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Catalytic Tomography of Ground States

Chi-Fang Chen, Robbie King

TL;DR

This work presents a catalytic tomography protocol for Δ-gapped ground states that enables nondestructive, quasi-local readout of observables using a single copy. By constructing a filtered operator Â_f from the system Hamiltonian and the observable, the authors implement high-confidence phase estimation that preserves the ground state up to a small trace distance. They establish two filter families—Gaussian and exact—providing rigorous bounds on block-diagonality and leakage, and they show optimal scaling: Hamiltonian-evolution time scales as $\tilde{O}(1/(\Delta\epsilon))$ while local readout incurs a patch radius that scales polylogarithmically with $1/\delta$ and $1/\epsilon$ and inversely with the gap. Complementary lower bounds demonstrate the fundamental limits for black-box single-copy tomography and the necessity of locality radius $\Omega(1/\Delta)$ for quasi-local filtering. The approach promises substantial tomography-overhead reductions in quantum simulations by enabling efficient, non-destructive, and locally implementable readouts of ground-state observables.

Abstract

We introduce a simple protocol for measuring properties of a gapped ground state with essentially no disturbance to the state. The required Hamiltonian evolution time scales inversely with the spectral gap and target precision (up to logarithmic factors), which is optimal. For local observables on geometrically local systems, the protocol only requires Hamiltonian evolution on a quasi-local patch of inverse-gap radius. Our results show that gapped ground states are algorithmically readable from a single copy without a recovery or rewinding procedure, which may drastically reduce tomography overhead in certain quantum simulation tasks.

Catalytic Tomography of Ground States

TL;DR

This work presents a catalytic tomography protocol for Δ-gapped ground states that enables nondestructive, quasi-local readout of observables using a single copy. By constructing a filtered operator Â_f from the system Hamiltonian and the observable, the authors implement high-confidence phase estimation that preserves the ground state up to a small trace distance. They establish two filter families—Gaussian and exact—providing rigorous bounds on block-diagonality and leakage, and they show optimal scaling: Hamiltonian-evolution time scales as while local readout incurs a patch radius that scales polylogarithmically with and and inversely with the gap. Complementary lower bounds demonstrate the fundamental limits for black-box single-copy tomography and the necessity of locality radius for quasi-local filtering. The approach promises substantial tomography-overhead reductions in quantum simulations by enabling efficient, non-destructive, and locally implementable readouts of ground-state observables.

Abstract

We introduce a simple protocol for measuring properties of a gapped ground state with essentially no disturbance to the state. The required Hamiltonian evolution time scales inversely with the spectral gap and target precision (up to logarithmic factors), which is optimal. For local observables on geometrically local systems, the protocol only requires Hamiltonian evolution on a quasi-local patch of inverse-gap radius. Our results show that gapped ground states are algorithmically readable from a single copy without a recovery or rewinding procedure, which may drastically reduce tomography overhead in certain quantum simulation tasks.

Paper Structure

This paper contains 12 sections, 7 theorems, 33 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup
  3. Results
  4. Key ideas
  5. Filtered operators
  6. Gaussian filters
  7. Exact filters
  8. Lower bounds
  9. Hamiltonian evolution time
  10. Locality of filtered operator
  11. Implementation
  12. Locality analysis

Key Result

Lemma 5.1

Consider a Hermitian $\bm{A}$ such that $\Vert {\bm{A}} \Vert\le 1.$ For the Gaussian function eq:gaussian, the filtered operator $\hat{\bm{A}}_f$ is almost block-diagonal in the energy basis for some operator $\bm{A}^{\perp}$ such that $\bm{A}^{\perp}\ket{\psi_0} = \bra{\psi_0}\bm{A}^{\perp}= 0.$

Figures (2)

  • Figure 1: (Left) Acting operator $\bm{A}$ on the ground state incurs transitions to excited states that may be difficult to undo. (Right) When we apply the filtered operator $\hat{\bm{A}}_f,$ the transition amplitudes to the excited state are highly suppressed, and the state remains at the ground state.
  • Figure 2: Block-encoding for a Riemann sum-LCU implementation of $\hat{\bm{A}}_f$, using state preparation circuit for $\sqrt{f}$ and a block-encoding $\bm{V}_{\bm{A}}$ for $\bm{A}$; see also the circuit for operator Fourier transform, which has an additional frequency register chen2023quantum. If $\bm{A}$ is unitary, we can simply replace $\bm{V}_{\bm{A}}$ with $\bm{A}$, implying $b=0$, i.e., the second registers can be omitted.

Theorems & Definitions (15)

  • Lemma 5.1: Almost block-diagonal
  • proof
  • Lemma 5.2: Compactly supported bump functions bachmann2012automorphic
  • Lemma 5.3: Exactly block-diagonal
  • proof
  • proof
  • Lemma 6.1: Quasi-local filtering implies decay of correlation
  • proof
  • Lemma A.1
  • proof
  • ...and 5 more