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Communications in Theoretical Physics

Direct measurement for general quantum states using parametric quantum circuits

Zhiyuan Wang, Zijing Zhang, Yuan Zhe Zhao

Published Nov 1, 2022• 2 citations
Publisher

Abstract

Abstract In the field of quantum information, the acquisition of information for unknown quantum states is very important. When we only need to obtain specific elements of a state density matrix, the traditional quantum state tomography will become very complicated, because it requires a global quantum state reconstruction. Direct measurement of the quantum state allows us to obtain arbitrary specific matrix elements of the quantum state without state reconstruction, so direct measurement schemes have obtained extensive attention. Recently, some direct measurement schemes based on weak values have been proposed, but extra auxiliary states in these schemes are necessary and it will increase the complexity of the practical experiment. Meanwhile, the post-selection process in the scheme will reduce the utilization of resources. In order to avoid these disadvantages, a direct measurement scheme without auxiliary states is proposed in this paper. In this scheme, we achieve the direct measurement of quantum states by using quantum circuits, then we extend it to the measurement of general multi-particle states and complete the error analysis. Finally, when we take into account the dephasing of the quantum states, we modify the circuits and the modified circuits still work for the dephasing case.

DOI
10.1088/1572-9494/ac9f0e

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Abstract In the field of quantum information, the acquisition of information for unknown quantum states is very important. When we only need to obtain specific elements of a state density matrix, the traditional quantum state tomography will become very complicated, because it requires a global quantum state reconstruction. Direct measurement of the quantum state allows us to obtain arbitrary specific matrix elements of the quantum state without state reconstruction, so direct measurement schemes have obtained extensive attention. Recently, some direct measurement schemes based on weak values have been proposed, but extra auxiliary states in these schemes are necessary and it will increase the complexity of the practical experiment. Meanwhile, the post-selection process in the scheme will reduce the utilization of resources. In order to avoid these disadvantages, a direct measurement scheme without auxiliary states is proposed in this paper. In this scheme, we achieve the direct measurement of quantum states by using quantum circuits, then we extend it to the measurement of general multi-particle states and complete the error analysis. Finally, when we take into account the dephasing of the quantum states, we modify the circuits and the modified circuits still work for the dephasing case.
Efficient acquiring information from a quantum state is important for research in fundamental quantum physics and quantum information applications. Instead of using standard quantum state tomography method with reconstruction algorithm, weak values were proposed to directly measure density matrix elements of quantum state. Recently, similar to the concept of weak value, modular values were introduced to extend the direct measurement scheme to nonlocal quantum wavefunction. However, this method still involves approximations, which leads to inherent low precision. Here, we propose a new scheme which enables direct measurement for ideal value of the nonlocal density matrix element without taking approximations. Our scheme allows more accurate characterization of nonlocal quantum states, and therefore has greater advantages in practical measurement scenarios.
A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. However, the direct measurement protocols to date are primarily based on weak or strong measurements with an ancillary pointer, which interacts with the investigated system to extract information about the specified elements. Here, we present a direct measurement scheme based on phase-shifting operations which do not need ancillary pointers. In this method, estimates of at most six expectation values of projective observables suffice to determine any specific element of an unknown quantum density matrix. A concrete quantum circuit to implement this direct measurement protocol for multiqubit states is provided, which is composed of just single-qubit gates and two multiqubit controlled-phase gates. This scheme is also extended for the direct measurement of the density matrix of continuous-variable quantum states. Our method can be used in quantum information applications where only partial information about the quantum state needs to be extracted, for example, problems such as entanglement witnessing, fidelity estimation of quantum systems, and quantum coherence estimation.
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)O(d^2)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.
New techniques based on weak measurements have recently been introduced to the field of quantum state reconstruction. Some of them allow the direct measurement of each matrix element of an unknown density operator and need only O(d) different operations, compared to d^{2} linearly independent projectors in the case of standard quantum state tomography, for the reconstruction of an arbitrary mixed state. However, due to the weakness of these couplings, these protocols are approximated and prone to large statistical errors. We propose a method which is similar to the weak measurement protocols but works regardless of the coupling strength: our protocol is not approximated and thus improves the accuracy and precision of the results with respect to weak measurement schemes. We experimentally apply it to the polarization state of single photons and compare the results to those of preexisting methods for different values of the coupling strength. Our results show that our method outperforms previous proposals in terms of accuracy and statistical errors.
The measurement of unknown quantum states is crucial in quantum information science. Standard quantum state tomography can retrieve information regarding unknown quantum states. However, as the number of qubits increases, the measurement basis required for standard tomography increases exponentially. When we are only interested in partial information of the quantum state, such as coherence and entanglement, global state reconstruction is not necessary. Therefore, direct characterization schemes are required to obtain the information of interest without complete state tomography. In this Letter, we present a circuit scheme with ancilla-only measurements for the direct characterization of multi-particle states that requires only four projection probabilities, independent of the system size. Ancilla-only measurements refer to a strategy where all readouts are on ancillas, never directly on the target qubits. Target state information is indirectly extracted via target–ancilla coupling. Moreover, our scheme can automatically reset the target qubits, thereby providing an immediately available quantum register for subsequent tasks and avoiding the target quantum system initialization required for the next task. Finally, numerical simulations and experimental verification were performed on a quantum cloud platform, confirming the validity of the theoretical scheme.
Recent work has revealed that the wave function of a pure state can be measured directly and that complementary knowledge of a quantum system can be obtained simultaneously by weak measurements. However, the original scheme applies only to pure states, and it is not efficient because most of the data are discarded by post-selection. Here, we propose tomography schemes for pure states and for mixed states via weak measurements, and our schemes are more efficient because we do not discard any data. Furthermore, we demonstrate that any matrix element of a general state can be directly read from an appropriate weak measurement. The density matrix (with all of its elements) represents all that is directly accessible from a general measurement.