Quantum T-design - Spherical Designs

Spherical Designs

Complex projective (t,t)-designs have been studied in quantum information theory as quantum 2-designs, and in t-designs of vectors in the unit sphere in which, when transformed to vectors in become complex projective (t/2,t/2)-designs.

Formally, we define a complex projective (t,t)-design as a probability distribution over quantum states if

Here, the integral over states is taken over the Haar measure on the unit sphere in

Exact t-designs over quantum states cannot be distinguished from the uniform probability distribution over all states when using t copies of a state from the probability distribution. However in practice even t-designs may be difficult to compute. For this reason approximate t-designs are useful.

Approximate (t,t)-designs are most useful due to their ability to be efficiently implemented. i.e. it is possible to generate a quantum state distributed according to the probability distribution in time. This efficient construction also implies that the POVM of the operators can be implemented in time.

The technical definition of an approximate (t,t)-design is:

If

and

then is an -approximate (t,t)-design.

It is possible, though perhaps inefficient, to find an -approximate (t,t) design consisting of quantum pure states for a fixed t.