D. W. Leung
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.
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With the general theorem of SQPT, we shall develop a scheme to detemine an arbitrary matrix element of $chi$, which is expanded with the Choi operators, in a scalbe way.
A minimal set of measurement operators for quantum state tomography has in the non-degenerate case ideally eigenbases which are mutually unbiased. This is different for the degenerate case. Here, we consider the situation where the measurement operators are projections on individual pure quantum states. This corresponds to maximal degeneracy. We present numerically optimized sets of projectors and find that they significantly outperform those which are taken from a set of mutually unbiased bases.
It is known that mutually unbiased bases, whenever they exist, are optimal in an information theoretic sense for the determination of unknown state of a quantum ensemble. These bases may not exist in most dimensions and some suboptimal choices have to be made. The present paper deals with estimates of the information loss in suboptimal choice of bases. The information is calculated directly in terms of transition probabilities. I give estimates for the information content of measurement in some approximate MUBs proposed recently.
Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States
该研究提出通过伪Choi态学习哈密顿量的新方法，利用时间演化幺正算符生成伪Choi态，通过经典阴影层析成像或量子均值估计高效估计哈密顿量系数，且过程对误差鲁棒。
Fast and Sure-ious Quantum Process Tomography
本文证明部分归一化是量子信道Choi态的精确保真度投影，从而将量子态层析算法提升为量子过程层析算法，并给出保真度投影最小二乘法的样本复杂度与数值优势。
The main goal of this paper is to extend and apply the principle of maximum entropy (MaxEnt) to incomplete quantum process estimation tasks. We will define a so-called process entropy function being the von Neumann entropy of the state associated with the quantum process via Choi-Jamiolkowski isomorphism. It will be shown that an arbitrary process estimation experiment can be reformulated in a unified framework and MaxEnt principle can be consistently exploited. We will argue that the suggested choice for the process entropy satisfies natural list of properties and it reduces to the state MaxE…
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.
In the present work, we propose a generalization of the confidence polytopes approach for quantum state tomography (QST) to the case of quantum process tomography (QPT). Our approach allows obtaining a confidence region in the polytope form for a Choi matrix of an unknown quantum channel based on the measurement results of the corresponding QPT experiment. The method uses the improved version of the expression for confidence levels for the case of several positive operator-valued measures (POVMs). We then demonstrate how confidence polytopes can be employed for calculating confidence intervals…
Quantum Tensor Product Decomposition from Choi State Tomography
针对量子算符层面纠缠性质难提取的问题，提出非平衡划分下的量子算法，通过小系统经典存储、环境量子资源访问实现酉算符张量积分解，可用于算符非定域性预测等，并在二维Heisenberg模型上验证。
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…
Full Quantum Process Tomography of a Universal Entangling Gate on an IBM's Quantum Computer
本研究在IBM量子计算硬件上对通用两比特纠缠门SQSCZ开展全量子过程层析成像，验证了其良好的保真度与噪声特性，为量子计算的理论与应用发展提供支撑。
Adaptive measurements were recently shown to significantly improve the performance of quantum state tomography. Utilizing information about the system for the on-line choice of optimal measurements allows to reach the ultimate bounds of precision for state reconstruction. In this article we generalize an adaptive Bayesian approach to the case of process tomography and experimentally show its superiority in the task of learning unknown quantum operations. Our experiments with photonic polarization qubits cover all types of single-qubit channels. We also discuss instrumental errors and the crite…
We present a tomographic scheme, based on spacetime symmetries, for the reconstruction of the internal degrees of freedom of a Dirac spinor. We discuss the circumstances under which the tomographic group can be taken as SU(2), and how this crucially depends on the choice of the gamma matrix representation. A tomographic reconstruction process based on discrete rotations is considered, as well as a continuous alternative.
The determination of the density matrix of an ensemble of identically prepared quantum systems by performing a series of measurements, known as quantum tomography, is minimal when the number of outcomes is minimal. The most accurate minimal quantum tomography of qubits, sometimes called a tetrahedron measurement, corresponds to projections over four states which can be represented on the Bloch sphere as the vertices of a regular tetrahedron. We investigate whether it is possible to implement the tetrahedron measurement of double slit qubits of light, using measurements performed on a single pl…
We introduce the concept of pairwise tomography networks to characterise quantum properties in many-body systems and demonstrate an efficient protocol to measure them experimentally. Pairwise tomography networks are generators of multiplex networks where each layer represents the graph of a relevant quantifier such as, e.g., concurrence, quantum discord, purity, quantum mutual information, or classical correlations. We propose a measurement scheme to perform two-qubit tomography of all pairs showing exponential improvement in the number of qubits $N$ with respect to previously existing methods…
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。
A system-ancilla bipartite state capable of containing the complete information of an unknown quantum channel acting on the system is called faithful. The equivalence between faithfulness of state and invertibility of the corresponding Jamiolkowski map proved by D'Ariano and Presti has been a useful characterization for ancilla-assisted process tomography albeit the proof was incomplete as they assumed trace nonincreasing quantum operations, not quantum channels. We complete the proof of the equivalence and introduce the generalization of faithfulness to various classes of quantum channels. We…
A Bayesian approach to quantum process tomography has yet to be fully developed due to the lack of appropriate probability distributions on the space of quantum channels. Here, by associating the Choi matrix form of a completely positive, trace preserving (CPTP) map with a particular space of matrices with orthonormal columns, called a Stiefel manifold, we present two parametric probability distributions on the space of CPTP maps that enable Bayesian analysis of process tomography. The first is a probability distribution that has an average Choi matrix as a sufficient statistic. The second is …
In this work we have explored few tools in Quantum State Tomography for Continuous Variable Systems. The concept of quantum states in phase space representation is introduced in a simple manner by using a few statistical concepts. Unlike most texts of Quantum information in which the Wigner function for a single mode is often more used, in this text the multi-modes state Wigner function is also approached. Our numerical investigations indicate that the reconstructed method using back-projection add some error due the choice of cutoff frequency, therefore it is necessary to use data post-proces…
Several Bayesian estimation based heuristics have been developed to perform quantum state tomography (QST). Their ability to quantify uncertainties using region estimators and include a priori knowledge of the experimentalists makes this family of methods an attractive choice for QST. However, specialized techniques for pure states do not work well for mixed states and vice versa. In this paper, we present an adaptive particle filter (PF) based QST protocol which improves the scaling of fidelity compared to nonadaptive Bayesian schemes for arbitrary multi-qubit states. This is due to the proto…
We investigate the performance of a Kennedy receiver, which is known as a beneficial tool in optical coherent communications, to the quantum state discrimination of the two superpositions of vacuum and single photon states corresponding to the $\hatσ_x$ eigenstates in the single-rail encoding of photonic qubits. We experimentally characterize the Kennedy receiver in vacuum-single photon two-dimensional space using quantum detector tomography and evaluate the achievable discrimination error probability from the reconstructed measurement operators. We furthermore derive the minimum error rate ob…
Parallel tomography of quantum non-demolition measurements in multi-qubit devices
本文提出QND测量并行层析协议，在7量子比特IBM-Q器件上重建测量Choi矩阵并评估保真度、QND性、破坏性与串扰，以表征单/双量子比特及广义测量。
Three-qubit W state tomography via full and marginal state reconstructions on ibm_osaka
通过全状态和边缘状态重建在ibm_osaka上实现三量子比特W态的高效成像，展示子系统方法在提高保真度方面的优势。
We propose and investigate a new method of quantum process tomography (QPT) which we call projected least squares (PLS). In short, PLS consists of first computing the least-squares estimator of the Choi matrix of an unknown channel, and subsequently projecting it onto the convex set of Choi matrices. We consider four experimental setups including direct QPT with Pauli eigenvectors as input and Pauli measurements, and ancilla-assisted QPT with mutually unbiased bases (MUB) measurements. In each case, we provide a closed form solution for the least-squares estimator of the Choi matrix. We propos…
Quantum entangled states have shown distinguished features beyond any classical state. Many methods like quantum state tomography have been presented to verify entanglement. In this work, we aim to identify unknown entanglements with partial information of the state space by developing a nonlinear entanglement witness. The witness consists of a generalized Greenberger-Horne-Zeilinger-like paradox expressed by Pauli observables, and a nonlinear inequality expressed by density matrix elements. First, we verify unknown bipartite entanglements and study the robustness of entanglement witnesses aga…
We propose a method to perform the quantum state tomography (QST) of an $\mathit{n}$-partite qudit state embedded in single photons using the Hong-Ou-Mandel (HOM) interference between the target state and probe state. This method requires only passive beam splitters for the HOM interference and removes all active optical devices in the target modes to control the measurement bases needed in conventional QST. Hence, it is applicable to various degree of freedom of the target state without altering the measurement setup. Moreover, a faithful estimation is realized even with classical probe light…
Prospects for quantum process tomography at high energies
本文提出将对撞机中的自旋和味道测量视为量子仪器，通过Choi矩阵的量子过程层析成像，在实验上重构量子通道，以探测标准模型扩展并检验量子力学基础，并以极化轻子对撞机上的顶夸克对产生过程为例。
Mixed state tomography reduces to pure state tomography
混合态成像通过纯态算法实现，首次在样本和门操作上都达到最优的成像算法。
Efficiently Learning Global Quantum Channels with Local Tomography
本文提出了一种基于局部到全局重建框架的方法，用于高效表征多量子比特通道，通过结合局部阴影全息法与凸优化获得的局部最优恢复映射，实现对全局属性的准确恢复。
Advancing quantum process tomography through quantum compilation
本文提出基于量子编译的QPT框架，通过优化Kraus算符和Choi矩阵，减少测量和计算开销，实现高精度的量子过程重构，适用于不同噪声环境下的设备验证。
Neural-Shadow Quantum State Tomography
本文提出NSQST，利用经典影子估计不忠度作为神经网络训练损失，实现无需误差缓解且样本高效、抗噪的量子态重建，并在相对相位学习中优于NNQST和直接影子估计。
Quantum tomography is the standard method of reconstructing the Wigner function of quantum states of light by means of balanced homodyne detection. The reconstruction quality strongly depends on the photodetectors quantum efficiency and other losses in the measurement setup. In this article we analyse in detail a protocol of enhanced quantum tomography, proposed by Leonhardt and Paul in 1994, which allows one to reduce the degrading effect of detection losses. It is based on phase sensitive parametric amplification, with the phase of the amplified quadrature being scanned synchronously with th…
Reducing Complexity of Shadow Process Tomography with Generalized Measurements
本文提出基于广义测量（POVMs）的影子过程层析框架（POVM-SPT），通过凸优化寻找最小化影子范数的最优 POVM，显著降低了样本复杂度，尤其在大规模量子系统中表现优异。
Recent advances in quantum computers and simulators are steadily leading us towards full-scale quantum computing devices. Due to the fact that debugging is necessary to create any computing device, quantum tomography (QT) is a critical milestone on this path. In practice, the choice between different QT methods faces the lack of comparison methodology. Modern research provides a wide range of QT methods, which differ in their application areas, as well as experimental and computational complexity. Testing such methods is also being made under different conditions, and various efficiency measur…
Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography
本文提出了可分希尔伯特空间上完全正算子的Kraus分解算法，结合过程层析成像给出强收敛的构造性证明，生成的Kraus算子序列可提供目标映射在递增子空间上限制的相干分解族。
The protocol of measurement the probability density of a rotated quadrature of microwave mode in an arbitrary quantum state is presented and numerically investigated. The protocol is adapted to the measuring procedure used in the CQED experiments. The method is based on measurement of integrated probability of detection of atoms - probes in the top working state from atoms flying via the cavity. Connection of random variables and is found, when is a qudrature phase. Connection of the tomogram and the tomogram is found. The protocol prescribes the rule of choice of problem parameters, such as i…
Optimal learning of quantum channels in diamond distance
本文提出并证明了一个算法，以O(d_in d_out k/ε^2)次通道使用实现任意Kraus秩不超过k的量子通道在金刚石距离下的ε-精度学习，并给出下界Ω(d_in d_out k)和Ω(1/ε^2)，将通道层析成像统一归约为纯态层析成像。
Spin-entaglement has been proposed and extensively used in the case of correlated triplet pairs which are intermediate states in singlet fission process in select organic semiconductors. Here, we employ quantum process tomography of polarization entangled photon-pairs resonant with the excited state absorption of these states to investigate the nature of the inherent quantum correlations and to explore for an unambiguous proof for the existence of exciton entanglement.
Physics-Constrained Conditional Generative Learning for Quantum State and Process Tomography
该研究提出物理约束条件生成式对抗网络，用于解决量子态与过程层析的高计算规模问题，通过嵌入乔列斯基层保证物理约束，结合张量网络可扩展至6比特以上系统，为噪声中等规模量子设备提供数据驱动层析方案。
The Optimal Rate Function in Covariant Quantum State Tomography
研究协变量子态层析协议的最优速率函数，证明Keyl基于Schur采样的协议在所有协变协议中具有最大速率函数，该函数是量子相对熵的退火版本。
We investigate quantum state reconstruction of a superconducting qubit threaded by an Aharonov-Bohm flux, with particular attention to the local geometric phase. A state reconstruction scheme is introduced with a proper account of the local geometric phase generated by Faraday's law of induction. Our scheme is based on measurement of three complementary quantities, that is, the extra charge and two local currents. Incorporating time-reversal symmetry and the Faraday's law, we show that the full density matrix can be reconstructed without ambiguity in the choice of gauge. This procedure clearly…
The design and operation of a quantum-mechanical device as a laboratory instrument puts models written in equations of quantum mechanics in contact with instruments. This contact is recordable in files of a Classical Digital Process-control Computer (CPC) used both to calculate with the equations and to manage the instruments. By noticing that equations and instruments make contact in a CPC, we rewrite equations of quantum mechanics to explicitly include functions of CPC-commands to the instruments. This sets up a proof that a scientist's choice in linking mathematical models to instruments is…
Semiconductor quantum dots are probably the preferred choice for interfacing anchored, matter spin qubits and flying photonic qubits. While full tomography of a flying qubit or light polarization is in general straightforward, matter spin tomography is a challenging and resource-consuming task. Here we present a novel all-optical method for conducting full tomography of quantum-dot-confined spins. Our method is applicable for electronic spin configurations such as the conduction-band electron, the valence-band hole, and for electron-hole pairs such as the bright and the dark exciton. We excite…
Quantum metrology uses quantum states with no classical counterpart to measure a physical quantity with extraordinary sensitivity or precision. Most metrology schemes measure a single parameter of a dynamical process by probing it with a specially designed quantum state. The success of such a scheme usually relies on the process belonging to a particular one-parameter family. If this assumption is violated, or if the goal is to measure more than one parameter, a different quantum state may perform better. In the most extreme case, we know nothing about the process and wish to learn everything.…
Experimental Hybrid Shadow Tomography and Distillation
本研究针对量子阴影层析成像估计非线性函数测量成本高的问题，基于单光子实现高保真Fredkin门并演示混合阴影层析成像，显著降低样本复杂度，还实现虚拟蒸馏提升量子计量参数估计精度。
Quantum state reconstruction involves measurement devices that are usually described by idealized models, but not known in full detail in experiments. For weak propagating microwaves, the detection process requires linear amplifiers which obscure the signal with random noise. Here, we introduce a theory which nevertheless allows one to use these devices for measuring all quadrature moments of propagating quantum microwaves based on cross-correlations from a dual-path amplification setup. Simultaneously, the detector noise properties are determined, allowing for tomography. We demonstrate the f…
Quantum processes involving single-photon states are of broad interest in particular for quantum communication. Extending to continuous values a recent proposal by Yuan et al \cite{YUAN16}, we show that single-photon quantum processes can be characterized using phase randomized coherent states (PRCS) as inputs. As a proof of principle, we present the experimental investigation of single-photon tomography using PRCS. The probability distribution of field quadratures measurements for single-photon states can be accurately derived from the PRCS data. As a consequence, the Wigner function and the …
Quantum state tomography, an important task in quantum information processing, aims at reconstructing a state from prepared measurement data. Bayesian methods are recognized to be one of the good and reliable choice in estimating quantum states~\cite{blume2010optimal}. Several numerical works showed that Bayesian estimations are comparable to, and even better than other methods in the problem of $1$-qubit state recovery. However, the problem of choosing prior distribution in the general case of $n$ qubits is not straightforward. More importantly, the statistical performance of Bayesian type es…
We report a proof-of-concept demonstration of a tunable infrared (IR) optical coherence tomography (OCT) technique with detection of only visible range photons. Our method is based on the nonclassical interference of frequency correlated photon pairs. The nonlinear crystal, introduced in the Michelson-type interferometer, generates photon pairs with one photon in the visible and another in the IR range. The intensity of detected visible photons depends on the phase and loss of IR photons, which interact with the sample under study. This enables us to perform imaging and characterize sample pro…
Quantum state tomography (QST) represents an essential tool for the characterization, verification, and validation (QCVV) of quantum processors. Only for a few idealized scenarios, there are analytic results for the optimal measurement set for QST. E.g., in a setting of non-degenerate measurements, an optimal minimal set of measurement operators for QST has eigenbases which are mutually unbiased. However, in other set-ups, dependent on the rank of the projection operators and the size of the quantum system, the optimal choice of measurements for efficient QST needs to be numerically approximat…
Optimal lower bounds for quantum state tomography
本文证明了量子态层析的样本复杂度下界为 $\Omega(rd/\varepsilon^2)$，与已知上界匹配，通过投影仪层析和 Bures 距离归约技术解决了混合态层析的样本复杂度问题。
Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers
本文提出ShadowQPT，一种基于经典阴影的量子过程层析成像方法，通过随机重建Choi矩阵，支持高效计算重叠、约化过程及哈密顿量学习，并在IonQ量子计算机上验证了其有效性。
Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means
该研究针对对数损失下量子态在线学习问题建立遗憾下界，推导样本最优量子态层析算法与矩阵均值置信序列算法，数值实验显示后者具竞争力。
Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition
本文研究了量子信道反演的最优查询复杂度，揭示了在不同τ值下查询复杂度的相变行为，发现τ=1时具有海森堡尺度，而τ≥1+Ω(1)时具有经典尺度。
Quantum channel tomography and estimation by local test
本文通过局部测试方法，在不同条件下提出了量子通道体素和估计的最优查询复杂度，展示了海森堡缩放的实现条件。
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