arXiv subjects

Christian B. Mendl

Publications and source records attributed to Christian B. Mendl.

At least 37 records · Page 2Linked to original sources

arXiv2026

Hessian-vector products for tensor networks via recursive tangent-state propagation

Optimizing tensor networks with standard first-order methods often leads to slow convergence and entrapment in local minima. Although second-order optimization offers enhanced robustness, explicitly constructing the full Hessian matrix is computationally prohibitive for large-scale systems. In this work, we bypass this bottleneck by introducing an analytical Hessian-vector product kernel designed for arbitrary compositions of linear maps. This two-pass algorithm leverages recursive tangent-state propagation with a bounded virtual bond dimension to guarantee scalability. We demonstrate the practical utility of this kernel by integrating it into a Riemannian trust-region framework for quantum circuit compression. Evaluated on time-evolution circuits for various spin chains, our second-order approach achieves up to a four-order-of-magnitude improvement in fidelity over naive Trotterization, while delivering significantly smoother, faster convergence than conventional first-order methods such as Riemannian ADAM.

quant-ph↗

arXiv2025

Belief propagation for general graphical models with loops

There is an increasing interest in scaling tensor network methods through belief propagation (BP), as well as increasing the accuracy of BP through tensor network methods. We develop a unification framework that takes an arbitrary graphical model with loops and provides message passing update rules and inference equations. We show that recent state-of-the-art methods regarding tensors and BP, like block belief propagation and tensor network message passing, are special instances of our framework. From a practical perspective, we discuss how our framework can be useful to understand the benefits of scheduling in BP, and show how it can be used for decoding purposes in quantum error correction. We simulate the computation of marginals, internal energy, Shannon entropy and the partition function on synthetic topologies (Kagome lattice and lattices resembling quantum error-correcting codes) and a real world topology of a power grid. The results show orders of magnitude accuracy increases for modest computational overheads. For the marginals, for example, we show that our framework can achieve an accuracy improvement of more than six orders of magnitude over tensor network BP.

quant-ph↗

arXiv2025

Protein Folding with Neural Ordinary Differential Equations

Recent advances in protein structure prediction, such as AlphaFold, have demonstrated the power of deep neural architectures like the Evoformer for capturing complex spatial and evolutionary constraints on protein conformation. However, the depth of the Evoformer, comprising 48 stacked blocks, introduces high computational costs and rigid layerwise discretization. Inspired by Neural Ordinary Differential Equations (Neural ODEs), we propose a continuous-depth formulation of the Evoformer, replacing its 48 discrete blocks with a Neural ODE parameterization that preserves its core attention-based operations. This continuous-time Evoformer achieves constant memory cost (in depth) via the adjoint method, while allowing a principled trade-off between runtime and accuracy through adaptive ODE solvers. Benchmarking on protein structure prediction tasks, we find that the Neural ODE-based Evoformer produces structurally plausible predictions and reliably captures certain secondary structure elements, such as alpha-helices, though it does not fully replicate the accuracy of the original architecture. However, our model achieves this performance using dramatically fewer resources, just 17.5 hours of training on a single GPU, highlighting the promise of continuous-depth models as a lightweight and interpretable alternative for biomolecular modeling. This work opens new directions for efficient and adaptive protein structure prediction frameworks.

cs.LG↗

arXiv2025

Wavefunction optimization at the complete basis set limit with Multiwavelets and DMRG

The density matrix renormalization group (DMRG) is a powerful numerical technique to solve strongly correlated quantum systems: it deals well with systems which are not dominated by a single configuration (unlike Coupled Cluster) and it converges rapidly to the Full Configuration Interaction (FCI) limit (unlike truncated Configuration Interaction (CI) expansions). In this work, we develop an algorithm integrating DMRG within the multiwavelet-based multiresolution analysis (MRA). Unlike fixed basis sets, multiwavelets offer an adaptive and hierarchical representation of functions, approaching the complete basis set limit to a specified precision. As a result, this combined technique leverages the multireference capability of DMRG and the complete basis set limit of MRA and multiwavelets. More specifically, we adopt a pre-existing Lagrangian optimization algorithm for orbitals represented in the MRA domain and improve its computational efficiency by replacing the original CI calculations with DMRG. Additionally, we substitute the reduced density matrices computation with the direct extraction of energy gradients from the DMRG tensors. We apply our method to small systems such H2, He, HeH2, BeH2 and N2. The results demonstrate that our approach reduces the final energy while keeping the number of orbitals low compared to FCI calculations on an atomic orbital basis set.

physics.chem-ph↗

arXiv2025

Enhanced Krylov Methods for Molecular Hamiltonians: Reduced Memory Cost and Complexity Scaling via Tensor Hypercontraction

We introduce an algorithm that is simultaneously memory-efficient and low-scaling for applying ab initio molecular Hamiltonians to matrix-product states (MPS) via the tensor-hypercontraction (THC) format. These gains carry over to Krylov subspace methods, which can find low-lying eigenstates and simulate quantum time evolution while avoiding local minima and maintaining high accuracy. In our approach, the molecular Hamiltonian is represented as a sum of products of four MPOs, each with a bond dimension of only 2. Iteratively applying the MPOs to the current quantum state in MPS form, summing and re-compressing the MPS leads to a scheme with the same asymptotic memory cost as the bare MPS and reduces the computational cost scaling compared to the Krylov method using a conventional MPO construction. We provide a detailed theoretical derivation of these statements and conduct supporting numerical experiments to demonstrate the advantage. Our algorithm is highly parallelizable and thus lends itself to large-scale HPC simulations.

cond-mat.str-el↗

arXiv2025

Quantum-Guided Cluster Algorithms for Combinatorial Optimization

Finding the ground state of Ising spin glasses is notoriously difficult due to disorder and frustration. Often, this challenge is framed as a combinatorial optimization problem, for which a common strategy employs simulated annealing, a Monte Carlo (MC)-based algorithm that updates spins one at a time. Yet, these localized updates can cause the system to become trapped in local minima. Cluster algorithms (CAs) were developed to address this limitation and have demonstrated considerable success in studying ferromagnetic systems; however, they tend to encounter percolation issues when applied to generic spin glasses. In this work, we introduce a novel CA designed to tackle these challenges by leveraging precomputed two-point correlations, aiming solve combinatorial optimization problems in the form of Max-Cut more efficiently. In our approach, clusters are formed probabilistically based on these correlations. Various classical and quantum algorithms can be employed to generate correlations that embody information about the energy landscape of the problem. By utilizing this information, the algorithm aims to identify groups of spins whose simultaneous flipping induces large transitions in configuration space with high acceptance probability -- even at low energy levels -- thereby escaping local minima more effectively. Notably, clusters generated using correlations from the Quantum Approximate Optimization Algorithm exhibit high acceptance rates at low temperatures. These acceptance rates often increase with circuit depth, accelerating the algorithm and enabling more efficient exploration of the solution space.

quant-ph↗

arXiv2025

A Riemannian Approach to the Lindbladian Dynamics of a Locally Purified Tensor Network

Tensor networks offer a valuable framework for implementing Lindbladian dynamics in many-body open quantum systems with nearest-neighbor couplings. In particular, a tensor network ansatz known as the Locally Purified Density Operator employs the local purification of the density matrix to guarantee the positivity of the state at all times. Within this framework, the dissipative evolution utilizes the Trotter-Suzuki splitting, yielding a second-order approximation error. However, due to the Lindbladian dynamics' nature, employing higher-order schemes results in non-physical quantum channels. In this work, we leverage the gauge freedom inherent in the Kraus representation of quantum channels to improve the splitting error. To this end, we formulate an optimization problem on the Riemannian manifold of isometries and find a solution via the second-order trust-region algorithm. We validate our approach using two nearest-neighbor noise models and achieve an improvement of orders of magnitude compared to other positivity-preserving schemes. In addition, we demonstrate the usefulness of our method as a compression scheme, helping to control the exponential growth of computational resources, which thus far has limited the use of the locally purified ansatz.

quant-ph↗

arXiv2025

Quantum circuit simulation with a local time-dependent variational principle

Classical simulations of quantum circuits are vital for assessing potential quantum advantage and benchmarking devices, yet they require sophisticated methods to avoid the exponential growth of resources. Tensor network approaches, in particular matrix product states (MPS) combined with the time-evolving block decimation (TEBD) algorithm, currently dominate large-scale circuit simulations. These methods scale efficiently when entanglement is limited but suffer rapid bond dimension growth with increasing entanglement and handle long-range gates via costly SWAP insertions. Motivated by the success of the time-dependent variational principle (TDVP) in many-body physics, we reinterpret quantum circuits as a series of discrete time evolutions, using gate generators to construct an MPS-based circuit simulation via a local TDVP formulation. This addresses TEBD's key limitations by (1) naturally accommodating long-range gates and (2) optimally representing states on the MPS manifold. By diffusing entanglement more globally, the method suppresses local bond growth and reduces memory and runtime costs. We benchmark the approach on five 49-qubit circuits: three Hamiltonian circuits (1D open and periodic Heisenberg, 2D 7x7 Ising) and two algorithmic ones (quantum approximate optimization, hardware-efficient ansatz). Across all cases, our method yields substantial resource reductions over standard tools, establishing a new state-of-the-art for circuit simulation and enabling advances across quantum computing, condensed matter, and beyond.

quant-ph↗

arXiv2025

Riemannian quantum circuit optimization based on matrix product operators

We significantly enhance the simulation accuracy of initial Trotter circuits for Hamiltonian simulation of quantum systems by integrating first-order Riemannian optimization with tensor network methods. Unlike previous approaches, our method imposes no symmetry assumptions, such as translational invariance, on the quantum systems. This technique is scalable to large systems through the use of a matrix product operator representation of the reference time evolution propagator. Our optimization routine is applied to various spin chains and fermionic systems described by the transverse-field Ising Hamiltonian, the Heisenberg Hamiltonian, and the spinful Fermi-Hubbard Hamiltonian. In these cases, our approach achieves a relative error improvement of up to four orders of magnitude for systems of 50 qubits, although our method is also applicable to larger systems. Furthermore, we demonstrate the versatility of our method by applying it to molecular systems, specifically lithium hydride, achieving an error improvement of up to eight orders of magnitude. This proof of concept highlights the potential of our approach for broader applications in quantum simulations.

quant-ph↗

arXiv2025

Optimizing Tensor Network Partitioning using Simulated Annealing

Tensor networks have proven to be a valuable tool, for instance, in the classical simulation of (strongly correlated) quantum systems. As the size of the systems increases, contracting larger tensor networks becomes computationally demanding. In this work, we study distributed memory architectures intended for high-performance computing implementations to solve this task. Efficiently distributing the contraction task across multiple nodes is critical, as both computational and memory costs are highly sensitive to the chosen partitioning strategy. While prior work has employed general-purpose hypergraph partitioning algorithms, these approaches often overlook the specific structure and cost characteristics of tensor network contractions. We introduce a simulated annealing-based method that iteratively refines the partitioning to minimize the total operation count, thereby reducing time-to-solution. The algorithm is evaluated on MQT Bench circuits and achieves an 8$\times$ average reduction in computational cost and an 8$\times$ average reduction in memory cost compared to a naive partitioning.

quant-ph↗

arXiv2025

Quantum Entrepreneurship Lab: Training a Future Workforce for the Quantum Industry

The Quantum Entrepreneurship Lab (QEL) is a one-semester, project-based course at the Technical University of Munich (TUM), designed to bridge the gap between academic research and industrial application in the quantum sector. As part of the Munich Quantum Valley (MQV) ecosystem, the course fosters interdisciplinary collaboration between technical and business students, equipping them with the skills necessary to contribute to or lead in the emerging quantum industry. The QEL curriculum integrates two complementary tracks. First, technical students form teams where they engage in cutting-edge, industry-relevant research topics under academic supervision. Meanwhile business students in a parallel course explore commercialization strategies, risks, and opportunities within the quantum technology landscape. Midway through the semester, a selection of the business students join the technical course to form interdisciplinary teams which assess the feasibility of transforming scientific concepts into viable business solutions. The course culminates in three key deliverables: a publication-style technical report, a white paper analyzing the business potential and financial requirements, and a startup pitch presented to the quantum community at a Demo Day. This work outlines the course structure, objectives, and outcomes, providing a model for other institutions seeking to cultivate a highly skilled, innovation-driven workforce in quantum science and technology.

physics.ed-ph↗

arXiv2025

Large-scale stochastic simulation of open quantum systems

Understanding the precise interaction mechanisms between quantum systems and their environment is crucial for advancing stable quantum technologies, designing reliable experimental frameworks, and building accurate models of real-world phenomena. However, simulating open quantum systems, which feature complex non-unitary dynamics, poses significant computational challenges that require innovative methods to overcome. In this work, we introduce the tensor jump method (TJM), a scalable, embarrassingly parallel algorithm for stochastically simulating large-scale open quantum systems, specifically Markovian dynamics captured by Lindbladians. This method is built on three core principles where, in particular, we extend the Monte Carlo wave function (MCWF) method to matrix product states, use a dynamic time-dependent variational principle (TDVP) to significantly reduce errors during time evolution, and introduce what we call a sampling MPS to drastically reduce the dependence on the simulation's time step size. We demonstrate that this method scales more effectively than previous methods and ensures convergence to the Lindbladian solution independent of system size, which we show both rigorously and numerically. Finally, we provide evidence of its utility by simulating Lindbladian dynamics of XXX Heisenberg models up to a thousand spins using a consumer-grade CPU. This work represents a significant step forward in the simulation of large-scale open quantum systems, with the potential to enable discoveries across various domains of quantum physics, particularly those where the environment plays a fundamental role, and to both dequantize and facilitate the development of more stable quantum hardware.

quant-ph↗

arXiv2025

Optimal Symbolic Construction of Matrix Product Operators and Tree Tensor Network Operators

This research introduces an improved framework for constructing matrix product operators (MPOs) and tree tensor network operators (TTNOs), crucial tools in quantum simulations. A given (Hamiltonian) operator typically has a known symbolic "sum of operator strings" form that can be translated into a tensor network structure. Combining the existing bipartite-graph-based approach and a newly introduced symbolic Gaussian elimination preprocessing step, our proposed method improves upon earlier algorithms in cases when Hamiltonian terms share the same prefactors. We test the performance of our method against established ones for benchmarking purposes. Finally, we apply our methodology to the model of a cavity filled with molecules in a solvent. This open quantum system is cast into the hierarchical equation of motion (HEOM) setting to obtain an effective Hamiltonian. Construction of the corresponding TTNO demonstrates a sub-linear increase of the maximum bond dimension.

quant-ph↗

arXiv2025

High-Performance Contraction of Quantum Circuits for Riemannian Optimization

This work focuses on optimizing the gates of a quantum circuit with a given topology to approximate the unitary time evolution governed by a Hamiltonian. Recognizing that unitary matrices form a mathematical manifold, we employ Riemannian optimization methods -- specifically the Riemannian trust-region algorithm -- which involves second derivative calculations with respect to the gates. Our key technical contribution is a matrix-free algorithmic framework that avoids the explicit construction and storage of large unitary matrices acting on the whole Hilbert space. Instead, we evaluate all quantities as sums over state vectors, assuming that these vectors can be stored in memory. We develop HPC-optimized kernels for applying gates to state vectors and for the gradient and Hessian computation. Further improvements are achieved by exploiting sparsity structures due to Hamiltonian conservation laws, such as parity conservation, and lattice translation invariance. We benchmark our implementation on the Fermi-Hubbard model with up to 16 sites, demonstrating a nearly linear parallelization speed-up with up to 112 CPU threads. Finally, we compare our implementation with an alternative matrix product operator-based approach.

quant-ph↗

arXiv2025

Optimization Framework for Reducing Mid-circuit Measurements and Resets

The paper addresses the optimization of dynamic circuits in quantum computing, with a focus on reducing the cost of mid-circuit measurements and resets. We extend the probabilistic circuit model (PCM) and implement an optimization framework that targets both mid-circuit measurements and resets. To overcome the limitation of the prior PCM-based pass, where optimizations are only possible on pure single-qubit states, we incorporate circuit synthesis to enable optimizations on multi-qubit states. With a parameter $n_{pcm}$, our framework balances optimization level against resource usage.We evaluate our framework using a large dataset of randomly generated dynamic circuits. Experimental results demonstrate that our method is highly effective in reducing mid-circuit measurements and resets. In our demonstrative example, when applying our optimization framework to the Bernstein-Vazirani algorithm after employing qubit reuse, we significantly reduce its runtime overhead by removing all of the resets.

quant-ph↗

arXiv2025

Dead Gate Elimination

Hybrid quantum algorithms combine the strengths of quantum and classical computing. Many quantum algorithms, such as the variational quantum eigensolver (VQE), leverage this synergy. However, quantum circuits are executed in full, even when only subsets of measurement outcomes contribute to subsequent classical computations. In this manuscript, we propose a novel circuit optimization technique that identifies and removes dead gates. We prove that the removal of dead gates has no influence on the probability distribution of the measurement outcomes that contribute to the subsequent calculation result. We implemented and evaluated our optimization on a VQE instance, a quantum phase estimation (QPE) instance, and hybrid programs embedded with random circuits of varying circuit width, confirming its capability to remove a non-trivial number of dead gates in real-world algorithms. The effect of our optimization scales up as more measurement outcomes are identified as non-contributory, resulting in a proportionally greater reduction of dead gates.

quant-ph↗

arXiv2025

Quantum Dissipative Search via Lindbladians

Closed quantum systems follow a unitary time evolution that can be simulated on quantum computers. By incorporating non-unitary effects via, e.g., measurements on ancilla qubits, these algorithms can be extended to open-system dynamics, such as Markovian processes described by the Lindblad master equation. In this paper, we analyze the convergence criteria and speed of a Markovian, purely dissipative quantum random walk on an unstructured classical search space. Notably, we show that certain jump operators make the quantum process replicate a classical one, while others yield differences between open quantum (OQRW) and classical random walks. We also clarify a previously observed quadratic speedup, demonstrating that OQRWs are no more efficient than classical search. Finally, we analyze a dissipative discrete-time ground-state preparation algorithm with a lower implementation cost. This allows us to interpolate between the dissipative and the unitary domain and thereby illustrate the important role of coherence for the quadratic speedup.

quant-ph↗

arXiv2025

Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering

Hamiltonian simulation is a domain where quantum computers have the potential to outperform their classical counterparts. One of the main challenges of such quantum algorithms is increasing the system size, which is necessary to achieve meaningful quantum advantage. In this work, we present an approach to improve the scalability of eigenspace filtering for the ground state preparation of a given Hamiltonian. Our method aims to tackle limitations introduced by a small spectral gap and high degeneracy of low energy states. It is based on an adaptive sequence of eigenspace filtering through Quantum Eigenvalue Transformation of Unitary Matrices (QETU) combined with spectrum profiling. By combining our proposed algorithm with state-of-the-art phase estimation methods, we achieved good approximations for the ground state energy with local, two-qubit gate depolarizing probability up to $10^{-4}$. To demonstrate the key results in this work, we ran simulations with the transverse-field Ising Model on classical computers using Qiskit. We compare the performance of our approach with the static implementation of QETU and show that we can consistently achieve three to four orders of magnitude improvement in the absolute error rate.

quant-ph↗