Jekyll2022-11-16T20:08:49+00:00https://everettyou.github.io/feed.xmlYou GroupYi-Zhuang You's GroupScalable and Flexible Classical Shadow Tomography2022-09-07T00:00:00+00:002022-09-07T00:00:00+00:00https://everettyou.github.io/2022/09/07/CST<p>Quantum state tomography is an essential task in quantum information technology. It aims to reconstruct a quantum state from repeated measurements of copies of the state. While reconstructing the full density matrix requires exponentially many samples in many-body systems, predicting a collection of (possibly exponentially many) properties of the quantum system can be efficiently achieved with only a polynomial number of samples under the name of <em>shadow tomography</em>. <a href="https://arxiv.org/abs/2002.08953">Huang, Kueng, Preskill</a> further propose a more experiment-friendly shadow tomography scheme, called the <em>classical shadow tomography</em>, which reduces the data acquisition and classical post-processing complexity while retaining the superior polynomial sample complexity.</p>
<p>However, the original proposal was limited to two measurement schemes: the single-qubit (local) Pauli measurement, which is well suited
for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. It is desired to go beyond these two limits and develop more flexible measurement schemes for classical shadow tomography.</p>
<p>In recent work <a href="https://arxiv.org/abs/2209.02093">arXiv:2209.02093</a>, we developed a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. For more details, see my talk
<a href="https://online.kitp.ucsb.edu/online/dynisq22/you/">Scalable Classical Shadow Tomography with Shallow Circuits and Quantum Dynamics - Yi-Zhuang You, UC San Diego</a> at the <a href="https://online.kitp.ucsb.edu/online/dynisq22/">KITP Program</a> of
Quantum Many-Body Dynamics and Noisy Intermediate-Scale Quantum Systems.</p>Quantum state tomography is an essential task in quantum information technology. It aims to reconstruct a quantum state from repeated measurements of copies of the state. While reconstructing the full density matrix requires exponentially many samples in many-body systems, predicting a collection of (possibly exponentially many) properties of the quantum system can be efficiently achieved with only a polynomial number of samples under the name of shadow tomography. Huang, Kueng, Preskill further propose a more experiment-friendly shadow tomography scheme, called the classical shadow tomography, which reduces the data acquisition and classical post-processing complexity while retaining the superior polynomial sample complexity.Symmetric Mass Generation2022-08-19T00:00:00+00:002022-08-19T00:00:00+00:00https://everettyou.github.io/2022/08/19/SMG<p>Mass is a basic property of matter in physics. The origin of mass for matter particles is one of the most fundamental questions about our universe. The standard mechanism for fermions to acquire a mass is the Yukawa-Higgs mechanism, which is based on the spontaneous symmetry breaking via the condensation of a scalar Higgs field (that couples to the fermion field as a bilinear mass via the Yukawa coupling). The Yukawa-Higgs mechanism is responsible for the mass generation of fundamental fermions (leptons and quarks) in the Standard Model, a significant theoretical discovery acknowledged by the Nobel Prize in Physics 2013.</p>
<p>In the last few years, it was gradually realized that there is a new mechanism of mass generation for fermions without symmetry breaking or condensing any (fermion bilinear) Higgs field. This new mechanism is referred to as <em>symmetric mass generation</em> (SMG). The SMG mechanism relies on the non-perturbative interaction effect of fermions and goes beyond the conventional mean-field description of fermion mass generation in the Yukawa-Higgs theory. It is realized that the SMG has deep connections with interacting topological insulators, quantum anomaly cancellations, and deconfined quantum criticality. It also has strong implications for the lattice regularization for chiral gauge theories, which has been a long-standing problem in the lattice gauge theory. SMG has generated a broad research interest in condensed matter and high-energy physics communities.</p>
<p>Together with Juven Wang, we wrote a <a href="https://www.mdpi.com/2073-8994/14/7/1475">review paper</a> on SMG, introducing SMG models, summarizing the current numerical results, unifying current field theory understandings, and presenting an overview of various features and applications of SMG.</p>
<p>I gave a talk about SMG on <a href="https://indico.cern.ch/event/1162387/">Paths to Quantum Field Theory 2022</a>. The talk video can be watched on YouTube: <a href="https://www.youtube.com/watch?v=o5OP5QFtjdk&list=PPSV">Symmetric Mass Generation - Yi-Zhuang You</a>.</p>Mass is a basic property of matter in physics. The origin of mass for matter particles is one of the most fundamental questions about our universe. The standard mechanism for fermions to acquire a mass is the Yukawa-Higgs mechanism, which is based on the spontaneous symmetry breaking via the condensation of a scalar Higgs field (that couples to the fermion field as a bilinear mass via the Yukawa coupling). The Yukawa-Higgs mechanism is responsible for the mass generation of fundamental fermions (leptons and quarks) in the Standard Model, a significant theoretical discovery acknowledged by the Nobel Prize in Physics 2013.Decipher the Holographic Universe with Artificial Intelligence2020-06-25T00:00:00+00:002020-06-25T00:00:00+00:00https://everettyou.github.io/2020/06/25/NeuralRG<p>The known universe consists of four fundamental forces: electromagnetic force, weak force, strong force, and gravity. The first three forces can be described theoretically. Gravity, however—which makes up the vast space of the universe—lacks a quantum theory. For three decades, scientists have tried to understand quantum gravity by using a model called the holographic universe.</p>
<p><img src="/assets/img/figures/virtual_reality.jpeg" alt="Artificial intelligence deciphers the holographic universe as virtual reality." /></p>
<p>According to University of California San Diego Assistant Professor of Physics Yi-Zhuang You, the holographic universe can be thought of as a universe in virtual reality. All the information about this three-dimensional universe is projected from a two-dimensional screen of quantum pixels. The quantum physics on the two-dimensional screen gives rise to the quantum gravity behavior in the virtual universe. Between the two—the quantum boundary and the bulk gravity—lie dual theories known as the “holographic duality.”</p>
<p>To reveal this elusive duality, You and his team developed a novel machine-learning algorithm that enables artificial intelligence (AI) to learn from the boundary quantum theory and propose the corresponding bulk gravity dual. The results of their study are published in Physical Review Research, an American peer-reviewed scientific journal established in 1893 and published by the American Physical Society. According to the scientists, their outcome may shed light on a better understanding of the holographic duality, and it could result in new tools for exploring quantum gravity.</p>
<p>You and Hong-Ye Hu, a physics graduate student at UC San Diego, collaborated with Shuo-Hui Li and Lei Wang (Institute of Physics, Chinese Academy of Sciences, Beijing) to develop the machine-learning algorithm based on a deep artificial neural network—an interconnected group of nodes that mimic simple brain neurons.</p>
<p>“By arranging the artificial neurons in a hierarchical structure, the neural network is able to generate quantum field configurations as an artist creating a painting, level-by-level, from general composition to specific details. In this way, the artificial intelligence learns to identify the features at different scales in the quantum system and present them at different depths in the holographic bulk such that it can serve as a translator that can interpret the quantum theory in terms of the gravity theory and vice versa,” explained You. “The artificial intelligence not only learns to propose physics theories but also learns to find the connection between different theories that are dual to each other.”</p>
<p>According to You and Hu, at first the neural network “does not have a clue,” so it proposes arbitrary gravity theories. It generates quantum field configurations on the holographic boundary based on the hypothesized bulk gravity theory. The likelihood of the generated quantum field configuration is then evaluated according to the given field theory action, which provides an indication of how well the gravity theory is describing the boundary quantum physics. Such an indication will be fed back to train the neural network parameters.</p>
<p>“In iterative rounds of trial-and-error, the neural network gradually learns to modify its proposal of the gravity theory to improve the overall correctness of the consequent predictions,” explained Hu.</p>
<p>The structure of the neural network is inspired by the idea of renormalization group—an iterative approach to remove detail features in the physical system and extract overall structures at progressively larger scales—which gives the artificial intelligence the power to grasp and analyze the information at different length scales.</p>
<p><img src="/assets/img/figures/holography.jpg" alt="One of the popular cartoon pictures about holographic universe from Google.One of the popular cartoon pictures about holographic universe from Google." /></p>
<p>“One of our epiphanies in this research was to realize that if the process of renormalization were reversed, it would constitute a generative procedure, which could reconstruct the physical system in a hierarchical manner,” said You.</p>
<p>The researchers tested the algorithm on a quantum system with a large number of interacting particles. The quantum system is in a state that exhibits self-similarity, meaning that the system looks similar to itself from small to large scales. In this case, a new dimension will emerge in the holographic universe, which corresponds to the different scales to observe the quantum system.</p>
<p>“We are amazed to observe that a beautiful hyperbolic geometry of the holographic universe emerged in the neural network under training,” said Hu. “This is exciting because scientifically it shows how artificial intelligence can help scientists to understand how space-time can emerge from interacting quantum objects and to establish the holographic duality on the field theory level. In the long term, the space-time fluctuation could also be incorporated in the framework, which may lead to a possible quantum description of gravity.”</p>
<p>Another usage of the holographic duality established by the AI is to simulate the strong-interacting boundary quantum system from its holographic bulk.</p>
<p>“In many cases, the particles in the holographic universe are almost free, which makes the bulk gravity theory much easier to simulate compared to the boundary quantum theory. Therefore, our approach can help condensed matter physicists to simulate quantum materials, and find new phases of matter in the future,” said You.</p>
<p>The UC San Diego researchers are supported by a startup fund from the university. Li and Wang are supported by the National Natural Science Foundation of China (grant no. 11774398) and the Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB28000000).</p>
<p><em>(By Cynthia Dillon, reprinted from <a href="https://physicalsciences.ucsd.edu/media-events/articles/2020/using-artificial-intelligence-to-paint-the-holographic-universe.html">UCSD Physical Science News</a>)</em></p>Cynthia DillonThe known universe consists of four fundamental forces: electromagnetic force, weak force, strong force, and gravity. The first three forces can be described theoretically. Gravity, however—which makes up the vast space of the universe—lacks a quantum theory. For three decades, scientists have tried to understand quantum gravity by using a model called the holographic universe.Emmy Noether looks at Deconfine Quantum Critical Point2019-04-16T00:00:00+00:002019-04-16T00:00:00+00:00https://everettyou.github.io/2019/04/16/DQCP<p>The <a href="https://en.wikipedia.org/wiki/Noether%27s_theorem">Noether theorem</a> is a profound theorem in physics, relating continuous symmetries to conservation laws. The German mathematician Emmy Noether first proved it. Some famous examples include space-time translation symmetry and momentum-energy conservation, rotation symmetry and angular momentum conservation, internal U(1) symmetry and charge conservation. In quantum many-body systems, the discussion of symmetry can be more involved. For example, new symmetry that does not exist in the system can even emerge at low energy. Such a phenomenon generally occurs when the symmetry breaking terms are all irrelevant under the renormalization group (RG) flow in the field theory description. Do emergent symmetries also lead to emergent conservation laws?</p>
<p>A great platform to test these ideas is the deconfined quantum critical point (DQCP). The DQCP is an exotic quantum critical point between two spontaneous symmetry breaking phases: the antiferromagnetic Neel phase and the valence bond solid (VBS) phase. Deconfined degrees of freedom, such as fractionalized spinons and emergent gauge fluctuations, appear at and only at the critical point. The fractionalization generally enlarges the scaling dimension of order parameters and other symmetry breaking anisotropy terms (making them more irrelevant under RG flow), which paves the way for larger symmetry to emerge at the DQCP. Depending on the model details, the emergent symmetry can be O(4) or SO(5), which both involve rotation of the Neel and VBS order parameters to each other.</p>
<p><img src="/assets/img/figures/DQCP.png" alt="Illustration of the DQCP phase diagram" /></p>
<p>If the proposed emergent symmetry is there, following the Noether theorem, we should expect to detect its associated emergent conserved currents. The Noether theorem further tells us: each generator of the continuous symmetry corresponds to a distinct type of conserved current. At the DQCP, the most incredible emergent symmetry generator is the one that generates the rotation between the Neel and VBS order parameters, which is not realizable on the lattice model level. The detection of its associated conserved current (the Neel-VBS current) would provide strong support for the emergent symmetry and hence the field theory description of the DQCP.</p>
<p>Two questions appear in front of us. First, where should we find the signal of the emergent conserved current in the low-energy excitation spectrum? Second, how to verify whether the observed current is conserved or not? In our recent work <a href="https://arxiv.org/abs/1811.08823">arXiv:1811.08823</a>, we can identify the Neel-VBS current to be the spin fluctuation at momentum $(\pi,0)$ and $(0,\pi)$ by symmetry analysis and field theory argument. We then check the conservation of the emergent current by measuring its scaling dimension. In our large-scale quantum Monte Carlo numerical simulation, we observe that the scaling dimension of the current fluctuation is precisely consistent with the requirement of the conservation law. Therefore we confirm the emergent symmetry at DQCP by measuring the associated emergent conserved current in the spin excitation spectrum.</p>
<p>The proposal of detecting emergent symmetry by conserved currents applies to more general scenarios, such as low-dimensional analog of DQCP. In another recent work <a href="https://arxiv.org/abs/1904.00021">arXiv:1904.00021</a>, we discover the U(1)xU(1) emergent symmetry of the Ising-DQCP in (1+1)D. Our theoretical and numerical study also guides the experimental search for the DQCP in quantum magnets. In particular, the Shastry-Sutherland magnet SrCu2(BO3)2 was proposed to be a candidate material that could potentially realize the DQCP under pressure. In our latest work <a href="https://arxiv.org/abs/1904.07266">arXiv:1904.07266</a>, we provide supporting evidence for the emergent O(4) symmetry and point out the unique spectral features originated from the corresponding emergent conserved currents. Experimental detection of these spectral signals would further deepen our understanding of exotic quantum phase transitions and quantum spin liquid.</p>The Noether theorem is a profound theorem in physics, relating continuous symmetries to conservation laws. The German mathematician Emmy Noether first proved it. Some famous examples include space-time translation symmetry and momentum-energy conservation, rotation symmetry and angular momentum conservation, internal U(1) symmetry and charge conservation. In quantum many-body systems, the discussion of symmetry can be more involved. For example, new symmetry that does not exist in the system can even emerge at low energy. Such a phenomenon generally occurs when the symmetry breaking terms are all irrelevant under the renormalization group (RG) flow in the field theory description. Do emergent symmetries also lead to emergent conservation laws?Rediscover Quantum Mechanics in Machine Learning2019-02-01T00:00:00+00:002019-02-01T00:00:00+00:00https://everettyou.github.io/2019/02/01/discoverQM<p>The emergent phenomenon is a central theme of condensed matter physics. Not only matter and forces can be emergent, spacetime and gravity could also be emergent. These exciting ideas are being actively explored in the frontier of physics research. But wait a moment, aren’t all these physics theories themselves also emergent phenomena? This is an interesting idea. Physics theories are indeed collective neural activations in human’s brain. It is unclear how the ideas of physics emerge in the neural network of a physicist, or more generally, in the physics community. Understanding the universal principles of emergent intelligence in complex networks should be one important goal of science.</p>
<p>We are probably still far away from a full understanding of intelligence. However, the recent development in machine learning allows us to make the first step towards this direction. We want to investigate whether artificial neural networks can be used to discover physical concepts and laws from experimental data. Let us take quantum mechanics for example. Suppose quantum mechanics has not been formulated so far, yet amazingly, physicists somehow know how to perform cold atom experiments to collect density distributions of Bose-Einstein condensate (BEC) in potential traps of different shapes. Can quantum mechanics be discovered as the most natural theory to explain the experimental data without the prior bias of human civilization? Or will the machine come up with an alternative form of quantum mechanics?</p>
<p>In our recent work <a href="https://arxiv.org/abs/1901.11103">arXiv:1901.11103</a>, we demonstrate how a machine learning algorithm can discover quantum mechanics in learning to predict the BEC density given the potential profile. The machine is only exposed to the data of potentials and densities, yet the quantum wave function can emergent as latent variables in the neural network. We are inspired by the development of machine translation, which is trained to map sequences of words from one language to another. The machine translator can develop a semantic space in its hidden layers, which holds the intrinsic representation of words or phrases that are universal to all languages. By analyzing the structure of the semantic space, we could gain understanding about the relations among words as perceived by the translator. To put our problem in this context, we treat the potential-to-density mapping as an example of sequence-to-sequence mapping which can be handled by the machine translation approach, such as the recurrent neural network.</p>
<p><img src="/assets/img/figures/introspective_RNN.png" alt="Introspective recurrent neural network" /></p>
<p>We train a recurrent neural network to translate the potential profile to the density profile along a one-dimensional trap. By learning to perform this translation, the machine must have also gain intuitions about the underlying physics. To extract what has been realized by the machine translator, we design a higher-level machine, called knowledge distiller, to learn from the neural activations (hidden states) of the lower-level translator. The knowledge distiller is an auto-encoder incorporated in another recurrent neural network structure. Its task is to compress the hidden states generated by the translator at each step as much as possible, without losing the prediction power to reconstruct subsequent hidden states. In this way, the knowledge distiller can identify the essential variables. Our study shows that the reconstruction loss of the knowledge distiller only increase abruptly when its latent space dimension is reduced below two. This implies that at least two real variables are required to describe the behavior of the potential-to-density translator. As we plot these two variables out, they correspond to the real and imaginary parts of the quantum wave function (up to basis freedom). Further inspection of the update rules for these variables shows that they are governed by a recurrent relation which precisely matches the discrete version of the Schrödinger equation. Knowledge about quantum mechanics indeed emerges in the neural network.</p>
<p>Moreover, if we relax the information bottleneck of the knowledge distiller, alternative forms of quantum mechanics, such as the density functional theory, could also emerge, but it requires at least three real variables to describe. It could be reassuring to know that human’s current formulation of quantum mechanics, in terms of the wave function and Schrödinger equation, is indeed the most parsimonious theory among all alternative theories of quantum mechanics that have been discovered in our neural network.</p>The emergent phenomenon is a central theme of condensed matter physics. Not only matter and forces can be emergent, spacetime and gravity could also be emergent. These exciting ideas are being actively explored in the frontier of physics research. But wait a moment, aren’t all these physics theories themselves also emergent phenomena? This is an interesting idea. Physics theories are indeed collective neural activations in human’s brain. It is unclear how the ideas of physics emerge in the neural network of a physicist, or more generally, in the physics community. Understanding the universal principles of emergent intelligence in complex networks should be one important goal of science.Moire Superlattice2018-05-21T00:00:00+00:002018-05-21T00:00:00+00:00https://everettyou.github.io/2018/05/21/Moire<p>When two periodic lattices lie on top of each other with a relative twist or a mismatched lattice constant, they can interfere to create a Moire pattern. As shown in the following figure, a Moire pattern has a spatial structure on a larger scale than either of the lattice along. The (quasi)periodic pattern forms a larger lattice, known as the Moire superlattice. Experimentally, the Moire superlattice can be realized by stacking 2D materials together, including graphene, hexagonal boron nitride (hBN), molybdenum disulfide and many others. Exciting new physics can emerge on Moire superlattices, including Mott insulators, unconventional superconductors and possibly new topological phases.</p>
<p><img src="/assets/img/figures/twisted_bilayer_graphene.png" alt="Twisted bilayer graphene" /></p>
<p>A simple intuition is that as the lattice length scale increases, the electronic matter wave also propagates at a longer wave length on the lattice, which corresponds to a lower momentum. Since the kinetic energy of the electron positively correlates with its momentum, the reduced momentum on the superlattice implies a reduced kinetic energy. This makes the interaction energy relatively more important compare to the kinetic energy, resulting in the enhanced interaction effect among the electrons. In fact, band structure calculations reveal that at certain twist angles (known as the magic angle), the electronic band near the Fermi energy can become very flat. This opens up a platform for strongly correlated quantum many-body physics to take place.</p>
<p>Typically there are two approaches to handle these correlated electronic systems: one is the weak coupling approach that start with the electronic band structure and add in interaction perturbatively, another is the strong coupling approach that start with the clusters of electrons with strong local interaction and try to connect them together virtual tunneling processes. In a recent work (<a href="https://arxiv.org/abs/1805.06867">arXiv:1805.06867</a>) with Ashvin Vishwanath, we propose a low-energy effective theory for twisted bilayer graphene and follow the weak coupling approach to analyze the leading ordering instability including the possible pairing symmetries. My group is also actively exploring from the strong coupling approach to gain a more comprehensive understanding of the correlated physics in these Moire superlattice systems.</p>When two periodic lattices lie on top of each other with a relative twist or a mismatched lattice constant, they can interfere to create a Moire pattern. As shown in the following figure, a Moire pattern has a spatial structure on a larger scale than either of the lattice along. The (quasi)periodic pattern forms a larger lattice, known as the Moire superlattice. Experimentally, the Moire superlattice can be realized by stacking 2D materials together, including graphene, hexagonal boron nitride (hBN), molybdenum disulfide and many others. Exciting new physics can emerge on Moire superlattices, including Mott insulators, unconventional superconductors and possibly new topological phases.Quantum Information Dynamics2018-03-28T00:00:00+00:002018-03-28T00:00:00+00:00https://everettyou.github.io/2018/03/28/DynQ<p>Quantum information dynamics is an emerging field that ties several topics together, including non-equilibrium and driven quantum systems, many-body localization and thermalization, quantum chaos and black holes, tensor network holography. Traditionally, one may think that physics is about the dynamics of matter and spacetime. Extending our scope to the dynamics of quantum information is a new trend, which in turn deepen our understanding of the dynamics of matter and spacetime.</p>
<p>A tool to quantify quantum information is the entanglement entropy, as well as various information measures that can be constructed out of it. A prominent difference between quantum and classical information lies in the phenomenon of quantum entanglement. Quantum entanglement is a non-local information storing scheme that can only be realized in quantum systems, where information is not stored in any particular entity that makes up the system but stored in the relationships among the the entities. Therefore any attempt to separate a subsystem out of an entangled quantum system would inevitably cut off its quantum entanglement with the rest of the system, which leads to an information loss, or equivalently an entropy growth. The amount of entropy associated with separating the subsystem is the entanglement entropy, which also quantifies the entanglement between the subsystem and its counterpart.</p>
<p>For a given quantum many-body state, each choice of the subsystem (which may be disconnected) is associated with an entanglement entropy. Many quantum information measures, such as mutual or tripartite information, can be constructed by adding and subtracting entanglement entropies over different subsystems. So one may attempt to give a full description of quantum information by the collection of entanglement entropies over all possible subsystems. However, the amount of data is enormous. For a system of \(N\) qubits, there are totally \(2^N\) different choices of subsystem and hence \(2^N\) potentially different entanglement entropies. How to arrange the exponentially large amount of data efficiently? One idea is to encode the entanglement entropies in the energy function of a statistical mechanics model.</p>
<p>The intuition is to consider each choice of the subsystem as an Ising configuration, such that the qubits within (outside) the subsystem is marked with a down (up) spin. In this way, we can think of the entanglement entropy of a subsystem as the energy associated to the corresponding Ising configuration. Once the energy function is define, we can follow the principles of statistical mechanics to define a statistical ensemble of subsystems, in which the probability of each choice of the subsystem is proportional to the Boltzmann weight \(W=e^{-S}\) set by the entanglement entropy \(S\). This Boltzmann weight was first proposed and named as the entanglement feature in <a href="https://arxiv.org/abs/1709.01223">arXiv:1709.01223</a>. The entire statistical ensemble encodes the entanglement entropies over all possible subsystems of a given quantum many-body state. For example, a disentangled product state would correspond to the paramagnetic limit of the Ising ensemble where all spin configurations appear with equal probability. On the other hand, a strongly entangled state would correspond to an Ising ensemble with strong ferromagnetic correlation.</p>
<p>As the many-body state evolves with time, the entanglement features also evolves. Therefore we can formulate the dynamics of quantum information as the dynamics of the entanglement features that describes the entanglement Ising ensemble. Our group is working to develop and apply this idea to understand quantum information dynamics in quantum chaotic <a href="https://arxiv.org/abs/1803.10425">arXiv:1803.10425</a> or periodic driven systems. This Ising formulation of quantum many-body entanglement turns to to be closely related to tensor network holography, which allows us to explore the corresponding dynamics in holographic bulk as well.</p>Quantum information dynamics is an emerging field that ties several topics together, including non-equilibrium and driven quantum systems, many-body localization and thermalization, quantum chaos and black holes, tensor network holography. Traditionally, one may think that physics is about the dynamics of matter and spacetime. Extending our scope to the dynamics of quantum information is a new trend, which in turn deepen our understanding of the dynamics of matter and spacetime.Entanglement Feature Learning2018-01-31T00:00:00+00:002018-01-31T00:00:00+00:00https://everettyou.github.io/2018/01/31/EFL<p>The holographic duality (AdS/CFT) was originally proposed as the correspondence between a \(d\)-dimensional quantum field theory and a \((d+1)\)-dimensional quantum gravity theory. The information about the space-time geometry in the higher-dimensional holographic bulk is encoded in the quantum many-body dynamics on the lower-dimensional holographic boundary, hence the name “holography”. The holographic duality reveals a deep connection between quantum entanglement and space-time geometry: “Entanglement is the fabric of space-time,” said Brain Swingle in <a href="https://www.quantamagazine.org/tensor-networks-and-entanglement-20150428/">Quantum Magazine</a>.</p>
<p><img src="/assets/img/figures/tensor_network.png" alt="Representing quantum many-body state as tensor network" /></p>
<p>One way to envision the structure of quantum entanglement in a many-body system is to represent the quantum many-body wave function in the form of a tensor network (as illustrated in the above figure). One can view each link in the network as an entangled pair state and each circle as a measurement that projects the local degrees of freedom to a state in a measurement basis. This way, the unmeasured degrees of freedom (the open ends of dangling links) will be entangled to form a quantum many-body state. Although the entangled pair states only possess local two-body entanglement, the resulting state can contain very complicated multi-body entanglement after the projective measurements. In adversary, the tensor network representation effectively resolves the non-local multi-body entanglement structures in the many-body state as local pairwise network connectivity at the price of introducing auxiliary tensors in the bulk of the network. The emergent bulk and its network geometry correspond to the holographic spacial geometry dual to the quantum many-body state.</p>
<p>Different bulk geometry (tensor network connection) typically leads to different entanglement features in the tensor network state. So given the entanglement features of a many-body state, can we determine the optimal tensor network connectivity that best reproduces the given entanglement features? This smells similar to the problem of training a neural network. Indeed, in this paper (<a href="https://arxiv.org/abs/1709.01223">arXiv:1709.01223</a>), we show that optimizing the connectivity of a random tensor network to fit the entanglement features can be mapped to training a deep Boltzmann machine. Using machine learning techniques, we demonstrate how the emergent holographic bulk grows deeper and deeper as we push the boundary quantum field theory towards the conformal limit. As we serve the entanglement “big-data” to the machine, the machine develops a neural network to capture the features of the data, and as the training goes on, the holographic spatial geometry emerges in the bulk as the network geometry.</p>The holographic duality (AdS/CFT) was originally proposed as the correspondence between a \(d\)-dimensional quantum field theory and a \((d+1)\)-dimensional quantum gravity theory. The information about the space-time geometry in the higher-dimensional holographic bulk is encoded in the quantum many-body dynamics on the lower-dimensional holographic boundary, hence the name “holography”. The holographic duality reveals a deep connection between quantum entanglement and space-time geometry: “Entanglement is the fabric of space-time,” said Brain Swingle in Quantum Magazine.