The debate over evidence for the many worlds interpretation is largely misconceived. Suppose that you take quantum mechanical equations of motion seriously and apply them to all physical systems, including macroscopic systems like detectors. When you do this, you find that it implies the existence of multiple versions of all those systems. Those multiple versions are bound together in layers, each of which resembles the universe described by classical physics to some approximation. In addition, taking this approach allows you to explain what is going in single particle interference, the EPR experiment and that sort of thing, see "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch for popular accounts and for more technical stuff, see
Theories that purport to explain those experiments without many worlds have to modify the equations of motion such as the Schrodinger or Heisenberg equations. These are often called interpretations, but they ought to be recognised and judged for what they are: alternatives to quantum mechanics. Some of these alternatives are completely ad hoc, like saying the wavefunction collapses (all the universes go away except one of them), but giving no equation of motion for collapse. These are not serious scientific theories since they make no specific independent predictions. They should be dismissed out of hand, but for some reason that nobody has explained they are not so dismissed. Other theories make non-ad-hoc modifications that may be independently testable but they do not solve any major problem with the theory (e.g. - the GRW theory), so the motivation for proposing them is a bit of a mystery.
In principle, an alternative to quantum theory might predict that other universes don't exist and that a quantum computer would not work. I am not aware of any alternative that makes a specific prediction about this issue. So it is a bit difficult to say why a quantum computer would be relevant to judging the issue.