Measuring entanglement without local addressing in quantum many-body simulators via spiral quantum state tomography

Abstract

Quantum state tomography serves as a key tool for identifying quantum states generated in quantum computers and simulators, typically involving local operations on individual particles or qubits to enable independent measurements. However, this approach requires an exponentially larger number of measurement setups as quantum platforms grow in size, highlighting the necessity of more scalable methods to efficiently perform quantum state estimation. Here, we present a tomography scheme that scales far more efficiently and, remarkably, eliminates the need for local addressing of single constituents before measurements. Inspired by the ``spin-spiral'' structure in magnetic materials, our scheme combines a series of measurement setups, each with different spiraling patterns, with compressed sensing techniques. The results of the numerical simulations demonstrate a high degree of tomographic efficiency and accuracy. Additionally, we show how this method is suitable for the measurement of specific entanglement properties of interesting quantum many-body states, such as entanglement entropy, under various realistic experimental conditions. This method offers a positive outlook across a wide range of quantum platforms, including those in which precise individual operations are challenging, such as optical lattice systems.

Figures (9)

• Figure 1: Spiral measurements. (a) Illustrations of the three spiral measurement sets. The blue arrows indicate the local measurement axis at each qubit. (b) Pitch angle $q$ in the spiral plane of $\tilde{M}^{XY}(q)$.
• Figure 2: Numerical tests of spiral QST. Sample-averaged fidelity and trace distance for compressed-sensing tomography of (a) random pure states (rank $r=1$) and (b) random mixed states (rank $r=3$) of 8 qubits, plotted as functions of the number of independent measurements $m$ scaled by $d^2=4^8$. The sampled states include 5 percent depolarizing noise ($\gamma=0.05$) and measurement Gaussian statistical noise with a standard deviation of $\sigma=0.1/d$. Spiral and Pauli measurements are compared in each case. For $r=3$, the results of Pauli-based tomography under the same conditions were also reported in previous work Gross2010-mz.
• Figure 3: Magnetic field gradient method for spiral QST. (a) Magnetic field gradient used to create spiral-spin settings. Typical experimental noise arises from fluctuations in the strength of the magnetic field over time. These fluctuations cause variations in the zero point of the gradient, modeled as Gaussian fluctuations with a mean of 0 and a standard deviation $\sigma_{\rm zp}$ (in units of lattice spacing). (b) Protocol for numerical simulations of spiral QST with magnetic field gradient. We simulate a realistic experimental process to measure observables by averaging the outcomes over many shots of repeatedly prepared target states for each experimental setting of spiral plane $\alpha\beta=XY$, $YZ$, or $ZX$ and pitch $q$. Each repetition is subject to fluctuations in the magnetic field gradient, introducing noise into the expectation values of the spiral operators.
• Figure 4: Spiral QST for the ground state of the NN Heisenberg chain. Sample-averaged fidelity and trace distance for spiral compressed-sensing QST of the ground state of the 8-site Heisenberg chain with NN antiferromagnetic spin-exchange coupling, plotted as functions of the number of measurements $m$ scaled by $d^2=4^8$. In (a), the results for different amplitudes of magnetic field fluctuations ($\sigma_{\rm zp}=0.01,0.1,0.5$) are compared, with the number of repetitions for evaluating each expectation value fixed at ${\rm reps}=500$. The dependence on $\sigma_{\rm zp}$ is essentially absent, which is supported by the symmetry argument given in the main text. In (b), the results for different numbers of repetitions (${\rm reps}=100, 500, 1000$) are compared (the amplitude of magnetic field fluctuations is chosen as $\sigma_{\rm zp}=0.1$ for concreteness). We adopt the strategy of progressively including expected relevant pitch angles in symmetric fashion; the data points with the fewest measurements correspond to the spiral QST with $\{l=0\}$, followed by $\{l = 0, 1,-1\}$, $\{l = 0, 1,-1, 2,-2\}$, and so on.
• Figure 5: Spiral QST for the ground state of the NN Heisenberg + DM chain. Sample-averaged fidelity and trace distance for spiral compressed-sensing QST of the ground state of the 8-site Heisenberg chain with NN ferromagnetic spin-exchange coupling and DM interactions of equal strength, plotted as functions of the number of measurements $m$ scaled by $d^2=4^8$. The results for different amplitudes of magnetic field fluctuations ($\sigma_{\rm zp}=0.01,0.1,0.5,1.0$) are compared, with the number of repetitions for evaluating each expectation value fixed at ${\rm reps}=500$; note that the data for $\sigma_{\rm zp}=0.01$ and $\sigma_{\rm zp}=0.1$ are nearly indistinguishable. The data points with the fewest measurements correspond to the spiral QST with $l=0$, followed by $l = 0, 1$, $l = 0, 1, 2$, and so on.