Tensor networks are a family of techniques designed to efficiently represent and manipulate specific families of high-dimensional data, for example quantum states possessing certain entanglement properties. Instead of storing an exponentially large object (like a many-body quantum state), a tensor network rewrites this quantum state as several small sized tensors. These tensors are connected by indices, forming a graph-like structure that captures the correlations between subsystems. Famous examples include Matrix Product States (MPS) for 1D systems, Projected Entangled Pair States (PEPS) for 2D systems, and Tree Tensor Networks (TTN), each tailored to different geometries. The key idea is that many physically relevant quantum states exhibit limited (often area-law) entanglement, which allows tensor network methods to efficiently (in polynomial time and memory) simulate properties of these states (such as expectation values of certain observables, or output probabilities). Furthermore, since noise degrades quantum entanglement, tensor network methods are useful to simulate noisy quantum systems. Thus, they provide a very solid benchmark to test potential claims of quantum-over-classical speedup in the presence of noise.
Frequently asked questions
- Why are tensor networks useful for quantum computing?
They significantly reduce the complexity of simulating certain quantum systems (like low entanglement, noise). By compressing the data (usually based on entanglement limits), they allow classical computers to bypass the exponential memory requirements that sometimes make the quantum simulation impossible.
- Where are they used?
Primarily in quantum many-body physics and quantum computing, but also increasingly in machine learning and optimization.
- When do efficient tensor network simulations fail ?
If they are applied in the wrong settings, for example to simulate systems where there is high entanglement. Indeed, tensor networks approximate the state by discarding mathematically insignificant data (a process known as truncating small singular values). On the other hand, if the state is in a volume law entanglement, these truncations often lead to significant deviations between the ideal expectation values and those predicted by the tensor network simulation.
- How do they relate to quantum computing?
They can simulate certain quantum systems efficiently and help understand which quantum computations are classically hard. Because of that, they can be considered as a solid benchmarking tool for quantum algorithms.