Error-mitigated quantum metrology via enhanced virtual purification

Abstract

Quantum metrology stands as a leading application of quantum science and technology, yet noise often constrains its precision and sensitivity. In near-term quantum metrology, existing protocols largely depend on virtual state purification, but significant noise accumulation and additional noise from the implementations of these protocols can impede their effectiveness. We propose enhanced virtual channel purification to address these problems, yielding enhanced virtual state purification as a by-product. Within sequential quantum metrology schemes, our error analysis reveals substantial bias reduction and quantum advantages in sampling cost when the number of encoding channels is ${\mathcal{O}}(p^{-1})$, where $p$ represents the error rate of encoding channels. In this range, our methods demonstrate significant improvements in parameter estimation precision and robustness against practical noise, as evidenced by numerical simulations for both single- and multi-parameter tasks. Particularly, these methods can naturally extend beyond quantum metrology, indicating their broad applicability in quantum information and quantum computation.

Figures (14)

• Figure 1: Sequential feedback scheme of quantum metrology. Green boxes represent quantum gates, while red circles indicate local noise occurring immediately after each quantum gate. Particularly, the first gray box, the blue box and the final gray box represent the state preparation stage, parameter encoding stage and measurement preparation stage, respectively. Subsequently, the output state is measured on a computational basis.
• Figure 1: Parallel scheme of quantum metrology. Green boxes represent quantum gates, while red circles indicate local noise occurring immediately after each quantum gate. Particularly, the first gray box, the blue box and the final gray box represent the state preparation stage, parameter encoding stage and measurement preparation stage, respectively. Subsequently, the output state is measured on a computational basis.
• Figure 2: Circuit implementations of virtual purification methods. The 2nd-order implementations for (a) virtual state purification (VSP), (b) virtual channel purification (VCP), and (c) $L$-layer VCP are illustrated. Green boxes denote quantum gates, while red circles indicate noise present in the target quantum circuit. Here, the implementation of VSP and VCP is assumed to be noise-free. Quantum gates $P_i$ in the orange boxes are the tensor product of single-qubit random Pauli unitaries.
• Figure 2: Parameter estimation gaps under varying noise levels $p$ for local (a) depolarizing noise, (b) dephasing noise, and (c) amplitude damping noise. The error rates for single-qubit and two-qubit gates are 0.001 and $p\in[0.001,0.02]$. The gaps in the original quantum circuit, as well as those after applying VSP and VCP-1 under different $p$, are represented by the blue solid line, orange and green dashed lines, respectively. Besides, when noisy cSWAP gates are introduced, with an error rate of $5p$, the gaps following VSP and VCP-1 for different $p$ are depicted by orange and green solid lines, respectively.
• Figure 3: Noise categories for the virtual channel purification circuit. Green boxes represent quantum gates, and red circles indicate local noise. Noise is classified into four types indicated by numbered light blue boxes: ① noise in the control subsystem, ② noise in the last two subsystems between two cSWAP layers and ③ (④) noise in the ancillary (target) subsystem after the second cSWAP layer.

Theorems & Definitions (2)

• Theorem 1
• Lemma 1