Variational Quantum Fidelity Estimation
Marco Cerezo, Alexander Poremba, Lukasz Cincio, Patrick J. Coles
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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>based on the ``truncated fidelity''<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, which is evaluated for a state<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ρ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>obtained by projecting<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math>onto its<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math>, (2) compute matrix elements of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math>in the eigenbasis of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math>, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math>is arbitrary and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math>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.