A quantum t-design is a probability distribution over either pure quantum states or unitary operators which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less.
Here the Haar measure is a uniform probability distribution over all quantum states or over all unitary operators.
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.
[1] A spherical design is a collection of points on the unit sphere for which polynomials of bounded degree can be averaged over to obtain the same value that integrating over surface measure on the sphere gives.
Spherical and projective t-designs derive their names from the works of Delsarte, Goethals, and Seidel in the late 1970s, but these objects played earlier roles in several branches of mathematics, including numerical integration and number theory.
[1] The theory of unitary 2-designs was developed in 2006 [1] specifically to achieve a practical means of efficient and scalable randomized benchmarking[3] to assess the errors in quantum computing operations, called gates.
Since then unitary t-designs have been found useful in other areas of quantum computing and more broadly in quantum information theory and applied to problems as far reaching as the black hole information paradox.
[4] Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators.
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.[citation needed] 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.
Since every operator in SU(2) is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere.
This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
An important class of complex projective t-designs, are symmetric informationally complete positive operator-valued measures (POVMs), which are complex projective 2-design.
elements, a SIC-POVM is a minimal sized complex projective 2-designs.
[6] These are closely related to spherical 2t-designs of vectors in the unit sphere in
give rise to complex projective t-designs.
[7] Formally, we define a probability distribution over quantum states
Here, the integral over states is taken over the Haar measure on the unit sphere in
Approximate t-designs are most useful due to their ability to be efficiently implemented.
This efficient construction also implies that the POVM of the operators
-approximate t-design consisting of quantum pure states for a fixed t. For convenience d is assumed to be a power of 2.
{0,...,d-1} the image under f, where f is chosen at random from S, is exactly the uniform distribution over tuples of N elements of {0,...,d-1}.
[1] The theory of unitary 2-designs was developed in 2006 [1] specifically to achieve a practical means of efficient and scalable randomized benchmarking[3] to assess the errors in quantum computing operations, called gates.
Since then unitary t-designs have been found useful in other areas of quantum computing and more broadly in quantum information theory and in fields as far reaching as black hole physics.
[4] Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators.
Observe that the space linearly spanned by the matrices
Using the permutation maps it is possible[6] to verify directly that a set of unitary matrices forms 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.
This imposes an upper bound on the size of a unitary design.
This bound is absolute meaning it depends only on the strength of the design or the degree of the code, and not the distances in the subset, X.
Specifically, a unitary code is defined as a finite subset