The investigation of strongly-correlated quantum matter is difficult due to the curse of dimensionality and intricate entanglement structures. These challenges are particularly pronounced in the vicinity of continuous quantum phase transitions, where quantum fluctuations manifest across all length scales. While quantum simulators give controlled access to a number of strongly correlated systems, the study of critical phenomena has been hampered by finite-size effects arising from diverging correlation lengths. Moreover, the experimental investigation of entanglement in many-body systems has been hindered by limitations in measurement protocols. To address these challenges, we employ the multiscale entanglement renormalization ansatz (MERA) and implement a holographic scheme for subsystem tomography on a fully-connected trapped-ion quantum computer. Our method accurately represents infinite systems and long-range correlations with few qubits, facilitating the efficient extraction of observables and entanglement properties, even at criticality. We observe a quantum phase transition with spontaneous symmetry breaking and reveal the evolution of entanglement properties across the critical point. For the first time, we demonstrate log-law scaling of subsystem entanglement entropies at criticality on a digital quantum computer. This achievement highlights the potential of MERA for the investigation of strongly-correlated many-body systems on quantum computers.
The investigation of strongly-correlated quantum matter is difficult due to the curse of dimensionality and intricate entanglement structures. These challenges are particularly pronounced in the vicinity of continuous quantum phase transitions, where quantum fluctuations manifest across all length scales. While quantum simulators give controlled access to a number of strongly correlated systems, the study of critical phenomena has been hampered by finite-size effects arising from diverging correlation lengths. Moreover, the experimental investigation of entanglement in many-body systems has been hindered by limitations in measurement protocols. To address these challenges, we employ the multiscale entanglement renormalization ansatz (MERA) and implement a holographic scheme for subsystem tomography on a fully-connected trapped-ion quantum computer. Our method accurately represents infinite systems and long-range correlations with few qubits, facilitating the efficient extraction of observables and entanglement properties, even at criticality. We observe a quantum phase transition with spontaneous symmetry breaking and reveal the evolution of entanglement properties across the critical point. For the first time, we demonstrate log-law scaling of subsystem entanglement entropies at criticality on a digital quantum computer. This achievement highlights the potential of MERA for the investigation of strongly-correlated many-body systems on quantum computers.
While standard approaches to quantum simulation require a number of qubits proportional to the number of simulated particles, current noisy quantum computers are limited to tens of qubits.With the technique of holographic quantum simulation, a D-dimensional system can be simulated with a (D -1)-dimensional subset of qubits, enabling the study of systems significantly larger than current quantum computers.Using circuits derived from the multiscale entanglement renormalization ansatz (MERA), we accurately prepare the ground state of an L = 32 critical, nonintegrable perturbed Ising model and measure long-range correlations on the ten-qubit Quantinuum trapped-ion computer.We introduce generalized MERA networks that interpolate between MERA and matrix product state networks and demonstrate that generalized MERA can capture far longer correlations than a MERA with the same number of qubits, at the expense of greater circuit depth.Finally, we perform noisy simulations of these two network ansatzes and find that the optimal choice of network depends on the noise level, available qubits, and the state to be represented.
Simulating strongly correlated quantum systems in continuous space belongs to the most challenging and long-concerned issues in quantum physics. This work investigates the quantum entanglement and criticality of the ground-state wave functions of infinitely many coupled quantum oscillators (iCQOs). The essential task involves solving a set of partial differential equations (Schr\"odinger equations in the canonical quantization picture) with infinitely many variables, which currently lacks valid methods. By extending the imaginary-time evolution algorithm with translationally invariant functional tensor network, we simulate the ground state of iCQOs with the presence of two- and three-body couplings. We determine the range of coupling strengths where there exists a real ground-state energy (dubbed as physical region). With two-body couplings, we reveal the logarithmic scaling law of entanglement entropy (EE) and the polynomial scaling law of correlation length against the virtual bond dimension $\ensuremath{\chi}$ at the dividing point of physical and nonphysical regions. These two scaling behaviors are signatures of criticality, according to the previous results in quantum lattice models, but were not reported in continuous-space quantum systems. The scaling coefficients result in a central charge c=1, indicating the presence of free boson conformal field theory (CFT). We further show that the presence of three-body couplings, for which there are no analytical or numerical results, breaks down the CFT description at the dividing point. Our work uncovers the scaling behaviors of EE in the continuous-space quantum many-body systems. These results provide solid numerical evidence supporting the high efficiency of TN in representing the continuous-space quantum many-body wave functions in the thermodynamic limit, and meanwhile suggest an efficient approach to study the entanglement properties and criticality in continuous space.
Although quantum mechanics underpins the microscopic behavior of all materials, its effects are often obscured at the macroscopic level by thermal fluctuations. A notable exception is a zero-temperature phase transition, where scaling laws emerge entirely due to quantum correlations over a diverging length scale. The accurate description of such transitions is challenging for classical simulation methods of quantum systems, and is a natural application space for quantum simulation. These quantum simulations are, however, not without their own challenges-representing quantum critical states on a quantum computer requires encoding entanglement of a large number of degrees of freedom, placing strict demands on the coherence and fidelity of the computer's operations. Using Quantinuum's H1-1 quantum computer, we address these challenges by employing hierarchical quantum tensor-network techniques, creating the ground state of the critical transverse-field Ising chain on 128-sites with sufficient fidelity to extract accurate critical properties of the model. Our results suggest a viable path to quantum-assisted tensor-network contraction beyond the limits of classical methods.
The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions.
The multiscale entanglement renormalization ansatz (MERA) is a tensor network based variational ansatz that is capable of capturing many of the key physical properties of strongly correlated ground states such as criticality and topological order. MERA also shares many deep relationships with the AdS/CFT (gauge-gravity) correspondence by realizing a UV complete holographic duality within the tensor networks framework. Motivated by this, we have repurposed the MERA tensor network as an analysis tool to study the real-time evolution of the 1D transverse Ising model in its low-energy excited state sector. We performed this analysis by allowing the ancilla qubits of the MERA tensor network to acquire quantum fluctuations, which yields a unitary transform between the physical (boundary) and ancilla qubit (bulk) Hilbert spaces. This then defines a reversible quantum circuit, which is used as a ``holographic transform'' to study excited states and their real-time dynamics from the point of the bulk ancillae. In the gapped paramagnetic phase of the transverse field Ising model, we demonstrate the holographic duality between excited states induced by single spin-flips (Ising ``magnons'') acting on the ground state and single ancilla qubit spin flips. The single ancillae qubit excitation is shown to be stable in the bulk under real-time evolution and hence defines a stable holographic quasiparticle, which we have named the ``hologron.'' Their bulk 2D Hamiltonian, energy spectrum, and dynamics within the MERA network are studied numerically. The ``dictionary'' between the bulk and boundary is determined and realizes many features of the holographic correspondence in a non-CFT limit of the boundary theory. As an added spin-off, this dictionary together with the extension to multihologron sectors gives us a systematic way to construct quantitatively accurate low-energy effective Hamiltonians.
This thesis reviews the multiscale entanglement renormalisation ansatz or MERA, a numerical tool for the study of quantum many-body systems and a discrete realisation of the AdS/CFT duality. The thesis covers an introduction to the necessary background concepts of entanglement, entanglement entropy and tensor network states, the structure and main features of MERA and its applications in condensed matter theory and holography. Also covered are details on the algorithmic implementation of MERA and some of its generalisations and extensions. MERA belongs to a class of variational ansatze for quantum many-body states known as tensor network states. It is especially well-suited for the study of scale invariant critical points. MERA is based on a real-space renormalisation group procedure called entanglement renormalisation, designed to systematically handle entanglement at different length scales along the coarse-graining flow. Entanglement renormalisation has be used for example to efficiently describe Kitaev states of the toric code, the prime example of topological order, and numerically study the ground state of the highly frustrated spin-1/2 Heisenberg model on a kagome lattice and various other one- and two-dimensional lattice models. The geometric and causal structure of MERA, which underlies its effectiveness as a numerical tool, also makes it a discrete version of the AdS/CFT duality. This duality describes a conformal field theory by a gravity theory in a higher dimensional space, and vice versa. The duality is manifest in the scaling of entanglement entropy in MERA, which is governed by a law highly analogous to the Ryu-Takayanagi formula for holographic entanglement entropy, in the connection between thermal states and a black-hole-like MERA and in the connection between correlation functions and holographic geodesics in a scale invariant MERA. The aim of this thesis is to lead the reader to an understanding of what MERA is, how it works and how it can be used. MERA's core features and uses are presented in a comprehensive and explicit way, and a broad view of possible applications and further directions is given. Plenty of references are also offered to direct the reader to further research on how MERA may relate to his/her interests.