Approximate Message Passing for Quantum State Tomography

TL;DR

The paper tackles the challenge of quantum state tomography (QST) in the regime where the state is low-rank, addressing the exponential growth of the Hilbert space with system size. It introduces an AMP-based approach tailored for QST by normalizing the sensing operator, enforcing positive semidefinite and unit-trace density matrices via a projected SVT denoiser, and applying damping to stabilize convergence. The authors demonstrate both numerical and experimental advantages: AMP achieves lower NMSE and higher state fidelity than baseline methods (MLE and MiFGD) across multiple 8-qubit states and realistic noise scenarios, and their measurement-setting strategy substantially reduces quantum runtime in IBM Kingston experiments. The work also provides insights into how noise models influence fidelity predictions and outlines pathways for extending AMP-QST to broader tomography tasks. Overall, AMP offers a scalable, structure-exploiting framework for accurate low-rank QST with practical implications for larger quantum systems.

Abstract

Quantum state tomography (QST) is an indispensable tool for characterizing many-body quantum systems. However, due to the exponential scaling cost of the protocol with system size, many approaches have been developed for quantum states with specific structure, such as low-rank states. In this paper, we show how approximate message passing (AMP), a compressed sensing technique, can be used to perform low-rank QST. AMP provides asymptotically optimal performance guarantees for large systems, which suggests its utility for QST. We discuss the design challenges that come with applying AMP to QST, and show that by properly designing the AMP algorithm, we can reduce the reconstruction infidelity by over an order of magnitude compared to existing approaches to low-rank QST. We also performed tomographic experiments on IBM Kingston and considered the effect of device noise on the reliability of the predicted fidelity of state preparation. Our work advances the state of low-rank QST and may be applicable to other quantum tomography protocols.

Figures (7)

• Figure 1: A comparison of different AMP approaches for QST. We reconstruct a rank-3 5-qubit random state ($M=384$ observables, $N=1024$ shots per observable) using variations of the AMP algorithm and show the reconstruction quality as measured by the normalized mean squared error (NMSE) and state fidelity. The baseline AMP approach (orange crosses) diverges. By replacing the sensing operator $\mathcal{A}$\ref{['eq:amp-alg-pseudodata']} with the normalized version $\widetilde{\mathcal{A}}$\ref{['eq:def-normalized-a']} (green squares), the estimator $\widehat{\rho}$ successfully reconstructs the true density matrix $\rho^*$. In order to incorporate the physical constraints of $\rho^*$, we use the projected singular value thresholding ($\mathop{\mathrm{PSVT}}\nolimits$) denoiser \ref{['eq:def-psvt']}. Without damping (purple pluses), the $\mathop{\mathrm{PSVT}}\nolimits$-based AMP algorithm does not recover $\rho^*$. With damping \ref{['eq:damping']} (blue circles), the $\mathop{\mathrm{PSVT}}\nolimits$-based denoiser recovers $\rho^*$ with lower NMSE and higher state fidelity than the $\mathop{\mathrm{SVT}}\nolimits$-based approach. The inset in the NMSE plot is a zoom-in on the normalized, projected, and projection-plus-damping results.
• Figure 2: Comparison in reconstruction quality for 8-qubit states between approximate message passing (AMP), maximum likelihood estimation (MLE), and momentum-inspired factored gradient descent (MiFGD). We consider the GHZ, Hadamard, and W states, along with a random rank-1 and random rank-3 state. The shot count for each observable is fixed at $N=1024$. The $M$ observables are randomly sampled from the $d^2$ Pauli observables. Shaded regions indicate minimum and maximum state infidelity over 10 trials. AMP consistently outperforms both MLE and MiFGD, in some cases reducing the state infidelity by almost two orders of magnitude.
• Figure 3: Comparison in reconstruction quality for 8-qubit states between approximate message passing (AMP), maximum likelihood estimation (MLE), and momentum-inspired factored gradient descent (MiFGD). We consider the GHZ, Hadamard, and W states, along with a random rank 1 and random rank 3 state. We fix $M = 16384$. The $M$ observables are randomly sampled from the $d^2$ Pauli observables. Shaded regions indicate minimum and maximum state infidelity over 10 trials. AMP consistently outperforms both MLE and MiFGD, in some cases reducing the state infidelity by over an order of magnitude.
• Figure 4: Recovering rank-$n$ states with AMP. We run AMP on random $n$-qubit states of rank $n$ for $n=6$, $n=7$, and $n=8$ with $N=4096$ shots per observable. As $n$ increases, so does the number of observables required to recover the random state. However, the fraction $M/4^n$ required to reach a reconstruction state fidelity of $10^{-1}$ decreases with increasing $n$. Shaded regions show the maximum and minimum state infidelity over 10 trials.
• Figure 5: Comparison between sampling circuits based on measurement settings and sampling circuits based on observables. Recovering a $\rho_{\text{Random}(n,1)}$ state with $N=4096$ shots per circuit. When sampling circuits based on measurement settings instead of observables, each circuit can be used to estimate the expectation values of $d$ observables, which reduces the number of circuits that need to be run in order to perform QST. We randomly sample circuits based on either measurement settings or observables and report the average state infidelity. Shaded regions show the minimum and maximum state infidelity. We observe a large reduction in the number of circuits needed to perform QST to a desired fidelity when circuits are sampled based on measurement settings instead of observables.