Rong Fu, Zhongling Su, Han-Sen Zhong, Xiti Zhao, Jianyang Zhang, Feng Pan, Pan Zhang, Xianhe Zhao, Ming-Cheng Chen, Chao-Yang Lu, Jian-Wei Pan, Zhiling Pei, Xingcheng Zhang, Wanli Ouyang Quantum Computational Superiority boasts rapid computation and high energy efficiency. Despite recent advances in classical algorithms aimed at refuting the milestone claim of Google's sycamore, challenges remain in generating uncorrelated samples of random quantum circuits. In this paper, we present a groundbreaking large-scale system technology that leverages optimization on global, node, and device levels to achieve unprecedented scalability for tensor networks. This enables the handling of large-scale tensor networks with memory capacities reaching tens of terabytes, surpassing memory space constraints on a single node. Our techniques enable accommodating large-scale tensor networks with up to tens of terabytes of memory, reaching up to 2304 GPUs with a peak computing power of 561 PFLOPS half-precision. Notably, we have achieved a time-to-solution of 14.22 seconds with energy consumption of 2.39 kWh which achieved fidelity of 0.002 and our most remarkable result is a time-to-solution of 17.18 seconds, with energy consumption of only 0.29 kWh which achieved a XEB of 0.002 after post-processing, outperforming Google's quantum processor Sycamore in both speed and energy efficiency, which recorded 600 seconds and 4.3 kWh, respectively.
Xian-He Zhao, Han-Sen Zhong, Feng Pan, Zi-Han Chen, Rong Fu, Zhongling Su, Xiaotong Xie, Chaoxing Zhao, Pan Zhang, Wanli Ouyang, Chao-Yang Lu, Jian-Wei Pan, Ming-Cheng Chen Random quantum circuit sampling serves as a benchmark to demonstrate quantum computational advantage. Recent progress in classical algorithms, especially those based on tensor network methods, has significantly reduced the classical simulation time and challenged the claim of the first-generation quantum advantage experiments. However, in terms of generating uncorrelated samples, time-to-solution, and energy consumption, previous classical simulation experiments still underperform the \textitSycamore processor. Here we report an energy-efficient classical simulation algorithm, using 1432 GPUs to simulate quantum random circuit sampling which generates uncorrelated samples with higher linear cross entropy score and is 7 times faster than \textitSycamore 53 qubits experiment. We propose a post-processing algorithm to reduce the overall complexity, and integrated state-of-the-art high-performance general-purpose GPU to achieve two orders of lower energy consumption compared to previous works. Our work provides the first unambiguous experimental evidence to refute \textitSycamore's claim of quantum advantage, and redefines the boundary of quantum computational advantage using random circuit sampling.
A simple method to calculate Wigner coupling coefficients and Racah recoupling coefficients for U(3) in two group-subgroup chains is presented. While the canonical U(3)->U(2)->U(1) coupling and recoupling coefficients are applicable to any system that respects U(3) symmetry, the U(3)->SO(3) coupling coefficients are more specific to nuclear structure studies. This new procedure precludes the use of binomial coefficients and alternating sums which were used in the 1973 formulation of Draayer and Akiyama, and hence provides faster and more accurate output of requested results. The resolution of the outer multiplicity is based on the null space concept of the U(3) generators proposed by Arne Alex et al., whereas the inner multiplicity in the angular momentum subgroup chain is obtained from the dimension of the null space of the SO(3) raising operator. A C++ library built on this new methodology will be published in a complementary journal that specializes in the management and distribution of such programs.
We introduce the spin dissymmetry factor, a measure of the spin-selectivity in the optical transition rate of quantum particles. This spin dissymmetry factor is valid locally, including at material interfaces and within optical cavities. We design and numerically demonstrate an optical cavity with three-fold rotational symmetry that maximizes spin dissymmetry, thereby minimizing the spin dephasing of a cavity-coupled quantum particle. Our approach emphasizes the difference between spin and chirality in the nearfield and reveals a classical parameter for designing more efficient quantum optical devices.
Efficient simulation of quantum circuits has become indispensable with the rapid development of quantum hardware. The primary simulation methods are based on state vectors and tensor networks. As the number of qubits and quantum gates grows larger in current quantum devices, traditional state-vector based quantum circuit simulation methods prove inadequate due to the overwhelming size of the Hilbert space and extensive entanglement. Consequently, brutal force tensor network simulation algorithms become the only viable solution in such scenarios. The two main challenges faced in tensor network simulation algorithms are optimal contraction path finding and efficient execution on modern computing devices, with the latter determines the actual efficiency. In this study, we investigate the optimization of such tensor network simulations on modern GPUs and propose general optimization strategies from two aspects: computational efficiency and accuracy. Firstly, we propose to transform critical Einstein summation operations into GEMM operations, leveraging the specific features of tensor network simulations to amplify the efficiency of GPUs. Secondly, by analyzing the data characteristics of quantum circuits, we employ extended precision to ensure the accuracy of simulation results and mixed precision to fully exploit the potential of GPUs, resulting in faster and more precise simulations. Our numerical experiments demonstrate that our approach can achieve a 3.96x reduction in verification time for random quantum circuit samples in the 18-cycle case of Sycamore, with sustained performance exceeding 21 TFLOPS on one A100. This method can be easily extended to the 20-cycle case, maintaining the same performance, accelerating by 12.5x compared to the state-of-the-art CPU-based results and 4.48-6.78x compared to the state-of-the-art GPU-based results reported in the literature.
We propose a general framework for decoding quantum error-correcting codes with generative modeling. The model utilizes autoregressive neural networks, specifically Transformers, to learn the joint probability of logical operators and syndromes. This training is in an unsupervised way, without the need for labeled training data, and is thus referred to as pre-training. After the pre-training, the model can efficiently compute the likelihood of logical operators for any given syndrome, using maximum likelihood decoding. It can directly generate the most-likely logical operators with computational complexity $\mathcal O(2k)$ in the number of logical qubits $k$, which is significantly better than the conventional maximum likelihood decoding algorithms that require $\mathcal O(4^k)$ computation. Based on the pre-trained model, we further propose refinement to achieve more accurately the likelihood of logical operators for a given syndrome by directly sampling the stabilizer operators. We perform numerical experiments on stabilizer codes with small code distances, using both depolarizing error models and error models with correlated noise. The results show that our approach provides significantly better decoding accuracy than the minimum weight perfect matching and belief-propagation-based algorithms. Our framework is general and can be applied to any error model and quantum codes with different topologies such as surface codes and quantum LDPC codes. Furthermore, it leverages the parallelization capabilities of GPUs, enabling simultaneous decoding of a large number of syndromes. Our approach sheds light on the efficient and accurate decoding of quantum error-correcting codes using generative artificial intelligence and modern computational power.
Measuring properties of quantum systems is a fundamental problem in quantum mechanics. We provide a simple method for estimating the expectation value of observables with an unknown quantum state. The idea is to use a data structure to sample the terms of observables based on the Pauli decomposition proportionally to their importance. We call this technique operator shadow as a shorthand for the procedure of preparing a sketch of an operator to estimate properties. Only when the numbers of observables are small for multiple local observables, the sample complexity of this method is better than the classical shadow technique. However, if we want to estimate the expectation value of a linear combination of local observables, for example the energy of a local Hamiltonian, the sample complexity is better on all parameters. The time complexity to construct the data structure is $2^{O(k)}$ for $k$-local observables, similar to the post-processing time of classical shadows.
Jiaojian Shi, Yuejun Shen, Feng Pan, Weiwei Sun, Anudeep Mangu, Cindy Shi, Amy McKeown-Green, Parivash Moradifar, Moungi G. Bawendi, William E. Moerner, Jennifer A. Dionne, Fang Liu, Aaron M. Lindenberg The development of many optical quantum technologies depends on the availability of solid-state single quantum emitters with near-perfect optical coherence. However, a standing issue that limits systematic improvement is the significant sample heterogeneity and lack of mechanistic understanding of microscopic energy flow at the single emitter level and ultrafast timescales. Here we develop solution-phase single-particle pump-probe spectroscopy with photon correlation detection that captures sample-averaged dynamics in single molecules and/or defect states with unprecedented clarity at femtosecond resolution. We apply this technique to single quantum emitters in two-dimensional hexagonal boron nitride, which suffers from significant heterogeneity and low quantum efficiency. From millisecond to nanosecond timescales, the translation diffusion, metastable-state-related bunching shoulders, rotational dynamics, and antibunching features are disentangled by their distinct photon-correlation timescales, which collectively quantify the normalized two-photon emission quantum yield. Leveraging its femtosecond resolution, spectral selectivity and ultralow noise (two orders of magnitude improvement over solid-state methods), we visualize electron-phonon coupling in the time domain at the single defect level, and discover the acceleration of polaronic formation driven by multi-electron excitation. Corroborated with results from a theoretical polaron model, we show how this translates to sample-averaged photon fidelity characterization of cascaded emission efficiency and optical decoherence time. Our work provides a framework for ultrafast spectroscopy in single emitters, molecules, or defects prone to photoluminescence intermittency and heterogeneity, opening new avenues of extreme-scale characterization and synthetic improvements for quantum information applications.
We study the problem of generating independent samples from the output distribution of Google's Sycamore quantum circuits with a target fidelity, which is believed to be beyond the reach of classical supercomputers and has been used to demonstrate quantum supremacy. We propose a new method to classically solve this problem by contracting the corresponding tensor network just once, and is massively more efficient than existing methods in obtaining a large number of uncorrelated samples with a target fidelity. For the Sycamore quantum supremacy circuit with $53$ qubits and $20$ cycles, we have generated one million uncorrelated bitstrings $\{\mathbf s\}$ which are sampled from a distribution $\hat P(\mathbf s)=|\hat \psi(\mathbf s)|^2$, where the approximate state $\hat \psi$ has fidelity $F\approx 0.0037$. The whole computation has cost about $15$ hours on a computational cluster with $512$ GPUs. The obtained one million samples, the contraction code and contraction order is made public. If our algorithm could be implemented with high efficiency on a modern supercomputer with ExaFLOPS performance, we estimate that ideally, the simulation would cost a few dozens of seconds, which is faster than Google's quantum hardware.
Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on the Jordan-Schwinger boson realization of the SU(2) algebra. It is shown that the solution of the Bethe ansatz equations can be obtained as zeros of the related extended Heine-Stieltjes polynomials. Symmetry properties of excited levels of the system and zeros of the related extended Heine-Stieltjes polynomials are discussed. As an example of an application of the theory, the two equal spin case is studied in detail, which demonstrates that the levels in each band are symmetric with respect to the zero energy plane perpendicular to the level diagram and that excited states are always well entangled.
Restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives an understanding of the expressive power of RBM and DBM using entanglement structures of the tensor networks, also provides an efficient tensor network contraction algorithm for the computing partition function of RBM and DBM. Using numerical experiments, we demonstrate that the proposed algorithm is much more accurate than the state-of-the-art machine learning methods in estimating the partition function of restricted Boltzmann machines and deep Boltzmann machines, and have potential applications in training deep Boltzmann machines for general machine learning tasks.
We propose a general tensor network method for simulating quantum circuits. The method is massively more efficient in computing a large number of correlated bitstring amplitudes and probabilities than existing methods. As an application, we study the sampling problem of Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. Using our method, employing a small computational cluster containing 60 graphical processing units (GPUs), we have generated one million correlated bitstrings with some entries fixed, from the Sycamore circuit with 53 qubits and 20 cycles, with linear cross-entropy benchmark (XEB) fidelity equals 0.739, which is much higher than those in Google's quantum supremacy experiments.
In this article, we investigate the problem of engineering synchronization in non-Markovian quantum systems. First, a time-convoluted linear quantum stochastic differential equation is derived which describes the Heisenberg evolution of a localized quantum system driven by multiple colored noise inputs. Then, we define quantum expectation synchronization in an augmented system consisting of two subsystems. We prove that, for two homogenous subsystems, synchronization can always be synthesized without designing direct Hamiltonian coupling given that the degree of non-Markovianity is below a certain threshold. System parameters are explicitly designed to achieve quantum synchronization. Also, a numerical example is presented to illustrate our results.
We present a general method for approximately contracting tensor networks with an arbitrary connectivity. This enables us to release the computational power of tensor networks to wide use in inference and learning problems defined on general graphs. We show applications of our algorithm in graphical models, specifically on estimating free energy of spin glasses defined on various of graphs, where our method largely outperforms existing algorithms including the mean-field methods and the recently proposed neural-network-based methods. We further apply our method to the simulation of random quantum circuits, and demonstrate that, with a trade off of negligible truncation errors, our method is able to simulate large quantum circuits that are out of reach of the state-of-the-art simulation methods.
Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian H, coupling operator L, and memory kernel function \gamma, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.
An exact solution of nuclear spherical mean-field plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits is presented. The extended one-variable Heine-Stieltjes polynomials associated to the Bethe ansatz equations of the solution are determined, of which the sets of the zeros give the solution of the model, and can be determined relatively easily. A comparison of the solution to that of the standard pairing interaction with constant interaction strength among pairs in any orbit is made. It is shown that the overlaps of eigenstates of the model with those of the standard pairing model are always large, especially for the ground and the first excited state. However, the quantum phase crossover in the non-separable pairing model cannot be accounted for by the standard pairing interaction.
We study a mixture of s-bosons and like-nucleon pairs with the standard pairing interaction outside a inert core. Competition between the nucleon-pairs and s-bosons is investigated in this scenario. The robustness of the BCS-BEC coexistence and crossover phenomena is examined through an analysis of pf-shell nuclei with realistic single-particle energies in which two configurations with Pauli blocking of nucleon-pair orbits due to the formation of the s-bosons is taken into account. When the nucleon-pair orbits are considered to be independent of the s-bosons, the BCS-BEC crossover becomes smooth with the number of the s-bosons noticeably more than that of the nucleonpairs near the half-shell point, a feature that is demonstrated in the pf-shell for several values of the standard pairing interaction strength. As a further test of the robustness of the BCS-BEC coexistence and crossover phenomena in nuclei, results are given for B(E2; 0^+_g->2^+_1) values of even-even 102-130Sn with 100Sn taken as a core and valence neutron pairs confined within the 1d5/2, 0g7/2, 1d3/2, 2s1/2, 1h11/2 orbits in the nucleon-pair orbit and the s-boson independent approximation. The results indicate that the B(E2) values are well reproduced.
We construct the tripartite Bell-type inequalities of product states for l1-norm of coherence, relative entropy of coherence and skew information. Some three-qubit entangled states violate these inequalities. Particulary, the tripartite Bell-type inequality for relative entropy of coherence is always violated by the W class pure or mixed states as well as the GHZ class pure or mixed states, being used as entanglement witness.
Bethe ansatz solutions of the two-axis countertwisting Hamiltonian for any (integer and half-integer) $J$ are derived based on the Jordan-Schwinger (differential) boson realization of the $SU(2)$ algebra after desired Euler rotations, where $J$ is the total angular momentum quantum number of the system. It is shown that solutions to the Bethe ansatz equations can be obtained as zeros of the extended Heine-Stieltjes polynomials. Two sets of solutions, with solution number being $J+1$ and $J$ respectively when $J$ is an integer and $J+1/2$ each when $J$ is a half-integer, are obtained. Properties of the zeros of the related extended Heine-Stieltjes polynomials for half-integer $J$ cases are discussed. It is clearly shown that double degenerate level energies for half-integer $J$ are symmetric with respect to the $E=0$ axis. It is also shown that the excitation energies of the `yrast' and other `yrare' bands can all be asymptotically given by quadratic functions of $J$, especially when $J$ is large.
We establish two complementarity relations for the relative entropy of coherence in quantum information processing, i.e., quantum dense coding and teleportation. We first give an uncertaintylike expression relating local quantum coherence to the capacity of optimal dense coding for bipartite system. The relation can also be applied to the case of dense coding by using unital memoryless noisy quantum channels. Further, the relation between local quantum coherence and teleportation fidelity for two-qubit system is given.
It is shown that the two-axis countertwisting Hamiltonian is exactly solvable when the quantum number of the total angular momentum of the system is an integer after the Jordan-Schwinger (differential) boson realization of the SU(2) algebra. Algebraic Bethe ansatz is used to get the exact solution with the help of the SU(1,1) algebraic structure, from which a set of Bethe ansatz equations of the problem is derived. It is shown that solutions of the Bethe ansatz equations can be obtained as zeros of the Heine-Stieltjes polynomials. The total number of the four sets of the zeros equals exactly to $2J+1$ for a given integer angular momentum quantum number $J$, which proves the completeness of the solutions. It is also shown that double degeneracy in level energies may also occur in the $J\rightarrow\infty$ limit for integer $J$ case except a unique non-degenerate level with zero excitation energy.
An extended Bose-Hubbard (BH) model with number-dependent multi-site and infinite-range hopping is proposed, which, similar to the original BH model, describes a phase transition between the delocalized super-fluid (SF) phase and localized Mott insulator (MI) phase. It is shown that this extended model with local Euclidean E2 symmetry is exactly solvable when on-site local potential or disorder is included, while the model without local potential or disorder is quasi-exactly solvable, which means only a part of the excited states including the ground state being exactly solvable. As applications of the exact solution for the ground state, phase diagram of the model in 1D without local potential and on-site disorder for filling factor rho = 1 with M = 6 sites and that with M = 10 are obtained. The probabilities to detect n particles on a single site, Pn, for n = 0, 1, 2 as functions of the control parameter U/t in these two cases are also calculated. It is shown that the critical point in Pn and in the entanglement measure is away from that of the SF-MI transition determined in the phase analysis. It is also shown that the the model-independent entanglement measure is related with Pn, which, therefore, may be practically useful because Pn is measurable experimentally.
A recursive method for construction of symmetric irreducible representations of O(2l+1) in the O(2l + 1) supset O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l + 1) supset U(2l- 1) x U(2), of which the symmetric irreducible representations are simply reducible. The basis vectors of the O(2l+1) supset O(2l-1) x U(1) can easily be constructed from those of U(2l + 1) supset U(2l-1) x U(2) supset O(2l-1) x U(1) with no boson pairs, from which one can construct symmetric irreducible representations of O(2l+1) in the O(2l-1) x U(1) basis when all symmetric irreducible representations of O(2l-1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the O1(3) x U(1) basis. Matrix representations of O(5) supset O1(3) x U(1), together with the elementary Wigner coe?cients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coe?cients of the O(5) supset O(3) basis vectors in terms of those of the O1(3) x U(1) is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of O(5) supset O(3). Formulae for evaluating the elementary Wigner coe?cients of O(5) supset O(3) are derived explicitly. Analytical expressions of some elementary Wigner coe?cients of O(5) supset O(3) for the coupling (tau, 0) x (1,0) with resultant angular momentum quantum number L = 2 tau+ 2 - k for k = 0, 2, 3,...,6 with a multiplicity 2 case for k = 6 are presented.
A new angular momentum projection for systems of particles with arbitrary spins is formulated based on the Heine-Stieltjes correspondence, which can be regarded as the solutions of the mean-field plus pairing model in the strong pairing interaction G ->Infinity limit. Properties of the Stieltjes zeros of the extended Heine-Stieltjes polynomials, of which the roots determine the projected states, and the related Van Vleck zeros are discussed. The electrostatic interpretation of these zeros is presented. As examples, applications to n nonidentical particles of spin-1/2 and to identical bosons or fermions are made to elucidate the procedure and properties of the Stieltjes zeros and the related Van Vleck zeros. It is shown that the new angular momentum projection for n identical bosons or fermions can be simpli?ed with the branching multiplicity formula of U(N) supset O(3) and the special choices of the parameters used in the projection. Especially, it is shown that the solutions for identical bosons can always be expressed in terms of zeros of Jacobi polynomials. However, unlike non-identical particle systems, the n-coupled states of identical particles are non-orthogonal with respect to the multiplicity label after the projection.
The O(N) invariant quartic anharmonic oscillator is shown to be exactly solvable if the interaction parameter satisfies special conditions. The problem is directly related to that of a quantum double well anharmonic oscillator in an external field. A finite dimensional matrix equation for the problem is constructed explicitly, along with analytical expressions for some excited states in the system. The corresponding Niven equations for determining the polynomial solutions for the problem are given.
A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For $k$ pairs of valence nucleons on $n$ different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one (k+1)x(k+1) and one (n-1)x(k+1) matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach thus provides with a new efficient and systematic way to solve the problem, which, by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems and to establish a new efficient angular momentum projection method for multi-particle systems.
New polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model corresponding to the standard two-site Bose-Hubbard model are derived based on the Stieltjes correspondence. It is shown that there is a one-to-one correspondence between zeros of this new polynomial and solutions of the Bethe ansatz equations for the LMG model.A one-dimensional classical electrostatic analogue corresponding to the special LMG model is established according to Stieltjes early work. It shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the Bethe ansatz equations for the LMG model, and the number of possible configurations of equilibrium positions of the charges equals exactly to the number of energy levels in the LMG model. Some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the LMG model are derived.
We discuss some of the challenges that future nuclear modeling may face in order to improve the description of the nuclear structure. One challenge is related to the need for A-body nuclear interactions justified by various contemporary nuclear physics studies. Another challenge is related to the discrepancy in the NNN contact interaction parameters for 3He and 3H that suggests the need for accurate proton and neutron masses in the future precision calculations. MSC2010 Classification: 17B81 Applications to physics, 17B80 Applications to integrable systems, 81R12 Relations with integrable systems, 81V70 Many-body theory, 81V35 Nuclear physics, 81U15 Exactly and quasi-solvable systems, 82B23 Exactly solvable models; Bethe ansatz.
A diagonalization scheme for the Rabi Hamiltonian, which describes a qubit interacting with a single-mode radiation field via a dipole interaction, is proposed. It is shown that the Rabi Hamiltonian can be solved almost exactly using a progressive scheme that involves a finite set of one variable polynomial equations. The scheme is especially efficient for lower part of the spectrum. Some low-lying energy levels of the model with several sets of parameters are calculated and compared to those provided by the recently proposed generalized rotating-wave approximation and full matrix diagonalization.
A new step-by-step diagonalization procedure for evaluating exact solutions of the nuclear deformed mean-field plus pairing interaction model is proposed via a simple Bethe ansatz in each step from which the eigenvalues and corresponding eigenstates can be obtained progressively. This new approach draws upon an observation that the original one- plus two-body problem in a $k$-particle Hilbert subspace can be mapped unto a one-body grand hard-core boson picture that can be solved step by step with a simple Bethe ansatz known from earlier work. Based on this new procedure, it is further shown that the extended pairing model for deformed nuclei [Phys. Rev. Lett. 92, 112503 (2004) ] is similar to the standard pairing model with the first step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is taken into consideration. Our analysis show that the standard pairing model with the first step approximation displays similar pair structures of first few exact low-lying states of the model, which, therefore, provides a link between the two models.
A simple Mathematica code based on the differential realization of hard-core boson operators for finding exact solutions of the periodic-N spin-1/2 systems with or beyond nearest neighbor interactions is proposed, which can easily be used to study general spin-1/2 interaction systems. As an example, The code is applied to study XXX spin-1/2 chain with nearest neighbor interaction in a uniform transverse field. It shows that there are [N/2] level-crossing points in the ground state, where N is the periodic number of the system and [x] stands for the integer part of x, when the interaction strength and magnitude of the magnetic field satisfy certain conditions. The quantum phase transitional behavior in the ground state of the system in the thermodynamic limit is also studied.
Jan 09 2007
quant-ph arXiv:quant-ph/0701028v4
Quantum phase transitional behavior of a finite periodic XX spin-1/2 chain with nearest neighbor interaction in a uniform transverse field is studied based on the simple exact solutions. It is found that there are [N/2] level-crossing points in the ground state, where N is the periodic number of the system and [x] stands for the integer part of x, when the interaction strength and magnitude of the magnetic field satisfy certain conditions. The quantum phase transitions are of the first order due to the level-crossing. The ground state in the thermodynamic limit will be divided into three distinguishable quantum phases.
The simple entanglement of N-body N-particle pure states is extended to the more general M-body or M-body N-particle states where $N\neq M$. Some new features of the M-body N-particle pure states are discussed. An application of the measure to quantify quantum correlations in a Bose-Einstien condensate model is demonstrated.
Oct 25 2005
quant-ph arXiv:quant-ph/0510178v2
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.
The quantum critical behavior of the Bose-Hubbard model for a description of two coupled Bose-Einstein condensates is studied within the framework of an algebraic theory. Energy levels, wavefunction overlaps with those of the Rabi and Fock regimes, and the entanglement are calculated exactly as functions of the phase parameter and the number of bosons. The results show that the system goes though a phase transition and that the critical behavior is enhanced in the thermodynamic limit.
Aug 03 2004
quant-ph arXiv:quant-ph/0408005v3
A complete analysis of entangled triqubit pure states is carried out based on a new simple entanglement measure. An analysis of all possible extremally entangled pure triqubit states with up to eight terms is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis presented are most helpful for finding different entanglement types in multipartite pure state systems.
May 25 2004
quant-ph arXiv:quant-ph/0405133v2
A simple entanglement measure for multipartite pure states is formulated based on the partial entropy of a series of reduced density matrices. Use of the proposed new measure to distinguish disentangled, partially entangled, and maximally entangled multipartite pure states is illustrated.
Jun 27 1997
quant-ph arXiv:quant-ph/9706057v2
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with some numerical examples.
Apr 09 1997
quant-ph arXiv:quant-ph/9704014v1
A complementary group to SU(n) is found that realizes all features of the Littlewood rule for Kronecker products of SU(n) representations. This is accomplished by considering a state of SU(n) to be a special Gel'fand state of the complementary group \cal U(2n-2). The labels of \cal U(2n-2) can be used as the outer multiplicity labels needed to distinguish multiple occurrences of irreducible representations (irreps) in the SU(n)\times SU(n)↓SU(n) decomposition that is obtained from the Littlewood rule. Furthermore, this realization can be used to determine SU(n)⊃SU(n-1)\times U(1) Reduced Wigner Coefficients (RWCs) and Clebsch-Gordan Coefficients (CGCs) of SU(n), using algebraic or numeric methods, in either the canonical or a noncanonical basis. The method is recursive in that it uses simpler RWCs or CGCs with one symmetric irrep in conjunction with standard recoupling procedures. New explicit formulae for the multiplicity for SU(3) and SU(4) are used to illustrate the theory.
Apr 09 1997
quant-ph arXiv:quant-ph/9704015v1
A general procedure for the derivation of SU(3)⊃U(2) reduced Wigner coefficients for the coupling (\lambda_1\mu_1)\times (\lambda_2\mu_2)↓(\lambda\mu)^\eta, where \eta is the outer multiplicity label needed in the decomposition, is proposed based on a recoupling approach according to the complementary group technique given in (I). It is proved that the non-multiplicity-free reduced Wigner coefficients of SU(n) are not unique with respect to canonical outer multiplicity labels, and can be transformed from one set of outer multiplicity labels to another. The transformation matrices are elements of SO(m), where m is the number of occurrence of the corresponding irrep (\lambda\mu) in the decomposition (\lambda_1\mu_1)\times (\lambda_2\mu_2)↓(\lambda\mu). Thus, a kind of the reduced Wigner coefficients with multiplicity is obtained after a special SO(m) transformation. New features of this kind of reduced Wigner coefficients and the differences from the reduced Wigner coefficients with other choice of the multiplicity label given previously are discussed. The method can also be applied to the derivation of general SU(n) Wigner or reduced Wigner coefficients with multiplicity. Algebraic expression of another kind of reduced Wigner coefficients, the so-called reduced auxiliary Wigner coefficients for SU(3)⊃U(2), are also obtained.
Braneshop, and
the US National Science Foundation.
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