Who can compete with quantum computers? Lecture notes on quantum inspired tensor networks computational techniques

TL;DR

This work surveys tensor-network techniques as quantum-inspired, classical tools for handling exponentially large vectors and matrices. It foregrounds $MPS$/$MPO$ representations, exact and approximate emulation via TEBD/DMRG, and learning-based approaches like Tensor Cross Interpolation ($TCI$) and the quantics framework, with applications to quantum circuits, the transverse-field Ising model, and high-dimensional PDEs. A central theme is that many challenging problems admit efficient TN representations when entanglement is limited or structure can be exploited, enabling fast, scalable simulations and novel PDE-solving strategies. The text also critically discusses the limits of quantum supremacy arguments and emphasizes the practical potential of quantum-inspired tensor networks for classical computation and beyond. Overall, the notes argue that TNs provide a versatile, scalable toolbox for linear algebra on ultra-large objects and for mapping diverse problems (including PDEs) into a TN framework through $TCI$ and quantics, with explicit constructions like the QFT MPO illustrating the approach.

Abstract

This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.

Figures (8)

• Figure 1: Time needed for the construction of a GHZ state $[|00\cdots{}0\rangle+|11\cdots{}1\rangle]/\sqrt{2}$ with $N$ qubits using the full state simulator (exponential scaling) and the exact MPO.MPS simulator (linear scaling per gate, here $\chi=2$ is exact). Contributed by Chen-How Huang.
• Figure 2: Fidelity $F$ versus depth $D$ for $N=20$ qubits, a random quantum circuit and various values of $\chi= 10,20,50$. The symbols correspond to an exact calculation of $F$ (possible for this small system), and the lines correspond to the right-hand side of Eq. \ref{['eq:fid']}. Adapted from zhou2020.
• Figure 3: Energy versus imaginary time $\tau$ for the TFI model using the imaginary-time TEBD algorithm. The energy converges to the ground state in the long-time limit. Calculations were performed with $N=30$ spins for three values of the time step $\eta$. Parameters are $h_Z=0$ and $h_X=1$. The maximum bond dimension used here is $\chi=40$ (relative precision better than $10^{-3}$), but $\chi=10$ is indistinguishable at the scale of the figure. The reference energy was obtained with the DMRG algorithm implemented using the Tenpy package hauschild2018. (Contributed by Chen-How Huang).
• Figure 4: Illustration of the cross interpolation (CI) of a matrix. The large red triangles indicate real pivots and the smaller red triangles indicate automatically generated pivots. The right-hand side only contains small sub-parts of the matrix. Adapted from Nunez et al., PRX 12, 041018 (2022).
• Figure 5: Error $|A_{ij}-[A_{\mathrm{CI}}]_{ij}|$ versus $i$ and $j$ at different stages of the Cross-Interpolation for a $M\times M$ matrix with $M=20$. In this toy example, $A_{ij}=\left(\frac{i/M}{i/M+1}\right)^{4}(1+e^{-(j/M)^{2}})\left[1+(j/M)\cos(j/M)e^{-(j/M)\frac{i/M}{(i/M)+1}}\right]$. The red dots indicate the pivots. The $x$ and $y$ axis have been rescaled to be in $[0,10]$. Adapted from Jeannin et al. PRB 110, 035124 (2024).