Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonstabilizerness Enhances Thrifty Shadow Estimation

Datong Chen, Huangjun Zhu

Abstract

Shadow estimation is a powerful approach for estimating the expectation values of many observables. Thrifty shadow estimation is a simple variant that is proposed to reduce the experimental overhead by reusing random circuits repeatedly. Although this idea is so simple, its performance is quite elusive. In this work we show that thrifty shadow estimation is effective on average whenever the unitary ensemble forms a 2-design, in sharp contrast with the previous expectation. In thrifty shadow estimation based on the Clifford group, the variance is inversely correlated with the degree of nonstabilizerness of the state and observable, which is a key resource in quantum information processing. For fidelity estimation, it decreases exponentially with the stabilizer 2-Rényi entropy of the target state, which endows the stabilizer 2-Rényi entropy with a clear operational meaning. In addition,we propose a simple circuit to enhance the efficiency, which requires only one layer of $T$ gates and is particularly appealing in the NISQ era.

Nonstabilizerness Enhances Thrifty Shadow Estimation

Abstract

Shadow estimation is a powerful approach for estimating the expectation values of many observables. Thrifty shadow estimation is a simple variant that is proposed to reduce the experimental overhead by reusing random circuits repeatedly. Although this idea is so simple, its performance is quite elusive. In this work we show that thrifty shadow estimation is effective on average whenever the unitary ensemble forms a 2-design, in sharp contrast with the previous expectation. In thrifty shadow estimation based on the Clifford group, the variance is inversely correlated with the degree of nonstabilizerness of the state and observable, which is a key resource in quantum information processing. For fidelity estimation, it decreases exponentially with the stabilizer 2-Rényi entropy of the target state, which endows the stabilizer 2-Rényi entropy with a clear operational meaning. In addition,we propose a simple circuit to enhance the efficiency, which requires only one layer of gates and is particularly appealing in the NISQ era.

Paper Structure

This paper contains 34 sections, 46 theorems, 215 equations, 8 figures, 2 tables.

Table of Contents

  1. Proofs of Theorem \ref{['thm:AverageShadow']} and Proposition \ref{['pro:VarNoiseShadow']}
  2. Two auxiliary lemmas
  3. Main proofs
  4. Pauli group and Clifford group
  5. Properties of characteristic and cross characteristic functions
  6. Proofs of Lemmas \ref{['lem:CrossChar']}-\ref{['lem:CharAvg']}
  7. Characteristic functions and stabilizer 2-Rényi entropies of some pure states
  8. Phased W states
  9. $|S_{n,k}(\theta)\rangle$
  10. Additional results on the variances $V(O,\rho)$ and $V_*(O,\rho)$
  11. Impact of depolarizing noise on $V(O,\rho)$ and $V_*(O,\rho)$
  12. Mean variances in fidelity estimation based on thrifty shadow
  13. The variances $V(O,\rho)$ and $V_*(O,\rho)$ associated with the ensembles $\mathrm{Cl}_n$, $\mathbb{U}_{k,l}$, and $\tilde{\mathbb{U}}_k$
  14. Proof of Lemma \ref{['lem:VDepolarization']}
  15. Proof of Proposition \ref{['pro:AverageVF']}
  16. ...and 19 more sections

Key Result

Theorem 1

Suppose $\mathcal{U}$ is a unitary 2-design on $\mathcal{H}$, $\rho\in \mathcal{D}(\mathcal{H})$, and $O\in\mathcal{L}^\mathrm{H}_0(\mathcal{H})$. Then

Figures (8)

  • Figure 1: Mean variances in fidelity estimation based on thrifty shadow and Clifford measurements. Here $O=|\phi\rangle\langle\phi|-\mathbbm{1}/d$, and each data point is the average over the Haar random pure state $|\phi\rangle$. The variance $V(O,\phi)$ is determined by Proposition \ref{['pro:VarNoiseShadow']} with $F=1$; the mean value of $V_*(O,\phi)$ is determined by Proposition \ref{['pro:AverageVF']} in Appendix \ref{['app:BoundsAdd']}, which coincides with the variance in Eq. (\ref{['eq:VarFHaar']}); $V_{10}(O,\phi)$ and $V_{10^3}(O,\phi)$ are determined by Eq. (\ref{['eq:VarMultiShot']}) with $R=10$ and $R=1000$, respectively.
  • Figure 2: The variance $V_*(O,\phi)$ in fidelity estimation based on thrifty shadow and the three unitary ensembles $\mathrm{Cl}_n$, $\mathbb{U}_{5, 1}$, and $\mathbb{U}_{10, 1}$. Here $O=|\phi\rangle\langle\phi|-\mathbbm{1}/d$, $|\phi\rangle$ is shown in each plot, and $V_*(O,\phi)$ is determined by Theorem \ref{['thm:VarFCl']} and Proposition \ref{['pro:VarUklFphi']} in Appendix \ref{['app:BoundsAdd']}, where $M_2(\phi)$ is determined in Proposition \ref{['pro:CharNormPhasedW']} and Eq. (\ref{['eq:SRESnk']}).
  • Figure 3: (a) Clifford circuit with $l=2$ interleaved layers of $T$ gates, which corresponds to the ensemble $\mathbb{U}_{k, l}$. (b) A simple circuit underlying the ensemble $\tilde{\mathbb{U}}_k$, which is equally effective as the interleaved Clifford circuit in thrifty shadow estimation.
  • Figure 4: The variance in fidelity estimation based on thrifty shadow, where $O=|\phi\rangle\langle\phi|-\mathbbm{1}/d$ with $|\phi\rangle=|S_{20,2}(\pi/4)\rangle$. The circles, triangles, and diamonds correspond to the unitary ensembles $\mathbb{U}_{k, 1}$, $\mathbb{U}_{1, k}$, and $\tilde{\mathbb{U}}_k$, respectively. For each ensemble, $50000$ unitaries are sampled and each one is reused $R$ times.
  • Figure 5: A scatter plot about the relation between $\|\Xi_{\phi, O}\|_2^2$ and $\tilde{\Xi}_{\phi, O}\cdot\Xi_{\phi, O}$ for a five-qubit quantum system. Here $\phi$ is a fixed random pure state, and $O$ is sampled (1000 times) from a unitary-invariant ensemble generated from a fixed Hermitian operator that is normalized with respect to the Hilbert-Schmidt norm.
  • ...and 3 more figures

Theorems & Definitions (78)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 1
  • ...and 68 more