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Which is the correct depiction of a qubit state showing the possibilities before it collapses? Is it like:


MODEL 1:

1000 0100 0010 0001


MODEL 2:

1000 0100 0010 0001

0111 1011 1101 1110


MODEL 3:

0000 0000 0000 0000

1000 1000 1000 1000

1100 1100 1100 1100

1110 1110 1110 1110

1111 1111 1111 1111


Or some other way?


I find the spherical diagrams, XOR gates and | difficult to conceptualize concretely. Thanks!

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  • 4
    $\begingroup$ It's not clear what model you're trying to convey in your question. $\endgroup$ – Craig Gidney May 25 at 1:52
  • $\begingroup$ @Mark S can you please show what Hadamaring only 1 qubit looks like with kets? $\endgroup$ – elsa Jun 26 at 16:12
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I'm going to go out on a limb and say that it sounds like you're having trouble to know what's meant by $4$ qubits in a uniform superposition.

Remember with $4$ qubits, the dimension of the Hilbert space is $2^4=16$. Thus we will be in a superposition of $16$ potential kets.

If we apply a Hadamard gate to each of the $4$ qubits initially in $\vert 0\rangle$, then after Hadamarding the entire qubits will be in the state:

$\frac{1}{4}(\vert 0000\rangle+\vert 0001\rangle+\vert 0010\rangle+\vert 0011\rangle+\vert 0100\rangle+\vert 0101\rangle+\vert 0111\rangle+\vert 1000\rangle+\vert 1001\rangle+\vert 1010\rangle+\vert 1011\rangle+\vert 1100\rangle+\vert 1101\rangle+\vert 1110\rangle+\vert 1111\rangle)$

That is, each potential $4$-bit vector is part of the superposition.

EDIT

Reading the above, the $\frac{1}{4}$ coefficient is the "normalizing coefficient." This is used to determine the probrability that any of the particular states are measured. According to Born's rule we square the amplitude to determine the probabilities. Because there are $16$ states, and because it sounds like the OP wants the states to be uniformly distributed with an equal chance of collapsing to any vector, we have $(\frac{1}{4})^2=\frac{1}{16}$.

Additionally the Dirac bra-ket notation, $\vert \cdots\rangle$, is how, in the quantum world, we describe vectors. The OP posted her/his vectors without the surrounding notation, which is maybe OK when you're just starting out and you want to grok the point of a superposition. But the Dirac notation has a lot of advantages later on when you start looking at things like inner/outer products and density matrices.

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  • $\begingroup$ This answer might be scary for someone that isn't familiar with Dirac notation $\endgroup$ – Arthur-1 May 25 at 21:01
  • $\begingroup$ @Mark S and so any 1 of those 16 Hadamar gates using hte Dirac notation, if compared to more basic math/comp sci, would be equivalent, or could be represented by its corresponding single, 2 * 2 matrix? $\endgroup$ – elsa May 28 at 16:39
  • $\begingroup$ @ Arthur-1 It's not based on models that exist, rather a way of my being able to communicate with people who are not physics experts but can comprehend the conversation given an adequate explanation for the purposes of what we need the info for-- most models are either way too simplistic or way too dense-- do you have names of models that are useful to computer science experts that are not physics scientists other than how I drew it out? $\endgroup$ – elsa May 28 at 16:55
  • $\begingroup$ @elsa I want to use the words that quantum computer-scientists like to use. When you say a model for "basic math/comp sci" I would like to say a model for "classical" bits. This is opposed to the superposition - a "model" - of "quantum bits," or qubits. When you ask "any 1 of those 16 ... [states], if compared to more basic math/comp sci, would be equivalent," I'm thinking you are asking what happens when you measure. After measuring, the state collapses to one of the given vectors, going from quantum to classical. After measuring, qubits become bits. $\endgroup$ – Mark S May 28 at 17:17
  • $\begingroup$ @Arthur-1 So I can phrase the use of vectors as such: 1 classical bit is denoted by a collapsed qubit. A qubit is the resulting collapsed state out of 16 possible states. The 16 possible states can be represented by the corresponding 2*2 vectors. The 2*2 vectors originate from one of the following probabilities: MODEL 1: 1000 0100 0010 0001... 2 ... 3... These models, known as Hadamar Gates using Dirac notation 1/4(|0000⟩...) are equivalent to classical XOR gates. Each 1 qubit represents 1 Hadamar gate, or 2 classic XOR gates. This is simplistic and pedantic but accurate, yes? $\endgroup$ – elsa May 28 at 17:33

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