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arXiv:2609.11811

Block entropy area based non-local fermionic mode optimization with gradient disentanglers

Miklós Antal Werner, Gero Friesecke, Andor Menczer, Kornél Kapás, Örs Legeza

cond-mat.str-elcond-mat.stat-mechphysics.comp-phquant-phSubmitted Sep 10, 2026• Announced Sep 11, 2026• 0 citations
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Abstract

We introduce a systematic block entropy area based mode optimization algorithm for many-body quantum states of interacting fermions represented by matrix product states. From the gradient of a global cost function, the block entropy area, a long-ranged, non-interacting effective disentangler Hamiltonian is formed. We then simulate the time-dependent Schrödinger equation driven by the disentangler Hamiltonian by employing the time-dependent variational principle based on projector splitting, and minimize the cost function. The combination of the density matrix renormalization group with this gradient-based entanglement minimization forms an efficient low-rank iterative ground-state algorithm that also provides an optimized single-particle basis for matrix product state representation. We demonstrate the method on two-dimensional lattice models of interacting fermions and the Fe${_4}$S${_4}$ cluster, and show its robustness and superiority over earlier protocols using nearest-neighbor mode rotations and reorderings.

Comments
6 pages, 3 figures

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We introduce a systematic block entropy area based mode optimization algorithm for many-body quantum states of interacting fermions represented by matrix product states. From the gradient of a global cost function, the block entropy area, a long-ranged, non-interacting effective disentangler Hamiltonian is formed. We then simulate the time-dependent Schrödinger equation driven by the disentangler Hamiltonian by employing the time-dependent variational principle based on projector splitting, and minimize the cost function. The combination of the density matrix renormalization group with this gradient-based entanglement minimization forms an efficient low-rank iterative ground-state algorithm that also provides an optimized single-particle basis for matrix product state representation. We demonstrate the method on two-dimensional lattice models of interacting fermions and the Fe4{_4}4​S4{_4}4​ cluster, and show its robustness and superiority over earlier protocols using nearest-neighbor mode rotations and reorderings.
An optimal choice of single-particle modes leads to an optimal tensor network-based compression of the many-body wave function for systems of indistinguishable particles, which compression can be essential for large, close-to-critical systems. The proposed mode optimization relies on the minimization of the block entropy area, a global quantity that measures entanglement of the wave function along the one-dimensional chain of modes. Extension of the two-site DMRG algorithm by nearest-neighbor mode optimizations is straightforward, but it performs poorly without systematic reordering of modes. Moreover, finding a stationary optimal point in the combination of the fixed-rank MPS manifold and the unitary group of mode rotations requires optimization for every generator, i.e., for every two-mode pair. Here, a systematic joint optimization protocol over the MPS manifold and the full unitary group is presented in which every two-mode pair is accessed by a carefully constructed sequence of two-qubit operations, including swap-gate controlled permutations. Large-scale DMRG simulations of strongly correlated two-dimensional fermionic lattice models and the multireference Fe4_44​S4_44​ transition metal cluster demonstrate rapid convergence towards the optimum, achieving low energy and low entanglement.
We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently. In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.
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Abstract 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 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.