Source-linked AI summary
Optimal strategies for estimating the average fidelity of quantum gates
Daniel M. Reich, Giulia Gualdi, Christiane P. Koch
TL;DR
Efficiently assessing how well a quantum device implements a desired operation is challenging because full process tomography scales exponentially. The paper classifies Monte Carlo strategies using two-designs and classical fidelities, reporting reduced experimental effort for general unitaries with trade-offs in fidelity estimates or input-state preparation.
Problem
Assessing how well a quantum device implements a desired operation is difficult, while full process tomography scales exponentially and supplies more information than the gate error alone.
Source: p. 1Method
The paper classifies available strategies and applies Monte Carlo estimation to two-designs and classical fidelities for general unitary gates.
Source: p. 1Results
The required average number of experiments scales as 2^n for general unitaries, a 2^n reduction versus Monte Carlo characterization.
Source: p. 4Takeaways & Limitations
The classification lets experimentalists choose a procedure according to available settings, measurement effort, and computational resources.
Source: pp. 1, 4The resource savings require either bounds from two classical fidelities or entangled input states for two-designs.
Source: p. 4
Abstract
from arXiv · showhide
We show that the minimum experimental effort to characterize the proper functioning of a quantum device scales as 2^n for n qubits and requires classical computational resources ~ n^2 2^{3n}. This represents an exponential reduction compared to the best currently available protocol, Monte Carlo characterization. The reduction comes at the price of either having to prepare entangled input states or obtaining bounds rather than the average fidelity itself. It is achieved by applying Monte Carlo sampling to so-called two-designs or two classical fidelities. For the specific case of Clifford gates, the original version of Monte Carlo characterization based on the channel-state isomorphism remains an optimal choice. We provide a classification of the available efficient strategies for device characterization in terms of the number of required experimental settings, average number of actual measurements and classical computational resources.
The relevance distribution for two-designs,
The two-design relevance distribution is normalized, with its normalization verified as in the preceding subsection using Eq. (2) and |Φ⟩ = U|Ψ_i⟩.
The relevance distribution for two-designs,
- Pr2des(i, k) is normalized.Source: The relevance distribution for… · p. 7
- Its normalization is verified analogously to the previous subsection.Source: The relevance distribution for… · p. 7
- The verification uses Eq. (2) with |Φ⟩ = U|Ψ_i⟩.Source: The relevance distribution for… · p. 7
The variance of Xl is smaller than one.
The section bounds the variance of the Monte Carlo random variable X_l by one, including for the two-design approach.
The variance of Xl is smaller than one.
- X_l is the top-level Monte Carlo variable, defined as a ratio of measurement outcomes for the actual and ideal states.Source: The variance of Xl is smaller than one · pp. 7–8
- For the classical fidelities, the proof bounds the variance using the range 0 ≤ F_j ≤ 1 and a density-matrix eigenbasis.Source: The variance of Xl is smaller than one · p. 8
- Var(X_l) is smaller than one because it is the difference between two values in [0, 1].Source: The variance of Xl is smaller than one · p. 8
- For two-designs, Var(X_l) ≤ 1 follows analogously using the normalization of Pr2des.Source: The variance of Xl is smaller than one · p. 8