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arXiv:2404.08548 · Physical Review A

Entanglement determination with a deep multiscale entanglement renormalization ansatz

Kushagra Garg, Zeeshan Ahmed, Andreas M. D. Thomasen

Published Apr 7, 2025• 0 citations
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Abstract

We study the deep multiscale entanglement renormalization Ansatz (DMERA) on quantum hardware and the causal cone of a subset of the qubits, which make up the Ansatz. This causal cone spans $O(M+logN)$ physical qubits on a quantum device, where $M$ and $N$ are the subset size and the total number qubits in the Ansatz, respectively. This allows for the determination of the von Neumann entanglement entropy of the $N$-qubit wave function using $O(M+logN)$ qubits by diagonalization of the reduced density matrix (RDM). We show this by randomly initializing a 16-qubit DMERA and diagonalizing the resulting RDM of the $M$-qubit subsystem using density-matrix simulation. As an example of practical interest, we also encode the variational ground state of the long-range transverse field Ising model (LRTIM) on eight spins using DMERA. We perform density-matrix simulation with and without noise to obtain entanglement entropies in separate experiments using only four qubits. Finally, we repeat the experiment on the IBM Kyoto back end reproducing simulation results.

DOI
10.1103/physreva.111.042411

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We study the deep multiscale entanglement renormalization Ansatz (DMERA) on quantum hardware and the causal cone of a subset of the qubits, which make up the Ansatz. This causal cone spans O(M+logN)O(M+logN)O(M+logN) physical qubits on a quantum device, where MMM and NNN are the subset size and the total number qubits in the Ansatz, respectively. This allows for the determination of the von Neumann entanglement entropy of the NNN-qubit wave function using O(M+logN)O(M+logN)O(M+logN) qubits by diagonalization of the reduced density matrix (RDM). We show this by randomly initializing a 16-qubit DMERA and diagonalizing the resulting RDM of the MMM-qubit subsystem using density-matrix simulation. As an example of practical interest, we also encode the variational ground state of the long-range transverse field Ising model (LRTIM) on eight spins using DMERA. We perform density-matrix simulation with and without noise to obtain entanglement entropies in separate experiments using only four qubits. Finally, we repeat the experiment on the IBM Kyoto back end reproducing simulation results.
Many simulation tasks require that one first prepare a system's Gibbs state. We present a family of quantum circuits for variational preparation of thermal Gibbs states on a quantum computer; we call them the thermal multi-scale entanglement renormalization ansatz (TMERA). TMERA circuits transform input qubits to wavepacket modes localized to varying length scales and approximate a systems Gibbs state as a mixed state of these modes. The TMERA is a based on the deep multi-scale entanglement renormalization ansatz (DMERA); a TMERA modifies a ground-state DMERA circuit by preparing each input qubit as a mixed state. The excitation probabilities for input qubits serve as variational parameters used to target particular temperature Gibbs states. Since a TMERA is a special case of the product spectrum ansatz for thermal states, it is simple to prepare, analyze, and optimize. We benchmark the TMERA on the transverse field Ising model in one dimension and find that for D=6D=6D=6 it produces global fidelities F>0.4\mathcal F > 0.4F>0.4 for 512-site systems across all temperatures.
We perform a detailed resource estimate for the prospect of using deep entanglement renormalization ansatz (DMERA) on a fault-tolerant quantum computer, focusing on the regime in which the target system is large. For probing a relatively large system size (64×6464\times 6464×64), we observe up to an order of magnitude reduction in the number of qubits, compared to the approaches based on quantum phase estimation (QPE). We discuss two complementary strategies to measure the energy. The first approach is based on a random sampling of the local terms of the Hamiltonian, requiring O(1/ε2)\mathcal{O}(1/ε^2)O(1/ε2) invocations of quantum circuits, each of which have depth of at most O(log⁡N)\mathcal{O}(\log N)O(logN), where εεε is the relative precision in the energy and NNN is the system size. The second approach is based on a coherent estimation of the expectation value of observables averaged over space, which achieves the Heisenberg scaling while incurring only a logarithmic cost in the system size. For estimating the energy per site of εεε, O(log⁡Nε)\mathcal{O}\left(\frac{\log N}ε \right)O(εlogN​) TTT gates and O(log⁡N)\mathcal{O}\left(\log N \right)O(logN) qubits suffice. The constant factor of the leading contribution is shown to be determined by the depth of the DMERA circuit, the gates used in the ansatz, and the periodicity of the circuit. We also derive tight bounds on the variance of the energy gradient, assuming the gates are random Pauli rotations.
Here we investigate the use of deep multiscale entanglement renormalization ansatz (DMERA) circuits as a variational ansatz. We use the exactly solvable one-dimensional critical transverse-field Ising model as a test bed. Numerically exact simulation of the quantum circuit ansatz can in this case be carried out to hundreds of qubits by exploiting efficient classical algorithms for simulating matchgate circuits. We find that, for this system, the DMERA strongly outperforms a standard quantum approximate optimization algorithm (QAOA)--style ansatz, and that a major source of systematic error in correlation functions approximated using the DMERA is the breaking of the translational and Kramers-Wannier symmetries of the transverse-field Ising model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging, without incurring additional cost in qubits or circuit depth. We propose that this technique for mitigating systematic error could be applied to noisy intermediate-scale quantum (NISQ) simulations of physical systems with other symmetries.
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.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
Density matrix renormalization group (DMRG) and its extensions in the form of matrix product states are arguably the choice for the study of one-dimensional quantum systems in the last three decades. However, due to the limited entanglement encoded in the wave-function ansatz, to maintain the accuracy of DMRG with the increase of the system size in the study of two-dimensional systems, exponentially increased resources are required, which limits the applicability of DMRG to only narrow systems. We introduce a new ansatz in which DMRG is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in the new ansatz scales as l for an l × l system). In the new method, the O(D 3) low computational cost of DMRG is kept (with an overhead of O(d 4) and d the dimension of the physical degrees of freedom). We perform benchmark calculations with this approach on the two-dimensional transverse Ising and Heisenberg models. This new ansatz extends the power of DMRG in the study of two-dimensional quantum systems.