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arXiv:2501.18546 · Physical Review A

Mitigating shot noise in local overlapping quantum tomography with semidefinite programming

Zherui Jerry Wang, David Dechant, Yash J. Patel, Jordi Tura

Published May 27, 2025• 1 citation
arXivHTMLPDF

Abstract

Reduced density matrices (RDMs) are fundamental in quantum information processing, allowing the computation of local observables, such as energy and correlation functions, without the exponential complexity of fully characterizing quantum states. In the context of near-term quantum computing, RDMs provide sufficient information to effectively design variational quantum algorithms. However, their experimental estimation is challenging, as it involves preparing and measuring quantum states in multiple bases, a resource-intensive process susceptible to producing nonphysical RDMs due to shot noise from limited measurements. To address this, we propose a method to mitigate shot noise by reenforcing certain physicality constraints on RDMs. While verifying RDM compatibility with a global state is quantum Merlin-Arthur complete, we relax this condition by enforcing compatibility constraints up to a certain level using a polynomial-size semidefinite program to reconstruct overlapping RDMs from simulated data. Our approach yields, on average, tighter bounds for the same number of measurements compared to tomography without compatibility constraints. We demonstrate the versatility and efficacy of our method by integrating it into an algorithmic cooling procedure to prepare low-energy states of local Hamiltonians. Simulations on frustrated Hamiltonians reveal notable improvements in accuracy and resource efficiency, highlighting the potential of our approach for practical applications in near-term quantum computing.

DOI
10.1103/physreva.111.052444

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We introduce a correlated measurement noise model that can be efficiently described and characterized, and which admits effective noise-mitigation on the level of marginal probability distributions. Noise mitigation can be performed up to some error for which we derive upper bounds. Characterization of the model is done efficiently using Diagonal Detector Overlapping Tomography -- a generalization of the recently introduced Quantum Overlapping Tomography to the problem of reconstruction of readout noise with restricted locality. The procedure allows to characterize kkk-local measurement cross-talk on NNN-qubit device using O(k2klog(N))O(k2^klog(N))O(k2klog(N)) circuits containing random combinations of X and identity gates. We perform experiments on 15 (23) qubits using IBM's (Rigetti's) devices to test both the noise model and the error-mitigation scheme, and obtain an average reduction of errors by a factor >22>22>22 (>5.5>5.5>5.5) compared to no mitigation. Interestingly, we find that correlations in the measurement noise do not correspond to the physical layout of the device. Furthermore, we study numerically the effects of readout noise on the performance of the Quantum Approximate Optimization Algorithm (QAOA). We observe in simulations that for numerous objective Hamiltonians, including random MAX-2-SAT instances and the Sherrington-Kirkpatrick model, the noise-mitigation improves the quality of the optimization. Finally, we provide arguments why in the course of QAOA optimization the estimates of the local energy (or cost) terms often behave like uncorrelated variables, which greatly reduces sampling complexity of the energy estimation compared to the pessimistic error analysis. We also show that similar effects are expected for Haar-random quantum states and states generated by shallow-depth random circuits.
Abstract Quantum state tomography is one of the core techniques in quantum information science, aiming to reconstruct the density matrix of an unknown quantum state from measurement data. To address the limitations of conventional quantum state reconstruction methods in terms of noise resistance and computational efficiency, this work proposes an innovative Residual-Convolutional Quantum Denoising Autoencoder model (RC-QDAE). The results demonstrate that the model can reconstruct physical density matrices with fidelities comparable to those achieved by traditional full-data methods while requiring only 80% of the tomographic measurement data, thereby significantly alleviating the dependence on measurement resources in large-scale quantum system characterization. In three-qubit systems, the reconstruction speed of the proposed model achieves an improvement of three orders of magnitude over the maximum likelihood estimation method, effectively overcoming the excessive computational overhead inherent to traditional approaches. Moreover, the model incorporates a joint training mechanism with multiple typical noise types during the training phase, exhibiting stronger robustness when processing cross-noise data. Further evaluations show that under different noise intensities, the model consistently produces density matrices with higher fidelities than those obtained by conventional methods, highlighting its superior noise resistance and generalization capability.