This is uncharted territory for Draper, whose previous best at a major was a fourth-round showing in New York last year.
It will also be the first time the Briton has played on the 23,771-capacity Arthur Ashe Stadium – a venue he has visited every day during his time in the city.
But this has been a breakthrough year for Draper, following up his first ATP Tour title in June with an eye-catching win over Wimbledon champion Alcaraz at Queen’s.
He is now at a career high of 25th in the world rankings.
“He’s managed to have a long period of time where he hasn’t missed months of being on tour, so he’s banking way more experience,” Great Britain Davis Cup captain Leon Smith told BBC Sport.
“He’s getting more experience. He’s in the bigger events more often. From that, he’s understanding how better to handle himself.”
Quarter-final opponent De Minaur is undoubtedly more experienced. He has nine ATP titles to his name and has reached the last eight in three straight Slams.
He also leads the head-to-head, having won all three of his matches against Draper.
But, like Draper, the 25-year-old Aussie has never made it to a major semi-final.
“You’re expecting De Minaur to make it awkward and tricky,” Bartoli said.
“Jack will need to hit through the court and keep on going at him.”