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arXiv:2409.05676

Circuit optimization of qubit IC-POVMs for shadow estimation

You Zhou, Qing Liu, You Zhou

Published Sep 9, 2024• 0 citations
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Abstract

Extracting information from quantum systems is crucial in quantum physics and information processing. Methods based on randomized measurements, like shadow estimation, show advantages in effectively achieving such tasks. However, randomized measurements require the application of random unitary evolution, which unavoidably necessitates frequent adjustments to the experimental setup or circuit parameters, posing challenges for practical implementations. To address these limitations, positive operator-valued measurements (POVMs) have been integrated to realize real-time single-setting shadow estimation. In this work, we advance the POVM-based shadow estimation by reducing the CNOT gate count for the implementation circuits of informationally complete POVMs (IC-POVMs), in particular, the symmetric IC-POVMs (SIC-POVMs), through the dimension dilation framework. We show that any single-qubit minimal IC-POVM can be implemented using at most 2 CNOT gates, while an SIC-POVM can be implemented with only 1 CNOT gate. In particular, we provide a concise form of the compilation circuit of any SIC-POVM along with an efficient algorithm for the determination of gate parameters. Moreover, we apply the optimized circuit compilation to shadow estimation, showcasing its noise-resilient performance and highlighting the flexibility in compiling various SIC-POVMs. Our work paves the way for the practical applications of qubit IC-POVMs on quantum platforms.

DOI
10.48550/arxiv.2409.05676

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Extracting information from quantum systems is crucial in quantum physics and information processing. Methods based on randomized measurements, like shadow estimation, show advantages in effectively achieving such tasks. However, randomized measurements require the application of random unitary evolution, which unavoidably necessitates frequent adjustments to the experimental setup or circuit parameters, posing challenges for practical implementations. To address these limitations, positive operator-valued measurements (POVMs) have been integrated to realize real-time single-setting shadow estimation. In this work, we advance the POVM-based shadow estimation by reducing the CNOT gate count for the implementation circuits of informationally complete POVMs (IC-POVMs), in particular, the symmetric IC-POVMs (SIC-POVMs), through the dimension dilation framework. We show that any single-qubit minimal IC-POVM can be implemented using at most 2 CNOT gates, while an SIC-POVM can be implemented with only 1 CNOT gate. In particular, we provide a concise form of the compilation circuit of any SIC-POVM along with an efficient algorithm for the determination of gate parameters. Moreover, we apply the optimized circuit compilation to shadow estimation, showcasing its noise-resilient performance and highlighting the flexibility in compiling various SIC-POVMs. Our work paves the way for the practical applications of qubit IC-POVMs on quantum platforms.
In quantum information processing, using positive operator--valued measurements (POVMs) enables the design of general measurement schemes and efficient estimation methods with fixed circuits, but the CNOT-gate count during compilation becomes a bottleneck. This study shows that any single-qubit minimal informationally complete POVM (IC POVM) can be realized with at most two CNOT gates---and a symmetric case (SIC POVM) requires only one. This is achieved by virtually inserting gates to adjust unrelated parameters in the compilation circuit. The authors also provide a concise compilation circuit for single-qubit SIC POVMs, paving the way for practical application.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather nontrivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Quantum computers solve ever more complex tasks using steadily growing system sizes.Characterizing these quantum systems is vital, yet becoming increasingly challenging.The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge.The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)-a daunting bottleneck given the scale of existing and nearterm quantum devices.Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs).We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion-without requiring ancillary ions to assist in measurements.More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot.Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination.SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods.The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state.Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running).We demonstrate this on an eight-qubit entangled state (which has 2 8 • 2 8 -1 = 65 535 degrees of freedom), as well as for fast state identification.All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose are symmetric, informationally complete POVMs. SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather non-trivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been emphasized by the Zeilinger group in Vienna. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Symmetric informationally complete positive operator-valued measurement (SIC-POVM) is one important class of quantum measurement which is crucial for various quantum information processing tasks. SIC-POVMs have the advantage of providing an unbiased estimator for quantum states with the minimal number of outcomes needed for full tomography. We present an experimental approach on a photonic quantum walk which can be used to implement SIC-POVMs on a single-qbubit. The projection measurements of single-photons' positions correspond to elements of SIC-POVM on the polarization of single-photons.
Current quantum computers have the potential to overcome classical computational methods, however, the capability of the algorithms that can be executed on noisy intermediate-scale quantum devices is limited due to hardware imperfections. Estimating the state of a qubit is often needed in different quantum protocols, due to the lack of direct measurements. In this paper, we consider the problem of estimating the quantum state of a qubit in a quantum processing unit without conducting direct measurements of it. We consider a parameterized measurement model to estimate the quantum state, represented as a quantum circuit, which is optimized using the quantum tomographic transfer function. We implement and test the circuit using the quantum computer of the Technical Research Centre of Finland as well as an IBM quantum computer. We demonstrate that the set of positive operator-valued measurements used for the estimation is symmetric and informationally complete. Moreover, the resources needed for qubit estimation are reduced when direct measurements are allowed, keeping the symmetric property of the measurements.