Metrologically advantageous states: long-range entanglement and asymmetric error correction

TL;DR

This work establishes a unified framework linking quantum metrology to long-range entanglement and quantum error correction. It shows that surpassing the standard quantum limit with local Hamiltonians requires long-range entanglement, and proves no-go theorems for metrological advantage in broad classes of codes, including nondegenerate and CSS QLDPC codes with growing distance. Importantly, it identifies asymmetric error-correction structures as a viable route to Heisenberg-limited scaling, with explicit constructions based on classical LDPC codes and asymmetric toric codes. The results clarify the fundamental tension between metrological power and noise robustness and provide practical guidelines for designing metrologically advantageous, yet robust, many-body states.

Abstract

Quantum metrology aims to exploit many-body quantum states to achieve parameter-estimation precision beyond the standard quantum limit. For unitary parameter encoding generated by local Hamiltonians, such enhancement is characterized by superlinear scaling of the quantum Fisher information (QFI) with system size. Despite extensive progress, a systematic understanding of which many-body quantum states can exhibit this scaling has remained elusive. Here, we develop a general framework that connects metrological performance to long-range entanglement, state-preparation complexity, and quantum error-correction properties. We prove that super-linear QFI scaling necessarily requires long-range entanglement by deriving rigorous complexity-dependent upper bounds on the QFI. We further show that, for two broad classes of quantum error-correcting codes, nondegenerate codes and Calderbank--Shor--Steane quantum low-density parity-check codes, a nonconstant code distance precludes super-linear QFI scaling for a wide class of local Hamiltonians, revealing a fundamental incompatibility between metrological sensitivity and protection against local noise. Finally, we identify constructive routes that evade this obstruction by exploiting asymmetric code structures. In particular, we show that states associated with classical low-density parity-check codes, as well as asymmetric toric code states, both having asymmetric logical distances, can achieve Heisenberg-limited scaling. Together, our results establish long-range entanglement and asymmetric error correction as the essential resource underlying quantum metrology and clarify the interplay among state complexity, error correction, and metrological power.

Figures (4)

• Figure 1: Overview of our results. The blue region indicates states that do not exhibit a metrological advantage. (a) We show that short-range entangled states, non-degenerate code states with nonconstant distance, and CSS QLDPC code states with nonconstant distance cannot provide a metrological advantage. (b) Focusing on quantum error-correcting code states, we find that when the code distances in all directions scale as $\omega(1)$, no metrological advantage is possible. In contrast, if the code distances are asymmetric, namely, the distance in at least one direction is $O(1)$ while the distances in other directions scale as $\omega(1)$, then the code states can exhibit metrological advantage. This regime includes our constructions based on classical LDPC codes and asymmetric toric codes.
• Figure 2: Two constructions of metrologically advantageous states. Circles represent qubits, red rectangles represent $Z$-checks, and blue rectangles represent $X$-checks. (a) An example based on a classical LDPC code. By identifying the red rectangles ($Z$-checks) as vertices and the circles (qubits) as edges, one obtains the graph $G$ defined in the main text. (b) An example of an asymmetric toric code with lattice dimensions $\Theta(1)\times\Theta(n)$ and periodic boundary conditions.
• Figure 3: (a) The Tanner graph of the $9$-qubit Shor code. Physical qubits are represented by circles, $X$-checks are represented by the squares at the bottom, and $Z$-checks are represented by the squares at the top. (b) The Tanner graph of the $9$-qubit Shor code without $X$-checks. (c) The qubit-side collapse of the Tanner graph for $Z$-checks. (d) The super Tanner graph with respect to $\vec{z}^*$. (e) The qubit-side collapse of the Tanner graph with respect to $\vec{z}^*$.
• Figure 4: (a) An instance for the chosen plaquettes. (b) An instance for the connected component covering one element in $\hat{H}_1$ (represented by the red line). (c) An instance for the connected component covering the entire $\hat{H}_1$ (represented by the red line and the blue line). By complementary component, we mean the set of all the white plaquettes.

Theorems & Definitions (23)

• Theorem 1
• Corollary 1
• Theorem 2
• proof : Proof sketch.
• Definition 1: Weak Systolic and Cosystolic Expansion (Informal)
• Theorem 3
• Definition 2
• Definition 3
• Definition 4
• Definition 5: Cosystolic expanders