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Efficient Many-Body Shadow Metrology via Clifford Lensing

Sooryansh Asthana, Conan Alexander, Anubhav Kumar Srivastava, T. S. Mahesh, Sai Vinjanampathy

Abstract

Quantum probes that enable enhanced exploration and characterization of complex systems are central to modern science, spanning applications from biology to astrophysics and chemical design. In large many-body quantum systems, interactions delocalize phase information across many degrees of freedom, dispersing it away from accessible measurements and limiting the scalability of quantum metrology. Here we show that experimentally accessible Clifford operations acting jointly on quantum states and observables can refocus this distributed information. These operations implement what we term {\it Clifford lensing}--transformations that coherently localize phase information onto a reduced set of degrees of freedom, mapping optimal measurements onto observables of reduced Pauli weight. We establish a correspondence between quantum error-correcting codes and interferometric constructions that enforce deterministic phase kickback, and generalize this to circuits that concentrate many-body phase information onto a controllable subset of qubits. We further develop partial shadow tomography protocols for estimating subsystem-supported phases. We experimentally demonstrate these principles in liquid-state nuclear magnetic resonance systems of up to fifteen qubits, achieving optimal sensing with constrained resources. Our results establish a scalable route to coherent control of information flow in interacting quantum systems, enabling many-body quantum sensing and multimode interferometry across complex architectures.

Efficient Many-Body Shadow Metrology via Clifford Lensing

Abstract

Quantum probes that enable enhanced exploration and characterization of complex systems are central to modern science, spanning applications from biology to astrophysics and chemical design. In large many-body quantum systems, interactions delocalize phase information across many degrees of freedom, dispersing it away from accessible measurements and limiting the scalability of quantum metrology. Here we show that experimentally accessible Clifford operations acting jointly on quantum states and observables can refocus this distributed information. These operations implement what we term {\it Clifford lensing}--transformations that coherently localize phase information onto a reduced set of degrees of freedom, mapping optimal measurements onto observables of reduced Pauli weight. We establish a correspondence between quantum error-correcting codes and interferometric constructions that enforce deterministic phase kickback, and generalize this to circuits that concentrate many-body phase information onto a controllable subset of qubits. We further develop partial shadow tomography protocols for estimating subsystem-supported phases. We experimentally demonstrate these principles in liquid-state nuclear magnetic resonance systems of up to fifteen qubits, achieving optimal sensing with constrained resources. Our results establish a scalable route to coherent control of information flow in interacting quantum systems, enabling many-body quantum sensing and multimode interferometry across complex architectures.
Paper Structure (24 sections, 11 theorems, 15 equations, 2 figures)

This paper contains 24 sections, 11 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Clifford lensing of many-body symmetric logarithmic derivatives
  3. Stabilizer-compatible metrology protocols
  4. Stabilizer-incompatible metrology protocols
  5. Metrologically sufficient channels
  6. Partial shadow tomography with collective measurements
  7. Partial shadow tomography with collective unitaries
  8. NMR Based shadow metrology
  9. Conclusion
  10. Complexity of measurement of symmetric logarithmic derivative
  11. Need for Clifford lensing
  12. Clifford lensing of SLDs
  13. Clifford lensing of stabilizer-compatible metrology protocols
  14. Clifford lensing of Ramsey metrology
  15. Clifford lensing of surface code based metrology protocol
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathcal{H}_P = (\mathbb{C}^2)^{\otimes m}$ be a physical Hilbert space. The following statements are equivalent:

Figures (2)

  • Figure 1: The HMPA molecular structure (left) and the NMR pulse sequence (right) for the metrology experiment. All pulses shown by rectangles are 90 degree rotations with phases as shown. The Clifford gates were designed with GRAPE technique. The CPMG pulses during evolution delays are not shown.
  • Figure 2: The experimental angular sensitivity vs coherence order $n$ at different ranges of embedded angles varying from $9.4^\circ$ to $10.6^\circ$. The blue dashed line represents the Heisenberg limit, while the red dashed line represents the shot-noise (standard quantum limit). The outliers at high coherence orders are due to a combination of gradient echoes, vulnerability of the weak NMR signal, and baseline errors in the NMR spectrum.

Theorems & Definitions (21)

  • Theorem 1: Deterministic Clifford phase kickback
  • Theorem 2: Characterization of Clifford lensing
  • Theorem 1: Deterministic Clifford phase kickback
  • proof
  • Theorem 2: Characterization of Clifford lensing
  • proof
  • Definition 1: Local metrological sufficiency
  • Theorem 3: Tangent-space injectivity
  • proof
  • Corollary 4: Two-twirl obstruction
  • ...and 11 more