Variational Quantum Fidelity Estimation

Marco Cerezo; Alexander Poremba; Lukasz Cincio; Patrick J. Coles

doi:10.22331/q-2020-03-26-248

Variational Quantum Fidelity Estimation

Marco Cerezo^1,2, Alexander Poremba³, Lukasz Cincio¹, and Patrick J. Coles¹

¹Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
²Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
³Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA.

Published:	2020-03-26, volume 4, page 248
Eprint:	arXiv:1906.09253v2
Doi:	https://doi.org/10.22331/q-2020-03-26-248
Citation:	Quantum 4, 248 (2020).

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Abstract

Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity $F(\rho,\sigma)$ based on the ``truncated fidelity'' $F(\rho_m, \sigma)$, which is evaluated for a state $\rho_m$ obtained by projecting $\rho$ onto its $m$-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with $m$. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize $\rho$, (2) compute matrix elements of $\sigma$ in the eigenbasis of $\rho$, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where $\sigma$ is arbitrary and $\rho$ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.

Popular summary

In the past few years, there has been tremendous progress towards the development of near-term quantum devices. Today's quantum computers have limited number of qubits to work with, and are prone to noise-induced error. Nevertheless, despite those limitations, it has been recently shown that they can outperform our current supercomputers, being able to prepare quantum states that cannot be classically simulated. Hence, a very important question becomes: How do we verify and characterize these states prepared on a quantum computer?

Estimating a state's fidelity to a target state provides a $mean$ to verify and characterize the state. However, estimating the fidelity is known to $\textit{be a computationally}$ difficult task for a classical or even a quantum computer. Specifically, no classical or quantum algorithm exists to efficiently compute such quantities. In this work we introduce a hybrid quantum-classical algorithm to estimate the fidelity for the practically important case when one of the two states is low rank. Specifically, we introduce new lower and upper bounds for the fidelity which can be refined to arbitrary tightness, and we show that these bounds can outperform other known computable state-of-the-art bounds for the fidelity. Second, we introduce a near-term algorithm for computing our bounds.

► BibTeX data

@article{Cerezo2020variationalquantum,
  doi = {10.22331/q-2020-03-26-248},
  url = {https://doi.org/10.22331/q-2020-03-26-248},
  title = {Variational {Q}uantum {F}idelity {E}stimation},
  author = {Cerezo, Marco and Poremba, Alexander and Cincio, Lukasz and Coles, Patrick J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {248},
  month = mar,
  year = {2020}
}

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