Variational Quantum Fidelity Estimation

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 $F(ρ,σ)$ based on the "truncated fidelity" $F(ρ_m, σ)$, which is evaluated for a state $ρ_m$ obtained by projecting $ρ$ 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 $ρ$, (2) compute matrix elements of $σ$ in the eigenbasis of $ρ$, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where $σ$ is arbitrary and $ρ$ is low rank, which we call low-rank fidelity estimation, and we prove that a classical algorithm cannot efficiently solve this problem. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.

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