Take this tab for example:
How can I learn to jump from the B to the D, to the E, and back to the B as fast as possible?
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Turn the bass amp volume up a few notches. Make things louder. You need to hear what you are doing, and make the mistakes sonically, painfully clear. From there, the improvements will also become sublimely, joyfully clear as well.
Subtle palm-muting with your right hand will aid immeasurably the quick, dexterous practice motions of your left hand. When playing swiftly, don't let any strings ring on you.
All that, and repetition. Work it. Muscle memory. That's the best advice I can offer.
Assuming you play standing up and/or use some kind of strap, make sure your strap setup is working properly.
You should be able to remove your hands entirely from the instrument without experiencing the neck diving to the floor, or other movement. In other words, the instrument should be balanced so that your hands can focus on playing the instrument, and not on holding it in position. The advantages of this aren't only felt when doing jumps, but jumps are something that is helped.
This may sound obvious but my experience is that a lot of instrument/strap combinations are far from ideal in this regard.
One great skill that helps leaps - and all playing on bass is using all of the digits. Using a pick inevitably slows this sort of playing. You have (in this case) four strings, and five digits with which to play them. If the pinky gets left out of the equation, there are still enough for one per string, and with the hand positioned, one hovers over each string, simplifying the jump.
That said, I totally endorse Todd's thoughts that the tab is poorly written. There's absolutely no point in playing the lower notes on the A string. That only gives problems that are solved by moving that B note (and C in the 2nd bar) to the E string, putting the fretting on 7th fret. Nice surprise! It's the same fret as the G string 1st note!
So, r.h. is sorted, now l.h. A sort of barre would give both notes on fret 7, I'd use index, and pinky for the fret 9 notes. There's a mini problem that the lower note may ring out, so don't hold that barre down on the E tring - just lift the index tip, muting the note, and keep it ready for the next time it's needed - the end of that bar. Or, as a lot of EB players do, touch the E string with the thumb after playing.
Moral - don't ever trust tab - the notes may be right, but a lot is not written well. Search for alternatives!
I’d like to add something to the good advice you have already received and offer some specific things you can practice. Some great exercises for learning to make positional shifts are practicing scales and patterns up and down on one string. Fingering depends on whether you are a 1 finger per fret (1-2-3-4) or upright style (1-2-4) player. As an example, take an A major scale on the G string. Dashes indicate a shift in hand position:
Frets: 2 4 6 7 9 11 13 14
Fingering (1 per fret) 1 3 - 1 2 4 - 1 3 4
Fingering (upright style) 1 4 - 2 4 - 1 4 - 2 4
You can do this on all strings as well as do other types of scales. Pentatonics are great too because they make you shift larger gaps since they have less notes. Knowing how to do this will also prepare you for shifting scale pattern positions more smoothly.