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arXiv:2111.04359

K-sparse Pure State Tomography with Phase Estimation

Burhan Gülbahar

Published Nov 8, 2021• 0 citations
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Abstract

Quantum state tomography (QST) for reconstructing pure states requires exponentially increasing resources and measurements with the number of qubits by using state-of-the-art quantum compressive sensing (CS) methods. In this article, QST reconstruction for any pure state composed of the superposition of $K$ different computational basis states of $n$ qubits in a specific measurement set-up, i.e., denoted as $K$-sparse, is achieved without any initial knowledge and with quantum polynomial-time complexity of resources based on the assumption of the existence of polynomial size quantum circuits for implementing exponentially large powers of a specially designed unitary operator. The algorithm includes $\mathcal{O}(2 \, / \, \vert c_{k}\vert^2)$ repetitions of conventional phase estimation algorithm depending on the probability $\vert c_{k}\vert^2$ of the least possible basis state in the superposition and $\mathcal{O}(d \, K \,(log K)^c)$ measurement settings with conventional quantum CS algorithms independent from the number of qubits while dependent on $K$ for constant $c$ and $d$. Quantum phase estimation algorithm is exploited based on the favorable eigenstructure of the designed operator to represent any pure state as a superposition of eigenvectors. Linear optical set-up is presented for realizing the special unitary operator which includes beam splitters and phase shifters where propagation paths of single photon are tracked with which-path-detectors. Quantum circuit implementation is provided by using only CNOT, phase shifter and $- π\, / \, 2$ rotation gates around X-axis in Bloch sphere, i.e., $R_{X}(- π\, / \, 2)$, allowing to be realized in NISQ devices. Open problems are discussed regarding the existence of the unitary operator and its practical circuit implementation.

DOI
10.48550/arxiv.2111.04359

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Quantum state tomography (QST) is a crucial technique for characterizing quantum states, forming the bedrock of quantum information processing and quantum metrology. However, traditional QST relies on a large number of measurements, making it computationally expensive and experimentally demanding. This work addresses this challenge by proposing a novel algorithm that integrates compressed sensing (CS) and Bayesian inference to achieve accurate quantum state reconstruction from limited measurement data. The core idea is to exploit the inherent sparsity of quantum states and leverage CS principles to significantly reduce the required measurement count. The algorithm utilizes a sparse dictionary learned from prior knowledge of the quantum state and employs Bayesian inference to estimate the state parameters, providing a robust and efficient reconstruction process. The results demonstrate the feasibility and effectiveness of this approach, offering a pathway towards more practical and accessible QST methods. Key mathematical formulations are presented, including the Bayesian inference framework and the CS reconstruction process.
Quantum state tomography (QST) is a crucial technique for characterizing the state of quantum systems. However, traditional QST requires a large number of measurements, leading to significant experimental overhead in terms of time, resources, and data acquisition. This work proposes a novel approach to QST by leveraging compressed sensing (CS) techniques. CS allows for accurate reconstruction of the quantum state from a significantly reduced set of measurements, thereby mitigating the experimental burden. We employ sparsity regularization, a key component of CS, to minimize the number of measurements needed. The theoretical framework and the underlying principles are presented, focusing on the mathematical formulation of the problem and the optimization strategies involved. This method offers a pathway to more efficient and cost-effective QST, particularly for complex quantum states or when limited experimental resources are available. We demonstrate the potential of this approach through a detailed mathematical analysis.
Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an nnn-qubit state supported on kkk basis elements, the protocol reconstructs all amplitudes using 1+2(k−1)1 + 2(k-1)1+2(k−1) measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.
Abstract Compressed sensing (CS) has been verified that it offers a significant performance improvement for large quantum systems comparing with the conventional quantum tomography approaches, because it reduces the number of measurements from O ( d 2 ) to O ( rd log( d )) in particular for quantum states that are fairly pure. Yet few algorithms have been proposed for quantum state tomography using CS specifically, let alone basis analysis for various measurement sets in quantum CS. To fill this gap, in this paper an efficient and robust state reconstruction algorithm based on compressive sensing is developed. By leveraging the fixed point equation approach to avoid the matrix inverse operation, we propose a fixed-point alternating direction method algorithm for compressive quantum state estimation that can handle both normal errors and large outliers in the optimization process. In addition, properties of five practical measurement bases (including the Pauli basis) are analyzed in terms of their coherences and reconstruction performances, which provides theoretical instructions for the selection of measurement settings in the quantum state estimation. The numerical experiments show that the proposed algorithm has much less calculating time, higher reconstruction accuracy and is more robust to outlier noises than many existing state reconstruction algorithms.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
Quantum computing shows promise for addressing computationally intensive problems but is constrained by the exponential resource requirements of general quantum state tomography (QST), which fully characterizes quantum states through parameter estimation. We introduce the QST with Chebyshev polynomials, an approximate tomography method for pure quantum states encoding complex-valued functions. This method reformulates tomography as the estimation of Chebyshev expansion coefficients, expressed as inner products between the target quantum state and Chebyshev basis functions, measured using the Hadamard test circuit. By treating the truncation order of the Chebyshev polynomials as a controllable parameter, the method provides a practical balance between efficiency and accuracy. For quantum states encoding functions dominated by large-scale features, such as those representing fluid flow fields, appropriate truncation enables faithful reconstruction of the dominant components via quantum circuits with linear depth, while keeping both measurement repetitions and post-processing independent of qubit count, in contrast to the exponential scaling of full measurement-based QST methods. Validation on analytic functions and numerically generated flow-field data demonstrates accurate reconstruction and effective extraction of large-scale features, indicating the method's suitability for systems governed by macroscopic dynamics.