Quantum simulation for chemistry and materials science
Simulating quantum systems is a key tool in chemistry and materials science, with applications in green energy and sustainability. We develop tensor network methods and quantum computing algorithms toward this goal. Achievements include orbital optimization using multiwavelets, methods for tree topologies and large-scale contraction, and optimized quantum circuits for time evolution.
Focus Group: Quantum Computing
Prof. Christian B. Mendl (TUM), Alumnus Rudolf Mößbauer Tenure Track Assistant Professor
image: TU Dresden
Sustainable and eco-friendly energy conversion, the exploration and optimization of chemical processes (such as nitrogen fixation and catalytic reactions), and improvements in materials science are urgent and relevant goals now and in the foreseeable future. A critical tool to make progress toward these goals is microscopic simulation, i.e., quantum simulations of atoms, electrons, and their interactions with light. This motivates the work of our group, which approaches this task using classical computational algorithms, tensor methods in particular, and quantum computing. In this context, tensor networks often emerge as an insightful mathematical framework for bridging quantum and classical computing.
Tensor network methods
Generalizing from vectors and matrices to higher-dimensional structures naturally leads to the concept of tensors. For example, a matrix can be interpreted as a two-dimensional table of numbers, and likewise, a tensor in three dimensions as a volume of numbers. This dimensionality is called the degree of a tensor; e.g., a matrix is a tensor of degree two. In a diagrammatic representation, the degree is equal to the number of tensor legs, i.e., the number of indices to address an entry. A tensor network can be regarded as a mathematical graph, where the nodes represent individual tensors and the edges represent logical contractions of the corresponding nodes. Tensor networks play an important role as computational tools for representing or approximating high-dimensional functions or data, particularly in quantum physics, where they are used to approximate the many-body quantum state.
In electronic structure calculations in the second quantization formulation corresponding to a given orbital basis set, the quantum molecular Hamiltonian governing the electron wavefunction is given by
where is the fermionic creation operator for the spatial orbital i and spin σ. The first term in the Hamiltonian captures the kinetic energy and nuclear attraction acting on the electrons, and the second term captures the Coulomb repulsion between them. The “full configuration interaction” quantum ground state formally consists of a linear combination of all possible occupation patterns of the orbitals, i.e., whether filling an orbital with an electron or not. For L spin orbitals, there are 2L combinatorial possibilities, illustrating the high dimensionality (or “curse of dimensionality”) of the problem.
The orbital basis set is conventionally constructed from predefined atomic orbitals (the s, p, d, … type orbitals of an atom). However, such basis sets are difficult to systematically improve or to adapt (e.g., to the presence of a strong magnetic field). Multiwavelets, which represent generic functions via a hierarchically refinable three-dimensional grid, offer an alternative with such capabilities. In collaboration with the group of Luca Frediani at UiT (The Arctic University of Norway), we have conducted a proof-of-principle study combining multiwavelets with DMRG calculations to approximate the ground-state wavefunction of small molecules [1]. DMRG is a widely used tensor network algorithm for optimizing a so-called matrix-product-state Ansatz. A particular feature of our study is an iterative loop between orbital optimization (represented as multiwavelets) and the quantum state in second-quantization form, as illustrated in Fig. 1.
Employing the DMRG algorithm for quantum chemistry is currently limited to about 100 orbitals. The restriction mainly stems from the large required virtual bond dimension (scaling as L2) in the matrix-product-operator representation of the electron-electron Coulomb repulsion (second term in the above equation). Nevertheless, it is generically possible to “compress” the coefficient tensor vijkl via the so-called tensor hypercontraction (THC) form, which can be interpreted as an interpolation of the Coulomb kernel. Interestingly, the THC form can be used for a memory-efficient, “matrix-free” application of the Hamiltonian to a matrix product state, as we have derived in a recent publication [2]. The algorithm developed there is highly parallelizable, rendering implementation and deployment on high-performance computing clusters a promising direction for future work.
Regarding tensor network algorithms in general, we have focused in several works on tree tensor network topologies, as well as large-scale, high-performance computing-oriented contractions of tensor networks [3]. We have bundled an implementation of these algorithms in the open-source software packages PyTreeNet, TNC and ChemTensor.
Figure 2
Quantum computing
Quantum computing, the idea of using the principle of quantum mechanics for calculations and simulations, has received significant excitement and research efforts in recent years, especially due to the transition from pure research to the industrial-scale construction of quantum computing hardware. As envisioned already in the beginning of the 1980s, quantum simulation, i.e., using a quantum computer to simulate another target quantum system of interest (from chemistry, materials science, or fundamental particle physics, for example), remains one of the most promising use-cases. A fundamental building block is implementing the time evolution operator e− i H t) governed by a quantum Hamiltonian H. Conventionally, this is realized using a Trotter splitting method, which specifies the gates in a brick-wall-type quantum circuit but can yield relatively deep circuits. Interestingly, it is feasible to further optimize the gates in this Ansatz via a classical preprocessing step to increase accuracy or reduce circuit depth. In line with similar works in the research community, we have recently published two papers that implement this idea. Fig. 2 shows a technical step from one of the publications [4], where we use tensor network techniques to represent short time steps.
A second direction in the field of quantum computing, which we actively pursue, is a sub-algorithm of the qubitization and singular value transformation framework, specifically, the block-encoding of a quantum Hamiltonian H. The overall goal is to implement the application of H to a quantum state as a quantum circuit. The difficulty is that H is not unitary in general, and hence not directly realizable as a quantum circuit. In our publication [5], we develop a quantum algorithm tailored to Hamiltonians given in matrix product operator form.
[1]
M. Nibbi, L. Frediani, E. Dinvay, and C. B. Mendl (2025).
[2]
Y. Wang, M. Luo, M. Reumann, and C. B. Mendl (2025).
[3]
M. Stoian, R. M. Milbradt, and C. B. Mendl (2024).
[4]
I. N. M. Le, S. Sun, and C. B. Mendl (2025).
[5]
M. Nibbi and C. B. Mendl (2024).
Selected publications
- M. Nibbi, L. Frediani, E. Dinvay, and C. B. Mendl, “Wavefunction Optimization at the Complete Basis Set Limit with Multiwavelets and DMRG,” The Journal of Physical Chemistry A, vol. 129, no. 47, pp. 11041–11052, Nov. 2025, doi: 10.1021/acs.jpca.5c05836.
- Y. Wang, M. Luo, M. Reumann, and C. B. Mendl, “Enhanced Krylov Methods for Molecular Hamiltonians: Reduced Memory Cost and Complexity Scaling via Tensor Hypercontraction,” Journal of Chemical Theory and Computation, vol. 21, no. 14, pp. 6874–6886, Jul. 2025, doi: 10.1021/acs.jctc.5c00525.
- M. Stoian, R. M. Milbradt, and C. B. Mendl, “On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond,” SIAM Journal on Scientific Computing, vol. 46, no. 5, pp. B647–B668, Oct. 2024, doi: 10.1137/23m161286x.
- I. N. M. Le, S. Sun, and C. B. Mendl, “Riemannian quantum circuit optimization based on matrix product operators,” Quantum, vol. 9, p. 1833, Aug. 2025, doi: 10.22331/q-2025-08-27-1833.
- M. Nibbi and C. B. Mendl, “Block encoding of matrix product operators,” Physical Review. A, vol. 110, no. 4, Oct. 2024, doi: 10.1103/physreva.110.042427.