Virtual purification complements quantum error correction in quantum metrology

Abstract

Quantum resources enable one to achieve quantum-enhanced estimation sensitivity beyond its classical counterpart. Many studies mainly focus on reducing statistical error, under the assumption that one can always set an unbiased estimator. However, setting an unbiased estimator is not always feasible, especially when one cannot fully characterize noise. Such incomplete noise characterization induces a bias and eventually makes it impossible to attain the enhanced-estimation. In this work, we explore two systematic approaches; quantum error correction (QEC) and the virtual purification (VP) to reduce the bias, and compare their performance. First, we show that when the noise is indistinguishable from the signal, QEC cannot reduce the bias since it is impossible to construct a QEC code that corrects the noise while preserving the signal. We then show that VP can mitigate indistinguishable error that eventually enable a more accurate estimation compared to QEC. Our findings reveal that VP offers a robust alternative to QEC in scenarios where indistinguishable errors pose significant challenges. We then demonstrate that VP with a stabilizer state probe can efficiently suppress the bias under local depolarizing noise, thereby yielding a significant improvement in estimation performance compared to the QEC-based approach.

Figures (11)

• Figure 1: To estimate the signal $\phi$, an $N$-qubit quantum probe $\ket{\psi_{0}}$ is prepared and the probe imprints an identical signal $\phi$ into each qubit mode. The corresponding signal state is then measured, and this process is repeated $M$ times to gather $M$ measurement outcomes. An estimator $\phi^{\mathrm{est}}$ is then derived from the outcomes, providing an estimated value of $\phi$.
• Figure 2: (a)-(c) Schematic illustrations of noisy quantum metrology, QEC-assisted metrology, and VP-assisted metrology are shown, respectively. We consider an $N$-qubit quantum probe prepared in an initial state $\ket{\psi_{0}}$, where the same unknown signal phase is imprinted identically onto each qubit; this signal imprinting process is represented by the boxed phase shift. (a) In the noisy quantum metrology scheme, Pauli noise acts on the probe both before and after the signal imprinting process. (b) In the quantum error correction (QEC)-assisted scheme, ideal QEC control is applied immediately after the occurrence of Pauli noise. To illustrate that QEC cannot efficiently correct the bias induced by unknown Pauli noise, we assume that the QEC procedure perfectly corrects the effect of the first noise process. (c) In the virtual purification (VP)-assisted scheme, multiple identical copies of the noisy quantum state are prepared simultaneously, and VP is applied to mitigate the bias arising from the noise.
• Figure 3: $\ket{\psi(\phi)}$ is the signal state and the ideal space is the space in which the signal state lies in. While the single Pauli $\hat{X}$ error can be suppressed by both methods, the single Pauli $\hat{Z}$ error can only be mitigated by virtual purification method.
• Figure 4: (a)-(b) The square of biases (shown on a logarithmic scale) in the estimation of $\phi \in \left[-\frac{\pi}{20}, \frac{\pi}{20}\right]$, obtained using (a) $5$-qubit GHZ state (b) logical $0$ state of the $7$-qubit Steane code as quantum probes, in the presence of local depolarizing noise with the noise strengths that render the largest eigenvalue as $\lambda=0.7$. Solid lines indicate the theoretical values of the square of biases, while circles, triangles, and squares represent the total estimation error, based on the maximum likelihood estimation obtained with $M = 10^9$ samples. The "Error" case corresponds to the scenario where none of the schemes are applied. We emphasize that VP enables a substantial improvement in estimation performance—namely, a significantly reduced total estimation error in the large sample limit, since in this regime, the bias becomes the dominant contribution to the total estimation error.
• Figure S5: (a)-(c) The square of biases (with a logarithmic scale) of the $\phi$ estimation, where $\phi \in \left[-\frac{\pi}{20}, \frac{\pi}{20}\right]$, exploiting $5$-qubit GHZ state as a quantum probe in the presence of local dephasing noise with different noise strengths (represented by different dominant eigenvalues). Solid lines indicate the theoretical values of the square of biases, while circles, squares, and up-triangles represent the squared differences between the estimated values and the actual values from simulations, i.e., $(\phi^{\text{est}} - \phi)^2$, where $\phi^{\text{est}}$ are MLE obtained with $M = 10^9$ samples. We emphasize that due to the large sample size, which significantly reduces fluctuations in $\phi^{\text{est}}$, the quantities $(\phi^{\text{est}} - \phi)^2$ can be interpreted as the square of bias derived from the simulations.