Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient Shadow Tomography of Thermal States

Chi-Fang Chen, András Gilyén

Abstract

We present a general protocol for estimating $M$ observables from only $\mathcal{O}(\log (M)/\varepsilon^2)$ copies of a Gibbs state whose Hamiltonian is accessible. The protocol uses single-copy, nonadaptive measurements and uses a total Hamiltonian simulation time of $\widetilde{\mathcal{O}}(βM/\varepsilon^2)$; we show that the sample complexity is optimal in a black-box setting where exponential time Hamiltonian simulation is prohibited. The key idea is a new interpretation of quantum Gibbs samplers as \textit{detailed-balance measurement channels}: measurements that preserve the Gibbs state when outcomes are marginalized. Consequently, shadow tomography of thermal states admits a general efficient algorithm when the Hamiltonian is known, substantially lowering the readout cost in quantum thermal simulation.

Efficient Shadow Tomography of Thermal States

Abstract

We present a general protocol for estimating observables from only copies of a Gibbs state whose Hamiltonian is accessible. The protocol uses single-copy, nonadaptive measurements and uses a total Hamiltonian simulation time of ; we show that the sample complexity is optimal in a black-box setting where exponential time Hamiltonian simulation is prohibited. The key idea is a new interpretation of quantum Gibbs samplers as \textit{detailed-balance measurement channels}: measurements that preserve the Gibbs state when outcomes are marginalized. Consequently, shadow tomography of thermal states admits a general efficient algorithm when the Hamiltonian is known, substantially lowering the readout cost in quantum thermal simulation.
Paper Structure (13 sections, 15 theorems, 78 equations, 1 figure, 1 table)

This paper contains 13 sections, 15 theorems, 78 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Prior work
  3. Problem setup
  4. Main results
  5. Proof ideas
  6. Techniques
  7. Detailed-balance measurement channels
  8. The proof of our main result: \ref{['thm:main']}
  9. Continuous-time approach
  10. Tight sample bounds
  11. Concentration of the average of subsequent measurements
  12. Learning distributions given by uniform sampling over a subset
  13. Hybrid method for bounding small probability advantages

Key Result

Theorem 1

Consider $M$ observables $\bm{A}_1, \cdots, \bm{A}_M$ such that $\Vert {\bm{A}_i} \Vert\le 1$, a Hamiltonian $\bm{H}$, and inverse temperature $\beta.$ Then, we can estimate all expectations $\mathrm{Tr}[ \bm{ \rho}\bm{A}_{i}]$ for the Gibbs state to error $\epsilon$, with failure probability $\delt Here, $\widetilde{\mathcal{O}}\left( \cdot \right)$ absorbs poly-logarithmic factor of $M,1/\epsilo

Figures (1)

  • Figure 1: Parallel measurements applied to multiple copies of $\bm{ \rho}$ to estimate $M$ expectation values $\mathrm{Tr}[\bm{ \rho} \bm{A}_1],\cdots,\mathrm{Tr}[\bm{ \rho} \bm{A}_M]$, where the outcome is denoted by $a$. For each column of $(a_i^{(1)},\cdots,a_i^{(r)})$, we estimate $\mathrm{Tr}[\bm{ \rho} \bm{A}_i]$ to precision $\epsilon$, with failure probability $\delta/M$. By the union bound, ensuring that all estimators are correct incurs only a logarithmic scaling in the number of copies.

Theorems & Definitions (28)

  • Theorem 1: Measuring many observables with very few samples of Gibbs states
  • Definition 1: Detailed-balanced measurement channel
  • Lemma 1: Efficient detailed-balance measurement channels from Gibbs sampling
  • proof
  • Remark 1
  • Lemma 2: Estimating one observable through a detailed-balanced measurement channel
  • proof : Proof of \ref{['lem:oneObs']}
  • proof : Proof of \ref{['thm:main']}
  • Definition 2: Detailed-balance continuous measurements
  • Lemma 3: Detailed-balanced continuous measurements from Gibbs sampling
  • ...and 18 more