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.
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.
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.
Ptychography is a computational imaging technique mainly used in optical and electron microscopy. Its quantum analog was recently introduced as a simple method for estimating unknown pure quantum states through projections onto partially overlapping subspaces, each one followed by a projective measurement in the Fourier basis. In the end, an iterative algorithm estimates the state from the collected data. Here, we theoretically describe how to implement this method for D -dimensional qudit states encoded in the angular position and orbital angular momentum (OAM) of single photons. For this purpose, we define the qudit by discretizing the spatial profile of a photon in cylindrical coordinates, using an array of D angular slits symmetrically distributed in the transverse plane. To apply ptychography to this encoding, we show that the intermediate projections will be carried out by simple binary spatial filters in the angular paths, while the measurement in the Fourier basis will be performed by postselecting D OAM modes compatible with the quantum Fourier transform of the path basis. We illustrate the effectiveness of this scheme through simulations and discuss its experimental feasibility.
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.
Extracting as much information as possible about an object when probing with a limited number of photons is an important goal with applications from biology and security to metrology. Imaging with a few photons is a challenging task as the detector noise and stray light are then predominant, which precludes the use of conventional imaging methods. Quantum correlations between photon pairs has been exploited in a so called 'heralded imaging scheme' to eliminate this problem. However these implementations have so-far been limited to intensity imaging and the crucial phase information is lost in these methods. In this work, we propose a novel quantum-correlation enabled Fourier Ptychography technique, to capture high-resolution amplitude and phase images with a few photons. This is enabled by the heralding of single photons combined with Fourier ptychographic reconstruction. We provide experimental validation and discuss the advantages of our technique that include the possibility of reaching a higher signal to noise ratio and non-scanning Fourier Ptychographic acquisition.