Quantum t-design

Information theory | Quantum information theory | Quantum information science

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. 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 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. Two particularly important types of t-designs in quantum mechanics are projective and unitary t-designs. 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. Particular examples of these objects have found uses in quantum information theory, quantum cryptography, and other related fields. Unitary t-designs are analogous to spherical designs in that they reproduce the entire unitary group via a finite collection of unitary matrices. The theory of unitary 2-designs was developed in 2006 specifically to achieve a practical means of efficient and scalable randomized benchmarking 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. Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators. (Wikipedia).

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Quantum t-design

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