⇦Chinese Physics Letters, 2022, Vol. 39, No. 10, Article code 100202Express Letter Measuring Quantum Geometric Tensor of Non-Abelian System in Superconducting Circuits Wen Zheng (郑文)1†, Jianwen Xu (徐建文)1†, Zhuang Ma (马壮)1, Yong Li (李勇)1, Yuqian Dong (董煜倩)1, Yu Zhang (张煜)1, Xiaohan Wang (王晓晗)1, Guozhu Sun (孙国柱)2, Peiheng Wu (吴培亨)2, Jie Zhao (赵杰)1, Shaoxiong Li (李邵雄)1, Dong Lan (兰栋)1*, Xinsheng Tan (谭新生)1*, and Yang Yu (于扬)1* Affiliations 1National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China 2School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China Received 9 August 2022; accepted manuscript online 30 August 2022; published online 8 September 2022 †These authors contributed equally to this work.
*Corresponding authors. Email: land@nju.edu.cn; tanxs@nju.edu.cn; yuyang@nju.edu.cn Citation Text: Zheng W, Xu J W, Ma Z et al. 2022 Chin. Phys. Lett. 39 100202    Abstract Topology played an important role in physics research during the last few decades. In particular, the quantum geometric tensor that provides local information about topological properties has attracted much attention. It will reveal interesting topological properties but have not been measured in non-Abelian systems. Here, we use a four-qubit quantum system in superconducting circuits to construct a degenerate Hamiltonian with parametric modulation. By manipulating the Hamiltonian with periodic drivings, we simulate the Bernevig–Hughes–Zhang model and obtain the quantum geometric tensor from interference oscillation. In addition, we reveal its topological feature by extracting the topological invariant, demonstrating an effective protocol for quantum simulation of a non-Abelian system.
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 DOI:10.1088/0256-307X/39/10/100202 © 2022 Chinese Physics Society Article Text Topology played an important role in physics research in the last few decades,[1] including the areas of high energy physics and condensed matter physics.[2-11] One of the most important quantities of topology is the quantum geometric tensor (QGT),[12-14] of which an imaginary part is the Berry curvature, which is closely related to many physical phenomena such as the Aharonov–Bohm phase[15] and topological orders in condensed matter.[16,17] The QGT can be used to classify various topological materials. Furthermore, the geometric phase is applied in the study of quantum computation and information,[18-20] and it holds the promise to improve the performance of quantum gates due to its immunity to local fluctuations.[21-24] The real part of the QGT, called the quantum metric tensor, characterizes the distance between states in a Hilbert space, which is closely related to quantum fluctuations.[25,26] In the recent period of time, it has attracted interest of researchers in quantum phase transitions[27-30] and quantum information theory.[31] Theoretical and experimental research has been conducted. For instance, similar to the well-known Chern number that characterizes the topological properties of the Dirac monopole,[32-34] the topological invariants of the tensor monopole are related to the geometric metric tensor.[35] The four-dimensional matter predicted by theory has been simulated in superconducting circuits[36] and NV-center,[37] while its quantum phase transition is well characterized by the metric tensor.[35] Compared to that of the Abelian quantum system, the quantum geometric tensor in non-Abelian systems has more complex mathematical structures and embodies more physics phenomena, including the four-dimensional Hall effect.[38] The topological order of the non-Abelian quantum systems can be extracted from the tensor, revealing the relationship between the physics phenomena and topological properties. For instance, the second Chern number, as the topological invariant of the Yang monopole, is theoretically and experimentally studied.[39-42] Furthermore, the Wilczek–Zee connection, which leads to the Holonomic properties in the generated system, allows the geometric gate to be widely used in quantum circuits.[43-46] This subject is extensively and thoroughly studied for the optimization of quantum gates.[47-50] Therefore, measuring the generalized non-Abelian quantum geometric tensor has theoretical and practical significance. However, detecting the geometric tensor of a complex system is tricky in practice. For an Abelian system, some approaches have been proposed, including modulation of Hamiltonian with sudden quench and periodic drivings.[51-53] These routines have been realized in various systems such as superconducting circuits[52] and NV-center.[53] For a non-Abelian system, several theoretical detection schemes have been proposed,[54] but there is still a lack of experimental demonstration. In this work, we construct a non-Abelian system in four-qubit superconducting circuits. Using longitudinal parametric modulation,[55-57] we demonstrate that the quantum geometric tensor, which depends on the geometry defined by a set of parameters, can be extracted by using geometric Rabi oscillations that are realized by periodically driving the system under a single parameter or under two parameters.[51,54] We implement this protocol to simulate the Bernevig–Hughes–Zhang model and reveal its physics feature such as $Z_2$ symmetry from the extracted topological invariant. The good agreement between experimental data and theoretical prediction confirms the validity of our approach.
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Fig. 1. Illustration of the quantum geometric tensor in a non-Abelian system. (a) The QGT can be obtained from two nearby states in the four-dimensional manifold spanned by the parameters $\lambda$, e.g., $\lambda = (\lambda_1, \lambda_2, \lambda_3, \lambda_4)$. (b) Schematic of our samples, which are four transmons capacitively coupled to each other. The simulated Hamiltonian can be realized by modulating the parameters of longitudinal driving applied to the superconducting circuits.
QGT in Non-Abelian Systems. We consider an $N$-band Hamiltonian with $n$-degeneracy dimensions, which can be written as $H_0(\lambda) = \sum_{\sigma=0}^{N} E_{\sigma} \sum_{i=1}^{n} |\psi_{i}^{\sigma}(\lambda) \rangle \langle \psi_{i}^{\sigma}(\lambda) |$ with a set of dimensionless parameters $\lambda = (\lambda_1, \lambda_2, \dots)$, while $|\psi_{i}^{0}(\lambda) \rangle$ is the ground state. Without loss of generality, we choose $N=1$ and $n=2$, so the dimension of the simulated Hamiltonian is four. Similar to that of the Abelian system, the geometric tensor can be derived from two nearby states with the formula defined as $dS:=\||\varPsi_0(\lambda_1 + d \lambda_1)\rangle-|\varPsi_0(\lambda_1)\rangle\|$ in a Hilbert space, as illustrated in Fig. 1. Here $|\varPsi_0(\lambda)\rangle = \sum_{i=1}^{n}|\psi_{i}^0 (\lambda)\rangle$. The matrix element of the QGT[14] can be found as follows: \begin{eqnarray} Q^{\mu \nu}_{i j} = \langle \partial_{\lambda_{\mu}} \psi_i^0 (\lambda) |[1 - \mathcal{P}(\lambda)]|\partial_{\lambda_{\nu}}\psi_j^0 (\lambda) \rangle, \tag {1} \end{eqnarray} where $\partial_{\lambda_{\mu, \nu}}\equiv\partial/\partial \lambda_{\mu,\nu}$ and $\mathcal{P}(\lambda)$ stands for the projection operator, defined as $\mathcal{P}(\lambda) = \sum_{i=1}^{n}|\psi_{i}^0(\lambda)\rangle \langle \psi_{i}^0(\lambda) |$. Notice that the QGT is a complex matrix, of which the real and imaginary parts are the Riemannian metric and Berry curvature respectively: $g^{\mu\nu} = (Q^{\mu\nu} + [Q^{\mu\nu}]^†)/2$, and $F^{\mu\nu} = i(Q^{\mu\nu} - [Q^{\mu\nu}]^†)$. To measure the QGT of the degenerate system, we construct the Hamiltonian $H_0$ and detect the interference oscillation with periodic weak parametric modulation,[54] which can be defined as $\lambda_{\mu} + 2A/\omega \cos(\omega t + \phi_{\mu})$, where $\phi_{\mu}$ is the initial phase of the modulation field, the modulating amplitude $A$ and the frequency $\omega$ satisfy $A \ll \omega$.[51,54] The amplitude $A$ is directly related to and in the scale of the oscillation frequency we measured (Rabi frequency $\varOmega_0$) according to Eq. (2). Then the values of the metric tensor and Berry curvature can be extracted from the measured Rabi frequencies by solving the Schrödinger equation in some basis such as the basis $\{|\psi_{i}^{\sigma}(\lambda)\rangle\}$. In this routine, the QGT elements are obtained with one- and two-parameter modulation, respectively. The corresponding Hamiltonians expanded to first order are $H_0(\lambda) + h_1(\lambda_{\mu})$ or $H_0(\lambda) + h_2(\lambda_{\mu}, \lambda_{\nu})$. Here $h_1(\lambda_{\mu}) = \frac{2A}{\omega} \cos{(\omega t + \phi_{\mu})} \partial_{\lambda_{\mu}} H_0 (\lambda)$ and $h_2(\lambda_{\mu}, \lambda_{\nu}) = \frac{2A}{\omega} [\cos{(\omega t+\phi_{\mu})} \partial_{\lambda_{\mu}} H_0 (\lambda) + \cos{(\omega t + \phi_{\nu})} \partial_{\lambda_{\nu}} H_0 (\lambda)]$. To simplify the experimental scheme and to improve measurement fidelity, we rotate the Hamiltonian with a unitary transformation, so the coupling terms between two-degeneracy dimensions will become zero (see the Supplementary Information for the diagonalization and measurement of non-Abelian quantum geometric tensor). In the modified Hamiltonian, the values of $Q^{\mu \nu}_{i j}$ can be extracted from the oscillation frequencies in the new eigenstates which correspond the modified Hamiltonian constructed from $H_0$ by Morris–Shore (MS) transformation.[54] The relation between the value $Q_{\rm QGT}$ and the Rabi oscillation is \begin{eqnarray} \varOmega_0 = \sqrt{A^2 Q_{\rm QGT} + \varDelta^2}, \tag {2} \end{eqnarray} for one- and two-parameter modulation with $Q_{\rm QGT}$ equal to $Q_{jj}^{\nu \nu}$ and $Q_{jj}^{\mu \mu} + Q_{jj}^{\nu \nu} + e^{-i\delta\phi}Q_{jj}^{\mu \nu} + e^{i\delta\phi}Q_{jj}^{\nu \mu}$ respectively. Here, $\delta \phi = \phi_{\nu} - \phi_{\mu}$ is the phase difference between the two periodic modulation pulses, $\varDelta$ is the detuning between the gap of eigenenergy levels and the driving frequency $\omega$.
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Fig. 2. The non-Abelian system realized by longitudinal parametric modulation. (a) Flow chart of the Hamiltonian simulation. The original theoretical model can be equivalently mapped to the Hamiltonian constructed by the four-qubit system with longitudinal parametric modulation, which can be further simplified to the two independent subspaces under a basis transformation. Here, $|\psi_{+,-}^{1,2}\rangle$ and $\{|0001\rangle,\,|0010\rangle,\,|0100\rangle,\,|1000\rangle\}$ in the left panel denote the states of the original model, and $E_{+,-}^{1,2}$ in the right panel indicate the eigenenergies of the transferred model. (b) Pulse sequence. To construct the two-band two-degeneracy Hamiltonian as denoted in the dashed boxes, $XY$ and $Z$ designed pulses are applied. The protocol is realized by applying the designed flux pulses through the $Z$ lines of tunable qubits TQ$_m$, which is followed by the quantum state tomography.
Building of the Non-Abelian System. The Hamiltonian constructed by four qubits can be written as $H_{\rm s} = \sum_{k} \omega_{k}a^†_k a_k + \frac{\alpha_k}{2}a^†_k a^†_k a_k a_k + \sum_{kl} J_{kl}(a_k + a_k^†)(a_l + a_l^†)$ with indices $k\in \{1,\,2,\,3,\,4\}$ and $kl \in \{12,\,23,\,34,\,41\}$ to mark qubits. Here, $a_k$ ($a^†_k$) is the annihilation (creation) operator and $J_{kl}$ is the nearest-neighbor coupling strength, while $\alpha_k$ denotes the anharmonicity of the $k$th transmon, of which the frequency in the lowest two levels is labeled as $\omega_{k}$ (see the Supplementary Information for the sample information and experimental setup). To realize the desired degenerate Hamiltonian $H_0(\lambda)$, whose level structure can be equivalent to a diamond shape as shown in the right panel of Fig. 2(a), we simultaneously introduce the longitudinal parametric modulation pulses $\omega_{m}(t) = \bar{\omega}_m + \omega^{T}_{m1}\cos{(2 \pi f_{m1} t+\beta_{m1})} + \omega^{T}_{m2}\cos{(2 \pi f_{m2} t+{\beta}_{m2})}$ by applying flux pulses to the tunable qubit TQ$_m$ (see the Supplementary Information for the realization of the non-Abelian system in superconducting circuits). Here $\bar{\omega}_m$ is the mean frequency of TQ$_m$ during the longitudinal parametric modulation, $\omega^{T}_{mi}$, $f_{mi}$, and $\beta_{mi}$ are the amplitude, frequency and phase of the applied longitudinal pulse. Two modulation pulses $\omega_m(t)$ ($m=1,\,3$) applied to TQ$_{m}$ realize the diamond-shape couplings as illustrated in Fig. 1(b). In this approach the parameters of each coupling term can be freely adjusted, so that the general non-Abelian Hamiltonian as shown in Fig. 2(a) can be directly realized in the Hilbert space spanned by $\{|0001\rangle,\,|0010\rangle,\,|0100\rangle,\,|1000\rangle\}$ (see the Supplementary Information). Experimental Results. In practice, we measure the corresponding QGT terms to study the topological properties of physical models, here we construct the Bernevig–Hughes–Zhang (BHZ) model as an example to demonstrate this scheme (see the Supplementary Information).[58,59] The BHZ model is closely related to the spin-Hall effect, which can be characterized by the spin Chern numbers. The Hamiltonian in momentum space can be written as \begin{align} &H_{\rm BHZ}\notag\\ =&\begin{bmatrix} B_{z} & 0 & B_{x} - i B_{y} & B_{g}\\ 0 & B_{z} & B_{g} & - B_{x} - i B_{y}\\ B_{x} + i B_{y} & B_{g} & - B_{z} & 0\\ B_{g} & - B_{x} + i B_{y} & 0 & - B_{z} \end{bmatrix}, \tag {3} \end{align} where $B_{x} = H_{xy} \sin{k_x}$, $B_{y} = H_{xy} \sin{k_y}$, $B_{z} = M - 2 H_{z}(2 - \cos{k_x} - \cos{k_y})$. The momentum $k_x, \, k_y$ belong to $[-\pi, \pi]$, while the parameters $H_{xy}$, $H_{z}$, and $B_{g}$ depend on the quantum well geometry, $M$ is related to the phase transition. In our superconducting circuit we can realize this BHZ Hamiltonian, which is a non-Abelian form in a four-dimensional parameter space. Its entire components of the QGT can be obtained by our approach (see the Supplementary Information for the BHZ model), while it is not necessary in practice. The BHZ model is equivalent to two copies of the Haldane Model,[59,60] which can be mapped to the Hilbert space of two degenerate systems. Therefore, the original Hamiltonian can be deformed to the individual degenerate subspace by a unitary transformation as shown in Fig. 2(a). The parametric modulation in experiment can be simplified to $\omega_{1}(t) = \bar{\omega}_1 + \omega^{T}_{1} \cos{ (2 \pi f_{1} t+\varphi)}$ and $\omega_{3}(t) = \bar{\omega}_3 + \omega^{T}_{3} \cos{(2 \pi f_{3} t-\varphi)}$, leading to the system Hamiltonian in spherical coordinate as follows (see the Supplementary Information): \begin{align} {H}=\,&{\varOmega_0}(\cos{\theta}|0001\rangle \langle 0001| - \cos{\theta}|0010\rangle \langle 0010| \notag\\ &+\cos{\theta}|0100\rangle \langle 0100| - \cos{\theta}|1000\rangle \langle 1000| \notag\\ &+\sin{\theta}e^{i\varphi}|0010\rangle \langle 0001| - \sin{\theta}e^{-i\varphi}|1000\rangle \langle 0100|\notag\\ &+ {\rm H.c.}),\tag {4} \end{align} where $\varOmega_0=\sqrt{\varOmega^2 + \varDelta^2}$. In our experiment, the modulated parameters $\lambda_{\mu}$ for measuring QGT in a quantum system are $\theta$ and $\varphi$. The weak periodic modulation fields can be written as $\theta(t) = \theta + \frac{2A}{\omega}\cos(\omega t + \phi_{\theta})$ and $\varphi(t) = \varphi + \frac{2A}{\omega}\cos(\omega t + \phi_{\varphi})$. Here $\varphi(t)$ is obtained by directly modulating the phase $\varphi$ of the longitudinal parametric modulation pulses, $\theta(t)$ is related to the experimental parameters $\omega_{1,3}^T$ and $f_{1,3}$ according to $\theta = \arctan{(\varOmega/\varDelta)}$ with Rabi oscillation frequencies $\varOmega = |J_{12} \mathcal{J}_1 (\frac{\omega_1^T}{2\pi f_1})|= |J_{34} \mathcal{J}_1 (\frac{\omega_3^T}{2\pi f_3})|$. $\mathcal{J}_1$ denotes the first kind 1st-order Bessel function. The QGT measurement, which contains initial state preparation, parametric modulation, and quantum state tomography, is demonstrated on four qubits by modulating the controlling parameters $\theta$ and $\varphi$.
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Fig. 3. Rabi oscillation with different parametric modulations. (a) Top panel: parametric modulation pulse applied to the subspace. From left to right: $\varphi$ modulation; $\theta$ modulation; $\theta$ and $\varphi$ modulation with $\delta\phi=0$; $\theta$ and $\varphi$ modulation with $\delta\phi=0.5\pi$. Bottom panel: measured Rabi oscillation at various ${\theta}$ obtained with corresponding routines of parametric modulation: ${\theta}=0.2 \pi$, $0.5 \pi$, and $0.8 \pi$. The amplitude and frequency of the modulated pulse are $A/2\pi=3 \,\rm{MHz}$ and $\omega/2\pi=13 \,\rm{MHz}$, respectively. (b) The Rabi oscillation frequency $\varOmega$ in subspace $\mathcal{S}_{11}$ as a function of the driving amplitude $A$ and detuning $\varDelta$, with $\theta=0.5\pi$ and $\omega = 13$ MHz. The slope of each dashed line indicates the corresponding QGT according to Eq. (2). Note that $h_1 (\theta)$ almost overlaps with $h_1 (\varphi)$. The error bars indicate standard deviation.
To obtain the complete quantum metric of the manifold, we need to execute the procedure by preparing different initial states throughout the entire Hilbert space. To improve measurement fidelity, we use a unitary transformation to rotate the frame axis. The eigenstates of the initial Hamiltonian remain at $|0001\rangle$ and $|0100\rangle$,[52] making the modulation pulse significantly simplified. With this routine, the weak periodic driving terms $h_1(\lambda_{\mu})$ and $h_2(\lambda_{\mu}, \lambda_{\nu})$ can be realized by accurately designing the frequency and amplitude of the flux pulses (see the Supplementary Information). For instance, in the measurement of $g^{\varphi\varphi}_{11}$, the parameter ${\varphi}$ in the modulated pulse simplified to $2A/\omega\cos(\omega t)$ with $A/2\pi=3$ MHz and $\omega/2\pi = 13$ MHz. Figure 3 shows the pulse schemes and measured Rabi patterns, with ${\theta}=0.2 \pi$, $0.5 \pi$, and $0.8 \pi$, respectively. Similarly, the Rabi patterns of the other QGT terms ($g^{\theta\theta}_{11}$, $g^{\varphi\theta}_{11}$, and $F^{\varphi\theta}_{11}$) can also be obtained. Furthermore, we can verify the relationship between the QGT and the parameters of the periodic weak field in Eq. (2). As shown in Fig. 3(b), we vary the amplitude $A$ and detuning $\varDelta$ as a double check. The data demonstrated in the figure are measured at ${\theta} = 0.5\pi$. Based on Eq. (2), in the top panel of Fig. 3(b), there is a linear relationship between $A$ and $\varOmega$ if $\varDelta/2\pi = 0\,\rm{MHz}$, so the values of the QGT can be extracted from the experimental results, in the bottom panel of Fig. 3(b), by setting $A/2\pi=3\,\rm{MHz}$, the relationship between $\varDelta$ and $\varOmega$ also agrees well with the predictions (dashed lines). Note that $h_1 (\theta)$ is almost identical to $h_1 (\varphi)$. With this modulating protocol, we can extract the QGT of the simulating Hamiltonian, revealing the corresponding geometric properties. As shown in Fig. 4(a), The experimental $Q_{\mu\nu}$ are in good agreement with theoretical predictions, which are \begin{align} & Q^{\theta\theta} = \frac{1}{4}\begin{bmatrix} &1 & 0\\ &0 & 1\\ \end{bmatrix}, ~~Q^{\varphi\varphi} = \frac{1}{4}\sin^2 \theta \begin{bmatrix} &1& 0\\ &0& 1\\ \end{bmatrix},\notag\\ & Q^{\theta\varphi} = \frac{i}{4}\sin \theta \begin{bmatrix} &1& 0\\ &0& -1\\ \end{bmatrix}.\tag {5} \end{align} It is worth noting that the MS transformation used in our scheme reduces the parameter dimension, hence the complete components of the original geometric tensor are obtained by repeating this approach with different unitary transformations (see the Supplementary Information for the BHZ model). Furthermore, we study the topological invariant in the BHZ model, which is the spin Chern number. The spin-up and spin-down charge carriers correspond to the individual subspace $\mathcal{S}_{11}$ and $\mathcal{S}_{22}$ we constructed, of which the corresponding topological invariant is the first Chern number $\mathcal{C_{\pm}}$. Since the manifold is covered by three-dimensional sphere surface $\mathbb{S}_2$, in spherical coordinates the Chern number of individual subspace can be written as \begin{eqnarray} \mathcal{C} = \frac{1}{2\pi} \int_{_{\scriptstyle \mathbb{S}_2}} F_{\theta\phi} d\theta d\phi. \tag {6} \end{eqnarray} The relation $\sqrt{{\rm{\det}}\,g}=\frac{1}{2}|F_{\theta\phi}|$[35,61-63] ensures that we can obtain the Chern number from the metric tensor $g^{\varphi\varphi}_{11(22)}$, $g^{\theta\theta}_{11(22)}$, $g^{\theta\varphi}_{11(22)}$ and Berry curvature $F^{\theta\varphi}_{11(22)}$ we measured, respectively. The accumulated values of the subspace $\mathcal{S}_{11}$ and $\mathcal{S}_{22}$ extracted from the Berry curvature are $\mathcal{C_+} = 1.019 \pm 0.005$ and $\mathcal{C_-} = -1.003 \pm 0.010$, which correspond to the positive and negative topological charges respectively, as illustrated in Fig. 4(b). The spin Chern number is obtained as $\mathcal{C}_{\rm scn}=(\mathcal{C}_+-\mathcal{C}_-)/2 = 1.011 \pm 0.005$ while $\mathcal{C}_++\mathcal{C}_- = 0.016 \pm 0.013$ due to the time-reversal symmetry, indicating the $Z_2$ symmetry of the Bernevig–Hughes–Zhang model.[64-68] Therefore, the topological properties of BHZ model can be explored directly with our approach.
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Fig. 4. Extraction of the quantum geometric tensor from Rabi oscillation. (a) Top panel: under the MS transformation, the manifold of the BHZ Hamiltonian transform from a sphere in a four-dimensional parameter space to a direct sum of two $\mathbb{S}_2$ spheres. Therefore, each measurement obtained by our approach is part of the original quantum geometric metric tensor. Bottom panel: the measured Riemannian metric, data in different colors correspond to different matrix elements. (b) Left: the illustration of the BHZ model, the spin-up (down) charge carriers correspond to the measured Berry curvature of subspace $\mathcal{S}_{11}$ ($\mathcal{S}_{22}$) shown in the right panel. The experimental data (solid points) agreed well with the theoretical predictions (dashed lines). The error bars indicate standard deviation.
In summary, we have demonstrated the simulation of a non-Abelian system in superconducting circuits. Using periodic driving, we measure the geometric tensor of this quantum system. In addition, we extract the topological invariant from the measured metric tensor and the Berry curvature, respectively, proving the feasibility and validity of our routine. Furthermore, our scheme is universal and can be extended to quantum simulations of other models. Acknowledgements. This work was supported by the Key R&D Program of Guangdong Province (Grant No. 2018B030326001), the National Natural Science Foundation of China (Grant Nos. 11474152, 12074179, U21A20436, and 61521001), the Natural Science Foundation of Jiangsu Province (Grant No. BE2021015-1). References Concept of nonintegrable phase factors and global formulation of gauge fieldsDirac monopole without strings: Monopole harmonicsColloquium : Topological insulatorsTopological insulators and superconductorsLight-induced gauge fields for ultracold atomsWeyl and Dirac semimetals in three-dimensional solidsTopological quantum matter with cold atomsTopological photonicsQuantum SimulatorsQuantum spin Hall effect of lightRiemannian structure on manifolds of quantum statesQuantum Critical Scaling of the Geometric TensorsAbelian and non-Abelian quantum geometric tensorSignificance of Electromagnetic Potentials in the Quantum TheoryMean-field theory of spin-liquid states with finite energy gap and topological ordersColloquium : Zoo of quantum-topological phases of matterHolonomic quantum computationNon-Abelian Berry connections for quantum computationUnconventional Geometric Quantum ComputationGeometric quantum gates that are robust against stochastic control errorsExperimental Demonstration of the Stability of Berry’s Phase for a Spin- 1 / 2 ParticleRydberg-atom-based scheme of nonadiabatic geometric quantum computationStatistical distance and Hilbert spaceStatistical distance and the geometry of quantum statesGround state overlap and quantum phase transitionsInformation-Theoretic Differential Geometry of Quantum Phase TransitionsGeometry of quantum phase transitionsInformation geometry and quantum phase transitions in the Dicke modelQuantised singularities in the electromagnetic field,Berry phase effects on electronic propertiesObservation of Dirac monopoles in a synthetic magnetic fieldRevealing Tensor Monopoles through Quantum-Metric MeasurementsExperimental Observation of Tensor Monopoles with a Superconducting QuditA synthetic monopole source of Kalb-Ramond field in diamondA Four-Dimensional Generalization of the Quantum Hall EffectGeneralization of Dirac’s monopole to SU2 gauge fieldsSecond Chern number of a quantum-simulated non-Abelian Yang monopoleMeasuring the Second Chern Number from Nonadiabatic EffectsSecond Chern Number and Non-Abelian Berry Phase in Topological Superconducting SystemsAppearance of Gauge Structure in Simple Dynamical SystemsGeometric Manipulation of Trapped Ions for Quantum ComputationWilson loop and Wilczek-Zee phase from a non-Abelian gauge fieldNon-Abelian anyons and topological quantum computationNon-adiabatic holonomic quantum computationNonadiabatic Holonomic Quantum Computation in Decoherence-Free SubspacesFault-Tolerant Holonomic Quantum ComputationHolonomic Quantum Computation in Decoherence-Free SubspacesExtracting the quantum metric tensor through periodic drivingExperimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting QubitExperimental measurement of the quantum geometric tensor using coupled qubits in diamondGeometrical Rabi oscillations and Landau-Zener transitions in non-Abelian systemsUniversal Gate for Fixed-Frequency Qubits via a Tunable BusDemonstration of universal parametric entangling gates on a multi-qubit latticeRealization of Superadiabatic Two-Qubit Gates Using Parametric Modulation in Superconducting CircuitsQuantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum WellsMeasurement of Spin Chern Numbers in Quantum Simulated Topological InsulatorsModel for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"Revealing Chern number from quantum metricMeasurement of topological order based on metric-curvature correspondenceRelating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers Z 2 Topological Order and the Quantum Spin Hall EffectQuantum Spin Hall Effect in GrapheneNondissipative Spin Hall Effect via Quantized Edge TransportQuantum Spin-Hall Effect and Topologically Invariant Chern NumbersSpin Chern numbers and time-reversal-symmetry-broken quantum spin Hall effect
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