I read in a book that ()  160 qubits (quantum bits) could hold $2^{160} \approx1.46\times 10^{48}$ bits while the qubits were involved in computation.&quot;  How does this calculation come about? The context of the statement is that a caffine molecule would require $10^{48}$ bits to be represented by a classical computer. However a quantum computer would require 160 qubits and is thus well suited for such representation. If I look at this question on Quora, a 512 bit computer (which I suppose are real) would give a largest 155 digit number (). Isn't that big enough to represent atoms, molecules etc.?
I recently read this , which stated:     For scientists trying to design a compound that will attach itself to,   and modify, a target disease pathway, the critical first step is to   determine the electronic structure of the molecule. But modeling the   structure of a molecule of an everyday drug such as penicillin, which   has 41 atoms at ground state, requires a classical computer with some   $10^{86}$ bits—more transistors than there are atoms in the observable   universe. Such a machine is a physical impossibility. But for quantum   computers, this type of simulation is well within the realm of   possibility, requiring a processor with 286 quantum bits, or qubits.   Along with this resource estimate for penicillin, I've also seen  of the number of qubits required to model the ground state of caffeine (160 qubits). Given that the above report offers no reference(s) (probably in the name of business intelligence) and much Internet searching and looking into the quantum chemistry literature has come up short, my question is: Where are these resource estimates coming from – is there a journal article that published these numbers? I would really like to identify the methodology and assumptions used in making these estimates.
An example of a group such that $G \cong G \times G$ I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group? Grant the OP a binding close as duplicate vote Since the  to the close as duplicate mechanism it is a lot more visible to the OP that there is/might be a duplicate to their question. Empirically, I've noticed a lot more people agreeing, in the comments under their question, that their question is a dupe.   For instance, :        I'm aware that allowing  is  (back in the stone age) but now Stack Exchange is so much larger, and Stack Overflow is groaning under the weight of questions, I think this should be partially revisited.  Based on the principle that the simplest way to keep a site clean is to have as many people as possible nibbling away, I propose that the OP gets granted a binding close-as-duplicate vote.  It should simply be an additional line in the banner, which is . It might be something like this:     This question may already have an answer here:   &nbsp; "" 3 answers   &nbsp; Does this fully answer your question?  Yes / No   I'm not wedded to the Yes / No thing I just can't think of a better way at the moment.  Clicking on Yes would take the OP to the close as duplicate dialogue with that question highlighted. If there are multiple duplicate questions you'd need to ask the OP multiple times. A No could be used to track the relative success of the project, i.e.   Whether the question subsequently gets closed. Whether the question gets reopened again. How many times the OP actually clicks. How many times the OP disagrees with the analysis of the community.   Potentially, a No could be used to open a comment dialogue asking the OP to explain why they think their question is not a duplicate thereby stopping them simply putting "this is not a duplicate" in the question and formalising this process somewhat. Not sure about this idea...  I do not think this should be restricted by reputation, anyone should be able to close their own question as a duplicate. It gives people with lower reputation a more positive experience of the closing process and emphasises the importance of duplicate finding.  Most importantly, self-closes should not, initially, count towards the question ban. We can help new users learn how to use a lot more of the site than they might otherwise see and encourage them to search for duplicates more thoroughly. If the OP then goes on to do the same thing 10 times in a row then you can start using it toward the ban as they're obviously using the users as a personalised search service but at least the first 3-4 times this should not have any effect.  This question is, deliberately, only requesting 20% of the linked question (from 2009) and then only based on the new UI. The linked question asks for binding migration votes etc, which I do not want to see and I am not requesting a binding close-as-duplicate for the OP until someone else has already voted to close (though I think if the OP has 3k rep this might be allowable). This, I hope, addresses the concern's raised in the linked question and explains why this is not a duplicate of that question. What, exactly, is transferred along supply lines? How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? Proof of Russian Peasant Multiplication I'm studying a combinatorics text () in preparation for taking discrete math this upcoming semester. I've taken a fair amount of math in college, through Calculus II, but this is my first 500+ level math course.    In chapter 2 of the  book, the author introduces an algorithm for multiplication called Russian Peasant Multiplication. I want to prove that this algorithm will always result in the product of two numbers.  I've been chewing on the proof for a while now, and just can't seem to get any traction. Thanks in advance for any help.  Description of Algorithm  I'll try to explain the algorithm clearly. The goal of the algorithm is to find the product of two numbers $m$ and $n$. The below table is filled out for $m$ and $n$:  $$\begin{array}{c} m&amp;&amp;n&amp;\\ \hline \bigl\lfloor \frac{m}{2} \bigr\rfloor&amp;&amp; 2n \\ \bigl\lfloor \frac{m}{4} \bigr\rfloor&amp;&amp; 4n \\ \bigl\lfloor \frac{m}{8} \bigr\rfloor&amp;&amp; 8n \\ \bigl\lfloor \frac{m}{16} \bigr\rfloor&amp;&amp; 16n \\ \bigl\lfloor \frac{m}{32} \bigr\rfloor&amp;&amp; 32n \\ \end{array}$$  The sequence of numbers in both columns is the sequence $a_n = 2a_{n-1}\text{ where }a_1=2$. The columns are extended down until the value in the left column is equal to the number one. After filling out the table, the product is found by adding the values of the right column where the left column is an odd number.  For example, to multiply 18 and 37 with the algorithm:  $$\begin{array}{c} m&amp;&amp;n&amp;\\ \hline 18 &amp;&amp; 37 \\ 9 &amp;&amp; 74 \\ 4 &amp;&amp; 148 \\ 2 &amp;&amp; 296 \\ 1 &amp;&amp; 592 \\ \end{array}$$  To find the product, cross out values of the left column where the value of the right column is even:  $$\begin{array}{c} m&amp;&amp;n&amp;\\ \hline \require{enclose} \enclose{horizontalstrike}{18} &amp;&amp; \enclose{horizontalstrike}{37} \\ 9 &amp;&amp; 74 \\ \enclose{horizontalstrike}{4} &amp;&amp; \enclose{horizontalstrike}{148} \\ \enclose{horizontalstrike}{2} &amp;&amp; \enclose{horizontalstrike}{296} \\ 1 &amp;&amp; 592 \\ \end{array}$$  And add the remaining numbers in the right column to get the result: $18\times37=74+592=666$  Alternate Description of Algorithm  As an aside, here is a shorter way to describe the algorithm, repeat the below steps until the left column == 1:  • Halve the last number in the first column • Write the result below (discard the remainder) • Double the last number in the first column. • Write the result below.   Now, for every even number in the first column, delete the adjacent number in the second column. Add the remaining numbers in the second column.  What I've done so Far  I haven't been able to get much of anywhere on the proof. One thing I did was create a bit of a different way of arriving at the same outcome:  $$\begin{array}{c} m&amp;&amp;n&amp;&amp;\text{index}&amp;&amp;\text{remainder}&amp;&amp;\text{if remainder == 1, }n\times2^{Index}&amp;\\ \hline 18 &amp;&amp; 37 &amp;&amp; 0 &amp;&amp; 0 \\ 9 &amp;&amp; 74 &amp;&amp; 1 &amp;&amp; 1 &amp;&amp; 74\\ 4 &amp;&amp; 148 &amp;&amp; 2 &amp;&amp; 0 \\ 2 &amp;&amp; 296 &amp;&amp; 3 &amp;&amp; 0 \\ 1 &amp;&amp; 592 &amp;&amp; 4 &amp;&amp; 1 &amp;&amp; 592\\ \end{array}$$  Which brings to light the relationship between this algorithm and binary numbers (the fourth column in the above table is the product in binary, read bottom to top).   I also wrote some Python functions to help me explore the problem, How do I add arrowheads to circular arcs? Protected Question does not show Add an Answer button This question is protected. To answer it, I must have at least 10 reputation.        Ignoring the association bonus of 100, I should .  Yet, the Android app does not show the blue "Add an Answer" button at the bottom. The app also does not show the protection status.  If I view the question in the desktop or mobile browser (and I'm signed in), I can add an answer there. Credit cards: How is a cash advance different from a purchase? Why are the fees so high? uint color;  bool parsedhex = uint.TryParse(TextBox1.Text, out color);  //where Text is of the form 0xFF0000 if(parsedhex)    //...   doesn't work. What am i doing wrong? How can I implement object oriented programming in Mathematica? The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$ (NBHM_2006_PhD Screening Test_Analysis)     The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the   point on the curve farthest from the line $x = y$   how should i solve this problem Eigenvalues and power of a matrix I've found plenty of posts about changing the visibility of a block programatically.  My problem is the opposite. Given a block's ID/delta, how do I find out if it should be displayed at the current path?  I have a field collection that allows you to append a block using the Block Reference module (I preprocess the field collection and use my own template, so the logic has been taken away from the Block Reference module). It works, but I only want it to kick in if the "Show block on specific pages" rule says that block should be displayed. How do I tell if a regular file does not exist in Bash? It may be something that will happen to the person who is having the feeling but it may also be to several persons, as might occur with a highly destructive earthquake, for instance.   The word or phrase would be used in the following sentence:      “I’m no spiritualist but I have a feeling of ___. I hope it’s just a feeling."   EDIT - The phrase I'm looking for conveys a much stronger feeling than "I have a bad hunch". It is about something catastrophic which the person feels is "going to happen in a few minutes". I want to know where I can find updated information about all the stable and in-development versions of Ubuntu, where I can find this information?     This question is meant to be a canonical question. It expects to answer the following questions:         Is X version stable yet?   Is X version in beta?   Is X version still supported?   What versions are still supported?   What are the versions in Beta/Alpha?   Until where X version will be supported?   Have X version ceased support?   When will be X version released?   Where I can download any release?   When will the next LTS version of Ubuntu be released?         The list is not exhaustive. In simple terms, what is the purpose of flush() in ostream Query with BETWEEN returns Zero or ONE Mysql resolving 127.0.0.1 to hostname Derivation of a generalization of Mertens' Third Theorem. One of Mertens' Theorems states $$\prod_{p\le x}(1-\frac{1}{p})\sim \frac{e^{-\gamma}}{\ln(x)}.$$ I have seen a generalized version that states  $$\prod_{m&lt;p\le x}(1-\frac{m}{p})\sim \frac{c(m)}{(\ln(x))^m},$$ where $m$ is a positive integer and $c(m)$ is a real number that depends on $m$. How can I derive this result? Is there a way to find what the values of $c(m)$ are? Thank you very much.