Quantum error-correcting codes are far from classical: a quantitative examination

IQC Special Colloquium - Zhi Li, Perimeter Institute

Quantum error-correcting codes play a pivotal role in enabling fault-tolerant quantum computation. These codes protect quantum information through intricately designed redundancies that encode the information in a global manner. Unlike classical objects, in a quantum error-correcting code, the knowledge of individual subregions, even when combined, reveals nothing about the overall state.

In this talk, we explore the quantification of how far quantum error-correcting code are from classical states. We examine this question from three different perspectives: circuit complexity (the mimimal number of circuit depth needed to prepare a quantum state), expansion number (the minimal number of terms needed to expand the wavefunction), and a quantity we termed product overlap, which characterizes the maximal overlap between a given state and any product state. We will demonstrate why any quantum error-correcting code states must exhibit exponentially small product overlap, and how it implies lower bounds for the circuit complexity and the expansion number.