Abstract Building upon the theoretical foundation of utilizing the eigenvectors of Pauli matrices for single-qubit quantum state preparation, we employ a set of easily preparable quantum states constructed from the eigenvectors of Pauli matrices and arbitrary real quantum logic gates. Using fidelity and trace norm as tools, we achieve the approximation of single-qubit quantum states. For different scenarios, we present complete exact solutions and analytical expressions for the optimal approximation. Furthermore, during the research, we investigated the comparative effectiveness of different distance metrics within the set B α Z for quantum state approximation. It was found that trace norm can approximate target state more easily than fidelity. This discovery significantly aids in the selection of measurement tools for quantum state approximation tasks.
Abstract Building upon the theoretical foundation of utilizing the eigenvectors of Pauli matrices for single-qubit quantum state preparation, we employ a set of easily preparable quantum states constructed from the eigenvectors of Pauli matrices and arbitrary real quantum logic gates. Using fidelity and trace norm as tools, we achieve the approximation of single-qubit quantum states. For different scenarios, we present complete exact solutions and analytical expressions for the optimal approximation. Furthermore, during the research, we investigated the comparative effectiveness of different distance metrics within the set B α Z for quantum state approximation. It was found that trace norm can approximate target state more easily than fidelity. This discovery significantly aids in the selection of measurement tools for quantum state approximation tasks.
Abstract We investigate the optimal convex approximation, optimally approximating a desired and unavailable qubit state by the convex mixing of a given set of available states. When the available states are the eigenvectors of three Pauli matrices, we present the complete exact solution for the optimal convex approximation of an arbitrary qubit state based on the fidelity distance. By the comparison of optimal states based on fidelity and trace norm, the advantages and disadvantages of the optimal convex approximation are identified. Several examples are provided to support this. We also analyze the geometrical properties of the target states which can be completely represented by a set of available states. Using the feature of convex combination, we derive the maximum range of completely represented states and clearly illustrate that any qubit state can be optimally prepared by at most three available states in known useable states.
Abstract Utilizing the l 1 norm as the quantum state distance metric, we consider the Pauli distance of arbitrary target quantum states, that is, the optimal convex approximation of any qubit mixed states relative to the available state of pairs of eigenstates in three Pauli matrices. We get the complete analytical expressions for B 1 -distance, B 2 -distance and B 3 -distance. We determine the parameter range for the optimal approximation of each B 2 -distance in eight distinct scenarios. Furthermore, some trade-off relations about B 2 -distances are also studied.
We investigate the computational complexity of estimating the operator norm distance T∞(ρ0,ρ1), defined via the operator norm ∥A∥∞=σmax(A), given poly(n)-size state-preparation circuits of n-qubit quantum states ρ0 and ρ1. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:
1. When one state is pure, we establish an optimal quantum estimator using Θ(1/ε) queries to the state-preparation circuits. Consequently, for constant additive error, say ε=1/5, our estimator runs in poly(n) time. Since the operator norm distance T∞(∣ψ⟩⟨ψ∣,ρ) is exactly half of the trace distance T(∣ψ⟩⟨ψ∣,ρ), our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with rank(ρ), which can be exp(n) in general.
2. For general quantum states, we also provide a quantum estimator using O(1/ε3/2) queries to the state-preparation circuits, which shows that the corresponding promise problem is BQP-complete and improves the QMA upper bound sketched by Liu and Wang (ESA 2025). Together with an Ω(1/ε) quantum query complexity lower bound, this leaves only square-root room for improvement.
The key intuition behind our estimators is that, when one state is pure, the pure state ∣ψ⟩ has overlap at least 1/2 with the top unit eigenvector of ∣ψ⟩⟨ψ∣−ρ, reflecting a structural feature specific to the operator norm distance.
We consider the problem of quantum 1-PCA: given copies of an unknown n-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an n-qubit state with top eigenvalue λ and spectral gap at least Δ>0, we give an algorithm that recovers the leading eigenvector to fidelity at least 1−ε with high probability using O~(2n⋅η2/Δ3ε3) copies and O~((2n/Δε)⋅poly(η/Δε)) time, where η=max(1−λ,ε). All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. ε=Ω(η), our runtime and copy complexity become O~(2n/ε). This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption η=0, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.
Complete optimal convex approximations of qubit states under B2 B 2 distance
We consider the optimal approximation of arbitrary qubit states with respect to an available states consisting the eigenstates of two of three Pauli matrices, the B2 -distance of an arbitrary target state. Both the analytical formulae of the B2 -distance and the corresponding complete optimal decompositions are obtained. The trade-off relations for both the sum and the squared sum of the B2 -distances have been analytically and numerically investigated.
Abstract Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory, offering critical guidance for experimental implementations. In this paper, we embark on an in-depth exploration of the use of a quantum state prepared by the convex combination of given qubit states to optimally approximate the l 1-norm of coherence of the target quantum state, striving to make the prepared state and the target state as similar as possible. Here, we present the analytical solution for the optimal distance for any N given quantum states. We find that the optimal approximation problem for any N > 4 quantum states can be transformed into an optimal approximation problem for no more than four quantum states, which not only significantly streamlines the problem but also proves advantageous for laboratories in terms of material conservation. Ultimately, a one-to-one comparison between the analytical and numerical solutions verifies the effectiveness of our approach.