I recently recorded a podcast on this topic, and provided the obvious answer: because they will be able to do things that conventional computers can never do. I offered huge numbers to convey the potential, and analogies to illustrate the key differences, since the math required to really understand quantum computing is not compatible with audio podcasts.
The podcast, called What is quantum computing and why should we care? was published on HPR.
My strategy was to use an analogy I got from Talia Gershon from IBM. She talks about qubits by having the audience imagine spinning coins. Normal bits in conventional computers are just coins that are laying on a surface with either heads or tails facing up, that we can consider to be like 0 or 1. A quantum bit, or qubit, is a spinning coin. It’s not exactly a heads or a tails, but some combination of both of those possibilities.
The entry level explanation of quantum computing is that two spooky properties of quantum mechanics are used to do things current computers cannot. I mention those two quantum effects by name: superposition and entanglement, and proceed to use the spinning coins on a tabletop analogy to explain them.
First I said that normally two coins would be laying on the table in one of four states: HH, HT, TH, or TT. This is like saying that with two conventional bits, a computer could represent one of four states: 00, 01, 10, or 11. But our spinning coins are in some sense representing all four possibilities, not just one of the four. Then for the astute I casualy thrown in the notion that qubits in superposition should be considered vectors that can be expressed as some linear combination of some basis vectors, like |0> and |1>.
I suggested that a quantum computer with 1000 logical qubits had the potential to do compute operations on a system with 21000 states at once, which is more than a classical computer could utilize even if it could store one bit of information in each particle in the visible universe for each nanosecond of time since the big bang. This is the awesome potential of the quantum effect called superposition, which is a fancy way of saying that if we can scale these up to millions or even thousands of logical qubits they will be extremely powerful devices to attack certain types of problems.
Next up, I went back to the two spinning coins analogy to talk about the other main quantum effect that quantum computers leverage: entanglement. I described again the classical situation, where two coins could end up in any of four possible combinations of heads and tails. Then I characterized entangled qubits as two spinning coins that can only end up as heads-heads, or tails-tails.
If superposition is what gives quantum computers their incredible potential, then entanglement is what enables quantum computers to do eye-opening, counterintuitive things like teleportation. I closed with an overview of various application areas for quantum technologies like networking and communications, cryptography, and quantum sensors.