Street Fighter III 3rd Strike: Fight for the Future - Secret Q Boss Fight

To fight Q, you must:

not lose any rounds.
Q should appear after you finish Stage 7.

not continue.
You're not supposed to lose rounds, anyway.

get a grade of at least a D or better on every stage.

get 2 or more "Special Points."
You get these from getting better grades each match.

get more than 5 Super Art Finishes.
If you change the maximum rounds in a match, this condition's formula is basically: [Number of Max Rounds in Each Match + 2 Super Art Finishes = Condition Met]
For example, if you set the matches to end after one round, you would need to use 3 Super Art Finishes throughout the game.

This is the only way you can fight him in Single Player, although he can be selected normally on the character select screen.