We propose a new method of finding the mutually unbiased bases for three qubits. The key element is the construction of the table of striation-generating curves in the discrete phase space. We derive a system of equations in the Galois field GF(8) and show that the solutions of these equations are sufficient for the construction of the general sets of complete mutually unbiased bases. A few examples are presented in order to show how our algorithm works in the following cases: striation table with three and two axes, and one and no axis in the discrete phase space.
For a system of
N qubits, living in a Hilbert space of dimension
d=2N, it is known that there exists
d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of
N qubits is outlined.
Mutually unbiased bases (MUBs) play a crucial role in numerous applications within quantum information science, such as quantum state tomography, error correction, entanglement detection, and quantum cryptography. Utilizing
2n+1 MUB circuits provides a minimal and optimal measurement strategy for reconstructing all
n-qubit unknown states. It significantly reduces the number of measurements compared to the traditional
4n Pauli observables, also enhancing the robustness of quantum key distribution (QKD) protocols. Previous circuit designs that rely on a single generator can result in exponential gate costs for some MUB circuits. In this work, we present an efficient algorithm to generate each of the
2n+1 quantum MUB circuits on
n-qubit systems within
O(n3) time. The algorithm features a three-stage structure, and we have calculated the average number of different gates for random sampling. Additionally, we have identified two linear properties: the entanglement part can be directly defined into
2n−3 fixed sub-parts, and the knowledge of
n special MUB circuits is sufficient to construct all
2n+1 MUB circuits. This new efficient and simple circuit construction paves the way for the implementation of a complete set of MUBs in diverse quantum information processing tasks on high-dimensional quantum systems.
Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q+1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].
For the complete estimation of arbitrary unknown quantum states by measurements, the use of mutually unbiased bases has been well established in theory and experiment for the past 20 years. However, most constructions of these bases make heavy use of abstract algebra and the mathematical theory of finite rings and fields, and no simple and generally accessible construction is available. This is particularly true in the case of a system composed of several qubits, which is arguably the most important case in quantum information science and quantum computation. In this paper, we close this gap by providing a simple and straightforward method for the construction of mutually unbiased bases in the case of a qubit register. We show that our construction is also accessible to experiments, since only Hadamard and controlled-phase gates are needed, which are available in most practical realizations of a quantum computer. Moreover, our scheme possesses the optimal scaling possible, i.e., the number of gates scales only linearly in the number of qubits.
We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.