TensorGrad: Differentiable Tensor-Network Optimization for Ground States and Entanglement Diagnostics

TensorGrad: Differentiable Tensor-Network Optimization for Ground States and Entanglement Diagnostics Aulia Octaviani1 1Independent Researcher, Computational Physics, Indonesia (Dated: October 10, 2025) We present TensorGrad, a differentiable tensor-network framework for variational optimization of quantum many-body ground states. The method combines finite-difference and autodifferentiation backends to optimize matrix-product-state (MPS) parameters using gradient descent. Applied to benchmark models such as the Transverse-Field Ising Model (TFIM) and the Heisenberg spin chain, TensorGrad efficiently converges to low-energy configurations using a minimal ansatz. Furthermore, an additional two-body entangler circuit introduces controlled quantum correlations, enabling non-trivial reductions in ground-state energy and measurable increases in entanglement entropy. The framework also computes entanglement spectra and von Neumann entropy profiles, offering an accessible platform to study the interplay between variational optimization and entanglement in differentiable physics. I. INTRODUCTION Tensor-network (TN) methods such as matrix product states (MPS) and projected entangled-pair states (PEPS) provide powerful representations for quantum many-body systems. Their compact expressivity arises from entanglement-limited parameterizations, making them ideal for modeling ground states of local Hamiltonians. Meanwhile, differentiable programming frameworks have revolutionized machine learning, enabling gradient-based optimization of high-dimensional parameter spaces. Bridging these ideas, TensorGrad provides a minimal and differentiable implementation of TN optimization for spin systems, illustrating how variational quantum simulations can be treated as gradient-based learning tasks. Our goal is to demonstrate that even a lightweight differentiable MPS framework can reproduce physically meaningful energy landscapes, entanglement structures, and convergence behaviors comparable to traditional variational algorithms, while remaining computationally simple. II. METHODS A. Variational ansatz We consider a product-state MPS for Nspin-1 2sites, |ψ(θ, ϕ)⟩= N O i=1 cos θi 2|0⟩+eiϕisin θi 2|1⟩.(1) To introduce minimal correlations, a small two-body entangler is applied: U(γ) = exp iγ N−1 X i=1 XiXi+1!,(2) where γcontrols the degree of entanglement. The full variational state is |Ψ⟩=U(γ)|ψ(θ, ϕ)⟩. B. Hamiltonians We test TensorGrad on two canonical lattice Hamiltonians: 1. Transverse-Field Ising Model (TFIM): HTFIM =−JX i ZiZi+1 −hX i Xi,(3) 2. Heisenberg Model: HHeis =JxX i XiXi+1 +JyX i YiYi+1 +JzX i ZiZi+1. (4) Finite chains with open boundary conditions are used throughout (N≤8). C. Optimization The objective function is the variational energy: E(θ, ϕ) = ⟨Ψ|H|Ψ⟩ ⟨Ψ|Ψ⟩.(5) Gradients are obtained via finite differences or the JAX autodiff backend: ∇f(x)≈f(x+ϵ)−f(x−ϵ) 2ϵ,(6) and parameters are updated using gradient descent: xt+1 =xt−η∇f(xt),(7) where ηis the learning rate. III. RESULTS A. Ground-state energy optimization Figure 1shows convergence of the variational energy for TFIM with N= 6 and N= 8. TensorGrad quickly 2 minimizes the energy to near ground-state levels within a few hundred steps. FIG. 1. Energy optimization for the TFIM model using gradient descent. Larger Nyields deeper minima as system complexity grows. For the Heisenberg chain with entangler γ= 0.02, the energy landscape becomes smoother and the minimum energy significantly lower (Fig. 2). B. Entanglement diagnostics We compute the von Neumann entanglement entropy, SA=−Tr(ρAlog ρA),(8) for bipartitions across the chain. Without entangler, the entropy remains close to zero (product state). When the entangler is applied, SAincreases, peaking near the system’s center (Fig. 3). The corresponding entanglement spectrum at the central cut also broadens (Fig. 4), demonstrating the entanglement’s nontrivial structure. FIG. 2. Energy convergence for the Heisenberg chain (N= 6) with entangler γ= 0.02. FIG. 3. Entanglement entropy profile for N= 6 with entangler γ= 0.02. The increase in SAindicates emergence of quantum correlations. FIG. 4. Entanglement spectrum at mid-cut (N= 6,γ= 0.02). 3 IV. DISCUSSION The results obtained with TensorGrad demonstrate that even a compact, differentiable implementation of matrix product states (MPS) can effectively reproduce the qualitative physics of quantum ground states. The successful convergence of the variational energy in both the TFIM and Heisenberg models confirms that gradientbased methods, when applied to tensor-network ansätze, can serve as an alternative to traditional optimization techniques such as the density matrix renormalization group (DMRG) or variational quantum eigensolvers (VQE). While TensorGrad operates at a small system size, its behavior reflects many of the characteristic features of these larger frameworks. The framework provides a bridge between physicsbased variational principles and modern differentiable programming paradigms. In this sense, it serves as a conceptual laboratory for exploring how automatic differentiation, loss-based training, and parameter-space geometry can be used to optimize quantum states directly within tensor-network manifolds. The inclusion of a simple XX entangler circuit plays a crucial role. This operator acts as a minimal generator of local correlations, analogous to two-body gates in variational quantum circuits or local projectors in PEPS. Its effect—lowering the variational energy and increasing the von Neumann entanglement entropy—illustrates how controlled entanglement can systematically improve variational expressivity. This observation is consistent with previous findings in tensor-network theory, where introducing short-range entanglement is known to enhance approximation accuracy. From a methodological standpoint, TensorGrad highlights that differentiable optimization does not necessarily require large-scale automatic differentiation frameworks; finite-difference gradients, when combined with clear tensor abstractions, are sufficient to explore variational landscapes and visualize optimization dynamics. This balance between simplicity and physical interpretability makes TensorGrad a useful pedagogical and research tool for studying differentiable tensor networks. Finally, the framework opens a path toward integrating differentiable optimization into tensor-network algorithms for more complex systems—such as chiral spin liquids, topologically ordered PEPS, or models with longrange interactions—where analytic gradients and entanglement diagnostics may reveal new structural insights into the nature of quantum correlations. V. CONCLUSION We introduced TensorGrad, a compact and differentiable tensor-network framework that successfully reproduces essential features of quantum many-body groundstate optimization. By combining numerical stability with conceptual transparency, TensorGrad demonstrates that gradient-based learning paradigms can be directly applied to physical variational problems without requiring large-scale symbolic algebra or high-performance contraction libraries. Its modular design encompassing MPS-based product states, differentiable entanglers, and automatic computation of observables such as energy and entanglement entropy provides a versatile educational and research testbed. Beyond a simple pedagogical tool, TensorGrad also establishes a conceptual link between differentiable programming and tensor-network variational principles. The observed lowering of the Heisenberg ground-state energy through a small XX entangler and the emergence of non-zero entanglement entropy show that even minimal differentiable circuits can mimic the physics of correlated quantum systems. This highlights the potential of hybrid physics-informed differentiable frameworks for exploring quantum correlations in low-dimensional lattice models. Future Work. Building upon this foundation, several research directions naturally emerge: •Implementing higher-dimensional tensor networks such as PEPS, including corner-transfer and boundary-MPS renormalization. •Integrating JAX or PyTorch backends for full automatic differentiation of tensor contractions and energy functionals. •Exploring the connection between differentiable entanglers and unitary parametrizations of topological order and chiral spin liquids. •Extending the framework toward differentiable quantum simulation pipelines, where loss functions are physically motivated quantities such as fidelity or mutual information. •Benchmarking TensorGrad against established DMRG and VQE algorithms to assess convergence scaling and variational expressivity. Overall, TensorGrad bridges numerical optimization and theoretical quantum information concepts, offering a small yet powerful prototype for the next generation of differentiable tensor-network research. Its simplicity makes it accessible to students, while its extensibility enables integration into modern machine-learning-based quantum simulation environments. 4 [1] A. Francuz, N. Schuch, B. Vanhecke, Stable and Efficient Differentiation of Tensor-Network Algorithms, Phys. Rev. Research 7, 013237 (2025). [2] F. Verstraete, J. I. Cirac, Matrix Product States Represent Ground States Faithfully, Phys. Rev. B 73, 094423 (2006). [3] R. Orús, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Ann. Phys. 349, 117 (2014). [4] U. Schollwöck, The Density-Matrix Renormalization Group in the Age of Tensor Networks, Ann. Phys. 326, 96 (2011). [5] J. Biamonte, M. Schuld, et al., Quantum Machine Learning, Nature 549, 195–202 (2017).