Construction of mutually unbiased bases with cyclic symmetry for qubit systems

Abstract

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 letter, 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.In quantum physics, the estimation of the state of a system is of high practical value [1]. It is well-known that for the complete estimation of a state, known as state tomography, a single measurement is not sufficient, even if performed many times to get the statistics of such measurement. It is necessary to measure a state in various different bases. The best choice of such bases for an arbitrary system is so-called mutually unbiased bases (MUBs), which offer the highest information outcome, as already stated by Wootters and Fields [2]: mutually unbiased bases provide an optimal means of determining an ensemble's state. Experimental results demonstrate the practicability of those schemes [3][4][5]. Different construction methods for MUBs are known [2, 6-10]. For a d-level system, i. e., a system described by a d × d density matrix, one would need d + 1 mutually unbiased bases, since any measurement statistically reveals d − 1 parameters. Unfortunately, it is not even known whether d + 1 such bases exist in every d-level system. Mutually unbiased bases are related to different topics in mathematics and physics, e. g., quantum cryptography, foundations of physics [11], orthogonal Latin squares or hidden-variable models [12,13] and even Feynman's path integral [14].In this letter, we want to focus on a system which is of particular interest in quantum information processing, namely, a quantum register built of qubits. We propose a complete set of mutually unbiased bases for quantum registers of size 1, 2, 4, 8, . . . , 256. In general, the construction of MUBs is quite involved and uses methods from abstract algebra and the mathematical theory of finite fields and rings [10], which are far apart from most methods that are commonly used in physics. We overcome this problem in such a way, that our construction (although not its proof) is very easy to follow and to apply by anyone with just basic knowledge of linear algebra. Moreover, our construction is applicable to experi...

Practical implementation of mutually unbiased bases using quantum circuits

“…We follow here the approach established in Refs. [28,29], that allows a cyclic generation of the MUBs, that is, the generators appearing in each class (2.5) can be expressed as…”

“…We revisit the MUB strategy, but capitalize on a recently developed construction which generates the corresponding MUBs in a cyclic way [28,29]. From an experimental viewpoint, the undeniable advantage of this approach is that a single unitary operation U is enough to create all the MUBs.…”

Graph-state formalism for mutually unbiased bases

“…First attempts in this direction have recently been made in Ref. [31] for a restricted class of qubit systems, and in Ref. [32] for bipartite systems of prime dimension.…”

“…In particular, our scheme does not only work for the special cases discussed in Refs. [31,32], but for all multipartite prime-dimensional systems. Besides the fact that our formalism is mathematically simple, it also allows to easily address questions related to the presence of entanglement in basis states of complete sets of MUBs.…”

Structure of the sets of mutually unbiased bases with cyclic symmetry

“…In even prime-power dimensions, complete sets of MUBs can be shaped in a cyclic manner, as the multiples of a single generating basis [69][70][71][72][73] . This procedure rests on the properties of the so-called Fibonacci polynomials 74 and leads directly to quantum circuits that can be used for a simple practical realization of these bases.…”