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I am a CS major trying to decipher quantum computing. I have done some elementary study on qubits and I always seem to get lost at the "infinitely many" states that the qubit can have. And I really want to understand what it actually means. My understanding is as follows and please correct me if I am wrong,

A classical bit has two states 0 and 1, which means n bits can represent 2^n different states, though each state is individually expressed. A Qubit could have "infinitely many" states between 0 and 1. This means that n qubits can represent m^n states, Where 'm' is the base and m > 2. This would mean we can represent larger numbers using fewer qubits when compared to the classical bit. If this is so, how does this lead to an increase in computation speeds?

If I am completely off track, what is the actual interpretation of a qubit?

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  • $\begingroup$ An "ok" start is: en.wikipedia.org/wiki/Qubit. But, that quickly devolves into an overly technical and not very useful exposition. (One of the answers, given below, is essentially a re-pasting of the Wikipedia information.) To understand quBits, you are going to have to delve into quantum mechanics and understand superposition. A discussion of "infinite states" has stepped past fundamental information, and won't be very helpful. $\endgroup$ Mar 10 at 20:09
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A qubit is a vector in a 2-state complex space. Its vector form is something like Qubit = $ax + by$, which by convention in Dirac notation is $|Q\rangle = a|0\rangle + b|1\rangle$, where $|0\rangle$ and $|1\rangle$ represent linearly independent basis vectors and the coefficients $a,b$ are complex numbers. $a^2$ and $b^2$ represent the probability of measuring 0 or 1, and they can take on any complex value as long as $|a|^2 + |b|^2 = 1$ (which must be true because we only have one qubit).

In this framework, the qubit acts like a vector pointing from the origin which naturally evolves or rotates around the infinitely many surface points of a ‘Bloch Sphere’ except when it’s measured it only ever tells you 0 or 1 (which represent the north and south poles of the sphere).

In short, a qubit is a vector.

The ‘infinitely many’ language comes from the ability of $a,b$ to take on any complex value so long as their squares equal 1. Measuring a qubit (meaning when you read it, touch it, look at it, record its value etc) only gives you 0 or 1, not both and not a decimal in between.

If you also think of the qubit as a wave, then the goal of quantum computing is to take a superposition of all possible results of a qubit string (000, 001, 010, 011 etc), weight the components appropriately to constructively and destructively interfere with one another, so that at the very end you’re biased to get the right result... with a nonzero probability that you will get the wrong answer even if you do everything right.

Incidentally a string of three qubits, you’ll notice, is the tensor product of three qubit vectors. This creates a complex space with $2^3 = 8$ basis vectors which our 3-string qubit can happily explore.

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A QuBit is an element of a quantum circuit.

Like regular bits, a QuBit has two states, 0 or 1. Unlike regular bits, a QuBit can be in a superposition of these states.

Point 1: This gives us a terminology problem: What should be meant by "state" when talking about quantum systems. This is a problem only if classical terminology is stretched too far. Accept that new understandings are needed for quantum systems, and don't get stuck in terminology issues.

Point 2: An understanding of "superposition" is needed. That would be a long topic to discuss by itself, and is not discussed here.

A single QuBit system isn't very useful. Using an operator to put a single QuBit into a state which assigns equal probability to either state then examining the state of the QuBit gives you the same result as a fair coin toss.

Point 3: "Operator" is another specific technical term. Loosely speaking, in a single bit system, operators might be "1" (set to the 1 state), "0" (set to the 0 state), "1/2" (set to the 1 state superimposed with the 0 state, with equal probabilities). Operators are more interesting when talking about more than one QuBit.

In a quantum circuit which has multiple QuBits, the states of QuBits can be entangled with each other. The circuit is programmed by applying quantum operators to pairs of QuBits. This puts the entire circuit in an entangled state. At this point, meaningful computation becomes possible.

Point 4: "Entangled" gives us the last technical term. This is another long topic which is not discussed here.

That's a start. To understand how computation is done using QuBits, all of the above should be understood. That will require understanding a number of computer science concepts as well as a number of concepts from quantum physics.

Point 5: Wait! What about complex numbers and probability amplitudes? Or 'ket notation?

All of that is important when talking concretely about superposition, entanglement, and operators. But one step at a time. This answer is to provide a brief description of what is a QuBit.

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In the previous answer:

"weight the components appropriately to constructively and destructively interfere with one another, so that at the very end you’re biased to get the right result... with a nonzero probability that you will get the wrong answer even if you do everything right."

Let's translate this.

a. weight the components appropriately.. hmm b. to constructively and destructively interfere with one another... c. so that at the very end you’re biased to get the right result... d. with a nonzero probability that you will get the wrong answer... e. even if you do everything right.

Let's think about this for a little more than a moment as you place your bet at a Roulette Table.

I'm at the roulette table with 37 slots including TWO double-Zeros. The final answer, the winner, is decided by elements of a. The roughness of the track the ball runs in... b. the amount of oil on the spinners fingers... c. the flip of his/her hand and how many times the ball has been flipped... d. The nearly infinite ways the ball can bounce when it falls out of the track... e. and.. where the wheel started, and where it was when the ball left the track...

You may notice the infinite amount of possible variations but really, only 37 answers. How hard can this be? These five listed variable are just the most obvious one out of a million others and is a micro description of "Weight the components correctly." But... The 'Vectors within a Qubit cover all that so Let's let that be handled by ONE Qubit.

Once the ball leaves the track and begins to bounce around inside the wheel... there is an infinite number of ways "to constructively and destructively interfere with one another." Lets let that be handled by the ONE MORE Qubit. Omy, the power we have here!

"so that at the very end you’re biased to get the right result." meaning... we need to let some or all of our Qubits be biased in such a way as to always (most always) predict an outcome that matches the expected answer.. 1-37. Looks like we have to let Qubit one interact with Qubit two.

Of course the non-zero probability of producing the right answer is STILL 1 out of 37 as the game is being played in the Casino and our money may be long gone before we can possibly change all the input required to help us win even one spin of the wheel.

As admitted.. "even if you do everything right." Odds are against being right the first time... Wanna take a Moon ride based on the output of this computer?

The idea is that if you can bias up the input and provide the right entanglement and the right forces of constructive and destructive interaction... you might somehow resolve a 64 encryption code with a minimum number of 'readings' of the Qubits and more quickly produce the actual code out of 1.8446744e+19 possibilities. There you go. Enjoy.

Finally... (I do have a little experience with the pre-cursors of Quantum computing.) I knew of a person that tried to simulate the mentioned inputs of our quantum computer by biasing the two sides of two flip-flops to simulate the multiple inputs of a given neuron in the human nervous system. The idea was to bias the two sides of a binary flip-flop and then power it up and see if it became a Binary One or Zero. The constructive/destructive part was simulated with positive and negative voltages. The interaction of Two flipflops was simulated with capacitive or inductive coupling between the inputs of the two flipflops. At power up time, the answer would be 00 01 10 or 11 from the two bits, exactly like the answer produced by two Qubits. That answer was completely dependent on INPUT and an expected result.

Think about it. What kind of input is needed to resolve a 64bit encryption code. See the problem?? There is NO KNOWN input to do that. Is there anything about entanglement that will help get us to the right required input? Simply stated. No. Admittedly, the possibilities ARE limitless but so are the number of wrong answers.

By the way... neurons are tiny little frequency generators... Each one only goes up or down in frequency.. with zero F being the unstimulated state and High F resulting when multiple inputs... of varying input frequency produce any one of an infinite number of output frequencies by the neuron. I would more likely spend my money in that direction if we are talking about how quickly the brain can identify a face in a crowd. The RGB sensors in our eyes can produce an infinite number of colors by producing just three frequency generators per pixel of color sensing. Effectively three bits per pixel. How many Qubits will be needed to tell if a car is maroon or beet colored? yikes! My daughter could do that when she was four. Just a thought.

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