To obtain a complete description of a quantum system, one usually employs standard quantum state tomography, which however requires exponential number of measurements to perform and hence is impractical when the system's size grows large. In this work, we introduce a self-learning tomographic scheme based on the variational hybrid quantum-classical method. The key part of the scheme is a learning procedure, in which we learn a control sequence capable of driving the unknown target state coherently to a simple fiducial state, so that the target state can be directly reconstructed by applying the control sequence reversely. In this manner, the state tomography problem is converted to a state-to-state transfer problem. To solve the latter problem, we use the closed-loop learning control approach. Our scheme is further experimentally tested using techniques of a 4-qubit nuclear magnetic resonance. {Experimental results indicate that the proposed tomographic scheme can handle a broad class of states including entangled states in quantum information, as well as dynamical states of quantum many-body systems common to condensed matter physics.
We develop a quantum process tomography method, which variationally reconstruct the map of a process, using noisy and incomplete information about the dynamics. The new method encompasses the most common quantum process tomography schemes. It is based on the variational quantum tomography method (VQT) proposed by Maciel et al. in arXiv:1001.1793[quant-ph].
Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with the missing information. This is the purpose of the Maximum Entropy estimation (MaxEnt) and the Variational Quantum Tomography (VQT). Here, we propose a variant of Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We pro…
Quantum process tomography is an experimental technique to fully characterize an unknown quantum process. Standard quantum process tomography suffers from exponentially scaling of the number of measurements with the increasing system size. In this work, we put forward a quantum machine learning algorithm which approximately encodes the unknown unitary quantum process into a relatively shallow depth parametric quantum circuit. We demonstrate our method by reconstructing the unitary quantum processes resulting from the quantum Hamiltonian evolution and random quantum circuits up to $8$ qubits. R…
We improve upon a recently introduced efficient quantum state reconstruction procedure targeted to states well-approximated by the multi-scale entanglement renormalization ansatz (MERA), e.g., ground states of critical models. We show how to numerically select a subset of experimentally accessible measurements which maximizes information extraction about renormalized particles, thus dramatically reducing the required number of physical measurements. We numerically estimate the number of measurements required to characterize the ground state of the critical 1D Ising (resp. XX) model and find th…
For any finite dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state dependen…
Centro Interdisciplinario de Telecomunicaciones Electrónica, Computación y Ciencia Aplicada (CONICET), Universidad Nacional de Rio Negro（里奥内格罗国立大学跨学科电信、计算与应用科学中心（阿根廷国家科学研究委员会））；Instituto de Ciencias de la Computación, CONICET - Universidad de Buenos Aires（布宜诺斯艾利斯大学计算机科学学院（阿根廷国家科学研究委员会））；Instituto de Física La Plata (CONICET-UNLP)（拉普拉塔物理研究所（阿根廷国家科学研究委员会-拉普拉塔国立大学））
We show that a generic $N$-qudit pure quantum state is uniquely determined by only $2$ of its $\lceil\frac{N+1}{2}\rceil$-particle reduced density matrices. Therefore we give a method to uniquely determine a generic $N$-qudit pure state of dimension $D=d^N$ with $O(D)$ local measurements, which is an improvement comparing with the previous known approach using $O(D\log^2 D)$ or $O(D\log D)$ measurements.
We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary $n$-qubit pure state among all quantum states. We show that only $11$ Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with $16$ measurements, and only $31$ Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with $64$ measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. …
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
We discuss quantum state tomography via a stepwise reconstruction of the eigenstates of the mixed states produced in experiments. Our method is tailored to the experimentally relevant class of nearly pure states or simple mixed states, which exhibit dominant eigenstates and thus lend themselves to low-rank approximations. The developed scheme is applicable to any pure-state tomography method, promoting it to mixed-state tomography. Here, we demonstrate it with machine learning-inspired pure-state tomography based on neural-network representations of quantum states. The latter have been shown t…
We introduce a new method to reconstruct unknown quantum states out of incomplete and noisy information. The method is a linear convex optimization problem, therefore with a unique minimum, which can be efficiently solved with Semidefinite Programs. Numerical simulations indicate that the estimated state does not overestimate purity, and neither the expectation value of optimal entanglement witnesses. The convergence properties of the method are similar to compressed sensing approaches, in the sense that, in order to reconstruct low rank states, it needs just a fraction of the effort correspon…
The success of quantum information processing applications relies on accurate and efficient characterization of quantum states, especially nearly-pure states. In this work, we investigate a procedure for adaptive qubit state tomography which achieves $O(1/N)$ scaling in accuracy for large $N$. We analyze the performance of the adaptive protocol on the characterization of pure, mixed and nearly-pure states, and clarify the interplay between the purity of the state and the statistical fluctuations of the measurement results. Our results highlight the subtle difference between the characterizatio…
The deployment of intermediate- and large-scale quantum devices necessitates the development of efficient full state tomographical techniques for quantum benchmarks. Here, we introduce a matrix filling-based method for tomography of pure quantum states, called the matrix-completion quantum state tomography. This method requires only 2n + 1 local Pauli operators and minimal post-processing for n-qubit states. Numerical results show that our method is highly efficient on superconducting real quantum devices and achieves better fidelity estimates of multiqubit quantum states as compared to contem…
Extracting information from quantum devices has long been a crucial problem in the field of quantum mechanics. By performing elaborate measurements, quantum state tomography, an important and fundamental tool in quantum science and technology, can be used to determine unknown quantum states completely. In this study, we explore methods to determine multi-qubit pure quantum states uniquely and directly. Two adaptive protocols are proposed, with their respective quantum circuits. Herein, two or three observables are sufficient, while the number of measurement outcomes is either the same as or fe…
Quantum state tomography (QST) aims at estimating a quantum state from averaged quantum measurements made on copies of the state. Most quantum algorithms rely on QST at some point and it is a well explored topic in the literature, mostly for mixed states. In this paper we focus on the QST of a pure quantum state using parallel unentangled measurements. Pure states are a small but useful subset of all quantum states, their tomography requires fewer measurements and is essentially a phase recovery problem. Parallel unentangled measurements are easy to implement in practice because they allow the…
Entangled pure-states, Werner-states and generalized mixed-states of any structure, spanning a 2x2 Hilbert space are created by a novel high-brilliance universal source of polarization-entangled photon pairs. The violation of a Bell inequality has been tested for the first time with a pure-state, indeed a conceptually relevant ideal condition, and with Werner-states. The generalized ''maximally entangled mixed states'' (MEMS) were also synthetized for the first time and their exotic properties investigated by means of a quantum tomographic technique.
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 a 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…
National University of Defense Technology（国防科技大学）；National Innovation Institute of Defense Technology（国防科技创新研究院）
Quantum state tomography is a key process in most quantum experiments. In this work, we employ quantum machine learning for state tomography. Given an unknown quantum state, it can be learned by maximizing the fidelity between the output of a variational quantum circuit and this state. The number of parameters of the variational quantum circuit grows linearly with the number of qubits and the circuit depth, so that only polynomial measurements are required, even for highly-entangled states. After that, a subsequent classical circuit simulator is used to transform the information of the target …
Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of $d=3,4$, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, w…
Recent work has revealed that the wave function of a pure state can be measured directly and that complementary knowledge of a quantum system can be obtained simultaneously by weak measurements. However, the original scheme applies only to pure states, and it is not efficient because most of the data are discarded by post-selection. Here, we propose tomography schemes for pure states and for mixed states via weak measurements, and our schemes are more efficient because we do not discard any data. Furthermore, we demonstrate that any matrix element of a general state can be directly read from a…
We study the performance of efficient quantum state tomography methods based on neural network quantum states using measured data from a two-photon experiment. Machine learning inspired variational methods provide a promising route towards scalable state characterization for quantum simulators. While the power of these methods has been demonstrated on synthetic data, applications to real experimental data remain scarce. We benchmark and compare several such approaches by applying them to measured data from an experiment producing two-qubit entangled states. We find that in the presence of expe…
By using a generalization of the optical tomography technique we describe the dynamics of a quantum system in terms of equations for a purely classical probability distribution which contains complete information about the system.
A powerful tool for studying geometrical problems in Hilbert space is developed. In particular, we study the quantum pure state tomography problem in finite dimensions from the point of view of dynamical systems and bifurcations theory. First, we introduce a generalization of the Hellinger metric for probability distributions which allows us to find a geometrical interpretation of the quantum state tomography problem. Thereafter, we prove that every solution to the state tomography problem is an attractive fixed point of the so-called physical imposition operator. Additionally, we demonstrate …
Universal compilation is a training process that compiles a trainable unitary into a target unitary and it serves vast potential applications from quantum dynamic simulations to optimal circuits with deep-compressing, device benchmarking, quantum error mitigation, and so on. Here, we propose a universal compilation-based variational algorithm for the preparation and tomography of quantum states in low-depth quantum circuits. We apply the Fubini-Study distance to be a trainable cost function under various gradient-based optimizers, including the quantum natural gradient approach. We evaluate th…
We study the problem of quantum-state tomography under the assumption that the state of the system is close to pure. In this context, an efficient measurements that one typically formulates uniquely identify a pure state from within the set of other pure states. In general such measurements are not robust in the presence of measurement noise and other imperfections, and therefore are less practical for tomography. We argue here that state tomography experiments should instead be done using measurements that can distinguish a pure state from {\em any} other quantum state, of any rank. We show t…
We develop a practical quantum tomography protocol and implement measurements of pure states of ququarts realized with polarization states of photon pairs (biphotons). The method is based on an optimal choice of the measuring scheme's parameters that provides better quality of reconstruction for the fixed set of statistical data. A high accuracy of the state reconstruction (above 0.99) indicates that developed methodology is adequate.
We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rd log^2 d) measurement settings, compared to standard methods that require d^2 settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable a…
We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole families of polarization states of qubits and ququaters including pure, mixed, entangled and separable.
Multipartite entanglement has been widely regarded as key resources in distributed quantum computing, for instance, multi-party cryptography, measurement based quantum computing, quantum algorithms. It also plays a fundamental role in quantum phase transitions, even responsible for transport efficiency in biological systems. Certifying multipartite entanglement is generally a fundamental task. Since an $N$ qubit state is parameterized by $4^N-1$ real numbers, one is interested to design a measurement setup that reveals multipartite entanglement with as little effort as possible, at least witho…
The article undertakes the problem of pure state estimation from projective measurements based on photon counting. Two generic frames for qubit tomography are considered -- one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases (MUBs). Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to hi…
We investigate quantum state tomography (QST) for pure states and quantum process tomography (QPT) for unitary channels via $adaptive$ measurements. For a quantum system with a $d$-dimensional Hilbert space, we first propose an adaptive protocol where only $2d-1$ measurement outcomes are used to accomplish the QST for $all$ pure states. This idea is then extended to study QPT for unitary channels, where an adaptive unitary process tomography (AUPT) protocol of $d^2+d-1$ measurement outcomes is constructed for any unitary channel. We experimentally implement the AUPT protocol in a 2-qubit nucle…
We report an experimental realization of an adaptive quantum state tomography protocol. Our method takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states we observe close to 1/N scaling of infidelity with overall number of registered events, while best non-adaptive protocols allow for $1/\sqrt{N}$ scaling only. Experiments are performed for polarization qubits, but the approach is readily adapted to any dimension.
Maximum likelihood estimation (MLE) is the most common approach to quantum state tomography. In this letter, we investigate whether it is also optimal in any sense. We show that MLE is an inadmissible estimator for most of the commonly used metrics of accuracy, i.e., some other estimator is more accurate for every true state. MLE is inadmissible for fidelity, mean squared error (squared Hilbert-Schmidt distance), and relative entropy. We prove that almost any estimator that can report both pure states and mixed states is inadmissible. This includes MLE, compressed sensing (nuclear-norm regular…
We introduce a fast and accurate heuristic for adaptive tomography that addresses many of the limitations of prior methods. Previous approaches were either too computationally intensive or tailored to handle special cases such as single qubits or pure states. By contrast, our approach combines the efficiency of online optimization with generally applicable and well-motivated data-processing techniques. We numerically demonstrate these advantages in several scenarios including mixed states, higher-dimensional systems, and restricted measurements.
We propose the simulation of quantum-optical systems in the ultrastrong-coupling regime using a variational quantum algorithm. More precisely, we introduce a short-depth variational form to prepare the groundstate of the multimode Dicke model on a quantum processor and present proof-of-principle results obtained via cloud access to an IBM device. We moreover provide an algorithm for characterizing the groundstate by Wigner state tomography. Our work is a first step towards digital quantum simulation of quantum-optical systems with potential applications to the spin-boson, Kondo and Jahn-Teller…
Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable tradeoffs inherent to quantum tomography.
We develop a technique for single qubit quantum state tomography using the mathematical setup of generalized quantization scheme for games. In our technique Alice sends an unknown pure quantum state to Bob who appends it with |0><0| and then applies the unitary operators on the appended quantum state and finds the payoffs for Alice and himself. It is shown that for a particular set of unitary operators these elements become equal to Stokes parameters for an unknown quantum state. In this way an unknown quantum state can be measured and reconstructed. Strictly speaking this technique is not a g…
The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance, and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography er…
A throughout study of statistical characteristics of fidelity in different protocols of quantum tomography is given. We consider protocols based on geometry of platonic solids and other polyhedrons with high degree of symmetry such as fullerene and its dual polyhedron. Characteristics of fidelity in different protocols are compared to the theoretical level of the minimum possible level of fidelity loss. Tomography of pure and mixed states in Hilbert spaces of different dimension is analyzed. Results of this work could be used for a better control of quantum gates and quantum states in quantum …
Using the existing classification of all alternatives to the measurement postulates of quantum theory we study the properties of bi-partite systems in these alternative theories. We prove that in all these theories the purification principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This allows us to derive the measurement postulates of quantum theory from the structure of pure states and reversible dynamics, and the requirement that the purification principle holds. The violation of the purification principle implies that there is so…
The need to perform quantum state tomography on ever larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. Both efficiently reconstruct nearly pure states in the 16-dimensional ground manifold, reaching average fidelities FCS = 0.92 and FLS = 0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experim…
We derive an expression for a density operator estimated via Bayesian quantum inference in the limit of an infinite number of measurements. This expression is derived under the assumption that the reconstructed system is in a pure state. In this case the estimation corresponds to an averaging over a generalized microcanonical ensemble of pure states satisfying a set of constraints imposed by the measured mean values of the observables under consideration. We show that via the ``purification'' ansatz, statistical mixtures can also be consistently reconstructed via the quantum Bayesian inference…
Compared with the conventional quantum state tomography (QST), the direct state tomography (DST) using weak value is easily manipulated in experiments. However, the efficiency of the DST is lower than that of the conventional QST, especially for high-dimensional systems. For a pure state, the DST is revised to improve the efficiency. In the revised DST, the real or imaginary parts of the weak values can be obtained by measuring only one system observable. We constructed a hybrid DST by combining the original DST and the revised DST. By using the appropriate measurement strength, the efficiency…
Modern day quantum simulators can prepare a wide variety of quantum states but the accurate estimation of observables from tomographic measurement data often poses a challenge. We tackle this problem by developing a quantum state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomia…
The dynamical equation of quantum mechanics are rewritten in form of dynamical equations for the measurable, positive marginal distribution of the shifted, rotated and squeezed quadrature introduced in the so called "symplectic tomography". Then the possibility of a purely classical description of a quantum system as well as a reinterpretation of the quantum measurement theory is discussed and a comparision with the well known quasi-probabilities approach is given. Furthermore, an analysis of the properties of this marginal distribution, which contains all the quantum information, is performed…
Quantum state tomography is an elementary tool to fully characterize an unknown quantum state. As the quantum hardware scales up in size, the standard quantum state tomography becomes increasingly challenging due to its exponentially growing complexity. In this work, we propose a scalable solution by considering state tomography as a language modeling task, where the unknown quantum state is treated as an unknown language, the correlation of the quantum state is interpreted as the semantic information specific to this language, and the measurement outcomes are simply the text instances generat…
