Quantum T-design

A Quantum t-design is a probability distribution over pure quantum states which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states. These designs are usually unique, and thus almost always calculable. Two particularly important types of t-designs in quantum mechanics are spherical and unitary t-designs.

Spherical t-designs are designs where points of the design (i.e. the points being used for the averaging process) are points on a unit sphere. Spherical t-designs and variations thereof have been considered lately and found useful in quantum information theory, quantum cryptography and other related fields.

Unitary designs are analogous to spherical designs in that they approximate the entire unitary group via a finite collection of unitary matrices. Unitary designs have been found useful in information theory and quantum computing. Unitary designs are especially useful in quantum computing since most operations are represented by unitary operators.