Practical implementation of mutually unbiased bases using quantum circuits

Abstract

The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits. We also analyze the quantum circuits implementing the various entanglement classes.

Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems

“…In the following we show how the matrices V (j) in ( 55) can be constructed from any sets of matrices U (k) i (i = 1, 2, 3) presented in Eqs. ( 36), (47), and (54), respectively, and we determine the corresponding linear transformations L (i) .…”

Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems

“…In the following we show how the matricesV (j) in (55) can be constructed from any sets of matrices U (k) i (i = 1, 2, 3) presented in Eqs. (36), (47), and (54), respectively, and we determine the corresponding linear transformations L (i) . Let us apply formula (17) for this casê…”

“…Wigner functions can be defined in this space using Hermitian operators connected to special mutually orthogonal sets of parallel lines called striations. There exist N + 1 different striations and the bases associated with them are mutually unbiased [4,40,46,47]. Such discrete Wigner functions have the same essential properties as their continuous counterparts.…”

Mutually unbiased bases with free parameters

“…The generation of the fixed set of bases, i.e. the set S 4 , requires different a circuit [42] which involves local, non-local controlled-phase and Toffoli gates [43]. Therefore, the sets of MUB with free parameters are generated by a composition of two different quantum circuits.…”

“…The generation of the set S 4 is given by the quantum circuit depicted in Figure 2. This circuit was recently derived [42]. The free parameters can be introduced by considering 7 quantum circuits (A to G).…”