Other approaches, such as matrix product state tomography, require relatively less measurementsscaling as a polynomial in n. In addition, the combination of convex optimization and quantum information may arouse some interesting applications such as neuralnetwork. Pdf quantum state tomography via compressed sensing. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using ord log2 d measurement settings. A robust compressive quantum state tomography algorithm using admm. A robust compressive quantum state tomography algorithm using. Optimal largescale quantum state tomography with pauli measurements by tony caiy, donggyu kimz, yazhen wangz, ming yuanzand harrison h.
Other approaches, such as matrix product state tomography 3, require relatively less measurementsscaling as a polynomial in n. Its aim is to find the solution to underdetermined linear systems, under the assumption that such a. Greenbaum 2015 introduction to quantum gate set tomography. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. Compressedsensing tomography the concept of cs 15 provides one approach to recover an unknown signal of known degree of sparsity by performing a small set of specialized compressive measurement outcomes, and has since been widely adopted in signal processing 20. We also give a theoretical construction of a confidence set for the density matrix of a quantum state that has optimal diameter in. In classical optics, pst involves measuring the intensity under spatial propagation 68 or time evolution. Novel schemes for measurementbased quantum computation. We call these methods indirect since they require quantumstate tomography in order to reconstruct a quantum process.
Even using advances such as compressed sensing, the effort for quantum state tomography requires at least 4 n measurements to accurately estimate an arbitrary state of n qubits. In other words, one can use a very simple setup for a characterization of a quantum state, which is robust to. Using the idea of compressed sensing, even full quantum state tomography can be made for low rank states with about the square root of the previously known. Aharonovbohminspired tomographic imaging via compressive. Second, we show that unknown lowrank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. Uncertainty quantification for matrix compressed sensing. Reconstruction of quantum channel via convex optimization. Here, we present a comprehensive analysis of experimental data from a multiphoton, multimode ghz state source using tools of compressed.
Quantum state tomography qst allows us to estimate the full state of a quantum system, providing an important diagnostic tool. Here, we present a comprehensive analysis of experimental data from a multi photon, multimode ghz state source using tools of compressed. We show that compressed tomography achieves nearly optimal sample complexity among all procedures using pauli measurements and, surprisingly, the sample complexity of compressed tomography is nearly independent of the number of measurement. As we shall see in section 3, in our case, the sparsity is due to the fact that each row of a is associated with weighted integration over a digital line in the image i and therefore a vector of weights.
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. Quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. With high probability, this stopping time is minimax optimal. In section 3 the compressive state tomography with quantum constraintsisformulatedformally,andtheproposedalgorithm is introduced and analyzed in detail. Quantum tomography, the standard tool used for this purpose. Quantum state tomography by continuous measurement and. Braket notations i are used to denote quantum states. The algorithm excels in the compressed sensing setting, where only a few data points are measured from a lowrank or highlypure quantum state of a highdimensional system. We establish methods for quantum state tomography based on compressed sensing. Zhoux university of pennsylvaniay, university of wisconsinmadisonzand yale universityx quantum state tomography aims to determine the state of a quantum system as represented by a density matrix. Flammia,3 stephen becker,4 and jens eisert5 1institute for theoretical physics, leibniz university hannover, 30167 hannover, germany 2institute for quantum information, california institute of technology, pasadena, california, usa 3perimeter institute for theoretical physics, waterloo, ontario, n2l 2y5 canada. We efficiently reconstruct these qudit states from a few scans with an intensified ccd camera. Provable quantum state tomography via nonconvex methods. First, we show that a lowrank density matrix can be estimated using fewer copies of the state, i.
In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using ordlog2d measurement settings, compared to standard methods. Multidimensional quantum entanglement with largescale. As an alternative to interferometry, phase space tomography pst is an elegant method to measure correlation functions. Full quantum state tomography has become the standard experimental tool for the complete characterization of small quantum systems comprised of up to 8 quantum particles 1. Gross et al 2018 quantum state tomography via compressed sensing. We discuss in particular the case of quantumstate tomography, where the aim is to recover the density matrix, a positive semide. Provable compressed sensing quantum state tomography via nonconvex methods. Maximum likelihood quantum state tomography is inadmissible.
Provable compressed sensing quantum state tomography via non. Quantum system identification quantum information theory. However, qst requires a large number of measurements, each derived from a different physical observable corresponding to a different experimental setup. Experimentally exploring compressed sensing quantum tomography a steffens et al 2017 quantum science and technology 2 025005. With nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for. The aharonovbohm effect is a wellestablished quantum phenomenon that relates the behaviour of an electron not only to the local electromagnetic field but also to the associated potentials. Simulations verify the e ectiveness of the proposed approach in section 4, and. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance. Reconstructing highdimensional twophoton entangled. This page is intended to be useful to a variety of visitors, from experimental research groups setting up quantum tomography systems, to students learning about the theory of characterizing quantum states.
Compressive sensing is a dataprocessing technique widely used in different signal reconstruction applications. In this letter, we investigate whether it is also optimal in any sense. Quantum state tomography via compressed sensing david gross,1 yikai liu,2 steven t. Since the measurement effort required grows exponentially with the number of qubits, this brute. The field of quantum information relies on the crucial issue of characterizing quantum states from measurements. Pdf quantum state tomography via compressed sensing yi. This list of publications is sorted by categories journal articles, conference articles, books, theses and presentations in the chronological order. In quantum mechanics, analogous techniques apply 10. As the article is a short letter only four pages it could.
Since pure states are specified by only od numbers, it seems plausible that one could be. The ability to completely characterize the state of a system is an essential element for the emerging quantum technologies. Korotkov1 1department of electrical engineering, university of california, riverside, california 92521, usa. In principle, quantum processes can be fully characterized using, for example, quantum process tomography 1 or gate set tomography 234. Here, we present a compressedsensinginspired method to ascertain any rankdeficient qudit state, which we experimentally encode in photonic orbital angular momentum. Provable compressed sensing quantum state tomography via nonconvex methods anastasios kyrillidis 1,2, amir kalev3, dohyung park4, srinadh bhojanapalli5, constantine caramanis 6and sujay sanghavi with nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for highdimensional systems.
Osa quantum state tomography with a single measurement setup. Jessen 1center for quantum information and control, college of optical sciences and department of physics. University of maryland the university of texas at austin ibm toyota technological institute at chicago 0 share. But they only work when an unknown state is uniquely. Provable compressed sensing quantum state tomography via. These methods are specialized for quantum states that are fairly. Quantum state tomography via compressed sensing laureline pinault january 5, 2017 abstract this report aims at presenting the article quantum state tomography via compressed sensing by david gross, yikai liu, steven t.
Theoretical description of compressive sensing and quantum tomography. Quantum state tomography via compressed sensing core. M cramer, mb plenio, st flammia, r somma, d gross, sd bartlett. We detail applications to quantum tomography problems where measurements arise from pauli observables. By applying the convex optimization, we will be able to extract the full and correct information from our measurement result. Quantum state tomography via compressed sensing arxiv vanity. Paper open access related content quantum tomography. This is performed through a process called quantum state tomography qst. In general, performing a complete tomography is an expensive task both in terms of the number of measurements and the computational time to reconstruct the density matrix from the data. Experimentally exploring compressed sensing quantum tomography. Characterising complex quantum systems is a vital task in quantum information science. An important consequence is that electron beams travelling through an arbitrary medium can carry information not only about the properties of the materials along with their trajectories but also about.
Optimal largescale quantum state tomography with pauli. Anastasios kyrillidis 1,2, amir kalev3, dohyung park4. Statistical analysis of compressive low rank tomography with random measurements. In this work, we introduce compressed sensingbased methods that facilitate the reconstruction of quantum channels of low kraus rank. Quantum state tomography by continuous measurement and compressed sensing a. The compressed quantum process tomography cqpt estimate of the 16.
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