Quantum Coherence as a Benchmark: $l_1$-Norm in Quantum Volume Random Circuits

George Biswas^1*, Shao Hua Hu², Jun-Yi Wu^1,3

¹Physics, Tamkank University, New Taipei, Taiwan
²Physics, National Tsing Hua University, Hsinchu, Taiwan
³Hon Hai (Foxconn) Research Institute, Taipei, Taiwan

* Presenter:George Biswas, email:georgebsws@gmail.com

We investigate the $l_1$-norm of coherence as a figure of merit for quantum device benchmarking. Using Quantum Volume (QV) circuits that approximate unitary 2-designs, we analytically derive that the average normalized $l_1$-norm of Haar-random states equals $\pi/4$ and decays as $(1-\varepsilon)\pi/4$ under global depolarizing noise. Numerical simulations with depolarizing noise applied after each two-qubit gate confirm that the scaled $l_1$-norm and return probability exhibit similar exponential decay behavior. This establishes the $l_1$-norm as a coherence-based indicator of circuit performance. The decay of return probability is connected to a general theoretical expression based on the weighted inner product between sampling frequency and noise vectors.

Keywords: Quantum benchmarking, Quantum coherence, Return probability, Depolarizing noise, Random circuits