# The difference between a bit and a Qubit

Ok, I have done a lot of research on Quantum computers. I understand that they are possibly the future of computers and may be commonplace in approximately 30-50 years time.

I know that a Binary is either 0 or 1, but a Qubit can be 0 or 1. But what I don't understand is how it can be anything other then 0 or 1. Surely a computer can only understand on and off, despite however fast it may be?

Probably the easiest analogy is to probabilities. If your computer can flip fair coins, you can think of each coin being in state Tails with probability 1/2 and in state Heads with probability half. So it's appropriate to think of a coin not as a bit but as a vector of two elements $(p_T, p_H)$, where $p_T$ is the probability of tails, $p_H$ is the probability of heads and we have that $p_T \geq 0$, $p_H \geq 0$, and $p_T + p_H = 1$. Once we sample a coin its state "collapses" to either tails or heads.

Just about the same happens with qubits. You can also think of quibit as being "in between" two states. It's also easiest to think of qubit not as a bit but as a vector of two elements $(q_0, q_1)$. Now, however, we allo $q_0$ and $q_1$ to be negative, and even further, to be any two complex numbers, as long as $|q_0|^2 + |q_1|^2 = 1$. Just as with sampling coins, once you measure a qubit, it collapses to either $0$ or $1$, and in fact you get $0$ with probability $|q_0|^2$ and $1$ with probability $|q_1|^2$. However, and this is the crucial part, these quantum probabilities can be made to cancel each other because they take negative values; this cannot happen with classical probability. This is the power of qubits.

BTW these complex valued quantities describe amplitudes and are related to the wave-particle duality, i.e. the discovery that particles in fact behave as waves, until you measure them, and then they behave as particles. The cancelation I am talking about is in fact interference between waves. So another answer to your question is that a qubit is not a discrete object that's either in one state or not, it's a wave..until you measure it.

• Ok I think I understand your question, but what still confuses me is that something this exotic would surely be too expensive to implement into every-day life. – clickonMe Apr 22 '13 at 15:36
• I don't understand what you are saying. What is exotic? I just described something very similar to classical probability. Modern desktops do quite exotic things, and they certainly sample random numbers all the time. – Sasho Nikolov Apr 23 '13 at 0:49

Here is the crux of the matter.

what I don't understand is how it can be anything other then 0 or 1.

This is actually a physics question in disguise, but I think that this forum is still a good place to address it. The two key facts are that information is physical at its root, and that physics is described well by quantum mechanics.

On the physical storage of information

What is a "bit"? It's a system with two easily distinguishable states. This could be a zero or a one written on paper; those are clearly distinguishable symbols, and importantly the ink and the paper are fairly stable with time, so that the ink on the paper can be used to "store" that information. (If it weren't stable, books would have to work differently, and the main reason why we have books is for storage of information.) In conventional electronics, we implement bits by having wires with two different voltage levels, "high" and "low" — separated by a gap of of other voltages, which we actively try to prevent the wire from having; but this gap ensures that the "high" and "low" voltages are easily distinguished from one another. In hard drives, the distinct states are represented by magnetic domains pointing in different directions. But in each case, a bit is represented by the states of a physical system which we can easily distinguish from one another.

A simple quantum-mechanical system

An example of a physical system with two easily distinguished states, among very small physical systems, is the orientation of the spin of a single particle, such as a proton. "Spin" is a vector quantity which is akin to angular momentum (hence the term), but does not actually arise from the particle revolving on an axis. Nevertheless, it has a magnetic moment (i.e. it acts like a microscopic bar magnet), and we can talk about which direction that bar magnet points: the direction in this case is what "spin" refers to. In particular, we can easily distinguish when it is pointing "up" from when it is pointing "down", e.g. using equipment such as in the Stern–Gerlach experiment.

There are other ways that the spin of a proton can point. If we use X, Y, and Z axes, and if "up" is in the +Z direction, we can also consider the spin pointing in the +X direction, the +Y direction, the −X direction, or in fact any direction described by a non-zero vector in three dimensions. Furthermore, the +X direction is distinguishable from the −X direction, and more generally any direction is clearly distinguishable from the opposite direction.

Measurement of a quantum system

In the usual representation of qubits, we identify 0 and 1 with the spin-directions +Z and −Z respectively. However, all of the other spin directions, such as +X, −X, +Y, −Y, and all of the other directions in 3D space are also legitimate possibilities for the proton's spin state. The point at which quantum mechanics rears its head is this: is the +X direction, the +Y direction, or any other direction than −Z clearly distinguishable from +Z?

Obviously, −Z ought to be the most easily distinguished state from +Z, but one might guess that we could detect if the spin were pointing in a slightly different direction than +Z. However, it turns out to be the case that if you attempt to measure the spin of a proton, a single measurement can only distinguish two opposite states: and that if you measure a particle whose spin is different from those states, you get an outcome which is (a) random and (b) indistinguishable from having randomly been in one of the two opposite outcomes in the first place. Furthermore, the spin of the particle changes to be consistent with the measurement outcome — if you measure it again, you will always get the same outcome — so that any information about the original state of the system (aside from the fact that it was not pointing opposite to the outcome which you obtained) is in effect destroyed; only if you have many copies of "the same state", stored by many other particles, can you get the statistical information required to determine what orientation the particle actually had to begin with.

Why this is the case I cannot tell you, and it's outside of the scope of this forum besides. I can only report that it apparently is the case, supported by something approaching a century of experimental observation.

The importance of quantum states as intermediate states

One simple reaction to this state of affairs would be to say: if the other spin directions don't seem to be distinguishable from +Z and −Z, then let's just not use them. This is one thing that you could do from the point of information storage: though avoiding them physically is not really practical, we can try to make fast transitions the way we currently do with voltages in electronics today.

However, we aren't really interested in quantum information as a way of describing how to store information, but for transformations of it — ways that you can use with it, either on your own (computation) or with others (communication) to evaluate functions and obtain the properties of structures represented by the information. While the information is being transformed, we don't necessarily have to care about how to describe the intermediate steps of the computation in terms of distinguishable states; and so this limitation of measurement, and the idea of distinguishing the possible states that a single "quantum bit" may have, is less important.

As far as we are able to determine, this freedom to ignore the distinguishability of the various states of a qubit in the midst of a computation is crucial: it is in effect the source of the (likely) power of quantum computation beyond classical computation. Ignoring the distinguishability of possible states of particles in mid-computation allows us to make use of the full state-space of a qubit; and importantly, also allows us to explore the full state-space of multi-particle states. Doing so allows us to perform computations in a manner which are popularly described as happening in "multiple worlds"; but given that we can only access the information computed by these multiple "worlds" if we can arrange for most of them to conspire to yield similar answers with high probability by so-called "interference", I think it is much better to recognise that the expanded state-space allows us more flexibility in how we transform information, allowing us literally to short-cut (i.e. find a shorter route than one might expect) through computational space to obtain solutions more quickly.

What are the nature of these short-cuts? Well, although we cannot perfectly distinguish +X from +Z, or −X from −Z, we can certainly perform operations on a quantum bit which transforms +Z to +X, and −Z to −X; or in fact any rotation of the direction of spin of a single qubit. This, together with controlled-not operations (which may be understood as essentially a reversible exclusive or operation on classical bits, where again we represent the classical bits 0 and 1 in terms of the +Z / −Z spin directions), is in principle enough to perform universal quantum computation.

But what do these single-qubit rotations mean in terms of the information stored in a single bit? Well: like a single-bit NOT operation, there isn't very much that you can do with them on their own. But the distinction between classical computation and quantum computation — and the distinction between mere shared randomness and entanglement — essentially boils down to the fact that these operations are sensible; that the +X state is as sensible a state in its own right as +Z, and that −X is as distinct from +X as +Z is from −Z. These other directions are sometimes described as being "in between" +Z and −Z, but it is equally correct to say that +Z and −Z are "in between" +X and −X. The different axes along which a qubits state may point are in themselves equally legitimate, and in their own way completely determined (as opposed to random) states which may represent pieces of information.

In short: that a quantum bit may meaningfully store information in a manner other than the states which you use to represent 0 and 1, in a way which is important when considering the intermediate steps of a computation. That is as much as I think anyone can tell you about quantum bits, without actually getting into physics, or linear algebra over the complex numbers.

Why are you so certain that "Surely a computer can only understand on and off"? Even classical (non-quantum) computers can deal with more than just "off" (no voltage) and "on" (high-voltage): look at Fuzzt Logic or Many valued logic in which the electrical circuit behaves differently according to the voltage level and not only "on" and "off". Analog devices (like microphones/headsets) use all the voltage scale, etc. (yet, later this voltage level will be sampled, since digital computers commonly use bits)

Quantum computers take this idea one step further, and instead of using "voltage" they use other features, that behave differently (as defined by quantum-mechanics). This gives a hardware that is capable of more than just processing "0"s and "1"s.

• Life used to be so much simpler before all this quantum computing stuff. – clickonMe Apr 21 '13 at 18:29

In Quantum theory, a Qubit has an extra state (superposition) which differentiates it from a regular bit. A regular bit can have states 0 ("off") or 1 ("on"). A Qubit, can have one extra state, which is a superposition of both 0 or 1. This is essentially a "do not know" state (or both), since we don't know whether it will be a 0 or a 1. However, there are probabilities in which it can be a 0, or can be a 1. When observed, it can be decided if it's a 0 or a 1.

• -1. There is not just one extra superposition state, and it definitely isn't useful to describe it as "I don't know". This is virtually the same as saying that there is one colour aside from blue and yellow, which represents a lack of certainty between blue and yellow. (Not only is there a very wide range of colours beside blue and yellow, they are colours in their own right.) – Niel de Beaudrap Apr 22 '13 at 7:43