What is the optimal strategy in Quantum Tic Tac Toe?
In Solving Quantum Tic-Tac-Toe, Ishizeki and Matsuura use a computer to search the game space to find solutions, but don't specifically speak to strategy. However, there are a few strategies we can glean from their results:


Go for a half-point victory, not a full point victory. In their search they found the first player, X, cannot guarantee a win by being the only player to get three in a row. But a win can be guaranteed where X and O both have 3 in a row, but X has the lower maximal subscript.
As player X, make your first marks split into opposite corners. If X plays perfectly, a half-point win can be guaranteed by playing the first pair of marks, X1, in opposite corners.
Don't play as O, the second player. Obviously this is not a fair strategy or one you're likely be able to follow in a real game. But since X can guarantee a half-point victory, it follows O can't guarantee a victory.


Since there is a guaranteed path to victory for X, the obvious strategy is to, after playing the first pair in opposite corners, always respond to a move by O by playing such that you are still guaranteed victory (or for O, always play the move that gives you the most options for victory, and hope X makes a mistake). Unfortunately the authors didn't include all of the paths in their search, so it would be necessary to reproduce their results to apply this strategy.

However, I have tried to glean some strategies from the one game they did include in the paper (see Figure 5. I've broken this out into each move in the image below):


Avoid entanglements. By not creating a cycle you avoid giving your opponent the advantage of choosing where your marks are fixed. Of course, if you can make a cycle such that either choice is a win for you, like the placement of X9, then you should do so. See both the resolution of X9 and the alternate resolution.
Play at least one mark of a pair in an empty square. This increases the chances of having the lowest maximum subscript in a tie breaker, as on each play you add another square where you have the mark with the lowest subscript. X does this up until the last move.
Play to increase the number of probable wins. For example on Move 3, after playing X1 in opposite corners, play one X3 in the middle and the other in a corner. When one X1 collapses, the other X1 that is now fixed will be in line with either X3. The same is true for X3 in relation to X1
Don't play on your own marks. X never does this until the final move when there is no other option.
Don't let your last mark be part of your three in a row. Of course your last move has the highest subscript, and can make you lose the tie-breaker. Making this actually happen may require some additional planning.