Worst-case Lower Bound Research Articles

Optimal two-qubit circuits for universal fault-tolerant quantum computation

We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS = diag(1, 1, 1, i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and outputs a Clifford+CS circuit for U, which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worst-case lower bound of 5{{rm{log}}}_{2}(frac{1}{epsilon })+O(1) on the number of CS gates required to ϵ-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.

Linear Regression Constrained to a Ball

Stoica, P., and Ganesan, G., Linear Regression Constrained to a Ball, Digital Signal Processing11 (2001), 80–90.A worst case lower bound (WCLB) result obtained by Nemirovskii suggests that a potentially significant estimation accuracy enhancement may be achieved provided the true parameter vector is known to belong to a ball. In this paper we discuss the many facets and implications of Nemirovskiirs result by using linear regression as a vehicle for illustration. In particular, we address briefly such issues as biased versus unbiased estimation, minimax optimal estimation, tightness of the WCLB, and comparison of WCLB with the performance of the least squares estimator constrained to the ball and that of the linear minimax estimator.