Xiaoguang Wang, Chang-Shui Yu, X. X. Yi
We give an alternative definition of quantum fidelity for two density operators on qudits in terms of the Hilbert-Schmidt inner product between them and their purity. It can be regarded as the well-defined operator fidelity for the two operators and satisfies all Jozsa's four axioms up to a normalization factor. One desire property is that it is not computationally demanding.
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We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in terms of minimal sets of pure state preparations alone, which may be interesting from the experimental point of view. These preparations can be seen as the outcomes of certain minimal positive operator valued measures. The connection of our results with these generalized measurements is briefly discussed.
The application of high-dimensional quantum systems (qudits) in quantum computing and communications seems to be a promising avenue due to the possibility of increasing the amount of information encoded in one physical carrier. In this work, we propose a method for implementing single-qudit gates for qudits based on light modes with orbital angular momentum. Method for logical qudits encoding, which ensures the quasi-cyclicity of operations, is introduced. Based on the protocol for converting the orbital angular momentum of light in the Raman quantum memory scheme [Vashukevich E.A. et. al. PRA…
Collateral coupling between superconducting resonators: Fast and high fidelity generation of qudit-qudit entanglement
本文提出利用超导谐振器间经量子比特电容耦合产生的距离无关相互作用，实现快速高保真的量子比特-量子比特纠缠，并以更少操作生成高保真NOON态，方案还可推广至光学腔原子晶格系统。
We study the conditions when mixtures of entangled pure states with maximally mixed one-qudit reduced density matrices remain entangled. We found that the resulting mixed state remains entangled when the number of entangled pure states to be mixed is less than or equal to the dimension of the pure states. For the latter case of mixing a number of pure states equal to their dimension, we found that the mixed state is entangled provided that the entangled pure states to be mixed are not equally weighted. We also found that one can restrict the set of pure states that one can mix from in order to…
We study the phase-covariant quantum cloning machine for qudits, i.e. the input states in d-level quantum system have complex coefficients with arbitrary phase but constant module. A cloning unitary transformation is proposed. After optimizing the fidelity between input state and single qudit reduced density opertor of output state, we obtain the optimal fidelity for 1 to 2 phase-covariant quantum cloning of qudits and the corresponding cloning transformation.
We extend the former matrix rearrangement approach of the entangling power to the general cases, without the requirement of the same dimensions of the subsystems. The entangling power of a unitary operator is completely determined by its realignment and partial transposition. As applications, we calculate the entangling power for the Ising interaction and the isotropic Heisenberg interaction in the mixed qudit system.
Toy model for the correlation of qudit bipartite states with maximally mixed marginals
针对高维 qudit 二部最大混合边缘态局域幺正分类难题，本文构造态族并给出不变量，还完整分类了一个双参数两 qutrit 族，将其与纯度和负性关联。
We report interferometric schemes to prepare arbitrary states of four-dimensional qudits (ququarts) based on biphoton states of ultrafast-pumped frequency-nondegenerate spontaneous parametric down-conversion. Preparation and tomographic characterization of a few examples of general single-ququart states, a pure state, a completely mixed state, and a partially-mixed state, are experimentally demonstrated.
Bounds, expressed in terms of d and N, on full Bell locality of a quantum state for $N\geq 3$ nonlocally entangled qudits (of a dimension $d\geq 2$) mixed with white noise are known, to our knowledge, only within full separability of this noisy N-qudit state. For the maximal violation of general Bell inequalities by an N-partite quantum state, we specify the analytical upper bound expressed in terms of dilation characteristics of this state, and this allows us to find new general bounds in $d, N$, valid for all $d\geq 2$ and all $N\geq 3$, on full Bell locality under generalized quantum measur…
We propose a quantum algorithm for the purification of a generic mixed state $ρ$ of a $n$-qudit system by using an ancillary $n$-qudit system. The algorithm is optimal in that (i) the number of ancillary qudits cannot be reduced, (ii) the number of parameters which determine the purification state $|Ψ>$ exactly equals the number of degrees of freedom of $ρ$, and (iii) $|Ψ>$ is easily determined from the density matrix $ρ$. Moreover, we introduce a quantum circuit in which the quantum gates are unitary transformations acting on a $2n$-qudit system. These transformations are determined by parame…
Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high fidelity two-qudit gates. We first describe how to implement parallel single qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth of the parallel gate sequence is equal to the height of an associated tree. The imp…
In a recent paper, Rungta et. al. [Phys. Rev. A, 64, 042315, 2001] introduced a measure of mixed-state entanglement called the I-concurrence for arbitrary pairs of qudits. We find an exact formula for an entanglement measure closely related to the I-concurrence, the I-tangle, for all mixed states of two qudits having no more than two nonzero eigenvalues. We use this formula to provide a tight upper bound for the entanglement of formation for rank-2 mixed states of a qubit and a qudit.
We consider the separability of various joint states of D-dimensional quantum systems, which we call "qudits." We derive two main results: (i) the separability condition for a two-qudit state that is a mixture of the maximally mixed state and a maximally entangled state; (ii) lower and upper bounds on the size of the neighorhood of separable states surrounding the maximally mixed state for N qudits.
The need for high fidelity quantum teleportation arises in a variety of quantum algorithms and protocols. Unfortunately, conventional continuous variable teleportation schemes rely on EPR states that yield a fidelity that approaches unity only in the limit of an unphysical amount of squeezing. A new method, which utilizes an ensemble of single photon entangled states to teleport continuous variable states with fidelity approaching unity with finite resources was recently proposed by Andersen and Ralph [Phys. Rev. Lett. 111, 050504 (2013)]. We extend these ideas to consider the general case of …
The unitary $ \mathrm{U}(d)$-equivalence relation between elements of the space $\mathfrak{P}_+\,$ of mixed states of $d$-dimensional quantum system defines the orbit space $ \mathfrak{P}_+/ \mathrm{U}(d)\,$ and provides its description in terms the ring $\mathbb{R}[\mathfrak{P}_+]^{\mathrm{U}(d)}\,$ of $\mathrm{U}(d)$-invariant polynomials. We prove that the semi-algebraic structure of $ \mathfrak{P}_+/ \mathrm{U}(d)\, $ is determined completely by two basic properties of density matrices, their semi-positivity and Hermicity. Particularly, it is shown that the Processi-Schwarz inequalities in…
We study the process of quantum telecloning of $d$-dimensional pure quantum states using partially entangled pure states as quantum channel. This process efficiently mixes optimal universal symmetric cloning with quantum teleportation. It is shown that it is possible to implement universal symmetric telecloning in a probabilistic way using unambiguous state discrimination and quantum state separation schemes. It is also shown that other strategies, such as minimum error discrimination, lead to a decrease in the fidelity of the copies and that certain partially entangled pure states with maxima…
For qubits, Monte Carlo estimation of the average fidelity of Clifford unitaries is efficient -- it requires a number of experiments that is independent of the number $n$ of qubits and classical computational resources that scale only polynomially in $n$. Here, we identify the requirements for efficient Monte Carlo estimation and the corresponding properties of the measurement operator basis when replacing two-level qubits by $p$-level qudits. Our analysis illuminates the intimate connection between mutually unbiased measurements and the existence of unitaries that can be characterized efficie…
Mixed-Dimensional Qudit State Preparation Using Edge-Weighted Decision Diagrams
本文研究混合维度 qudit 系统的量子态制备，提出基于边加权决策图的自动电路构造与近似方法，在精度、内存和操作数之间实现可控权衡，并集成于 MQT Qudits 工具包中。
Nearly all modern solid-state quantum processors approach quantum computation with a set of discrete qubit operations (gates) that can achieve universal quantum control with only a handful of primitive gates. In principle, this approach is highly flexible, allowing full control over the qubits' Hilbert space without necessitating the development of specific control protocols for each application. However, current error rates on quantum hardware place harsh limits on the number of primitive gates that can be concatenated together (with compounding error rates) and remain viable. Here, we report…
Wootters [PRL 80, 2245 (1998)] has derived a closed formula for the entanglement of formation (EOF) of an arbitrary mixed state in a system of two qubits. There is no known closed form expression for the EOF of an arbitrary mixed state in any system more complicated than two qubits. This paper, via a relatively straightforward generalization of Wootters' original derivation, obtains a closed form lower bound on the EOF of an arbitary mixed state of a system composed of a qubit and a qudit (a d-level quantum system, with d greater than or equal to 3). The derivation of the lower bound is detail…
Quantum tomography is an important tool for obtaining information about the quantum state from experimental data. In this study, we conduct a comparative analysis of various quantum tomography protocols, including protocols based on highly symmetric sets of state vectors, on mutually unbiased bases (MUB), and two-level transformations. Using the universal distribution for the fidelity of quantum state tomography, we estimate the fidelity and efficiency of quantum measurements for qudits of various dimensions as applied to a set of random states uniformly distributed over the Haar measure.
The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterised by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other N-m qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3 qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller t…
Based on a microwave-photon quantum processor with two superconducting resonators coupled to one transmon qutrit, we construct the controlled-phase (c-phase) gate on microwave-photon-resonator qudits, by combination of the photon-number-dependent frequency-shift effect on the transmon qutrit by the first resonator and the resonant operation between the qutrit and the second resonator. This distinct feature provides us a useful way to achieve the c-phase gate on the two resonator qudits with a higher fidelity and a shorter operation time, compared with the previous proposals. The fidelity of ou…
We examine, in correlated mixed states of qudit-qubit systems, the set of all conditional qubit states that can be reached after local measurements at the qudit based on rank-1 projectors. While for a similar measurement at the qubit, the conditional post-measurement qudit states lie on the surface of an ellipsoid, for a measurement at the qudit we show that the set of post-measurement qubit states can form more complex solid regions. In particular, we show the emergence, for some classes of mixed states, of sets which are the convex hull of solid ellipsoids and which may lead to cone-like and…
We derive the optimal N to M phase-covariant quantum cloning for equatorial states in dimension d with M=kd+N, k integer. The cloning maps are optimal for both global and single-qudit fidelity. The map is achieved by an ``economical'' cloning machine, which works without ancilla.
A novel approach to multi-image quantum encryption/decryption using qudits
提出基于qudits和空间填充曲线的多图像量子加密方法，通过推广量子baker map扩大参数空间提升安全性，并定义混合扰乱与混合扩散新概念及多种方案。
Stimulated Raman adiabatic passage (STIRAP) is a widely used protocol to realize high-fidelity and robust quantum control in various quantum systems. However, further application of this protocol in superconducting qubits is limited by population leakage caused by the only weak anharmonicity. Here we introduce an optimally controlled shortcut-to-adiabatic (STA) technique to speed up the STIRAP protocol in a superconducting qudit. By modifying the shapes of the STIRAP pulses, we experimentally realize a fast (32 ns) and high-fidelity quantum state transfer. In addition, we demonstrate that our …
To gain an advantage, quantum technologies utilize phenomena particular to quantum mechanics. Two such phenomena are squeezing and entanglement. Having generated states that exhibit these features, verification of their generation with local measurements can be a difficult process. Here we consider the states that are generated using the two-qudit single-axis squeezing Hamiltonian, that not only produces entangled two-qudit squeezed states but also results in various forms of interesting entangled states. We show how one can use local measurements to both efficiently verify and directly estima…
We focus on characterizing entanglement of high dimensional bipartite states using various statistical correlators for two-qudit mixed states. The salient results obtained are as follows: (a) A scheme for determining the entanglement measure given by Negativity is explored by analytically relating it to the widely used statistical correlators viz. mutual predictability, mutual information and Pearson Correlation coefficient for different types of bipartite arbitrary dimensional mixed states. Importantly, this is demonstrated using only a pair of complementary observables pertaining to the mutu…
Fidelity and Entanglement of Random Bipartite Pure States: Insights and Applications
本研究探究了 Haar 随机 bipartite 纯态相对于固定参考量子态的保真度与纠缠，发现其分布非均匀且依赖参考态纠缠，不同维度下狭窄纠缠范围内平均保真度为 $\frac{1}{d^2}$，乘积态保真度分布可用于分布式量子计算设备基准测试。
Generalising multipartite concentratable entanglement for practical applications: mixed, qudit, and optical states
本文将受控SWAP测试与可浓缩纠缠度量推广至混合态、量子比特高维态和光学态，提出对c-SWAP误差鲁棒的混合态CE上界猜想，使其适用于实际实验中的纠缠刻画。
Mixed-Dimensional Quantum Circuit Simulation with Decision Diagrams
针对混合维度qudit量子系统的模拟需求，本文提出基于决策图的qudit模拟器，可高效处理混合维度量子电路，单电路最多支持超100个qudit，代码已开源。
Distillation of continuous variable qudits from single photon sources: A cascaded approach
本文提出了一种利用单光子源和探测器的级联方法，实现高保真度的连续变量qudits态生成，包括高光子态、猫态和GKP资源态。
Agnostic learning of qudit stabilizer states
研究量子位稳定器状态的无偏学习问题，成功将稳定器自举框架推广到量子位系统，提出高效量子算法，能以高概率输出接近未知态保真度的稳定器状态，还给出高保真度简化算法。
Quantum state tomography (QST) is an essential tool for characterizing an unknown quantum state. Recently, QST has been performed for entangled qudits based on orbital angular momentum, time-energy uncertainty, and frequency bins. Here, we propose a QST for time-bin qudits, with which the number of measurement settings scales linearly with dimension $d$. Using the proposed scheme, we performed QST for a four-dimensional time-bin maximally entangled state with 16 measurement settings. We successfully reconstructed the density matrix of the entangled qudits, with which the average fidelity of th…
We show the propagation of entangled states of high-dimensional quantum systems. The qudits states were generated using the transverse correlation of the twin photons produced by spontaneous parametric down-conversion. Their free-space distribution was performed at the laboratory scale and the propagated states maintained a high-fidelity with their original form. The use of entangled qudits allow an increase in the quantity of information that can be transmitted and may also guarantee more privacy for communicating parties. Therefore, studies about propagating entangled states of qudits are im…
MQT Qudits: A Software Framework for Mixed-Dimensional Quantum Computing
本文提出开源工具MQT Qudits，作为慕尼黑量子工具包的一部分，通过标准化语言、电路编译与高效仿真，支持混合维度qudit量子计算应用的设计与实现。
Telecloning and its reverse process, referred to as remote information concentration(RIC), have attracted considerable interest because of their potential applications in quantum-information processing. We here present a general scheme for RIC in d-level systems (qudits), in which the quantum information initially distributed in many spatially separated qudits can be remotely and deterministically concentrated to a single qudit via an entangled channel without performing any global operations. We show that the entangled channel of RIC can be different types of entangled states, including mixed…
Qudit-Generalization of the Qubit Echo and Its Application to a Qutrit-Based Toffoli Gate
本文提出了qubit echo向一般qudits的直接推广——basis cycling技术，用于解决含噪量子计算机中无法通过校准纠正的动态门相位误差问题，并在IBM量子计算机上利用包含qutrit的Toffoli门分解实现了高达93.8±0.1%的CCZ门保真度。
Experimental realization of universal high-dimensional quantum gates with ultra-high fidelity and efficiency
本文在四维光学qudit的偏振-空间自由度上实验实现了通用高维量子门及受控门，保真度达99.73%、效率达99.47%，为大规模高维容错量子计算奠定基础。
We consider the unambiguous discrimination between two unknown qudit states in $n$-dimensional ($n\geqslant2$) Hilbert space. By equivalence of unknown pure states to known mixed states and with the Jordan-basis method, we demonstrate that the optimal success probability of the discrimination between two unknown states is independent of the dimension $n$. We also give a scheme for a physical implementation of the programmable state discriminator that unambiguously discriminate between two unknown states with optimal probability of success.
We study, by means of the stabilizer formalism, a quantum error correcting code which is alternative to the standard block codes since it embeds a qubit into a qudit. The code exploits the non-commutative geometry of discrete phase space to protect the qubit against both amplitude and phase errors. The performance of such code is evaluated on Weyl channels by means of the entanglement fidelity as function of the error probability. A comparison with standard block codes, like five and seven qubit stabilizer codes, shows its superiority.
A qudit ($d$-level quantum systems) has a large Hilbert space and thus can be used to achieve many quantum information and communication tasks. Here, we propose a method to transfer arbitrary $d$-dimensional quantum states (known or unknown) between two superconducting qudits coupled to a single cavity. The state transfer can be performed fast because of employing resonant interactions only. In addition, quantum states can be deterministically transferred without measurement. Numerical simulations show that high-fidelity transfer of quantum states between two superconducting transmon qudits ($…
Universal pulses for superconducting qudit ladder gates
本文提出一种通用脉冲构造方法，用于超导量子比特阶梯门，通过解析方案抑制控制误差，实现接近量子速度极限的高保真门操作，并在transmon电路上验证了显著改进。
Nonlinearity of the Fidelity in Open Qudit Systems: Gate and Noise Dependence in High-dimensional Quantum Computing
本文在Lindblad形式下建立了单qudit系统平均门不保真度的微扰理论框架，揭示了AGI在强耦合区域从线性到非线性的转变及其对维度、门选择和噪声强度的依赖，为高维量子计算的门设计与纠错提供理论依据。
A short review on qudit quantum machine learning
本文综述了qudit（多能级量子系统）在量子机器学习中的应用，重点分析其通过更大希尔伯特空间减少电路深度和参数数量的优势，并讨论了软件生态与面临的挑战。
Sparse Phase Ansatzes for Resource-Efficient Qudit State Preparation via the SNAP-Displacement Protocol
本文研究了利用SNAP位移协议在玻色子量子比特系统中实现资源高效的量子态制备，提出三种逐步更一般的稀疏假设，通过标量多目标优化在保真度与相位数量或假设持续时间之间进行权衡，展示了在理想和噪声环境下稀疏假设相较于完全参数化SD协议的优势。
In this paper, we prove that it is impossible to construct a convariant quantum feedback control instruments which aim to correct the channel noise imposed on an unknown qubit. The proof is based on the searching for the optimal quantum control protocol and it turns out there exist no better %\textit{Complete Positive Trace %Preserving}(CPTP) maps complete positive and covariant quantum operations which provide a higher fidelity than the trivial Identity operators. The generalization of the investigation to bipartite entangled pure state is also included.
Qudit Dynamical Decoupling on a Superconducting Quantum Processor
本文针对多能级量子位系统易退相干的问题，基于海森堡-魏尔群提出动态解耦协议，在超导处理器上实验验证了其能有效抑制退相干和交叉克尔耦合，显著提升贝尔态保真度，为可扩展量子计算铺路。
Fast Machine Learning for Quantum Control of Microwave Qudits on Edge Hardware
本文提出利用机器学习模型在边缘硬件上快速确定微波腔量子比特SNAP门的控制脉冲参数，通过数据生成、模型开发、设计空间探索和量化，实现了接近$10^{-3}$的低门迹保真度并高效利用可编程逻辑资源。
Pure entangled states of two spatial qudits have been produced by using the momentum transverse correlation of the parametric down-converted photons [Phys. Rev. Lett. \textbf{94} 100501]. Here we show a generalization of this process to enable the creation of mixed states of spatial qudits and by using the technique proposed we generate mixed states of spatial qubits. We also report how the process of quantum tomography is experimentally implemented to characterize these states. This tomographic reconstruction is based on the free evolution of spatial qubits, coincidence detection and a filter…
We show that it is possible to control the trade-off between information gain and disturbance in generalized measurements of qudits by utilizing the programmable quantum processor. This universal quantum machine allows us to perform a generalized measurement on the initial state of the input qudit to construct a Husimi function of this state. The trade-off between the gain and the disturbance of the qudit is controlled by the initial state of ancillary system that acts as a program register for the quantum-information distributor. The trade-off fidelity does not depend on the initial state of …
The logarithmic derivative (or, quantum score) of a positive definite density matrix appearing in the quantum Fisher information is discussed, and its exact expression is presented. Then, the problem of estimating the parameters in a class of the Werner-type N-qudit states is studied in the context of the quantum Cramér-Rao inequality. The largest value of the lower bound to the error of estimate by the quantum Fisher information is shown to coincide with the separability point only in the case of two qubits. It is found, on the other hand, that such largest values give rise to the universal f…
Dynamics of qudit gates and effects of spectator modes on optimal control pulses
针对高维量子计算中额外旁观模式引发串扰降低量子比特门脉冲有效性的问题，通过模拟cQED系统动力学，给出保真度衰减边界的标度公式，明确了频移需满足的条件。
Implementation of the Quantum Fourier Transform on a molecular qudit with full refocusing and state tomography
本文研究了利用分子稀土复合物实现量子三态的量子傅里叶变换，通过全重聚焦协议减少不均匀展宽，实现了高保真度的态恢复，并通过态成像验证了算法性能。
We present a general scheme for multiparty-controlled teleportation of an arbitrary m-qudit ($d$-dimensional quantum system) state by using non-maximally entangled states as the quantum channel. The sender performs m generalized Bell-state measurements on her 2$m$ particles, the controllers take some single-particle measurements with the measuring basis $X_{d}$ and the receiver only need to introduce one auxiliary two-level particle to extract quantum information probabilistically with the fidelity unit if he cooperates with all the controllers. All the parties can use some decoy photons to se…
Control Protocol for Dynamic Synthesis of Qubit and Qudit Gates Using Photonic Pulses and Magnetic Fields
本文提出利用光脉冲和磁场实现量子比特和量子拟态门的高保真度动态合成控制方法。
Fidelity Analysis of Adiabatically Driven Donor Spins as Two-Qubit and Ququart Systems
本文通过绝热驱动技术分析供体自旋系统，发现四量子比特操作在误差率上优于编码双量子比特操作。
For certain quantum operations acting on qubits, there exist bases of measurement operators such that estimating the average fidelity becomes efficient. The number of experiments required is then independent of system size and the classical computational resources scale only polynomially in the number of qubits. Here we address the question of how to optimally choose the measurement basis for efficient gate characterization when replacing two-level qubits by $d$-level qudits. We define optimality in terms of the maximal number of unitaries that can be efficiently characterized. Our definition …
We present a comprehensive framework for quantum state tomography (QST) of time-bin qudits sent through a fiber. Starting from basic assumptions, we define a positive-operator valued measure (POVM) which is then applied to the quantum state reconstruction problem. A realistic scenario is considered where the time uncertainty of the detector is treated as a source of experimental noise. The performance of the quantum tomography framework is examined through a series of numerical simulations conducted for different parameters describing the apparatus. The quality of state recovery, quantified by…
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