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. Quantum t-designs are so called because they are analogous to t-designs in classical statistics, which arose historically in connection with the problem of design of experiments. Quantum t-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.
Contents
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.
Motivation
In a d-dimensional Hilbert space when averaging over all quantum pure states the natural group is SU(d), the special unitary group of dimension d. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries.
A particularly widely used example of this is the spin
Another recent application is the fact that a symmetric informationally complete POVM is also a spherical 2-design. Also, since a 2-design must have more than
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
Formally, we define a complex projective (t,t)-design as a probability distribution over quantum states
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
The technical definition of an approximate (t,t)-design is:
If
and
then
It is possible, though perhaps inefficient, to find an
Construction
For convenience N is assumed to be a power of 2.
Using the fact that for any N there exists a set of
Let
Using this and Gaussian quadrature we can construct
Unitary Designs
Elements of the unitary design are elements of the unitary group, U(d), the group of
Suppose
Formally define a unitary t-design, X, if
Observe that the space linearly spanned by the matrices
Using the permutation maps it is possible to verify directly that a set of unitary matrices forms a t-design.
One direct result of this is that for any finite
With equality if and only if X is a t-design.
1 and 2-designs have been examined in some detail and absolute bounds for the dimension of X, |X|, have been derived.
Bounds for unitary designs
Define
then X is a unitary t-design.
We further define the inner product for functions
and
it follows that X is a unitary t-design iff
From the above it is demonstrable that if X is a t-design then
A unitary code is a finite subset of the unitary group in which a few inner product values occur between elements. Specifically, a unitary code is defined as a finite subset
It follows that