# Why are simulated qubits less effective?

"We could map the whole Universe — all of the information that has existed since the Big Bang — onto 300 qubits"

I've seen statements like this over the years coupled with the same explanation many times.

"qubits can be 1,0 or both."

Now I think using quantum particles as computer components is a great idea for a variety of reasons, but I've never understood how the above statement requires qubits. Apparently there's some error in my thinking or the descriptions. To the point.

Considering a programming construction such as

class qubit{
bool real;
signed char getValue(){
if( real && real ) return 1;
else if( !real && !real ) return -1;
else return 0;
}
}


Ok great, now we had to use 2 bits instead of one to represent 3 choices, but there we have true, false, superposition. Interactions can be determined by wave functions and so on to emulate true behavior.

qubit Cube;
doMatrixCalc(Cube);
outputState(Cube);


So what's no good about these 512 qubits? They only need 1024 classic bits. Are there other physical processes going on that this sort of logic can't do? (Note, the superposition here is 0 with false as -1, but can be interpretted as desired by conventional gates)

Another way of stating my scenario is that 2 bits can represent 4 states. Qubits only seem to represent 3 states. If quantum computations are useful because of the additional superposition state, why not just use 2 bits? Seemingly something else is the root of usefulness of qubits or we are not using classic bits properly now.

Conclusion: The quote "qubits can be 1,0 or both." is not accurate enough.

• Simple answer: because we do, there is a dozens of dozens of quantum architecture and algorithms simulation. – m0nhawk Nov 7 '15 at 22:13
• What you describe is a three-state equivalent of a bit, sometimes called a trit. In 300 trits you can store what you can store in 476 bits, not even a bit per cubic Gigalightyear of the observable universe. – doetoe Nov 7 '15 at 22:17
• @m0nhawk hmm you are right about that but still leaves the question I'm intending to ask. I need to rephrase this somehow. – Garet Claborn Nov 7 '15 at 22:17
• A single qubit can already be in an infinity of different superposition states. When combining qubits, we don't get the product of these superpositions, but superpositions of the products, e.g. with 10 bits you have 1024 different states, but with 10 qubits you have superpositions of 1024 different states. If you perform a logical operation implemented as qubit operations on the qubits, it acts on each summand of the superposition simultaneously, which sometimes is interpreted as parallel execution. The difficult part is to extract the individual summands of interest. – doetoe Nov 7 '15 at 22:25
• More precisely, the physically difficult part is to be able to stably realize these qubits and the operators, and the algorithmically difficult part is to put them to good use. – doetoe Nov 7 '15 at 22:29

`"qubits can be 1,0 or both" - the accurate statement is that a qubit is in a superposition of the two states $\left|0\right>$ and $\left|1\right>$:

$$\left|q\right> = \alpha\left|0\right> + \beta\left|1\right>$$

where $\alpha,\beta$ are two complex numbers with $|\alpha|^2+|\beta|^2=1$. The probabillity to measure either $0$ ($1$) is given by $|\alpha|^2$ ($|\beta|^2$).

That was one single qubic. When we have $n$ qubits the total state is a superposition of all the possible combinations of each qubit being $0$ and $1$:

$$\left|q\right> = \alpha_1\left|00\ldots 0\right> + \alpha_2\left|10\ldots 0\right> + \alpha_3\left|01\ldots 0\right> + \ldots + \alpha_{2^n}\left|11\ldots 1\right>$$

This amounts to $2^n$ distinct terms and we would therefore need $2^{n+1}$ complex numbers to fully specify the state. For your example with $n=512$ we would need $2^{512} \sim 10^{154}$ numbers. This is way beyond reach for normal computers.

• Is the "superposition" then, not a radial range of positions? This sounds more like a 2D float condensed into a bit imho. Which would indeed be much more useful. – Garet Claborn Nov 7 '15 at 22:40
• @GaretClaborn: Yes, in a sense it's a 2D float. However, you can't extract that information. All you'll ever be able to extract from one qubit is one bit of information. This asymmetry is at the heart of quantum computing and the conclusion that it can be more powerful than classical computation is not at all obvious. – Martin Nov 9 '15 at 10:40
• @Martin well thanks. that explains a great deal about miscommunication. I've always wondered where all those angles of spin and polarity went when folks discuss qubits. If all that information is 'eaten' by uncertainty; I understand roughly how the computer as a whole would play off the basic principles. – Garet Claborn Nov 9 '15 at 21:52
• @GaretClaborn: Yes, that is a crucial point. However, In some sense, the angles are still there: in principle, you can of course choose the axis in which you measure (in experiment, you usually can't, but "choosing the axis" is equivalent to rotating the axis first and then measuring and that's possible). But the measurement result will still be one bit. – Martin Nov 9 '15 at 22:08

The statement: "We could map the whole Universe — all of the information that has existed since the Big Bang — onto 300 qubits" is severely misleading to the point of actually simply not being true. The issue comes in with measurements.

How many bits can I get from one qubit when I measure it? The answer is one. So perhaps one can store all lot more information in a bunch of qubits, but you won't be able to get all that information out of it again.

So what is the point of doing quantum computing then? It is the fact that one can do multiple different calculation simultaneously. The catch is that once the computation is done and the answers to all these computations are stored in one quantum state, you better choose very carefully how you measure it, because you can only get one of these answers. There are particular cases where the quantum computation only gives one answer (with high probability). In such cases there is a clear benefit in quantum computing over its classical counter part.