Bosonic entanglement renormalization circuits from wavelet theory
TL;DR: This work shows how to construct Gaussian bosonic quantum circuits that implement entanglement renormalization for ground states of arbitrary free bosonic chains and explains how the continuum limit emerges naturally from the wavelet construction.
Abstract: Entanglement renormalization is a unitary real-space renormalization scheme. The corresponding quantum circuits or tensor networks are known as MERA, and they are particularly well-suited to describing quantum systems at criticality. In this work we show how to construct Gaussian bosonic quantum circuits that implement entanglement renormalization for ground states of arbitrary free bosonic chains. The construction is based on wavelet theory, and the dispersion relation of the Hamiltonian is translated into a filter design problem. We give a general algorithm that approximately solves this design problem and provide an approximation theory that relates the properties of the filters to the accuracy of the corresponding quantum circuits. Finally, we explain how the continuum limit (a free bosonic quantum field) emerges naturally from the wavelet construction.
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Posted Content•
TL;DR: In this paper, the ability of entanglement renormalization (ER) to generate a proper real-space group (RG) flow in extended quantum systems is analyzed in the setting of harmonic lattice systems in D=1 and D=2 spatial dimensions.
Abstract: The ability of entanglement renormalization (ER) to generate a proper real-space renormalization group (RG) flow in extended quantum systems is analysed in the setting of harmonic lattice systems in D=1 and D=2 spatial dimensions. A conceptual overview of the steps involved in momentum-space RG is provided and contrasted against the equivalent steps in the real-space setting. The real-space RG flow, as generated by ER, is compared against the exact results from momentum-space RG, including an investigation of a critical fixed point and the effect of relevant and irrelevant perturbations.
28 citations
14 Mar 2022
TL;DR: In this paper , a summary of recent progress and remaining challenges in applying the methods and ideas of quantum information theory to the study of quantum field theory and quantum gravity is presented.
Abstract: : We present a summary of recent progress and remaining challenges in applying the methods and ideas of quantum information theory to the study of quantum field theory and quantum gravity. Important topics and themes include: entanglement entropy in QFTs and what it reveals about RG flows, symmetries, and phases; scrambling, information spreading, and chaos; state preparation and complexity; classical and quantum simulation of QFTs; and the role of information in holographic dualities. We also highlight the ways in which quantum information science benefits from the synergy between the fields.
26 citations
TL;DR: This work uses multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way, which could be a starting point for constructing quantum circuit approximations for more general conformal field theories.
Abstract: The multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories.
16 citations
TL;DR: A rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras is presented, and it is shown that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field.
Abstract: We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies' wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies' scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations.
13 citations
DOI•
17 Jan 2022
TL;DR: In this paper , a multi-scale representation of free scalar bosonic and Ising model fermionic QFTs using wavelets is presented, making use of the orthogonality and self similarity of the wavelet basis functions.
Abstract: Quantum field theory (QFT) describes nature using continuous fields, but physical properties of QFT are usually revealed in terms of measurements of observables at a finite resolution. We describe a multi-scale representation of free scalar bosonic and Ising model fermionic QFTs using wavelets. Making use of the orthogonality and self similarity of the wavelet basis functions, we demonstrate some well known relations such as scale dependent subsystem entanglement entropy and renormalisation of correlations in the ground state. We also find some new applications of the wavelet transform as a compressed representation of ground states of QFTs which can be used to illustrate quantum phase transitions via fidelity overlap and holographic entanglement of purification.
4 citations
References
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Book•
01 Jan 1998
TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
17,693 citations
"Bosonic entanglement renormalizatio..." refers background in this paper
...(11) it holds that 1 √ 2 φ(x2 ) = ∑ n hs[n]φ h(x− n) [14]....
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...The continuous wavelet transform [14] can be defined for a much broader class of wavelet functions ψ, and if ψ is a biorthogonal wavelet the CWT can be discretized to a discrete wavelet transform....
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...Here we give a brief account of the theory of biorthogonal wavelet filters, see [14] for an introduction....
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...A vanishing moment of the scaling function φh corresponds to a factor (1 + eik) in the scaling filter hs [14]....
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TL;DR: A generalization of the numerical renormalization-group procedure used first by Wilson for the Kondo problem is presented and it is shown that this formulation is optimal in a certain sense.
Abstract: A generalization of the numerical renormalization-group procedure used first by Wilson for the Kondo problem is presented. It is shown that this formulation is optimal in a certain sense. As a demonstration of the effectiveness of this approach, results from numerical real-space renormalization-group calculations for Heisenberg chains are presented.
5,625 citations
"Bosonic entanglement renormalizatio..." refers background in this paper
...(a1 ) −1 = ( hs[0] hw[0] hs[1] hw[1] )...
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...In one spatial dimensional, prominent examples are the Density Matrix Matrix Renormalization Group [1], with the associated tensor network class of Matrix Product States (MPS) [2] and entanglement renormalization [3], with the corresponding Multiscale Entanglement Renormalization Ansatz (MERA) tensor network states [3]....
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...a1 = ( gs[0] gw[0] gs[1] gw[1] ) ....
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TL;DR: It is shown that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors and are robust against errors from photon loss and detector inefficiency.
Abstract: Quantum computers promise to increase greatly the efficiency of solving problems such as factoring large integers, combinatorial optimization and quantum physics simulation. One of the greatest challenges now is to implement the basic quantum-computational elements in a physical system and to demonstrate that they can be reliably and scalably controlled. One of the earliest proposals for quantum computation is based on implementing a quantum bit with two optical modes containing one photon. The proposal is appealing because of the ease with which photon interference can be observed. Until now, it suffered from the requirement for non-linear couplings between optical modes containing few photons. Here we show that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors. Our methods exploit feedback from photo-detectors and are robust against errors from photon loss and detector inefficiency. The basic elements are accessible to experimental investigation with current technology.
5,236 citations
"Bosonic entanglement renormalizatio..." refers background in this paper
...For details about quadratic bosonic Hamiltonians and Gaussian states see, for instance, [11, 15–17]....
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...A natural application of a quantum computer based on bosonic variables [11] is to simulate bosonic quantum field theories [12] and wavelets are a very efficient choice of basis to discretize a quantum field theory for this purpose [13]....
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TL;DR: Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future as mentioned in this paper, which will be useful tools for exploring many-body quantum physics, and may have other useful applications.
Abstract: Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away --- we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.
3,898 citations
Book•
01 Jan 1996
3,808 citations
"Bosonic entanglement renormalizatio..." refers background in this paper
...If we take x = 0 we can nevertheless define φh,l(n) for integer n, see Chapters 6 and 7 in [20] for technical details, so we get in any case K descendants in the spectrum....
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