Quantum Advantage: a Tensor Network Perspective

Abstract

We review the recent quantum advantage experiments by IBM, D-Wave, and Google, focusing on cases where efficient classical simulations of the experiment were demonstrated or attempted using tensor network methods. We assess the strengths and limitations of these tensor network-based approaches and examine how the interplay between classical simulation and quantum hardware has advanced both fields. Our goal is to clarify what these results imply for the next generation of quantum advantage experiments. We identify regimes and system features that remain challenging for current tensor network approaches, and we outline directions where improved classical methods could further raise the standard for claiming quantum advantage. By analyzing this evolving competition, we aim to provide a clear view of where genuine, scalable quantum advantage is most likely to emerge.

Figures (5)

• Figure 1: (a) Matrix Product States (MPS), (b) Matrix Product Operators (MPOs), (c) MPS wrapped in a snake pattern to simulate a 2d lattice, (d) Projected Entangled Pair States (PEPS), (e) Projected Entangled Pair Operators (PEPO), (f) Example of an orthogonality centre in a 2d isometric Tensor network, (g) Self consistent equation for defining the message tensors in Belief Propagation (BP) algorithm.
• Figure 2: (a) Connectivity structure of IBM's Eagle quantum processor with 127 qubits in a heavy hexagon lattice, (b) Corresponding Tensor network structure used to simulate the processor directly mimicking the lattice structure of the quantum processor. A notable feature of this lattice --- which has been used in belief-propagation–enabled PEPS simulations --- is its locally tree-like structure: starting from a given site, one can move several steps in any direction without encountering a loop.
• Figure 3: Random circuit sampling via tensor networks. (a) The tensor network representing the probability amplitude for a product state $|{\phi}\rangle$ in the evolved quantum state $U|\psi\rangle$ where $U$ is expressed as a brickwall circuit of (possibly, random) two-body gates. (b) Each layer of circuit can be regrouped as an MPO. (c) This yields a square-lattice tensor network whose horizontal and vertical directions correspond to space and time, respectively. The resulting network admits multiple contraction strategies, including boundary-MPS propagation along (d) time or (e) space, as well as (f–g) coarse-graining–based contraction methods.
• Figure 4: A basic decision tree for selecting an appropriate tensor network method for a quantum advantage experiment. Three main decision axes are illustrated (left): (1) the effective dimensionality of the system (quasi-1D, higher-dimensional lattice, or irregular), which guides the choice of tensor network dimensionality; (2) the correlation structure or geometry (tree, tree-like, or containing extensive loops), which informs the network connectivity and the potential effectiveness of message-passing methods; and (3) the presence or absence of causal structure (eigenstates vs. circuit dynamics), which determines the relevant contraction directions and highlights key physical features such as entanglement growth, area laws, chaos, localization, and integrability.
• Figure 5: Computing expectation values of local observables in a time-evolved quantum state does not require reconstructing the full final state. Instead, it reduces to contracting a tensor network, where only the tensors inside the causal cone of the operator contribute. The resulting reduced circuit can be contracted in several directions---for example, along the time or spatial directions, or orthogonally to the light-cone boundary, or using coarse-graining methods.