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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 fidelityF(ρ,σ)based on the ``truncated fidelity''F(ρm,σ), which is evaluated for a stateρmobtained by projectingρonto itsm-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically withm. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalizeρ, (2) compute matrix elements ofσin the eigenbasis ofρ, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case whereσis arbitrary andρis 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.
Highlights
In the near future, quantum computers will become quantum state preparation factories
We propose a variational hybrid quantum-classical algorithm [1, 13,14,15,16,17,18,19,20,21] for Low-Rank Fidelity Estimation called Variational Quantum Fidelity Estimation (VQFE)
We first verify that the sub- and super-fidelity bounds (SSFB) and the truncated fidelity bounds (TFB) present a pronounced dip near h = 1, implying that they can detect the presence of the zero-temperature transition
Summary
It is reasonable to suspect that estimating F (ρ, σ) is hard even for quantum computers This does not preclude the efficient estimation of fidelity for the practical case when one of the states is low rank. Our bounds tighten monotonically in m, and eventually they equal the fidelity when m = rank(ρ) This is in contrast to the state-of-the-art quantum algorithm to bound the fidelity, which employs. Since the SSFB are expressed as traces of products of density matrices, they can be efficiently measured on a quantum computer [26,27,28,29] These bounds are generally looser when both ρ and σ have high rank, and the SSFB likewise perform better when one of the states is low rank. All proofs of our results are delegated to the Appendix
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