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 \(2^n + 1\) 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 \(4^n\) 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 \(2^n + 1\) quantum MUB circuits on \(n\)-qubit systems within \(O(n^3)\) 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 \(2^n + 1\) 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.
An Efficient Quantum Circuit Construction Method for Mutually Unbiased Bases in n-Qubit Systems
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 (2^n + 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 (4^n) 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 (2^n + 1) quantum MUB circuits on (n)-qubit systems within (O(n^3)) 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 (2^n + 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.
Classical shadow tomography, harnessing randomized informationally complete (IC) measurements, provides an effective avenue for predicting many properties of unknown quantum states. In n-qubit systems, projections onto 2n+1 mutually unbiased bases (MUBs) are widely recognized as minimal and optimal IC measurements for full-state tomography. Mutually unbiased base circuits, structured as $\text{\ensuremath{-}}\mathrm{CZ}\text{\ensuremath{-}}S\text{\ensuremath{-}}H\text{\ensuremath{-}}$, form the minimal subset of the whole ${2}^{{n}^{2}+2n}{\ensuremath{\prod}}_{j=1}^{n}({4}^{j}\ensuremath{-}1)$ Clifford circuit ensemble. Each of them can be generated by n special MUB circuits. We study how to use MUB circuits as the ensemble in classical shadow tomography. For general observables, the variance to predict their expectation value is shown to be exponential to the number of qubits n. However, for a special class referred to as appropriate MUB-average (AMA) observables, the variance decreases to poly(n). Additionally, we find that through biased sampling of MUB circuits, the variance for non-AMA observables can again be reduced to poly(n) with the MUB-sparse condition. The performance and complexity of using the MUB and Clifford circuits as the ensemble in the classical shadow tomography are compared.
Predicting properties of large-scale quantum systems is crucial for the development of quantum science and technology. Shadow estimation is an efficient method for this task based on randomized measurements, where many-qubit random Clifford circuits are used for estimating global properties like quantum fidelity. Here we introduce the minimal Clifford measurement (MCM) to reduce the number of possible random circuits to the minimum, while keeping the effective postprocessing channel in shadow estimation. In particular, we show that MCM requires 2n+1 distinct Clifford circuits, and it can be realized by mutually unbiased bases, with n as the total qubit number. By applying the Z-tableau formalism, this ensemble of circuits can be synthesized to the $\ensuremath{-}\mathrm{S}\ensuremath{-}\mathrm{CZ}\ensuremath{-}\mathrm{H}\ensuremath{-}$ structure, which can be decomposed to $2n\ensuremath{-}1$ fixed circuit modules, and the total circuit depth is at most n+1. Compared to the original Clifford measurements, our MCM reduces the circuit complexity and the compilation costs. In addition, we find the sampling advantage of MCM on estimating off-diagonal operators, and extend this observation to the biased-MCM scheme to enhance the sampling improvement further.
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].
Certifying entanglement is an important step in the development of many quantum technologies, especially for higher-dimensional systems, where entanglement promises increased capabilities for quantum communication and computation. A key feature distinguishing entanglement from classical correlations is the occurrence of correlations for complementary measurement bases. In particular, mutually unbiased bases (MUBs) are a paradigmatic example that is well-understood and routinely employed for entanglement certification. However, implementing unbiased measurements exactly is challenging and not generically possible for all physical platforms. Here, we extend the entanglement-certification toolbox from correlations in MUBs to arbitrary bases. This practically significant simplification paves the way for efficient characterizations of high-dimensional entanglement in a wide range of physical systems. Furthermore, we introduce a simple three-MUBs construction for all dimensions without using the Wootters-Fields construction, potentially simplifying experimental requirements when measurements in more than two MUBs are needed, especially in high-dimensional settings.
Abstract From a practical perspective it is advantageous to develop methods that verify entanglement in quantum states with as few measurements as possible. In this paper we investigate the minimal number of mutually unbiased bases (MUBs) needed to detect bound entanglement in bipartite ( d × d ) -dimensional states, i.e. entangled states that are positive under partial transposition. In particular, we show that a class of entanglement witnesses (EWs) composed of MUBs can detect bound entanglement if the number of measurements is greater than d / 2 + 1 . This is a substantial improvement over other detection methods, requiring significantly fewer resources than either full quantum state tomography or measuring a complete set of d + 1 MUBs. Our approach is based on a partial characterisation of the (non-)decomposability of EWs. We show that non-decomposability is a universal property of MUBs, which holds regardless of the choice of complementary observables, and we find that both the number of measurements and the structure of the witness play an important role in the detection of bound entanglement.
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.