Alessandro Bisio, Giulio Chiribella, Giacomo Mauro D'Ariano, Stefano Facchini, Paolo Perinotti
The present short review article illustrates the latest theoretical developments on quantum tomography, regarding general optimization methods for both data-processing and setup. The basic theoretical tool is the informationally complete measurement. The optimization theory for the setup is based on the new theoretical approach of quantum combs.
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We present the first complete optimization of quantum tomography, for states, POVMs, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes.
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].
Optimal Quantum State Tomography via Weak Value
本文针对任意 d 维量子系统，通过搜索系统与测量装置的最优耦合强度，并利用均方误差分析，提出基于弱值的最优量子态层析策略，推导出重构密度矩阵的最小均方误差。
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance the estimation precision, we also propose a two-step adaptive detector tomography algorithm to optimize the probe states adapt…
Practical quantum state tomography is usually performed by carrying out repeated measurements on many copies of a given state. The accuracy of the reconstruction depends strongly on the dimensionality of the system and the number of copies used for the measurements. We investigate the accuracy of an experimental implementation of a minimal and optimal tomography scheme for one- and two-qubit states encoded in the polarization of photons. A suitable statistical model for the attainable accuracy is introduced.
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.
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…
Optimal qudit overlapping tomography and optimal measurement order
本文提出针对qudit系统的最优重叠测角方法，通过构建本地测量设置并优化测量顺序，减少实验开销，提升量子系统表征效率。
Given an experimental set-up and a fixed number of measurements, how should one take data in order to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [arXiv:quant-ph/0411093v1]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of 'minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias.…
Sample Optimal and Memory Efficient Quantum State Tomography
本文提出一种基于酉Schur采样的量子态层析算法，在保持样本最优性的同时实现内存高效，仅需流式访问样本。
Quantum tomography is an essential method of the photonic technology toolbox and is routinely used for evaluation of experimentally prepared states of light and characterization of devices transforming such states. The tomography procedure consists of many different sequentially performed measurements. We present considerable tomography speedup by optimally arranging the individual constituent measurements, which is equivalent to solving an instance of the traveling salesman problem. As an example, we obtain solutions for photonic systems of up to five qubits, and conclude that already for sys…
Nearly Time-Optimal Pure State Tomography with Pauli Measurements
本文提出一种近最优的纯态量子态重构算法，利用单量子比特测量，在时间复杂度和复制复杂度上接近最优，输出高保真度的量子态估计。
We present a method for quantum state tomography that enables the efficient estimation, with fixed precision, of any of the matrix elements of the density matrix of a state, provided that the states from the basis in which the matrix is written can be efficiently prepared in a controlled manner. Furthermore, we show how this algorithm is well suited for quantum process tomography, enabling to perform selective and efficient quantum process tomography.
Near-Optimal Parallel Unitary Process Tomography in Diamond Distance
本文证明在金刚石距离下，通过新颖的并行程序可实现近最优查询复杂度的幺正过程层析成像，弥补了与顺序电路的差距。
Mutually unbiased bases determine an optimal set of measurements to extract complete information about the quantum state of a system. However, quite often a priori information about the state exist, making some of the measurement bases superfluous. This is, for example, the case when multiqubit states belong to the permutationally invariant subspace. In this paper we derive the minimal sets of mutually unbiased bases needed to tomographically reconstruct such states.
Optimal Quantum Overlapping Tomography
本文通过将重叠层析问题映射到团覆盖模型，提出最优量子重叠层析统一框架，显著降低测量成本，并在核自旋与超导量子处理器上验证了其可行性与鲁棒性。
Balanced homodyning, heterodyning and unbalanced homodyning are the three well-known sampling techniques used in quantum optics to characterize all possible photonic sources in continuous-variable quantum information theory. We show that for all quantum states and all observable-parameter tomography schemes, which includes the reconstructions of arbitrary operator moments and phase-space quasi-distributions, localized sampling with unbalanced homodyning is always tomographically more powerful (gives more accurate estimators) than delocalized sampling with heterodyning. The latter is recently k…
It is a well-known fact that the optimal POVM for quantum state tomography is the symmetric, informationally complete, positive operator valued measure (SIC-POVM). We investigate the same problem only in the case when there are some a priori information about the state, specifically when some parameters are known. In this paper we mainly focus on solving a 3-dimensional optimization problem, which gives us a non-trivial example for the so-called conditional SIC- POVMs, a straightforward generalization of the concept of SIC-POVMs. We also present other special cases to show further applications…
We show that weighted unitary 2-designs define optimal measurements on the system-ancilla output state for ancilla-assisted process tomography of unital quantum channels. Examples include complete sets of mutually unbiased unitary-operator bases. Each of these specifies a minimal series of optimal orthogonal measurements. General quantum channels are also considered.
Using the the convex semidefinite programming method and superoperator formalism we obtain the finite quantum tomography of some mixed quantum states such as: qudit tomography, N-qubit tomography, phase tomography and coherent spin state tomography, where that obtained results are in agreement with those of References \cite{schack,Pegg,Barnett,Buzek,Weigert}.
Quantum state tomography--the practice of estimating a quantum state by performing measurements on it--is useful in a variety of contexts. We introduce "gentle tomography" as a version of tomography that preserves the measured quantum data. As an application of gentle tomography, we describe a simple polynomial-time method for universal source coding.
It is shown that quantum tomography can detect and correct unlimited number of errors during the evaluation of quantum algorithms on quantum computer.
Post-processing optimization and optimal bounds for non-adaptive shadow tomography
本文提出通过凸minimax问题优化后处理，以获得非适应影成像中固定POVM和可观察量下的最优方差界，并通过数值实验展示了其在减少采样复杂度和改善系统规模缩放方面的优势。
Quantum process tomography --- a primitive in many quantum information processing tasks --- can be cast within the framework of the theory of design of experiment (DoE), a branch of classical statistics that deals with the relationship between inputs and outputs of an experimental setup. Such a link potentially gives access to the many ideas of the rich subject of classical DoE for use in quantum problems. The classical techniques from DoE cannot, however, be directly applied to the quantum process tomography due to the basic structural differences between the classical and quantum estimation …
Amongst the multitude of state reconstruction techniques, the so-called "quantum tomography" seems to be the most fruitful. In this letter, I will start by developing the mathematical apparatus of quantum tomography and, later, I will explain how it can be applied to various quantum systems.
The efficiency of quantum state tomography is discussed from the point of view of quantum parameter estimation theory, in which the trace of the weighted covariance is to be minimized. It is shown that tomography is optimal only when a special weight is adopted.
We report on the experimental implementation of a polarimeter based on a scheme known to be optimal for obtaining the polarization vector of ensembles of spin-1/2 quantum systems, and the alignment procedure for this polarimeter is discussed. We also show how to use this polarimeter to estimate the polarization state for identically prepared ensembles of single photons and photon pairs and extend the method to obtain the density matrix for generic multi-photon states. State reconstruction and performance of the polarimeter is illustrated by actual measurements on identically prepared ensembles…
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…
We argue that the claimed optimality of a new process tomography method suggested in [quant-ph/0601033] and [quant-ph/0601034] is based on not completely fair comparison that does not take into account the available information in an equal way. We also argue that the method is not a new process tomography scheme, but rather represents an interesting modification of ancilla assisted process tomography method. In our opinion these modifications require deeper understanding and further investigation.
Experimental demonstration of Quantum Overlapping Tomography
针对量子层析的指数时间复杂度问题，本研究开发贝叶斯态估计方法并实验演示量子重叠层析方案，证实其能以对数时间复杂度和更少测量次数获取多体量子系统准确关键信息。
Shadow process tomography of quantum channels
本文将阴影态层析思想推广到量子过程表征，利用 Choi 同构建立测量次数界限并提出实现算法，显著降低理解大规模量子系统所需的测量开销。
The intimate connection between the Banach space wavelet reconstruction method on homogeneous spaces with both singular and nonsingular vacuum vectors, and some of well known quantum tomographies, such as: Moyal-representation for a spin, discrete phase space tomography, tomography of a free particle, Homodyne tomography, phase space tomography and SU(1,1) tomography is explained. Also both the atomic decomposition and banach frame nature of these quantum tomographic examples is explained in details. Finally the connection between the wavelet formalism on Banach space and Q-function is discuss…
In this paper, we derive analytic expressions for the starting (initial) values of the parameters of the T-matrix that is frequently employed in the construction of a theoretical density matrix in a Maximum Likelihood Estimate (MLE) procedure; this optimization procedure is used to fit the experimental data pertaining to single qubit state tomography. Appropriate starting values are critical in achieving a global minimum in the fitting process. We also indicate an analytic way of making the experimentally determined density matrix physical without resorting to the MLE process, if optimality in…
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…
Optimal and robust experiment design for quantum state tomography of star-topology register
本文针对星型拓扑寄存器中外围自旋不可独立寻址的约束，利用星型对称性分解希尔伯特空间，提出最少测量次数的紧凑层析策略，并通过优化参数化量子电路提升抗测量误差鲁棒性，在10自旋系统上验证了其表征大规模系统的能力。
Unified formalism and adaptive algorithms for optimal quantum state, detector and process tomography
本文统一了量子态、探测器和过程层析成像的保真度损失度量，提出自适应算法以实现最优标度，并通过实验首次验证了辅助辅助过程层析成像的最优性能。
The paper is devoted to the mathematical foundation of the quantum tomography using the theory of square-integrable representations of unimodular Lie groups.
Boosting projective methods for quantum process and detector tomography
本文提出结合解析投影与交替投影的量子过程及探测器层析成像方法，在保持效率的同时将最近量子信道逼近精度提升三个数量级，并经多量子比特数值模拟验证。
Tomography of parametrized quantum states
本研究拓展量子态层析成像基础至参数化量子态场景，提出统一框架与结合信号处理、压缩感知的即插即用算法，可实现带性能保证的参数化量子态及信道层析成像，并通过NMR等哈密顿量演化的阴影层析成像示例验证。
A number of problems in quantum state and system identification are addressed. Specifically, it is shown that the maximum likelihood estimation (MLE) approach, already known to apply to quantum state tomography, is also applicable to quantum process tomography (estimating the Kraus operator sum representation (OSR)), Hamiltonian parameter estimation, and the related problems of state and process (OSR) distribution estimation. Except for Hamiltonian parameter estimation, the other MLE problems are formally of the same type of convex optimization problem and therefore can be solved very efficien…
In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantum-state and qua…
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…
Optimal Single Qubit Tomography: Realization of Locally Optimal Measurements on a Quantum Computer
本文从量子计量学角度研究单量子比特状态的最优表征问题，提出在良好先验知识下的局部最优测量方法，并在超导量子计算机上实验验证其可达到 Nagaoka--Hayashi 理论极限，同时通过模拟证明该方法对不同先验知识量下的任意量子比特状态具有鲁棒性。
Quantum process tomography is a procedure by which an unknown quantum operation can be fully experimentally characterized. We reinterpret Choi's proof of the fact that any completely positive linear map has a Kraus representation [Lin. Alg. and App., 10, 1975] as a method for quantum process tomography. Furthermore, the analysis for obtaining the Kraus operators are particularly simple in this method.
Query-optimal unitary channel tomography in diamond distance with parallel access
针对并行查询下酉信道层析成像的效率问题，提出新协议以O(d²/ε)次并行查询达到金刚石距离ε精度，匹配下界，弥合并行与顺序策略差距，并优化了边界区域及费米子线性光学层析成像的查询复杂度。
The performance of a quantum teleportation algorithm implemented on an ion trap quantum computer is investigated. First the algorithm is analyzed in terms of the teleportation fidelity of six input states evenly distributed over the Bloch sphere. Furthermore, a quantum process tomography of the teleportation algorithm is carried out which provides almost complete knowledge about the algorithm.
Quantum tomography is a process of quantum state reconstruction using data from multiple measurements. An essential goal for a quantum tomography algorithm is to find measurements that will maximize the useful information about an unknown quantum state obtained through measurements. One of the recently proposed methods of quantum tomography is the algorithm based on rank-preserving transformations. The main idea is to transform a basic measurement set in a way to provide a situation that is equivalent to measuring the maximally mixed state. As long as tomography of a fully mixed state has the …
We study the possibility of performing quantum state tomography via equidistant states. This class of states allows us to propose a non-symmetric informationally complete POVM based tomographic scheme. The scheme is defined for odd dimensions and involves an inversion which can be analytically carried out by Fourier transform.
Sample-optimal single-copy quantum state tomography with shallow-depth measurements
研究通过浅深度测量实现单拷贝量子态层析成像，引入基于对数深度局部电路的无辅助单拷贝QST协议，对特定态实现样本最优，满秩混合态消除对数开销，表明可通过浅深度测量实现样本最优的QST。
Quantum tomography is a widely applicable tool for complete characterization of quantum states and processes. In the present work, we develop a method for precision-guaranteed quantum process tomography. With the use of the Choi-Jamiolkowski isomorphism, we generalize the recently suggested extended norm minimization estimator for the case of quantum processes. Our estimator is based on the Hilbert-Schmidt distance for quantum processes. Specifically, we discuss the application of our method for characterizing quantum gates of a superconducting quantum processor in the framework of the IBM Q E…
Robust, accurate and efficient quantum tomography is key for future quantum technologies. Traditional methods are impractical for even medium sized systems and are not robust against noise and errors. Here we report on an experimental demonstration of self-guided quantum tomography; an autonomous, fast, robust and precise technique for measuring quantum states with significantly less computational resources than standard techniques. The quantum state is iteratively learned by treating tomography as a projection measurement optimization problem. We experimentally demonstrate robustness against …
We present an experimental demonstration of quantum optical coherence tomography (QOCT). The technique makes use of an entangled twin-photon light source to carry out axial optical sectioning. QOCT is compared to conventional optical co1herence tomography (OCT). The immunity of QOCT to dispersion, as well as a factor of two enhancement in resolution, are experimentally demonstrated.
Using a Hong-Ou-Mandel interferometer, we apply the techniques of quantum process tomography to characterize errors and decoherence in a prototypical two-photon operation, a singlet-state filter. The quantum process tomography results indicate a large asymmetry in the process and also the required operation to correct for this asymmetry. Finally, we quantify errors and decoherence of the filtering operation after this modification.
We characterized unital quantum channels of single photon polarization qubits. The channels are composed of two birefringent crystals and wave-plates, where their decoherence properties are controlled. An experimental comparison between two different depolarizing configurations was performed using a quantum process tomography procedure. The results are with a high fidelity to the theoretical predictions.
The polarizing multi-photon quantum states tomography with non-unit quantum efficiency of detectors is considered. A new quantum tomography protocol is proposed. This protocol considers events of losing photons of multi-photon quantum state in one or more channels among with n-fold coincidence events. The advantage of the proposed protocol compared with the standard n-fold coincidence protocol is demonstrated using the methods of statistical analysis.
We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characteriz…
Optimal Overlapping Tomography
本研究针对重叠层析成像的性能极限问题，提出基于图论的最优Pauli测量设置方案和通用投影测量下的k体重构方法，并通过六光子实验验证了实用性，为量子系统表征提供了高效途径。
We introduce quantum tomography on locally compact Abelian groups $G$. A linear map from the set of quantum states on the $C^*$-algebra $A(G)$ generated by the projective unitary representation of $G$ to the space of characteristic functions is constructed. The dual map determining symbols of quantum observables from $A(G)$ is derived. Given a characteristic function of a state the quantum tomogram consisting a set of probability distributions is introduced. We provide three examples in which $G={\mathbb R}$ (the optical tomography), $G={\mathbb Z}_n$ (corresponding to measurements in mutually…
Annealing approach to quantum tomography is theoretically proposed. First, based on the maximum entropy principle, we introduce classical parameters to combine "quantum models (or quantum states)" given a prior for potentially representing the unknown target state. Then, we formulate the quantum tomography as an optimization problem on the classical parameters, by employing relative entropy of the parametrized state with the target state as the objective function to be minimized. We show that the objective function is physically implementable, in a theoretical sense at least, as an effective H…
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.
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