A method of quantum-state tomography for quantum-information processing is described. The method is based on the use of the Fourier-transform technique and involves detection of all the diagonal elements of the density matrix in a one-dimensional experiment and all the off-diagonal elements by a two-dimensional experiment. The method is efficient for a large number of qubits $(>~5).$ The proposed method is outlined using a two-qubit system and demonstrated using simulations by tomographing arbitrary complex density matrices of two- and four-qubit systems.
A method of quantum-state tomography for quantum-information processing is described. The method is based on the use of the Fourier-transform technique and involves detection of all the diagonal elements of the density matrix in a one-dimensional experiment and all the off-diagonal elements by a two-dimensional experiment. The method is efficient for a large number of qubits (>5). The proposed method is outlined using a two-qubit system and demonstrated using simulations by tomographing arbitrary complex density matrices of two- and four-qubit systems.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.
In this paper we describe in detail and generalize a method for quantum process tomography that was presented by Bendersky et al. [Phys. Rev. Lett. 100, 190403 (2008)]. The method enables the efficient estimation of any element of the $\ensuremath{\chi}$ matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required implementing it scale polynomially with the number of qubits of the system. The estimation of all diagonal elements of the $\ensuremath{\chi}$ matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, which is based on efficient estimation by random sampling over a set of states forming a 2-design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.
We propose a technique for performing quantum state tomography of photonic polarization-encoded multiqubit states. Our method uses a single rotating wave plate, a polarizing beam splitter, and two photon-counting detectors per photon mode. As the wave plate rotates, the photon counters measure a pseudocontinuous signal which is then Fourier transformed. The density matrix of the state is reconstructed using the relationship between the Fourier coefficients of the signal and the Stokes' parameters that represent the state. The experimental complexity, i.e., different wave plate rotation frequencies, scales linearly with the number of qubits.
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.
In quantum information transformation and quantum computation, the most critical issues are security and accuracy. These features, therefore, stimulate research on quantum state characterization. A characterization tool, Quantum state tomography, reconstructs the density matrix of an unknown quantum state. Theoretically, reconstructing an unknown state using this method can be arbitrarily accurate. However, this is less practical owing to the huge burden of measurements and data processing for large numbers of qubits. Even comprising an efficient estimator and a precise algorithm, an optimal tomographic framework can also be overburdened owing to the exponential growth of the measurements. Moreover, the consequential postprocessing of huge amounts of data challenges the capacity of computers. Thus, it is crucial to build an efficient framework that requires fewer measurements but yields an expected accuracy. To this end, we built a tomography schema by which only a few Pauli measurements enable an accurate tomographic reconstruction. Subsequently, this schema was verified as efficient and accurate through numerical simulations on the tomography of multi-qubit quantum states. Furthermore, this schema was proven to be robust through numerical simulations on a noisy superconducting qubit system. Therefore, the tomography schema paves an alternatively effective way to reconstruct the density matrix of a quantum state owing to its efficiency and accuracy, which are essential for quantum state tomography.
Quantum state tomography (QST) is a crucial tool for characterizing quantum states. However, performing full QST becomes impractical for reconstructing multiqubit density matrices since datasets and computational costs grow exponentially with qubit number. To bypass full QST, we introduce selective and efficient QST (SEEQST)—a method that enables efficient estimation of multiple selected elements of an arbitrary multiqubit density matrix. We show that any N -qubit density matrix can be partitioned into 2 N disjoint subsets, each containing 2 N elements. With SEEQST, any such subset can be accurately estimated from just two experiments with only single-qubit measurements. The complexity for estimating any subset remains constant regardless of Hilbert-space dimension, so, if desired, SEEQST can reconstruct the full density matrix, using 2 N + 1 − 1 experiments, where standard methods would use 3 N experiments. We provide a circuit decomposition for the SEEQST experiments, demonstrating that their maximum circuit depth scales logarithmically with N assuming all-to-all connectivity.