Recent advances in quantum simulator experiments enable unprecedented access to quantum many-body states through snapshot measurements of individual many-body configurations. Here, we introduce an exact renormalization group (RG) transformation that can be directly applied to any such snapshot dataset. Our SnapshotRG operates in real space, but can also be directly translated to an RG in the abstract dataspace of measurement configurations, providing a framework for the characterization of quantum many-body systems on a more general level. We demonstrate that snapshot datasets in dataspace exhibit self-similarity at continuous phase transitions, providing an explanation for the recently observed scale-freeness of so-called wavefunction networks. As a consequence, scale invariance extends beyond traditional low-order correlation functions to encompass the full statistical structure of quantum states as contained in their snapshot datasets. Our SnapshotRG can be readily implemented with snapshot data generated by numerical method such as neural quantum states or any quantum simulation platform, offering a versatile tool for characterizing quantum phase transitions and critical phenomena in quantum matter.
Recent advances in quantum simulator experiments enable unprecedented access to quantum many-body states through snapshot measurements of individual many-body configurations. Here, we introduce an exact renormalization group (RG) transformation that can be directly applied to any such snapshot dataset. Our SnapshotRG operates in real space, but can also be directly translated to an RG in the abstract dataspace of measurement configurations, providing a framework for the characterization of quantum many-body systems on a more general level. We demonstrate that snapshot datasets in dataspace exhibit self-similarity at continuous phase transitions, providing an explanation for the recently observed scale-freeness of so-called wavefunction networks. As a consequence, scale invariance extends beyond traditional low-order correlation functions to encompass the full statistical structure of quantum states as contained in their snapshot datasets. Our SnapshotRG can be readily implemented with snapshot data generated by numerical method such as neural quantum states or any quantum simulation platform, offering a versatile tool for characterizing quantum phase transitions and critical phenomena in quantum matter.
Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms. Published by the American Physical Society 2024
Quantum computers and simulators offer unparalleled capabilities of probing quantum many-body states, by obtaining snapshots of the many-body wave function via collective projective measurements. The probability distribution obtained by such snapshots (which are fundamentally limited to a negligible fraction of the Hilbert space) is of fundamental importance to determine the power of quantum computations. However, its relation to many-body collective properties is poorly understood. Here, we develop a theoretical framework to link quantum phases of matter to their snapshots, based on a combination of data complexity and network theory analyses. The first step in our scheme consists of applying Occam's razor principle to quantum sampling: given snapshots of a wave function, we identify a minimal-complexity measurement basis by analyzing the information compressibility of snapshots over different measurement bases. The second step consists of analyzing arbitrary correlations using network theory, building a wave-function network from the minimal-complexity basis data. This approach allows us to stochastically classify the output of quantum computers and simulations, with no assumptions on the underlying dynamics, and in a fully interpretable manner. We apply this method to quantum states of matter in one-dimensional translational invariant systems, where such classification is exhaustive, and where it reveals an interesting interplay between algorithmic and computational complexity for many-body states. Our framework is of immediate experimental relevance, and can be further extended both in terms of more advanced network mathematics, including discrete homology, as well as in terms of applications to physical phenomena, such as time-dependent dynamics and gauge theories.
Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with in situ access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms.
We propose a real-space renormalization group method to explicitly decouple into independent components a many-body system that, as in the phenomenon of spin-charge separation, exhibits separation of degrees of freedom at low energies. Our approach produces a branching holographic description of such systems that opens the path to the efficient simulation of the most entangled phases of quantum matter, such as those whose ground state violates a boundary law for entanglement entropy. As in the coarse-graining transformation of Vidal [Phys. Rev. Lett. 99, 220405 (2007)], the key ingredient of this decoupling transformation is the concept of entanglement renormalization, or removal of short-range entanglement. We demonstrate the feasibility of the approach, both analytically and numerically, by decoupling in real space the ground state of a critical quantum spin chain into two. Generalized notions of renormalization group flow and of scale invariance are also put forward.
Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. \n \nFirst, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. \n \nSecond, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. \n \nFinally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.
Abstract We develop a machine-learning renormalization group (MLRG) algorithm to explore and analyze many-body lattice models in statistical physics. Using the representation learning capability of generative modeling, MLRG automatically learns the optimal renormalization group (RG) transformations from self-generated spin configurations and formulates RG equations without human supervision. The algorithm does not focus on simulating any particular lattice model but broadly explores all possible models compatible with the internal and lattice symmetries given the on-site symmetry representation. It can uncover the RG monotone that governs the RG flow, assuming a strong form of the c-theorem. This enables several downstream tasks, including unsupervised classification of phases, automatic location of phase transitions or critical points, controlled estimation of critical exponents, and operator scaling dimensions. We demonstrate the MLRG method in two-dimensional lattice models with Ising symmetry and show that the algorithm correctly identifies and characterizes the Ising criticality.