Our first conference on the subject successfully held in December 2016 happened to celebrate the victory of condensed matter physics, in particular topological quantum matter, in 2016. The emergent phenomena in topological orders not only play a fundamental role in condensed matter physics but also point to a new way of thinking about spacetime, namely as an emergent entity of more fundamental discrete degrees of freedom. On the other hand, the anyonic excitations in topological orders also lead to topological quantum computation---a quantum computing scheme robust again local errors. Recent advances have shown that topological orders, quantum computation, and emergent gravity are intimately and deeply related.
In order to efficiently exchange the rapid developments and cultivate collaborations in the interdisciplinary studies of the subject, we once again aim to assemble the leading theorists and experimentalists of topological orders, quantum information and computation, and emergent gravity in this second conference.
薛其坤 —— 清华大学
俞大鹏 —— 南方科技大学/北京大学
David G. Cory —— University of Waterloo
王正汉 —— 美国微软研究院
John Barrett —— University of Nottingham
吴咏时 —— 复旦/University of UTAH
叶如纲 —— UCSB/中国科学技术大学
施郁 —— 复旦
李晓鹏 —— 复旦
万贤纲 —— 南京大学
周杰 —— Perimeter Institute
Akira Oiwa —— 大阪大学
尤力 —— 清华大学
孔良 —— University of New Hamphire
汪忠 —— 清华大学高等研究院
许岑珂 —— 加州大学圣巴巴拉分校(UCSB)
简超明 —— 加州大学圣巴巴拉分校(UCSB)
鹿芫鸣 —— 俄亥俄州立大学
陈谐 —— California Institute of Technology
Xun Gao —— 清华大学
Yichen Huang —— IQIM,California Institute of Technology
张鹏飞 —— 清华大学
姚宏 —— 清华大学
Guoxin Miao —— IQC, University of Waterloo
吴孝松 —— 北京大学
戚杨 —— 复旦
于海明 —— 北京航空航天大学
Mircea Trif —— 清华大学
Dieter Suter —— University of Dortmund
Domenico D'Alessandro —— Iowa State University
Chair: Yidun Wan (Fudan)
Introduction for next two conference (HIT/HSU)
Charles Bennett (IBM)
Lunch and Free Discussion
Chengzhi Peng (USTC) Quantum Satellite
Panel Discussion: Future of Quantum Technology
Charles Bennett (IBM)
Dapeng Yu (SUSTech)
Xincheng Xie (PKU)
Yongshi Wu (Fudan/UTAH)
Dieter Suter (University of Dortmund)
Chengzhi Peng (USTC)
A: Summer Palace C Hall
Chair: Guodong Kang (Jishou)
Chair: Guixin Tang (HIT)
(University of Nottingham)
Yongshi Wu (Fudan/UTAH)
Dawei Lu (SUSTech/UW)
Meng Cheng (Yale)
(Iowa State University)
Chair: Sirui Lu (THU)
Chair: Cheng Guo (THU)
Xiangang Wan (Nanjing)
Zhong Wang (THU)
Cenke Xu (UCSB)
Chair: Keren Li (THU)
Chair: Man Hong Yung(SUSTech)
Yuanming Lu (Ohio)
Sadashige Matsuo (Kyoto)
Shaoming Fei (CNU)
Xin Wang (CityU_HK)
Nengkun Yu (UTS)
Chair: Dawei Lu (SUSTech/UW)
Chair: Nengkun Yu (UTS)
Keren Li (THU)
Sirui Lu (THU)
Akira Oiwa (Osaka)
Ce Wang (THU)
Xiaopeng Li (Fudan)
Man Hong Yung (SUSTech)
Yunlong Xiao (UCalgary)
Xun Gao (THU)
6 & 7 July
Boltzmann's Brain, and Wigner's Friend
Modern cosmology has given new urgency to some early 20th century puzzles that had seemed to be more in the realm of unanswerable philosophy than science: the Boltzmann’s brain problem of whether we might be merely a rare statistical fluctuation in an old dead universe, rather than inhabitants of a thriving young one, and the Wigner’s friend problem, of what it feels like to be inside an unobserved quantum superposition.
Quantum Non-Commutative Geometry
The talk will outline an approach to quantum gravity by integration over a space of non-commutative geometries. The non-commutative structure enables a Planck-scale cutoff on the geometries in a consistent way. It also allows the definition of quantum geometries using a finite-dimensional version of the functional integral. Numerical results show that it is plausible that a space-time structure emerges at a phase transition.
Sub-100 nm-wavelength spin wave propagation in metal/insulator magnetic nanostructures
Spin waves  have the potential to provide beyond-CMOS applications, e.g. parallel processing in cellular networks . However, magnonic devices realized so far using metallic ferromagnets such as permalloy suffer from the relatively short spin wave decay length. Insulating magnetic oxides provide a solution. Here we report on spin excitations in the insulating thin-film ferrimagnet yttrium iron garnet (YIG) that offers decay length of several 100s μm at room temperature . Based on such YIG thin film, we excite propagating spin waves at GHz frequency with wavelengths λ down to 88 nm, when applying the recently reported nanomagnonic grating coupler . The grating coupler consists of an array of Permalloy or CoFeB nanodisks with large saturation magnetization that are fabricated on top of the thin-film YIG. A large spin-wave signal strength is obtained when we excite the grating coupler at its own ferromagnetic resonance frequency. The wavelength of the propagating spin waves is smaller by five orders of magnitude compared to the electromagnetic wave that is used for excitation (in free space). The findings are straightforwardly extended to even smaller wavelengths by further optimizing nanomagnets and array parameters as well as the material of the naonodisks. The results are important for the implementation of nanomagnonic devices combined with conventional microwave technology.
Financial support by the German Excellence Cluster Nanosystems Initiative Munich (NIM), the DFG via project GR1640/5-2 in the priority programme SPP1538, ANR-12-ASTR-0023 Trinidad and NSF China under Grant No. 11444005 are gratefully acknowledged. .
V. Chumak, V. I. Vasyuchka, A. A. Serga & B. Hillebrands “Magnon spintronics” Nat. Phys. Vol 11, 453, 2015
Khitun, A., Bao, M. and Wang, K. L., “Magnonic logic circuits”, Phys. D: Appl. Phys., Vol. 43, 264005, 2010.
Yu, H. et al., “Magnetic thin-film insulator with ultra-low spin-wave damping for coherent nanomagnonics”, Scientific Reports, Vol. 4, 6848, 2014.
Yu, H. et al., “Omnidirectional spin-wave nanograting coupler”, Commun., Vol 4, 2704, 2013.
*corresponding author: email@example.com
Quantum entanglement in high energy particle accelerators
Quantum entangled states of neutral mesons can be exploited to study discret symmetry breaking in particle physics. Specifically, I will descibe our recent work on genuine signals of time-reversal symmetry breaking as well as constraints on possible CPT violation in entangled mesons.
Diamond-NV centers as hybrid quantum registers
The continuing evolution of quantum information processing unveils many new targets, mostly for information processing and sensing. In many of thes applications, it has been discovered that the combination of different types of quantum systems into hybrid quantum registers result in highly flexible and powerful devices that offer possibilites that are not found in systems consisting of a single type of qubits. An example in case is the nitrogen-vacancy (NV)-center in diamond. Since the electron spin of the center is coupled to different types of nuclear spins, the system can be used as an ideal test-bed for applications of hybrid quantum registers. Possibilities include to use the nuclear spin for long-term storage of the information, while the electron spin offers faster processing. As with other hybrid systems, these strengths do not come without cost. In particular, the electron spin is more susceptible to environmental noise than the nuclear spin and may therefore suffer significant dephasing while gate operations are applied to the relatively slow nuclear spins. Dynamical decoupling can help to protect such systems by reducing the decoherence due to the environmental noise, but the protection must be designed such that it does not interfere with the control fields driving the logical operations. Another possibility is to store the information in subspaces of the total Hilbert space that are less susceptible to environmental noise and therefore provide higher fidelity.
Optimal Control of a Class of Quantum System and the Reduction of Symmetry
We consider the time optimal control of a class of quantum systems. These systems' dynamics is determined by a set of vector fields spanning the P part of a Cartan K-P Lie algebra decomposition of su(n). We call these systems K-P systems. This class includes a two level quantum system (quantum bit) controlled via two components of an electromagnietic field (after a rotating wave approximation). We provide an explicit solution for the time optimal control of a quantum bit and use this example as a guide for the general theory of optimal control on K-P systems. Such an analysis leads to the consideration of the role of symmetries in optimal control problems. There is in fact in this case a Lie group G action on a manifold M which transforms to equivalent dynamics, i.e., a group of symmetries. The K-P structure implies that such an action is not free and therefore the underlying quotient space M/G is not a manifold but it has the more general structure of a stratified space. We give general results about the optimal control of these systems in particular concerning the geometry of critical locus, cut locus and the reachable sets.
Topological semimetals have attracted widespread attentions in condensed matter physics. In these materials, the conduction band and valence band meet at certain lower-dimensional subspace of the Brillouin zone. Weyl semimetals and Dirac semimetals are examples of topological nodal-point semimetals. In this talk, I will give a brief introduction to the topological nodal-line semimetals , and then talk about our recent work on “nodal-link semimetals”, which host topologically linked nodal lines in the Brillouin zone. A general method to construct models of nodal-link semimetals will be explained.
[Reference: Yan, et al, Nodal-link semimetals, arXiv:1704.00655]
Finite Quantum Gravity & Black Holes
We compute the area term contribution to the black holes' conical entropy for a class of weakly non-local super-renormalizable gravitational theories coupled to matter. We explicitly prove that all the beta functions in the proposed theory, except for the cosmological constant, are identically zero in cut-off regularization scheme and not only in dimensional regularization scheme. In particular, we show that there is no quadratic in cut-off divergent contribution to the beta function for the Newton constant. As a consequence of this result, we argue that in these theories conical entropy is a sensible definition of physical entropy, in particular, it is positive-definite and gauge independent. On top of this the conical entropy, being expressed only in terms of the classical Newton constant, turns out to be finite and naturally coincides with Bekenstein-Hawking entropy. Therefore, we are able to remove the tension between the finite Wald Entropy and the quantum entropy, which is generically divergent in quantum field theory.
Non-Fermi Liquid and quantum chaos.
The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties and quantum chaos. In this work, after a brief introduction for SYK model, I will discuss two generalizations of the SYK model that contains two SYK models with different number of Majorana modes. In the first model, they are coupled by quadratic terms and the solution shows a quantum phase transition between two non-Fermi liquid chaotic phases. In the second model, they are coupled by random interaction and we find the chaotic behavior could be tuned.
arXiv:1705.03406 "Competition between Chaotic and Non-Chaotic Phases in a Quadratically Coupled Sachdev-Ye-Kitaev Model" by Xin Chen, Ruihua Fan, Yiming Chen, Hui Zhai, PZ.
arXiv:1705.09818 "Tunable Quantum Chaos in the Sachdev-Ye-Kitaev Model Coupled to a Thermal Bath" by Yiming Chen, Hui Zhai, PZ;
Solvable SYK models in higher dimensions: a new type of many-body localization transition
Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibits an MBL transition. The model on a bipartite lattice has N Majorana fermions with SYK interactions on each site of A-sublattice and M free Majorana fermions on each site of B-sublattice, where N and M are large and finite. For r≡M/N<r_c=1, it describes a diffusive metal exhibiting maximal chaos. Remarkably, its diffusive constant D vanishes [D∝(rc −r)1/2] as r→r_c, implying a dynamical transition to an MBL phase. It is further supported by numerical calculations of level statistics which changes from Wigner-Dyson (r<r_c) to Poisson (r>r_c) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent \nu=0, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.
Lieb-Schultz-Mattis theorems for symmetry protected topological phases
The Lieb-Schultz-Mattis (LSM) theorem and its descendants represent a class of powerful no-go theorems that rule out any short-range-entangled (SRE) symmetric ground state irrespective of the specific Hamiltonian, based only on certain microscopic inputs such as symmetries and particle filling numbers. In this work, we introduce and prove a new class of LSM-type theorems, where any symmetry-allowed SRE ground state must be a symmetry-protected topological (SPT) phase with robust gapless edge states. The key ingredient is to replace the lattice translation symmetry in usual LSM theorems by magnetic translation symmetry. These theorems provide new insights into numerical models and experimental realizations of SPT phases in interacting bosons and fermions.
Effective Hamiltonian approach to adiabatic anomaly inflow in 2D topological orders with boundaries
Whether two boundary conditions of a two-dimensional topological order can be adiabatically connected without a phase transition in between remains a challenging question. We tackle this challenge by constructing an effective Hamiltonian that realizes such an adiabatic deformation. At any point along the deformation, the model remains a fixed point model describing a gapped topological order with gapped boundaries. That the deformation retains the gap is due to the anomaly cancelation between the boundary and bulk. Such anomaly inflow is quantitatively studied using our effective Hamiltonian. We apply our method of effective Hamiltonian to the extended twisted quantum double model with boundaries. We show that for a given gauge group $G$ and a three-cocycle in $H^3[G,U(1)]$ in the bulk, any two gapped boundaries for a fixed subgroup $K\subseteq G$ on the boundary can be adiabatically connected via an effective Hamiltonian. Our results can be straightforwardly generalized to the extended Levin-Wen model with boundaries.
The folding trick for topological orders enriched by the mirror symmetry
We develop a folding trick to study the classification of two-dimensional (2D) mirror-symmetry-enriched topological (SET) orders. We fold the 2D system along the mirror axis, such that it becomes a double-layer system, where the mirror symmetry acts as an onsite unitary symmetry that exchanges the two layers. In the folded picture, the mirror enrichment is fully encoded as a anyon-condensation boundary condition on the mirror axis, where the bulk of the double-layer system becomes universal and only depends on the nature of the intrinsic topological order. Hence, the classification of mirror enrichment is formulated as a classification of symmetric boundary conditions of the universal double-layer system. Using the folding trick, we develop an algorithm to enumerate possible mirror SETs, and to detect which ones are anomalous, using only the modular matrices of the intrinsic topological order.
Quantum Spacetime on a Quantum Simulator
We experimentally simulate the fundamental building blocks—quantum tetrahedra—of quantum spacetime at the Planck level and their interactions in the framework of spin-networks. In this initial attempt to study quantum spacetime by quantum information processing, on a four-qubit nuclear magnetic resonance quantum simulator, we simulate the basic module—comprising five quantum tetrahedra—of the interactions of quantum spacetime. This basic module represents the Feynman diagram vertex in the spin-network formulation of quantum spacetime.
Quantum Computing in Nuclear Magnetic Resonance
Quantum computing exploits fundamentally new models of computation based on quantum mechanical properties instead of classical physics, and it is believed that quantum computers are able to dramatically improve computational power for particular tasks. At present, nuclear magnetic resonance (NMR) has been one of the most successful platforms amongst all current implementations. It has demonstrated universal controls on the largest number of qubits, and many advanced techniques developed in NMR have been adopted to other quantum systems successfully. In this talk, I will show how NMR quantum processors can satisfy the general requirements of a quantum computer, and describe advanced techniques developed towards this target. Additionally, I will review some recent NMR quantum processor experiments. These experiments include benchmarking protocols, quantum algorithms, quantum simulation, and exploring the foundations of quantum mechanics. Finally I will summarize the concepts and comment on future prospects.
Bosonic Topological Insulator and Duality between 2+1d Quantum
Recently great progresses have been made in understanding 2+1d conformal field theories without supersymmetry. It was proposed that seemingly different Lagrangians may correspond to the same CFT in the infrared, a property called duality. Within all these proposals, one theory is particularly interesting to condensed matter: the 2+1d N=2 QED with conserved gauge flux. This theory describes the transition between a bosonic topological insulator and a trivial Mott insulator. We
demonstrate that this theory has an unexpected self-duality, which grants it an enlarged O(4) symmetry at the critical point. Further studies show that this theory is also dual to the N=2 bosonic QED (the so called NCCP1 model in condensed matter literature), which describes the phase transition between the easy-plane Neel order to valence bond solid order in spin-1/2 system. We design a lattice model to realize
this desired bosonic topological phase transition, we also make quantitative predictions based these dualities that can be tested by numerical works.
Kondo effect and superconducting transport in SiGe self-assembled quantum dot transistors
In group IV quantum dots (QDs), manipulation and detection of single electron spins have been rapidly developed as fundamental technologies of spin-based quantum computing . In contrast to III-Ⅴ-based QDs, hyperfine interaction with nuclear spins of host crystal is strongly suppressed in isotopically enriched 28Si QDs, resulting in a long electron spin coherence time . Here, we focus on a hole system in SiGe self-assembled QDs (SAQDs) because the confined holes are expected to have very weak contact hyperfine interaction with nuclei and strong SOI due to their p-orbital nature . In this work, we present the superconducting transport and Kondo effect in SiGe SAQDs coupled to superconducting electrodes.
SiGe SAQDs grown by molecular beam epitaxy were contacted by aluminum electrodes as shown in Fig. 1. At magnetic fields higher than 1 T, the source-drain electrodes enters in the normal state. In a negative back-gate voltage region, zero bias conductance peak is observed, which can be attributed to Kondo effect (see Fig. 2). The estimated Kondo temperature was about 5 K. By further increasing magnetic field, the zero bias peak splits owing to Zeeman splitting. From the split Kondo features, we evaluate g-factor for perpendicular magnetic field to be 2.03 similar to the value reported previously . In the superconducting regime at low magnetic fields, the zero-bias peak is observed in the entire measured back-gate voltage region, suggesting a Josephson current. Quasi-particle tunneling peaks at ± 2D are seen almost independent of the back-gate voltage. In the Kondo region, additional subgap transport features associated with Andreev transport are visible at ± D (see Fig. 2).
 J. J. Pla et al., Nature 496, 334 (2013).
 M. Veldhorst et al., Nature Nanotech. 9, 981 (2014).
 G. Katsaros et al., Nature Nanotech. 5, 458 (2010).
 N. Ares et al., Phys. Rev. Lett. 110, 046602 (2013).
Machine learning approaches to entangled quantum states
Artificial neural networks play a prominent role in the rapidly growing field of machine learning and are recently introduced to quantum many-body systems. This talk will focus on using a machine-learning model, the restricted Boltzmann machine (RBM) to describe entangled quantum states. Both short- and long-range coupled RBM will be discussed. For a short-range RBM, the associated quantum state satisfies an entanglement area law, regardless of spatial dimensions. I will present our recently constructed exact RBM models for nontrivial topological phases, including a 1d cluster state and a 2d toric code. For a long-range RBM, the captured entanglement entropy scales linearly with the number of variational parameters in the RBM model, in sharp contrast to the log-scaling in matrix product state representation.
Incompatibility of Observables as State-Independent Bound of Uncertainty Relations
For a pair of observables, they are called “incompatible”, if and only if the commutator between them does not vanish, which represents one of the key features in quantum mechanics. The question is, how can we characterize the incompatibility among three or more observables? Here we explore one possible route towards this goal through Heisenberg’s uncertainty relations, which impose fundamental constraints on the measurement precisions for incompatible observables. Our measure of incompatibility represents a versatile tool for applications such as testing entanglement of bipartite states, and EPR-steering criteria. We also give a brief overview of some results in entropic uncertainty relations in the presence of quantum memory.
Experimental study of exotic transport in superconductor-semiconductor hybrid systems
Recent progress of topological superconductivity promotes experimental studies of superconductor-semiconductor hybrid systems. In this talk, I will explain our recent experimental results of Cooper pair splitting in the junctions of Al and parallel double InAs nanowires and also Andreev reflection in junctions of NbTi and InAs quantum Hall bulk state. These works are collaborative research with A. Oiwa, K. Li, H. Q. Xu, C. Juenger, A. Baumgartner C. Shonenberger, J. Shabani, and C. Palmstrom.
On Quantum Uncertainty Relations and Related
We study entropic, probabilistic, variance and standard deviation based, and statistical distance based quantum uncertainty relations, as well as their experimental verifications. The related quantum coherence and quantum correlations like quantum entanglement, quantum steering, Bell non-locality are also investigated.
On the advantage of barrier over tilt control of a singlet-triplet spin qubit
Overcoming the charge noise is key to the realization of scalable quantum computation using spin qubits. It has been recently demonstrated that the effects of charge noise can be suppressed if operations of a singlet-triplet qubit are implemented using barrier control instead of the traditional tilt control. We have found, however, that for certain gates involving extensive x-rotations, barrier control offers little or no improvement when the nuclear noise is significant. Nevertheless, we introduce a new set of composite pulses that reduce gate times by up to 90%. Using these optimized pulses, the barrier control shows great advantages in randomized benchmarking simulations, with the coherence time extended by about two orders of magnitude for experimentally relevant noises . We have also performed a microscopic calculation of a singlet-triplet qubit under the influence of an impurity. We have found that, the relative charge noise (charge noise divided by the exchange interaction), while generally believed to increase with increasing exchange interaction, actually decreases when the barrier control is implemented . This is understood as a combined consequence of the greatly suppressed detuning noise when the two dots are symmetrically operated, as well as an enhancement of the inter-dot hopping energy of an electron when the barrier is lowered.
 C. Zhang, R.E. Throckmorton, X.-C. Yang, X. Wang, E. Barnes, S. Das Sarma, arXiv:1701.03796 (Phys. Rev. Lett. in press)
 X.-C. Yang, X. Wang, arXiv:1704.07975
Sample-optimal tomography of quantum states
It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error ϵ in trace distance required $O(dr^2/ϵ^2)$ copies for a $d$-dimensional density matrix of rank r. Here, we give a theoretical measurement scheme (POVM) that requires $O(dr/δ)ln(d/δ)$ copies of $ρ$ to error $δ$ in infidelity, and a matching lower bound up to logarithmic factors. This implies $O((dr/ϵ^2)ln(d/ϵ))$ copies suffice to achieve error ϵ in trace distance. We also prove that for independent (product) measurements, $Ω(dr^2/δ^2)/ln(1/δ)$ copies are necessary in order to achieve error $δ$ in infidelity. For fixed $d$, our measurement can be implemented on a quantum computer in time polynomial in $n$.
A Separability-Entanglement Classier via Machine Learning
The problem of determining whether a given quantum state is entangled lies at the heart in quantum information processing. Despite the many methods -- such as the positive partial transpose (PPT) criterion and the聽$k$-symmetric extendibility criterion -- to tackle this problem, none of them enables a general, practical solution due to the problem's NP-hard complexity. Explicitly, states that are separable form a high-dimensional convex set of vastly complicated structure. In this talk, I will present our construction of a new separability-entanglement classifier underpinned by machine learning techniques. Our method outperforms the existing methods in generic cases in terms of both speed and accuracy, opening up the avenues to explore quantum entanglement via the machine learning approach.
 Sirui Lu, Shilin Huang, Keren Li, Jun Li, Jianxin Chen, Dawei Lu, Zhengfeng Ji, Yi Shen, Du- anlu Zhou, and Bei Zeng. A Separability-Entanglement Classier via Machine Learning, 2017. arXiv:1705.01523v1
Machine Learning for Frustrated Classical Spin Models.
In this talk, we will apply the machine learning method to study classical XY model on frustrated lattices, such as triangle lattice and UnionJack lattice. The low temperature phases of these frustrated models exhibit both U(1) and Z2 chiral symmetry breaking, and therefore they are characterized by two order parameters, and consequently, two successive phase transitions as lowering the temperature. By using classical Monte Carlo to generate a large number of data to feed computer, we use methods such as the principle component analysis (PCA) to analyze these data. We find that the PCA method can distinguish all different phases and locate phase transitions, without prior knowledge of order parameters. Our analysis pave a way to machine learning studies of more sophisticated models.
Transforming Bell's Inequalities into State Classifiers with Machine Learning
Man Hong Yung
Quantum information science has profoundly changed the ways we understand, store, and process information. A major challenge in this field is to look for an efficient means for classifying quantum state. For instance, one may want to determine if a given quantum state is entangled or not. However, the process of a complete characterization of quantum states, known as quantum state tomography, is a resource-consuming operation in general. An attractive proposal would be the use of Bell's inequalities as an entanglement witness, where only partial information of the quantum state is needed. The problem is that entanglement is necessary but not sufficient for violating
Bell's inequalities, making it an unreliable state classifier. Here we aim at solving this problem by the methods of machine learning. More precisely, given a family of quantum states, we randomly picked a subset of it to construct a quantum-state classifier, accepting only partial information of each quantum state. Our results indicated that
these transformed Bell-type inequalities can perform significantly better than the original Bell's inequalities in classifying entangled states. We further extended our analysis to three-qubit and four-qubit systems, performing classification of quantum states into multiple species. These results demonstrate how the tools in machine learning can be applied to solving problems in quantum information science.
Efficient Representation of Quantum Many-body States with Deep Neural Networks
The challenge of quantum many-body problems comes from the difficulty to represent large-scale quantum states, which in general requires an exponentially large number of parameters. Various variational approaches have been proposed to give efficient representation of quantum many-body states under certain configurations. Recently, a connection has been made between quantum many-body states and the neural network representation. An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to popularity of the deep learning methods. Here, we give a rigorous proof that a deep neural network (deep Boltzmann machine) can efficiently represent most physical states, including those generated from dynamics or ground states of complicated Hamiltonians, while a shallow network through a restricted Boltzmann machine, using complexity theory in computer science, cannot efficiently represent those states even without constraint on network architecture. Then we discuss briefly how to train a deep Boltzmann machine. Since related research establish a connection between neural network and quantum many-body physics, we discuss the possibility to introduce physics concepts into machine learning community.
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