The development of second-generation quantum technologies (quantum sensors, quantum computers and quantum internet) has been carried out by large companies and has received state investments in the order of billions of dollars in several countries. Among these technologies, the quantum computer gained greater prominence in the media due to its promise to solve problems much faster and more efficiently than classical computers, which could produce numerous scientific advances with impacts on society and industry. In recent years, companies such as IBM, Google, Microsoft, among others, have made great progress in building the first prototypes of quantum hardware and software. However, these prototypes are devices that operate with few qubits and are heavily affected by noise, which still prevents them from performing truly useful computation. In particular, IBM has provided remote access to some of its machines available to the general public, as well as special access to selected researchers in its research program. Using this resource, in this work we study a method for reconstructing pure states of a multi-qubit system in a quantum computer. In the context of quantum computing, this is a very useful tool, for example, to determine input or output states of a given algorithm, or to characterize the logical operations that constitute the algorithm. The method we study is known as quantum ptychography, where a pure state is subjected to a set of projections with some degree of intersection between them, followed by a projective measurement on a single orthonormal basis. The results generated in the process serve as input to an iterative algorithm that estimates the state. Our study was divided between simulations and experiments. The simulations were carried out in ideal scenarios without noise and more realistic scenarios with noise, where the method was successfully demonstrated for a variety of states of up to 6 qubits. In experiments, we obtained good results for reconstructing states of up to 4 qubits; the results for a larger number were strongly affected by noise in the devices. The advantages and limitations of the ptychographic method are discussed in comparison with other state reconstruction methods. In addition to demonstrating quantum ptychography on a quantum computer, we also achieved some advances in the method. The first was a modification to the iterative reconstruction algorithm, which improves its convergence and, consequently, the state estimation. The second was the use of measurement bases that are less affected by noise in a quantum computer. This second advance is still part of a preliminary study but points to promising avenues for future investigations.
The development of second-generation quantum technologies (quantum sensors, quantum computers and quantum internet) has been carried out by large companies and has received state investments in the order of billions of dollars in several countries. Among these technologies, the quantum computer gained greater prominence in the media due to its promise to solve problems much faster and more efficiently than classical computers, which could produce numerous scientific advances with impacts on society and industry. In recent years, companies such as IBM, Google, Microsoft, among others, have made great progress in building the first prototypes of quantum hardware and software. However, these prototypes are devices that operate with few qubits and are heavily affected by noise, which still prevents them from performing truly useful computation. In particular, IBM has provided remote access to some of its machines available to the general public, as well as special access to selected researchers in its research program. Using this resource, in this work we study a method for reconstructing pure states of a multi-qubit system in a quantum computer. In the context of quantum computing, this is a very useful tool, for example, to determine input or output states of a given algorithm, or to characterize the logical operations that constitute the algorithm. The method we study is known as quantum ptychography, where a pure state is subjected to a set of projections with some degree of intersection between them, followed by a projective measurement on a single orthonormal basis. The results generated in the process serve as input to an iterative algorithm that estimates the state. Our study was divided between simulations and experiments. The simulations were carried out in ideal scenarios without noise and more realistic scenarios with noise, where the method was successfully demonstrated for a variety of states of up to 6 qubits. In experiments, we obtained good results for reconstructing states of up to 4 qubits; the results for a larger number were strongly affected by noise in the devices. The advantages and limitations of the ptychographic method are discussed in comparison with other state reconstruction methods. In addition to demonstrating quantum ptychography on a quantum computer, we also achieved some advances in the method. The first was a modification to the iterative reconstruction algorithm, which improves its convergence and, consequently, the state estimation. The second was the use of measurement bases that are less affected by noise in a quantum computer. This second advance is still part of a preliminary study but points to promising avenues for future investigations.
Quantum ptychography is a method for estimating an unknown pure quantum state by subjecting it to overlapping projections, each one followed by a projective measurement on a single prescribed basis. Here, we present a comprehensive study of this method applied for estimating n-qubit states in a circuit-based quantum computer, including numerical simulations and experiments carried out on an IBM superconducting quantum processor. The intermediate projections are implemented through Pauli measurements on one qubit at a time, which sets the number of ptychographic circuits to 3n (in contrast to the 3n circuits for standard Pauli tomography); the final projective measurement in the computational basis is preceded by the quantum Fourier transform (QFT). Due to the large depth and number of two-qubit gates of the QFT circuit, which is unsuitable for noisy devices, we also test the approximate QFT (AQFT) and separable unitary operations. Using the QFT and AQFT of degree 2, we obtained high estimation fidelities in all tests with separable and entangled states for up to three and four qubits, respectively; on the other hand, the separable unitaries in this scenario provided good estimations only for separable states, in general. Our results compare favorably with recent results in the literature, and we discuss further alternatives to make the ptychographic method scalable for the current noisy devices.
Abstract Ptychography is an imaging technique in which a localized illumination scans overlapping regions of an object and generates a set of diffraction intensities used to computationally reconstruct its complex-valued transmission function. We propose a quantum analogue of this technique designed to reconstruct d-dimensional pure states. A set of n rank-r projectors “scans” overlapping parts of an input state and the moduli of the d Fourier amplitudes of each part are measured. These nd outcomes are fed into an iterative phase retrieval algorithm that estimates the state. Using d up to 100 and r around d / 2, we performed numerical simulations for single systems in an economic (n = 4) and a costly (n = d) scenario, as well as for multiqubit systems (n = 6logd). This numeric study included realistic amounts of depolarization and poissonian noise, and all scenarios yielded, in general, reconstructions with infidelities below 10−2. The method is shown, therefore, to be resilient to noise and, for any d, requires a simple and fast postprocessing algorithm. We show that the algorithm is equivalent to an alternating gradient search, which ensures that it does not suffer from local-minima stagnation. Unlike traditional approaches to state reconstruction, the ptychographic scheme uses a single measurement basis; the diversity and redundancy in the measured data—key for its success—are provided by the overlapping projections. We illustrate the simplicity of this scheme with the paradigmatic multiport interferometer.
In the context of quantum information theory, the use of certain photonic degrees of freedom allows the creation of high-dimensional quantum states, called qudits, which offer advantages over qubits in applications such as communication, cryptography, and fundamental tests of quantum mechanics. In many of these applications, it is necessary to reconstruct the quantum state to be used, since it contains all the measurable properties of a physical system. Recently, a simple method for reconstructing pure quantum states, called quantum ptychography, has been proposed. It is based on projections of the input state into various partial overlapping subspaces, each one followed by a projective measurement in the Fourier basis, which is generated by the quantum Fourier transform (QFT) of the computational basis. At the end, an iterative algorithm estimates the state from the collected data. In this work, we propose to apply this method to pure states of qudits encoded in angular position and orbital angular momentum (OAM) of single photons. For this purpose, we define these qudits based on the discretization of the spatial profile of a photon in cylindrical coordinates, through an arrangement of D angular slits, and describe their representation in the OAM basis. To apply ptychography to these qudits, we show that the intermediate projections will be implemented by simple binary spatial filters, while the measurement in the Fourier basis of the post-projection states will be performed by measuring in a basis defined by D OAM modes. Finally, we demonstrate the applicability of the ptychographic method by simulating the reconstruction of angular qudits for D = 5 and D = 12, and evaluate its experimental feasibility.
The quantum analogue of ptychography, a powerful coherent diffractive imaging technique, is a simple method for reconstructing d -dimensional pure states. It relies on measuring partially overlapping parts of the input state in a single orthonormal basis and feeding the outcomes to an iterative phase retrieval algorithm for postprocessing. We provide a proof of concept demonstration of this method by determining pure states given by superpositions of d transverse spatial modes of an optical field. A set of n rank- r projectors, diagonal in the spatial mode basis, is used to generate n partially overlapping parts of the input, and each part is projectively measured in the Fourier transformed basis. For d up to 32, we successfully reconstructed hundreds of random states using n = 5 and n = d rank- ⌈ d / 2 ⌉ projectors. The extension of quantum ptychography for other types of photonic spatial modes is outlined.
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 computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.