DMRG Preprints

The realization of fault-tolerant quantum computers remains a challenging endeavor, forcing state-of-the-art quantum hardware to rely heavily on noise mitigation techniques. Standard quantum error mitigation is typically based on post-processing strategies. In contrast, the present work explores a pre-processing approach, in which the effects of noise are mitigated before performing a measurement on the output state. The main idea is to find an observable $Y$ such that its expectation value on a noisy quantum state $\mathcal{E(ρ)}$ matches the expectation value of a target observable $X$ on the noiseless quantum state $ρ$. Our method requires the execution of a noisy quantum circuit, followed by the measurement of the surrogate observable $Y$. The main enablers of our method in practical scenarios are Tensor Networks. The proposed method improves over Tensor Error Mitigation (TEM) in terms of average error, circuit depth, and complexity, attaining a measurement overhead that approaches the theoretical lower bound. The improvement in terms of classical computation complexity is in the order of $\sim 10^6$ times when compared to the post-processing computational cost of TEM in practical scenarios. Such gain comes from eliminating the need to perform the set of informationally complete positive operator-valued measurements (IC-POVM) required by TEM, as well as any other tomographic strategy.

Frustration on the triangular lattice has long been a source of intriguing and often debated phases in many-body systems. Although symmetry analysis has been employed, the role of the seemingly trivial parity symmetry has received little attention. In this work, we show that phases induced by frustration are systematically shaped by an implicit rule of thumb associated with spontaneous parity breaking. This principle enables us to anticipate and rationalize the regimes and conditions under which nontrivial phases emerge. For the spin-$S$ antiferromagnetic XXZ model, we demonstrate that a controversial parity-broken phase appears only at intermediate values of $S$. In bilayer systems, enhanced frustration leads to additional phases, such as supersolids, whose properties can be classified by their characteristic parity features. Benefiting from our improved tensor network contraction techniques, we confirm these results through large-scale tensor-network calculations. This study offers an alternative viewpoint and a systematic approach for examining the interplay between spin, symmetry, and frustration in many-body systems.

Improving the computational efficiency of quantum many-body calculations from a hardware perspective remains a critical challenge. Although field-programmable gate arrays (FPGAs) have recently been exploited to improve the computational scaling of algorithms such as Monte Carlo methods, their application to tensor network algorithms is still at an early stage. In this work, we propose a fine-grained parallel tensor network design based on FPGAs to substantially enhance the computational efficiency of two representative tensor network algorithms: the infinite time-evolving block decimation (iTEBD) and the higher-order tensor renormalization group (HOTRG). By employing a quad-tile partitioning strategy to decompose tensor elements and map them onto hardware circuits, our approach effectively translates algorithmic computational complexity into scalable hardware resource utilization, enabling an extremely high degree of parallelism on FPGAs. Compared with conventional CPU-based implementations, our scheme exhibits superior scalability in computation time, reducing the bond-dimension scaling of the computational cost from $O(D_b^3)$ to $O(D_b)$ for iTEBD and from $O(D_b^6)$ to $O(D_b^2)$ for HOTRG. This work provides a theoretical foundation for future hardware implementations of large-scale tensor network computations.

This report documents the program of the second Toulouse Tensor Workshop which took place at the University of Toulouse on September 17-19, 2025, and summarizes the main points of discussion. This workshop follows the first Workshop (CECAM workshop on Tensor Contraction Library Standardization), which took place in Toulouse one year earlier, on May 24-25, 2024 and led to the formation of a tensor standardization working group, which has since specified a low-level standard interface for tensor operations available freely on GitHub. The 2025 workshop brought together developers of applications which rely extensively on tensor computations such as quantum many-body simulations in chemistry and physics (material science and electronic structure calculations), as well as developers and experts of tensor software who have the know-how to provide the technical support for such applications. The workshop enabled the community to provide feedback on the specified low-level interface and how it can be further refined. It also initiated a discussion on how the standardization efforts should be oriented in the near feature, in particular on what should be higher-level interfaces and how to tackle other requirements of the community such as tensor decompositions, symmetric tensors and structured sparsity support.

Characterizing the ground-state properties of disordered systems, such as spin glasses and combinatorial optimization problems, is fundamental to science and engineering. However, computing exact ground states and counting their degeneracies are generally NP-hard and #P-hard problems, respectively, posing a formidable challenge for exact algorithms. Recently, Tensor Networks methods, which utilize high-dimensional linear algebra and achieve massive hardware parallelization, have emerged as a rapidly developing paradigm for efficiently solving these tasks. Despite their success, these methods are fundamentally constrained by the exponential growth of space complexity, which severely limits their scalability. To address this bottleneck, we introduce the Branch-and-Bound Tensor Network (BBTN) method, which seamlessly integrates the adaptive search framework of branch-and-bound with the efficient contraction of tropical tensor networks, significantly extending the reach of exact algorithms. We show that BBTN significantly surpasses existing state-of-the-art solvers, setting new benchmarks for exact computation. It pushes the boundaries of tractability to previously unreachable scales, enabling exact ground-state counting for $\pm J$ spin glasses up to $64 \times 64$ and solving Maximum Independent Set problems on King's subgraphs up to $100 \times 100$. For hard instances, BBTN dramatically reduces the computational cost of standard Tropical Tensor Networks, compressing years of runtime into minutes. Furthermore, it outperforms leading integer-programming solvers by over 30$\times$, establishing a versatile and scalable framework for solving hard problems in statistical physics and combinatorial optimization.

We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.

Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.

Cold atoms have become a powerful platform for quantum-simulating lattice gauge theories in higher spatial dimensions. However, such realizations have been restricted to the lowest possible truncations of the gauge field, which limit the connections one can make to lattice quantum electrodynamics. Here, we propose a feasible cold-atom quantum simulator of a $(2+1)$-dimensional U$(1)$ lattice gauge theory in a spin $S=1$ truncation, featuring dynamical matter and gauge fields. We derive a mapping of this theory onto a bosonic computational basis, stabilized by an emergent gauge-protection mechanism through quantum Zeno dynamics. The implementation is based on a single-species Bose--Hubbard model realized in a tilted optical superlattice. This approach requires only moderate experimental resources already available in current ultracold-atom platforms. Using infinite matrix product state simulations, we benchmark real-time dynamics under global quenches. The results demonstrate faithful evolution of the target gauge theory and robust preservation of the gauge constraints. Our work significantly advances the experimental prospects for simulating higher-dimensional lattice gauge theories using larger gauge-field truncations.

We report the discovery of a generalized Luther-Emery liquid phase characterized by incommensurate pair-density-wave (iC-PDW) correlations in the two-leg $t$-$J$-$J_\perp$ ladder model. By tuning the potential difference between the legs, we explore the regime of intermediate layer polarization $P$. Combining density-matrix renormalization group (DMRG) simulations with bosonization analysis, we identify a spin-gapped phase at finite $P$, where the interlayer and intralayer pair correlations both oscillate, but with distinct periodicities. The interlayer correlations exhibit FFLO-like oscillations, driven by pairing between layers with mismatched Fermi momenta, with a period determined by their momentum difference. In contrast, the intralayer pair correlations arise from the coupling between charges on one layer and spin fluctuations on the opposite layer, with a momentum equal to twice the Fermi momentum of the opposite layer. The iC-PDW state is robust across a wide range of doping and polarization, although finite interlayer hopping eventually destabilizes it toward a state with charge-$4e$ correlations. We conclude by discussing the experimental realization of this model in optical lattice platforms and its relevance to the bilayer nickelate La$_3$Ni$_2$O$_7$.

Variational quantum algorithms are practical approaches to prepare ground states, but their potential for quantum advantage remains unclear. Here, we use differentiable 2D tensor networks (TN) to optimize parameterized quantum circuits that prepare the ground state of the transverse field Ising model (TFIM). Our method enables the preparation of states with high energy accuracy, even for large systems beyond 1D. We show that TN pre-optimization can mitigate the barren plateau issue by giving access to enhanced gradient zones that do not shrink exponentially with system size. We evaluate the classical simulation cost evaluating energies at these warm-starts, and identify regimes where quantum hardware offers better scaling than TN simulations.