Randomized measurements for multi-parameter quantum metrology

Abstract

The optimal quantum measurements for estimating different unknown parameters in a parameterized quantum state are usually incompatible with each other. Traditional approaches to addressing the measurement incompatibility issue, such as the Holevo Cramér--Rao bound, suffer from multiple difficulties towards practical applicability, as the optimal measurement strategies are usually state-dependent, difficult to implement and also take complex analyses to determine. Here we study randomized measurements as a new approach for multi-parameter quantum metrology. We show quantum measurements on single copies of quantum states given by $3$-designs perform near-optimally when estimating an arbitrary number of parameters in pure states and more generally, {approximately low-rank well-conditioned states}, whose metrological information is largely concentrated in a low-dimensional subspace. The near-optimality is also shown in estimating the maximal number of parameters for three types of mixed states that are well-conditioned on their supports. Examples of fidelity estimation and Hamiltonian estimation are explicitly provided to demonstrate the power and limitation of randomized measurements in multi-parameter quantum metrology.

Figures (3)

• Figure 1: Schematics of randomized measurements for multi-parameter quantum metrology. The measurement is taken on individual copies of the same parameterized state $\rho_\theta:=\rho_{\theta_1,\cdots,\theta_m}$. The goal is to obtain estimates $\hat{\theta}=(\hat{\theta}_1,\cdots,\hat{\theta}_m)$ with the highest precision as the number of copies approaches infinity. In all of our theorems showing the near-optimality of randomized measurements, it suffices to take the measurement to be a $3$-design.
• Figure 2: Bloch sphere. Any projective measurement onto the basis in the $x$-$y$ plane is optimal in the task of estimating parameter $\theta$ in $\ket{\psi_\theta} = \frac{\ket{0}+e^{i\theta}\ket{1}}{\sqrt{2}}$, including the $x$-basis $\{\ket{\pm} = \frac{\ket{0}\pm\ket{1}}{\sqrt{2}}\}$ and the $y$-basis $\{\ket{\pm i} = \frac{\ket{0}\pm i\ket{1}}{\sqrt{2}}\}$.
• Figure 3: Simulation of a $3$-qubit pure state fidelity estimation task comparing standard and local shadow estimators. The target state $\ket{\phi}$ and four unknown states with different infidelities all come from a $3$-parameter family of pure states described in Sec. \ref{['sec:numerics']}. For each unknown state, we build a dataset by uniformly sampling 100,000 random Clifford and measuring each for 10,000 shots. We then subsample $500$ batches from the dataset, each consists of $N=5000$ random Clifford and a single shot of measurement. (Left.) Comparison of theoretical (square roots of \ref{['eq:local_fid_var']} and \ref{['eq:shadow_fid_var']}, divided by $\sqrt{N}$) and empirical (averaged over $500$ batches of subsamples) root mean squared error (RMSE) of estimators. Error bars depict one standard error estimated via $200$ round of bootstrap sampling efron1994introduction. (Right.) Violin plots (smoothed histograms hintze1998violin) of estimation errors from $500$ batches of subsamples for the four unknown states labeled by the corresponding infidelity. The variance of our local shadow estimator decreases as the true infidelity decreases, whereas the variance of the standard shadow estimator remains nearly constant.

Theorems & Definitions (15)

• Definition 1: Near-optimality
• Definition 2: Weak near-optimality
• Theorem 1
• proof : Proof of \ref{['thm:pure']}
• Definition 3: Approximately low-rank well-conditioned state
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4