How to build a quantum computer in your house?

A serious answer: you pretty much can't. It's not that you in particular can't, it's that no one can. Huge companies pour in huge amounts of money to try and make a proof-of-concept quantum computer (there is actually no 'proper' quantum computer yet).

A slightly less serious answer: some odd $10-100$ Million would get you started I would say. It all depends on what type of quantum bits you want to build; most of them require the need of cooling them down to single digits (or much, much less) above absolute zero. You need 'special' refrigerators for that will do this, and they are pricey.

Oh, and do you have a nanofabrication lab or a clean room adjecent to your garage? Because you're probably gonna need one of those as well:)

This is not intended to shoot you down, but more as a light joke:)

Edit: Actually, what might be in reach for a personal investment is some very rudimentary version of a photonic quantum computer. Basically you need some glass fiber, a couple of beam splitters, a lot of attenuators and some lasers. (And a idiotically expensive single-photon detector, but we'll sweep that under the rug for now).

With current technology, there's not much of a chance to build a true quantum computer, but you may be able to build some interesting quantum circuits with a fairly sizable (but still on the scale of "self-funded" for the ordinary person) budget, using the optical photon model. For instance, one could use the linear optical quantum computing model. Using beamsplitters as $hat{y}$ rotations and phase shifters as $hat{z}$ rotations, one can construct arbitrary one-qubit quantum gates. With the addition of a nonlinear medium (like Kerr Media), one can construct some interesting circuits. For reference, here are the transformations that each of those optical tools performs: $$ mathrm{B}_theta = begin{bmatrix} text{cos}(theta) & -text{sin}(theta) text{sin}(theta) & text{cos}(theta) end{bmatrix} $$ $$ mathrm{P}_phi = begin{bmatrix} e^{i phi} & 0 0 & 1 end{bmatrix} $$ $$ mathrm{K}_mathcal{E} = begin{bmatrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & e^{i mathcal{E}} end{bmatrix} $$ The parametrizations on the first two matrices have to do with the properties of the materials used in creating the optical tools, and don't necessarily have anything to do with the physical angle which they are placed.

Note that while you probably can't build a quantum computer at home, you can simulate one with a classical computer, at the cost of merely an exponential slowdown. There's a rather long list of available software at https://www.quantiki.org/wiki/list-qc-simulators.

It's actually an interesting question. And the previous answers ("strictly speaking, you can't," "you can simulate quantum computers," and "photonics-based processing holds some promise") are all true.

According to a pioneer in photonics computing we won't see real quantum computing until around 2035. I've not yet seen anything to justify thinking to the contrary. Some of the downstream work of these photonic computing pioneers has yielded the fastest data interconnects that scale perfectly in parallel b/c there's no photon-electron conversion (and because there's an ingenious breakthrough at the core). There's also a really compelling set of demonstrations using these interconnects to run QC simulations , like running Grover's algorithm and QFT, fast. See https://www.datavortex.com/research/quantum-simulation/.

There's nothing wrong with simulating, there's nothing wrong with waiting 15 years, and there's nothing wrong with jumping in and trying to make it happen in one's garage. All one needs is some advanced mathematical training, some electricity, some money, and a refusal to quit.

All previous answers are true. I would like to contribute with a serious question and a comment.

Would it make sense to build a quantum computer in your house?

They are and will be hard to maintain. In my opinion, having them in the cloud is and will be the best solution.

Beecause of economy of scale, the cost of a home-made quantum computer will be always greater than the cost of industrial quantum devices deployed in server farms located in cold countries (e.g., Norway) and exposed as cloud services.

He has basically described how to build the Cirac-Zoller quantum computer.

In the most simple approach the qubit states are electronic states of the trapped ions. Two qubit gates are possible thanks to the coupling of the qubit states to the vibrational modes of the trap -- this is cool as in superconducting architectures the two qubits that are far apart on the chip are not coupled directly.

The trapping of ions was already demonstrated a while ago e.g. here or here. As he has mentioned the next big step in his design will be placing the trap inside an ultra high vacuum (UHV) chamber i.e. less than 100nPa. This is necessary as otherwise the atoms from the air would bump into the ions and would cause decoherence.

But the really challenging step is implementation of the lasers as they do the whole job -- this might be really difficult to achieve at home, however, judging on his current progress, I hope that he will pull it off.

A real competition to his start-up is a company called ionQ that is building quantum processors with a use of this architecture. The ion trap based quantum computers are known for having qubits and gates of v high fidelity. The drawback is that they are difficult to scale up. This is the most commonly attributed to anomalous heating.

Here you can check out other architectures with their strengths and weaknesses.