Quantum tomography-characterising a quantum system by assigning it a quantum state-is an essential building block for future technologies like quantum metrology or quantum computation. Ideal quantum tomography is accurate and robust to noise while being quick and efficient. The standard technique for decades, full quantum tomography (FQT), only fulfils parts of these criteria and quickly becomes unfeasible, even for medium-sized systems. As the number of possible states increases, the amount of measurements to characterise the system grows exponentially. The very feature that gives quantum physics immense possibilities hinders our ability to describe it. Additionally, FQT is neither robust to noise nor experimental errors and generally fails in the presence of these environmental perturbations unless additional resources are used.
Quantum tomography-characterising a quantum system by assigning it a quantum state-is an essential building block for future technologies like quantum metrology or quantum computation. Ideal quantum tomography is accurate and robust to noise while being quick and efficient. The standard technique for decades, full quantum tomography (FQT), only fulfils parts of these criteria and quickly becomes unfeasible, even for medium-sized systems. As the number of possible states increases, the amount of measurements to characterise the system grows exponentially. The very feature that gives quantum physics immense possibilities hinders our ability to describe it. Additionally, FQT is neither robust to noise nor experimental errors and generally fails in the presence of these environmental perturbations unless additional resources are used.
Quantum tomography—characterising a quantum system by assigning it a quantum state—–is an essential building block for future technologies like quantum metrology or quantum computation. Ideal quantum tomography is accurate and robust to noise, while simultaneously being quick and efficient. The standard technique used for decades, full quantum tomography (FQT), only fulfills parts of these criteria and quickly becomes unfeasible, even for medium sized systems. As the number of possible states increases, the requirements on the amount of measurements to characterise the system and the storage of the results grow with it exponentially. The very feature that gives quantum mechanics its immense possibilities, hinders our capacity to describe it. Additionally, FQT is neither robust to noise nor to experimental errors, and generally fails in the presence of these environmental perturbations unless additional resources are used at the cost of efficiency.
The exponential growth in Hilbert space with increasing size of a quantum system means that accurately characterizing the system becomes significantly harder with system dimension d. We show that self-guided tomography is a practical, efficient, and robust technique of measuring higher-dimensional quantum states. The achieved fidelities are over 99.9% for qutrits (d=3) and ququints (d=5), and 99.1% for quvigints (d=20)-the highest values ever realized for qudit pure states. We also show excellent performance for mixed states, achieving average fidelities of 96.5% for qutrits. We demonstrate robustness against experimental sources of noise, both statistical and environmental. The technique is applicable to any higher-dimensional system, from a collection of qubits through to individual qudits, and any physical realization, be it photonic, superconducting, ionic, or spin.
Quantum computation - the use of quantum systems as bits, or qubits, to perform computation - has been proven to be exponentially faster than what is thought possible with classical computation for several tasks. Quantum computers have the unique ability to store an amount of information that is exponential in the number of constituent particles used to encode that information. This property, which is arguably where quantum computers derive their power, makes it very difficult to characterize quantum systems. In order to fully characterize an unknown quantum state, a series of measurements must be performed on it and the state deduced from the results of those measurements. This process, called quantum tomography, is known to require exponential resources in the number of qubits. Both the number of measurements - the measurement complexity - and the number of times each measurement must be repeated - the sample complexity - present serious challenges when performing tomography. In addition, the measurement apparatus itself, which is usually a quantum system, must be characterized leading to even more experimental complexity. Here, I present three experiments which aim to reduce the experimental complexity of quantum tomography. The first aims to reduce the number of measurements needed to locate the decoherence-free subspaces of a noisy quantum channel. The process of locating these subspaces, which would normally require full process tomography, is performed with quadratically fewer measurement settings. The second experiment aims to characterize both an uncalibrated measurement apparatus and an unknown quantum state at the same time. With a specific model of measurement, it is found that an uncalibrated parameter of the measurement can be directly extracted using a single measurement. In the third experiment, I perform an experiment in which the optimal sample complexity for quantum state tomography is obtained using adaptivity: the ability to change measurement settings part way through an experiment. All three experiments were performed using the polarization degree of freedom of photons generated by spontaneous parametric downconversion. The source of photon pairs and the measurement techniques used are described in detail, along with the theory of quantum distinguishability and its relevance to tomography.
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.
We introduce a self-learning tomographic technique in which the experiment guides itself to an estimate of its own state. Self-guided quantum tomography uses measurements to directly test hypotheses in an iterative algorithm which converges to the true state. We demonstrate through simulation on many qubits that Self-guided quantum tomography is a more efficient and robust alternative to the usual paradigm of taking a large amount of informationally complete data and solving the inverse problem of postprocessed state estimation.
In quantum information transformation and quantum computation, the most critical issues are security and accuracy. These features, therefore, stimulate research on quantum state characterization. A characterization tool, Quantum state tomography, reconstructs the density matrix of an unknown quantum state. Theoretically, reconstructing an unknown state using this method can be arbitrarily accurate. However, this is less practical owing to the huge burden of measurements and data processing for large numbers of qubits. Even comprising an efficient estimator and a precise algorithm, an optimal tomographic framework can also be overburdened owing to the exponential growth of the measurements. Moreover, the consequential postprocessing of huge amounts of data challenges the capacity of computers. Thus, it is crucial to build an efficient framework that requires fewer measurements but yields an expected accuracy. To this end, we built a tomography schema by which only a few Pauli measurements enable an accurate tomographic reconstruction. Subsequently, this schema was verified as efficient and accurate through numerical simulations on the tomography of multi-qubit quantum states. Furthermore, this schema was proven to be robust through numerical simulations on a noisy superconducting qubit system. Therefore, the tomography schema paves an alternatively effective way to reconstruct the density matrix of a quantum state owing to its efficiency and accuracy, which are essential for quantum state tomography.