Impressive progress has been made in the past decade in the study of technological applications of varied types of quantum systems. With industry giants like IBM laying down their roadmap for scalable quantum devices with more than 1000-qubits by the end of 2023, efficient validation techniques are also being developed for testing quantum processing on these devices. The characterization of a quantum state is done by experimental measurements through the process called quantum state tomography (QST) which scales exponentially with the size of the system. However, QST performed using incomplete measurements is aptly suited for characterizing these quantum technologies especially with the current nature of noisy intermediate-scale quantum (NISQ) devices where not all mean measurements are available with high fidelity. We, hereby, propose an alternative approach to QST for the complete reconstruction of the density matrix of a quantum system in a pure state for any number of qubits by applying the maximal entropy formalism on the pairwise combinations of the known mean measurements. This approach provides the best estimate of the target state when we know the complete set of observables, which is the case of convergence of the reconstructed density matrix to a pure state. Our goal is to provide a practical inference of a quantum system in a pure state that can find its applications in the field of quantum error mitigation on a real quantum computer that we intend to investigate further.
Impressive progress has been made in the past decade in the study of technological applications of varied types of quantum systems. With industry giants like IBM laying down their roadmap for scalable quantum devices with more than 1000-qubits by the end of 2023, efficient validation techniques are also being developed for testing quantum processing on these devices. The characterization of a quantum state is done by experimental measurements through the process called quantum state tomography (QST) which scales exponentially with the size of the system. However, QST performed using incomplete measurements is aptly suited for characterizing these quantum technologies especially with the current nature of noisy intermediate-scale quantum (NISQ) devices where not all mean measurements are available with high fidelity. We, hereby, propose an alternative approach to QST for the complete reconstruction of the density matrix of a quantum system in a pure state for any number of qubits by applying the maximal entropy formalism on the pairwise combinations of the known mean measurements. This approach provides the best estimate of the target state when we know the complete set of observables, which is the case of convergence of the reconstructed density matrix to a pure state. Our goal is to provide a practical inference of a quantum system in a pure state that can find its applications in the field of quantum error mitigation on a real quantum computer that we intend to investigate further.
Quantum computation has been growing rapidly in both theory and experiments.In particular, quantum computing devices with a large number of qubits have been developed by IBM, Google, IonQ, and others.The current quantum computing devices are noisy intermediate-scale quantum devices, and so approaches to validate quantum processing on these quantum devices are needed.One of the most common ways of validation for an n-qubit quantum system is quantum tomography, which tries to reconstruct a quantum system's density matrix by a complete set of observables.However, the inherent noise in the quantum systems and the intrinsic limitations pose a critical challenge to precisely know the actual measurement operators that make quantum tomography impractical in experiments.Here, we propose an alternative approach to quantum tomography, based on the maximal information entropy, that can predict the values of unknown observables based on the available mean measurement data.This can then be used to reconstruct the density matrix with high fidelity even though the results for some observables are missing.Of additional contexts, a practical approach to the inference of the quantum mechanical state using only partial information is also needed.
Quantum state tomography (QST) is a fundamental technique for characterizing quantum states, traditionally employed in research laboratories with highly controlled environments. However, the burgeoning field of quantum computing faces the significant challenge of deploying and verifying quantum hardware in the presence of noise and decoherence, characteristics inherent in current Noisy Intermediate-Scale Quantum (NISQ) devices. This paper proposes an adapted QST algorithm designed to overcome these limitations. The core idea is to systematically reconstruct the density matrix of a qubit or a small quantum system using a series of carefully designed measurements. We address the impact of noise by incorporating error mitigation strategies within the reconstruction process. The algorithm's performance is evaluated through theoretical analysis and conceptual implementation, highlighting its potential to provide a practical and efficient method for hardware verification of NISQ computers. This work establishes a novel approach for characterizing and debugging NISQ hardware, a critical step toward realizing the full potential of quantum computation.
A significant amount of research is currently devoted to quantum technologies, i.e., technologies that harness the complex features of quantum systems for practical purposes. Remarkably, these have the potential to shape the way we will live in the future by, for example, providing new ways of breaking the current cryptographic security measures or by exponentially enhancing the computational capability of computers. This, in turn, could serve as a means for studying phenomena so complex that they render classical computers completely unusable. With these promises at hand, methods that efficiently certify the correct functioning of quantum devices and computations become of paramount importance. Specifically, it is often required to assess a characterization of the quantum states that come out from a specific computation or experiment. The traditional method of quantum state tomography relies on a rather straightforward but brute-force approach to reconstruct the full density matrix that encodes a quantum state. This method is highly demanding because of the curse of dimensionality, according to which the number of required measurements scales with the system’s dimension, and thus exponentially with the number of qubits. As the quantum protocols demand a progressively higher number of qubits, overall, a complete characterization of quantum states through tomography loses its effectiveness, thus calling for a partial but simpler reconstruction of states. Furthermore, an additional obstacle must be taken into account. In particular, the same complex features of the quantum realm that would help to navigate through the sea of potential applications hinder a growth of the current technological state of the art, which indeed is limited within the so-called NISQ (Noisy Intermediate-Scale Quantum) devices. These, as the name suggests, are characterized by inevitable noises and imperfections, and as such are not advanced enough for achieving a real quantum advantage in the near term. In this Thesis we are going to discuss new techniques for the estimation of properties of unknown quantum states in the context of quantum information science. The protocols that we will present are flexible enough to not only include state tomography within them, but to also adapt to noises and imperfections present in the computational devices. Ivan Palmisano was funded by the CITI-GENS project. This project received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Actions COFUND Action (Grant Agreement No 945231). All views expressed in this article are strictly those of the author and may under no circumstances be regarded as an official position of the European Research Executive Agency or the European Commission.
Quantum state tomography is an integral part of quantum computation and offers the starting point for the validation of various quantum devices. One of the central tasks in the field of state tomography is to reconstruct, with high fidelity, the quantum states of a quantum system. From an experiment on a real quantum device, one can obtain the mean measurement values of different operators. With such data as input, in this report we employ the maximal entropy formalism to construct the least biased mixed quantum state that is consistent with the given set of expectation values. Even though, in principle, the reported formalism is quite general and should work for an arbitrary set of observables, in practice we shall demonstrate the efficacy of the algorithm on an informationally complete (IC) set of Hermitian operators. Such a set possesses the advantage of uniquely specifying a single quantum state from which the experimental measurements have been sampled and hence renders the rare opportunity not only to construct a least-biased quantum state but even replicate the exact state prepared experimentally within a preset tolerance. The primary workhorse of the algorithm is reconstructing an energy function which we designate as the effective Hamiltonian of the system, and parameterizing it with Lagrange multipliers, according to the formalism of maximal entropy. These parameters are thereafter optimized variationally so that the reconstructed quantum state of the system converges to the true quantum state within an error threshold. To this end, we employ a parameterized quantum circuit and a hybrid quantum-classical variational algorithm to obtain such a target state, making our recipe easily implementable on a near-term quantum device.
Quantum state tomography (QST) aims at reconstructing the state of a quantum system. However, in conventional QST, the number of measurements scales exponentially with the number of qubits. Here, we propose a QST protocol, in which the introduction of a threshold allows one to drastically reduce the number of measurements required for the reconstruction of the state density matrix without compromising the result accuracy. In addition, one can also use the same approach to reconstruct an approximated density matrix tailoring the number of measurements on the available resources. We experimentally demonstrate this protocol by performing the tomography of states up to 7 qubits. We show that our approach can lead to results in agreement with those obtained by QST even when the number of measurements is reduced by more than two orders of magnitude.
Quantum state tomography (QST) is the gold standard for estimating the unknown state of quantum systems but suffers from the longstanding problem of exponentially growing measurements. Recent deep learning approaches alleviate the problem by training neural networks in a data-driven manner without theoretical convergence guarantees. They lack reliability in tomography quality and are restricted by retraining when generalized to extensive quantum systems with varying realizations in different environments. To address this issue, we propose the first generalized deep learning method for QST, named HD-QST, that leverages hybrid denoising priors with well-established convergence guarantees to fit extensive quantum states and environments. Hybrid denoising priors are achieved with the convex combination of analytic low-rank denoiser and neural network-based smooth denoiser dynamically determined via reinforcement learning. We demonstrate in theory that, when characterizing a quantum system, HD-QST guarantees a geometric convergence rate under the relaxed condition that the true density matrix is the fixed point of any one denoiser rather than the shared fixed point of multiple denoisers. Extensive experiments show that, compared with existing methods, HD-QST consistently obtains superior fidelity for pure and mixed quantum states in both single-device simulations and cross-device applications. Remarkably, it achieves state-of-the-art fidelity in QST of real-world W states by trapped ions and precise evaluation and comparison on cross-device QST on IBM quantum computers.