[ExI] teachers

Sat Sep 16 17:09:33 UTC 2023

I asked the question using Bing's chat. I don't know how  this AI compares
to the others, but these are the answers it provided for:

*Copenhagen Interpretation: *
(426 words, 3 links)

Shor’s algorithm is a quantum algorithm that can factor large numbers
exponentially faster than classical algorithms. It does so by finding the
period of a function, which is a problem that is believed to be hard for
classical computers. The algorithm uses a quantum computer to represent
every number from 0 to 2^4000 - 1 using qubits, and then performs a
distinct mathematical calculation against each one to get a distinct
result. Finally, it uses interference to cancel out the wrong answers to
produce a correct result <https://young.physics.ucsc.edu/150/shor.pdf>1
<https://young.physics.ucsc.edu/150/shor.pdf>.

In quantum mechanics, the Copenhagen interpretation is one of the most
widely accepted interpretations. According to this interpretation, the wave
function of a quantum system describes all possible states of the system. When
a measurement is made, the wave function collapses into one of these states
with a probability given by the Born rule
<https://young.physics.ucsc.edu/150/period.pdf>2
<https://young.physics.ucsc.edu/150/period.pdf>.

In Shor’s algorithm, the quantum computer uses qubits to represent all
possible states of the system. These states are represented by complex
numbers, which are stored in the amplitudes of the qubits. The number of
possible states that can be represented by n qubits is 2^n. In other words,
each qubit can represent two possible states (0 or 1), and n qubits can
represent 2^n possible states <https://young.physics.ucsc.edu/150/rsa.pdf>3
<https://young.physics.ucsc.edu/150/rsa.pdf>.

The number of qubits required to factor a number of size N using Shor’s
algorithm is proportional to log(N). For example, if N has 4000 bits, then
approximately 8000 qubits are required . While this may seem like an
enormous number of qubits, it is important to note that these qubits do not
represent physical objects like atoms. Rather, they are abstract
mathematical objects that exist in a superposition of states until they are
measured.

When a measurement is made on a quantum system, the wave function collapses
into one of its possible states with a probability given by the Born rule.
In Shor’s algorithm, measurements are made on the qubits at various stages
in order to extract information about the period of the function being
analyzed. The interference between different paths in the quantum
computation allows for incorrect answers to cancel out and correct answers
to be amplified .

In summary, Shor’s algorithm uses a quantum computer to represent all
possible states of a system using qubits. These qubits do not represent
physical objects like atoms, but rather abstract mathematical objects that
exist in a superposition of states until they are measured. The
interference between different paths in the quantum computation allows for
incorrect answers to cancel out and correct answers to be amplified.

*Many-Worlds:*
(234 words, 0 links)

Shor’s algorithm is a quantum algorithm that can factor large numbers
exponentially faster than classical algorithms. It works by finding the
period of a function, which is used to factor the number. The algorithm
requires a quantum computer with a sufficient number of qubits to represent
the number being factored.

In Shor’s algorithm, the qubits are used to represent every number from 0
to 2^4000 - 1, and a distinct mathematical calculation is performed against
each one to get a distinct result. The qubits are then measured, and the
result is used to determine the period of the function. Finally,
interference is used to cancel out the wrong answers to produce a correct
result.

The question of where these computations are being performed is an
interesting one. In the many-worlds interpretation of quantum mechanics,
there are an infinite number of parallel universes, each containing a
different version of reality. In this interpretation, the computations are
being performed in all possible universes simultaneously. The interference
that occurs in Shor’s algorithm is due to the fact that some universes will
produce correct results while others will not. The correct results are then
selected by interference between these universes.

It’s important to note that this is just one interpretation of quantum
mechanics, and there are other interpretations that do not involve parallel
universes. However, the many-worlds interpretation is one of the most
popular and widely studied interpretations.

*Quantum Bayesianism:*
(376 words, 4 links)

Quantum computers are designed to harness the principles of quantum
mechanics to perform complex calculations. Unlike classical computers that
use bits to represent information as either 0 or 1, quantum computers use
qubits, which can represent multiple states simultaneously through a
phenomenon called *superposition*
<https://www.technologyreview.com/2019/01/29/66141/what-is-quantum-computing/>
1
<https://www.technologyreview.com/2019/01/29/66141/what-is-quantum-computing/>.
This allows quantum computers to perform computations on a vast number of
possibilities in parallel.

In the case of Shor’s algorithm, the quantum computer uses its qubits to
represent every number from 0 to 2^4000 - 1 simultaneously
<https://learn.microsoft.com/en-us/azure/quantum/overview-understanding-quantum-computing>
2
<https://learn.microsoft.com/en-us/azure/quantum/overview-understanding-quantum-computing>.
This is made possible by exploiting the superposition property of qubits.
By performing a distinct mathematical calculation against each possible
number, the quantum computer obtains a set of results that contain both
correct and incorrect answers.

Here’s where *interference* comes into play. Interference is another
fundamental principle of quantum mechanics. It allows the quantum computer
to manipulate the probability amplitudes associated with each possible
result <https://www.ibm.com/topics/quantum-computing>3
<https://www.ibm.com/topics/quantum-computing>. Through carefully designed
operations, the quantum computer can enhance the probability amplitudes of
correct answers while suppressing those of incorrect answers. This process
is analogous to constructive and destructive interference in wave phenomena.

The final step involves *measurement*, which collapses the superposition of
qubits into a single state. The measurement process reveals one of the
possible results, and with high probability, it will be a correct
factorization of the input number
<https://www.technologyreview.com/2019/01/29/66141/what-is-quantum-computing/>
4 <https://en.wikipedia.org/wiki/Shor%27s_algorithm>. The computations are
not performed in a physical sense on atoms or particles but rather through
the manipulation of qubits and their associated probability amplitudes.

It’s important to note that while Shor’s algorithm demonstrates the
potential power of quantum computers for certain tasks, building practical
large-scale quantum computers is still an ongoing scientific and
engineering challenge. Researchers are actively working on improving qubit
stability, reducing errors, and developing error-correcting codes to make
quantum computers more reliable and scalable.

Regarding Quantum Bayesianism, it is an interpretation of quantum mechanics
that emphasizes subjective probabilities and Bayesian reasoning. It
provides an alternative perspective on how to interpret and understand
quantum phenomena but does not directly address the technical aspects of
how quantum computers work.

I hope this explanation helps shed some light on how quantum computers can
represent and process an enormous number of possibilities simultaneously
while leveraging principles from quantum mechanics. Let me know if you have
any further questions!

I leave it to the reader to consider which answer explained the most and
was the least obfuscatory.

I note that only one answer even addresses my question of "where are the
computations performed?"

Jason

On Sat, Sep 16, 2023 at 11:34 AM BillK via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

> On Sat, 16 Sept 2023 at 15:44, Jason Resch via extropy-chat
> <extropy-chat at lists.extropy.org> wrote:
> >
> > Interesting results Bill.
> > If you are interested, you might try Deutsch's question on it. For
> example, prompting it with something like:
> >
> > ------
> > "A quantum computer of 8000 qubits can, using Shor's algorithm, factor a
> 4000-bit number. This algorithm does so by using the qubits to represent
> every number from 0 to 2^4000 - 1, and performing a distinct mathematical
> calculation against each one to get a distinct result. Finally, it uses
> interference to cancel out the wrong answers to produce a correct result.
> >
> > My question is: how is it that the quantum computer can represent 2^4000
> distinct numbers, and perform 2^4000 distinct computations as it performs
> Shor's algorithm? Our universe only has some 2^260 atoms in it, there's not
> enough atoms in the universe to represent all these values. So then, where
> are all these numbers being represented? Where in reality are all these
> computations being performed? They must be done somewhere given we get the
> correct result, where are they all happening?
> >
> > Please explain as best you can, answers to these questions assuming that
> XXXXX is the correct description of quantum mechanics."
> > ------
> >
> > You could experiment having the AI answer this question in different
> sessions but change XXXXX to different interpretations, such as:
> >
> > "The Copenhagen Interpretation"
> > "Many-Worlds"
> > "Pilot-wave Theory"
> > "Quantum Bayesianism"
> >
> > The great thing about AI is we can have it answer these questions, where
> someone who believes in CI, for example, might refuse to answer or ignore
> the question.
> >
> > Jason
> > _______________________________________________
>
>
>
> Well, the AI has already said that the different interpretations are
> not relevant, as the quantum computer just calculates regardless. It
> doesn't care what nearby humans might think is happening during the
> calculation.
> A bit like we use GPS without bothering about the continual
> Einsteinian relativistic adjustments that need to be applied.
>
> I fear that I am getting to the stage where I don't understand either
> the question asked or the answer received!  :)
> Daniel has my sympathies.  :)
>
> BillK
>
> Anyway, I asked the AI -
> How can a quantum computer represent 2^4000 distinct numbers, and
> perform 2^4000 distinct computations as it performs Shor's algorithm?
>
> The AI immediately replied (no thinking time required!) —
> A quantum computer can represent 2^4000 distinct numbers and perform
> 2^4000 distinct computations due to its unique property of leveraging
> quantum bits, or qubits, which can exist in multiple states
> simultaneously. Unlike classical bits that can only be in a state of 0
> or 1, qubits can be in a superposition of both states at the same
> time. This allows quantum computers to perform parallel computations
> on a massive scale.
>
> To understand how a quantum computer can represent such a large number
> of distinct numbers, we need to delve into the concept of qubit
> superposition. In classical computing, an n-bit register can represent
> 2^n distinct numbers by assigning each bit a value of either 0 or 1.
> In contrast, a quantum computer with n qubits can represent 2^n
> complex numbers simultaneously. This is because each qubit can be in a
> superposition of both 0 and 1 states, resulting in an exponential
> increase in the number of possible combinations.
>
> In the case of representing 2^4000 distinct numbers, a quantum
> computer would require a register consisting of 4000 qubits. Each
> qubit can independently exist in a superposition of both 0 and 1
> states, allowing for an astronomical number of possible combinations.
> The total number of distinct numbers that can be represented is given
> by 2^4000.
>
> Similarly, when performing Shor's algorithm on a quantum computer, the
> ability to represent and manipulate such a large number of distinct
> numbers becomes crucial. Shor's algorithm is a quantum algorithm that
> efficiently factors large integers, which has significant implications
> for cryptography.
>
> The key component of Shor's algorithm is the quantum Fourier transform
> (QFT), which enables the efficient computation of periodicity. By
> representing the input number as a superposition of different values,
> the QFT allows for simultaneous evaluation of multiple potential
> factors. This parallelism is what gives quantum computers their
> exponential speedup over classical computers for certain problems.
>
> During the execution of Shor's algorithm, the quantum computer
> performs a series of operations on the qubits, including modular
> exponentiation and QFT. These operations exploit the unique properties
> of qubits to efficiently find the factors of the input number. The
> final result is obtained through a measurement process that collapses
> the superposition into a single outcome.
>
> It is important to note that while a quantum computer can represent
> and perform computations on an exponentially large number of states,
> extracting useful information from these superpositions requires
> careful design and implementation. Quantum error correction techniques
> are necessary to mitigate the effects of noise and decoherence, which
> can disrupt the delicate quantum states.
>
> In summary, a quantum computer can represent 2^4000 distinct numbers
> and perform 2^4000 distinct computations by utilizing qubit
> superposition. The ability of qubits to exist in multiple states
> simultaneously allows for an exponential increase in the number of
> possible combinations, enabling quantum computers to tackle complex
> problems like factoring large integers using algorithms such as Shor's
> algorithm.
> ----------------------
>
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