us to accomplish approximately 80% error reduction without changing the scaling of O(1/N) in numerical simulations and realize approximately 90% error reduction in superconducting circuit QED experiments.
Adaptive tomography has been widely investigated to achieve faster state tomography processing of quantum systems. Infidelity of the nearly pure states in a quantum information process generally scales as O(1/sqrt(N) ), which requires a large number of statistical ensembles in comparison to the infidelity scaling of O(1/N) for mixed states. One previous report optimized the measurement basis in a photonic qubit system, whose state tomography uses projective measurements, to obtain an infidelity scaling of O(1/N). However, this dramatic improvement cannot be applied to weak-value-based measurement systems in which two quantum states cannot be distinguished with perfect measurement fidelity. We introduce in this work a new optimal measurement basis to achieve fast adaptive quantum state tomography and a minimum magnitude of infidelity in a weak measurement system. We expect that the adaptive quantum state tomography protocol can lead to a reduction in the number of required measurements of approximately 33.74% via simulation without changing the O(1/sqrt(N)) scaling. Experimentally, we find a 14.81% measurement number reduction in a superconducting circuit system.
Recent work has revealed that the wave function of a pure state can be measured directly and that complementary knowledge of a quantum system can be obtained simultaneously by weak measurements. However, the original scheme applies only to pure states, and it is not efficient because most of the data are discarded by post-selection. Here, we propose tomography schemes for pure states and for mixed states via weak measurements, and our schemes are more efficient because we do not discard any data. Furthermore, we demonstrate that any matrix element of a general state can be directly read from an appropriate weak measurement. The density matrix (with all of its elements) represents all that is directly accessible from a general measurement.
Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.
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.