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Novel strategies for quantum state property estimation

Ivan Palmisano

Published Oct 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.17034/32640006

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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.
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 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.
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.
Quantum state tomography (QST) is a fundamental technique in quantum information science, aiming to reconstruct the density matrix of an unknown quantum state from experimental measurements. However, traditional QST methods often suffer from significant computational demands and are highly susceptible to measurement noise, hindering their practical application, particularly with near-term quantum computers. This work proposes a novel approach to QST utilizing variational quantum algorithms (VQAs). VQAs iteratively optimize a parameterized quantum circuit to minimize a cost function related to the fidelity between the desired state and the state predicted by the circuit. This approach dramatically reduces the number of required measurements and offers the potential for achieving high-fidelity state reconstruction with lower quantum resource requirements. We detail the theoretical framework, outlining the core algorithms and mathematical formulations. The key advantage lies in the ability to learn the state directly, mitigating the limitations of traditional methods. We explore the optimization landscape and discuss strategies for improving convergence and robustness. The results demonstrate a promising path toward scalable and robust QST on quantum devices.
The pace of innovation in quantum information science has recently exploded due to the hope that a quantum computer will be able to solve a multitude of problems that are intractable using classical hardware. Current quantum devices are in what has been termed the ``noisy intermediate-scale quantum'', or NISQ stage. Quantum hardware available today with 50-100 physical qubits may be among the first to demonstrate a quantum advantage. However, there are many challenges to overcome, such as dealing with noise, lowering error rates, improving coherence times, and scalability. \n \nWe are at a time in the field where minimization of resources is critical so that we can run our algorithms sooner rather than later. Running quantum algorithms ``at scale'' incurs a massive amount of resources, from the number of qubits required to the circuit depth. A large amount of this is due to the need to implement operations fault-tolerantly using error-correcting codes. \n \nFor one, to run an algorithm we must be able to efficiently read in and output data. Fault-tolerantly implementing quantum memories may become an input bottleneck for quantum algorithms, including many which would otherwise yield massive improvements in algorithm complexity. We will also need efficient methods for tomography to characterize and verify our processes and outputs. Researchers will require tools to automate the design of large quantum algorithms, to compile, optimize, and verify their circuits, and to do so in a way that minimizes operations that are expensive in a fault-tolerant setting. Finally, we will also need overarching frameworks to characterize the resource requirements themselves. Such tools must be easily adaptable to new developments in the field, and allow users to explore tradeoffs between their parameters of interest. \n \nThis thesis contains three contributions to this effort: improving circuit synthesis using large-scale parallelization; designing circuits for quantum random-access memories and analyzing various time/space tradeoffs; using the mathematical structure of discrete phase space to select subsets of tomographic measurements. For each topic the theoretical work is supplemented by a software package intended to allow others researchers to easily verify, use, and expand upon the techniques herein.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2)O(d^2)O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.