Robbie Neilson’s men lost 1-0 in the second leg at Tynecastle on Thursday night, with the Swiss champions progressing to the Europa League group stage 3-1 on aggregate.
There is, however, a consolation prize for Hearts and the raucous support that cheered them on at Tynecastle Park.
Last season’s Premiership third-place finishers will qualify for the Conference League group stage.
As Celtic and Rangers face off in the Champions League, Hearts will be the first Scottish side to take part in the group stage of UEFA’s new third competition and the first Scottish side to surpass the Old Firm to play group stage football since Aberdeen in 2007/08.
Even though this is the third tournament, there are still some big names waiting in the wings for Hearts to draw such as Partizan Belgrade, Dundee conquerors United AZ, Villarreal and West Ham United.
Here are all the details you need to know…
How many teams are there in the Conference League?
There will be 32 teams in total, divided into eight groups of four.
When will the Conference League group stage draw take place?
The draw will take place on Friday August 26. The ceremony takes place in August and is scheduled to take place at 1.30pm (UK).
How can I watch the draw?
UEFA will broadcast the draw live from HERE. BT Sport will also provide live coverage.
In which jar are the hearts?
The pots are decided by the club coefficient rankings established at the start of the season. The hearts are in Pot 3.
Who are in the other Pots?
Pot 1: Villarreal, Basel, Slavia Prague, AZ Alkmaar, Ghent, Istanbul Basaksehir, Partizan Belgrade, West Ham United, CFR Cluj.
Pot 2: Molde, Steaua Bucharest, Fiorentina, FC Cologne, Apollon, Hapoel Beer Sheva, Slovan Bratislava.
Pot 3: Nice, Anderlecht, Zalgiris Vilnius, Austria Vienna, Hearts, Shamrock, Sivasspor, Vaduz.
Pot 4: Dnipro-1, Lech Poznan, Slovacko, Silkeborg, Djurgårdens IF, Pyunik Yerevan, RFS, Ballkani.
When are the meeting dates?
Confirmation of the match, in terms of when Hearts are playing home and away, is expected to arrive within the next 24 hours. The key dates are:
1st day: September 8 2nd day: September 15 3rd day: October 6 4th day: October 13 5th day: October 27 6th day: November 3