Investigating Pure State Uniqueness in Tomography via Optimization

• URL: http://arxiv.org/abs/2501.00327v1
• Date: Tue, 31 Dec 2024 07:57:03 GMT
• Title: Investigating Pure State Uniqueness in Tomography via Optimization
• Authors: Jiahui Wu, Zheng An, Chao Zhang, Xuanran Zhu, Shilin Huang, Bei Zeng,
• Abstract summary: Quantum state (QST) is crucial for understanding and characterizing quantum systems through measurement data.<n>Traditional QST methods face scalability challenges, requiring $mathcalO(d2) measurements for a generaldimensionald state.<n>We develop a unified framework based on the Augmented Lagrangian Method (ALM) to address these issues.
• Score: 4.396311564396993
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring $\mathcal{O}(d^2)$ measurements for a general $d$-dimensional state. This complexity can be substantially reduced to $\mathcal{O}(d)$ in pure state tomography, indicating that full measurements are unnecessary for pure states. In this paper, we investigate the conditions under which a given pure state can be uniquely determined by a subset of full measurements, focusing on the concepts of uniquely determined among pure states (UDP) and uniquely determined among all states (UDA). The UDP determination inherently involves non-convexity challenges, while the UDA determination, though convex, becomes computationally intensive for high-dimensional systems. To address these issues, we develop a unified framework based on the Augmented Lagrangian Method (ALM). Specifically, our theorem on the existence of low-rank solutions in QST allows us to reformulate the UDA problem with low-rank constraints, thereby reducing the number of variables involved. Our approach entails parameterizing quantum states and employing ALM to handle the constrained non-convex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems and four-qubit symmetric states not only validate theoretical findings but also reveal the complete distribution of quantum states across three uniqueness categories: (A) UDA, (B) UDP but not UDA, and (C) neither UDP nor UDA. This work provides a practical approach for determining state uniqueness, advancing our understanding of quantum state reconstruction.

Related papers

• Measurement incompatibility and quantum steering via linear programming [18.995314010885533]
  We develop a hierarchy of linear programs that compute upper and lower bounds on the robustness of quantum measurements.<n>We show how our methods can be used to construct local hidden state models for states, or conversely to certify that a given state exhibits steering.
  arXiv  Detail & Related papers  (2025-06-03T16:28:25Z)
• Resource-Efficient Quantum Correlation Measurement: A Multicopy Neural Network Approach for Practical Applications [0.0]
  We propose an alternative strategy that reduces the required information by combining multicopy measurements with artificial neural networks. We have successfully measured two-qubit quantum correlations of Bell states subjected to a depolarizing channel. Our experiments, conducted with transmon qubits on IBMQ processors, quantified the violation of Bell's inequality and the negativity of two-qubit entangled states.
  arXiv  Detail & Related papers  (2024-11-08T18:07:23Z)
• Sufficient condition for universal quantum computation using bosonic circuits [44.99833362998488]
  We focus on promoting circuits that are otherwise simulatable to computational universality. We first introduce a general framework for mapping a continuous-variable state into a qubit state. We then cast existing maps into this framework, including the modular and stabilizer subsystem decompositions.
  arXiv  Detail & Related papers  (2023-09-14T16:15:14Z)
• A Variational Approach to Unique Determinedness in Pure-state Tomography [4.886657218460546]
  This study presents a new variational approach to examining unique determinedness (UD) in quantum state tomography. We put forward an effective algorithm that minimizes a specially defined loss function, enabling the differentiation between UD and non-UD measurement schemes. We discern an alignment between uniquely determined among pure states (UDP) and uniquely determined among all states (UDA) in qubit systems.
  arXiv  Detail & Related papers  (2023-05-18T08:49:11Z)
• End-to-end resource analysis for quantum interior point methods and portfolio optimization [63.4863637315163]
  We provide a complete quantum circuit-level description of the algorithm from problem input to problem output. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm.
  arXiv  Detail & Related papers  (2022-11-22T18:54:48Z)
• Regression of high dimensional angular momentum states of light [47.187609203210705]
  We present an approach to reconstruct input OAM states from measurements of the spatial intensity distributions they produce. We showcase our approach in a real photonic setup, generating up-to-four-dimensional OAM states through a quantum walk dynamics.
  arXiv  Detail & Related papers  (2022-06-20T16:16:48Z)
• Improved Quantum Algorithms for Fidelity Estimation [77.34726150561087]
  We develop new and efficient quantum algorithms for fidelity estimation with provable performance guarantees. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation. We prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general.
  arXiv  Detail & Related papers  (2022-03-30T02:02:16Z)
• Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States [11.58897666429261]
  We develop robustness intervals which are guaranteed to contain the output in the ideal setting. We demonstrate our results in the context of the variational quantum eigensolver (VQE) on noisy and noiseless simulations.
  arXiv  Detail & Related papers  (2021-10-19T08:13:03Z)
• Quantum verification and estimation with few copies [63.669642197519934]
  The verification and estimation of large entangled systems represents one of the main challenges in the employment of such systems for reliable quantum information processing. This review article presents novel techniques focusing on a fixed number of resources (sampling complexity) and thus prove suitable for systems of arbitrary dimension. Specifically, a probabilistic framework requiring at best only a single copy for entanglement detection is reviewed, together with the concept of selective quantum state tomography.
  arXiv  Detail & Related papers  (2021-09-08T18:20:07Z)
• Variational Quantum Algorithms for Trace Distance and Fidelity Estimation [7.247285982078057]
  We introduce hybrid quantum-classical algorithms for two distance measures on near-term quantum devices. First, we introduce the Variational Trace Distance Estimation (VTDE) algorithm. Second, we introduce the Variational Fidelity Estimation (VFE) algorithm.
  arXiv  Detail & Related papers  (2020-12-10T15:56:58Z)
• Amortized Conditional Normalized Maximum Likelihood: Reliable Out of Distribution Uncertainty Estimation [99.92568326314667]
  We propose the amortized conditional normalized maximum likelihood (ACNML) method as a scalable general-purpose approach for uncertainty estimation. Our algorithm builds on the conditional normalized maximum likelihood (CNML) coding scheme, which has minimax optimal properties according to the minimum description length principle. We demonstrate that ACNML compares favorably to a number of prior techniques for uncertainty estimation in terms of calibration on out-of-distribution inputs.
  arXiv  Detail & Related papers  (2020-11-05T08:04:34Z)
• Gaussian Process States: A data-driven representation of quantum many-body physics [59.7232780552418]
  We present a novel, non-parametric form for compactly representing entangled many-body quantum states. The state is found to be highly compact, systematically improvable and efficient to sample. It is also proven to be a universal approximator' for quantum states, able to capture any entangled many-body state with increasing data set size.
  arXiv  Detail & Related papers  (2020-02-27T15:54:44Z)
• Quantum State Interferography [0.0]
  In this letter, we present an interferometric method, in which, any qubit state, whether mixed or pure, can be inferred from the visibility, phase shift and average intensity of an interference pattern using a single shot measurement. We experimentally implement our method with high fidelity using the polarisation degree of freedom of light.
  arXiv  Detail & Related papers  (2020-02-18T09:32:47Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.