In connection to  question, I am wondering how to calculate value $\langle \psi|\phi \rangle$ for arbitrary quantum states $|\psi\rangle$ and $|\phi\rangle$. A swap test is able to return only $|\langle \psi|\phi \rangle|^2$ which means that sign of the product is forgotten in case the inner product is real. Moreover, information about real and imaginary parts is lost in case of complex result. Do you know about any more general method how to calculate the inner product than swap test?
I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number. One can get $|\langle\phi|\psi\rangle|^2$ through thte SWAP test for multiple qubits, but this is not really the inner product as the information of the real and imaginary parts are lost. I came across this , which claims that the task can be done by simply using CZ gates. I'm not sure if I am mistaken, but it seems to me that such a circuit works for computational basis states, but not for general quantum states. Could somebody help me confirm if this is actually the case? If the above algorithm doesn't work for general quantum states, what algorithm would you suggest?
I have  today and one member was voted as duplicate and private feedback was there, I choose no and submitted the form. When I revisit the question or refresh the page same notice reappears with yes/no options.  In-network tab I got 200 OK status and response as "success":true  Environment: OS: Windows 10 Browser: Chrome 79.0.3945.117 (Official Build) (64-bit)  From the comments, the same issue is happing with   (might be with whole StackExchange network)      From network tab Add notes under the table I'm using the latex package  because the  package isn't using APA style version 6 yet.  Now I'm trying to add a table with notes right underneath it, like in this table for instance:    Since the \caption{} is used as a title above the table, I have to add an extra row for the notes that goes over the whole table width. The problem is that the notes won't fit into one line so I need multi-line cell. How can I accomplish that? Or am I doing it totally wrong and there is a much easier way? I guess footnotes don't work here.  \documentclass{apa6e}  \begin{document} \shorttitle{} \begin{table} \caption{title} \begin{tabular}{cc} foo &amp; bar\\ \hline \multicolumn{2}{l}{Notes. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Cras lacus orci, porttitor et posuere sit amet, tempor non justo. Ut quis aliquet turpis. Ut dictum nulla eget purus elementum a lacinia libero varius. Donec malesuada nulla ut odio rhoncus convallis. Aenean faucibus sollicitudin sapien, a vestibulum ante rhoncus sit amet. In vitae mi justo. In arcu metus, porta et lacinia ut, ultricies sed eros.} \end{tabular}  \end{table} \end{document} Finding an angle within an 80-80-20 isosceles triangle Newtonian mechanics: massless particles Object not visible in render I have a model which you can see here (camera perspective).    In the second image, you can see how the render turned out.    I'm at my wit's end trying to figure our why it's not showing up in the render. If anyone needs more information, just ask and I'll get it to you. I've already checked the normals and they're not the cause. Reduce spacing between subequations After update, get "error: file not found." followed by "grub rescue> _" :  Not ten minutes ago, I received this error, twice:  And also an &quot;Oops, something bad happened&quot; error.  I'm also getting question posting errors.  Can we fix this, please? I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:      "The rank of a matrix A is the number   of non-zero rows in the reduced   row-echelon form of A".   The lecturer then explained that if the matrix $A$ has size $m  \times n$, then $rank(A) \leq m$ and $rank(A) \leq n$.   The way I had been taught about rank was that it was the smallest of    the number of rows bringing new information the number of columns bringing new information.    I don't see how that would change if we transposed the matrix, so I said in the lecture:  "then the rank of a matrix is the same of its transpose, right?"   And the lecturer said:   "oh, not so fast! Hang on, I have to think about it".   As the class has about 100 students and the lecturer was just substituting for the "normal" lecturer, he was probably a bit nervous, so he just went on with the lecture.  I have tested "my theory" with one matrix and it works, but even if I tried with 100 matrices and it worked, I wouldn't have proven that it always works because there might be a case where it doesn't.  So my question is first whether I am right, that is, whether the rank of a matrix is the same as the rank of its transpose, and second, if that is true, how can I prove it?  Thanks  :) In An Introduction to Statistical Learning (James et al.), in section 3.7 exercise 5, it states that the formula for $\hat{\beta}_1$ assuming linear regression without an intercept is  $$\hat{\beta}_1 = \dfrac{\displaystyle\sum\limits_{i=1}^{n}x_iy_i}{\displaystyle \sum\limits_{i=1}^{n}x_i^2}\text{,}$$ where $\hat{\beta}_0 = \bar{y}-\hat{\beta}_1\bar{x}$ and $\hat{\beta}_1 = \dfrac{\displaystyle S_{xy}}{S_{xx}}$ are the usual estimates under OLS for simple linear regression ($S_{xy} = \displaystyle\sum\limits_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})$).  This is not the actual exercise; I am merely wondering how to derive the equation. Without using matrix algebra, how do I derive it?   My attempt: with $\hat{\beta}_0 = 0$, we have $\hat{\beta}_1 = \dfrac{\bar{y}}{\bar{x}} = \dfrac{S_{xy}}{S_{xx}}$.  After some algebra, it can be shown that $\displaystyle S_{xy} = \sum\limits_{i=1}^{n}x_i^2 - n\bar{x}\bar{y}$ and $\displaystyle S_{xx} = \sum\limits_{i=1}^{n}x_i^2 - n\bar{x}^2$. From here, I'm stuck. Predicting the number of decimal digits needed to express a rational number The number $1/6$ can be expressed with only two digits (and a repeat sign denoted as $^\overline{}$), $$ \frac{1}{6} = \,.1\overline{6}$$  Meanwhile, it takes 49 digits to express the number $1/221$, since a string of 49 digits repeats: $$\frac{1}{221} = .\overline{004524886877828054298642533936651583710407239819}$$ Yet for $1/223$, 222 digits repeat, giving a total of 224 digits needed to express the number.  If $f:\mathbb{Q}\rightarrow\mathbb{N}$ is a function that gives the smallest number of digits needed to express a rational number in decimal notation, what can we say about $f$?  For example, if we do not consider the negative sign to be a digit, then $f$ is an odd function.  Other than that, is there any pattern to it at all? Expected value notation in GAN loss I am reading Goodfellow's original  on GANs.  What I struggle to understand is his notation of the subscript in expected values.  $\mathbb{E}_{\boldsymbol{x} \sim p_{data}(\boldsymbol{x})}$ ....  If I understand it correctly then $\boldsymbol{x}$ is a realization of some random variable $\mathbf{x}$, but how can there be an expectation wrt. $\boldsymbol{x}$?  Or is $\boldsymbol{x}$ a random variable?  Thank you.  Edit: I do not think this is duplicate, as the referenced question does not answer what $\boldsymbol{x}$ means. Can unconscious characters be willing? This situation came up last week during our session. The party was facing a pretty difficult mob of enemies and one of the party members was knocked unconscious. Seeing the situation worsening, one of the players decided to use the  (DMG p. 197). In the text, it says:     The rod then instantly transports you and up to 199 other willing creatures you can see to a paradise that exists in an extraplanar space.   At first glance I thought this was actually a pretty good time to use the rod. The question came up though, what about the party member who is unconscious? In the end I made the call that the character went with them to keep things moving and more fun.  However, I'm feeling like back pedaling a bit and notifying my players that for future uses, the line "... willing creatures..." will be more strictly enforced as I don't feel like an unconscious creature could be willing.  Did I handle this the right way? X, along with Y, "were" or "was"? I was interested in the following sentence which appeared in a   in The Washington Post, Politics, (WHO RUNS GOV).     She, along with Mark Childress, were set to be Daschle's deputies at   the White House before he stepped down.   Can someone clarify if the fragment "She, along with Mark Childress, were set to" is ungrammatical, as I think it is?  I would reword were with was because I think that the subject of the sentence - obviously, in grammatical terms - is "She", not "She" and "Mark Childress", but I'm not sure on this correction.  More precisely, I'm not so sure that "She, along with Mark Childress" is equivalent to She and Mark Childress with regard to the effects on the grammatical numbers. Changing default width of blocks in beamer How to split numbers into columns with negative and positive values, without empty cells? Integral around unit sphere of inner product For arbitrary $n\times n$ matrices M, I am trying to solve the integral  $$\int_{\|v\| = 1} v^T M v.$$  Solving this integral in a few low dimensions (by passing to spherical coordinates) suggests the answer in general to be  $$\frac{A\,\mathrm{tr}(M)}{n}$$ where $A$ is the surface area of the $(n-1)$-dimensional sphere. Is there a nice, coordinate-free approach to proving this formula? Can massive damage knock out rather than causing instant death? Suppose, as happened last night, a character intends to subdue rather than kill an opponent. She made a melee attack, hit, declared her intent to knock out. The damage roll was enough to reduce the opponent to zero and the remaining damage exceeded the opponent's maximum HP. Now we would seem to have two rules invoked:  Instant Death. Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum. (PHB p.197) Knocking a Creature Out. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable. (PHB p.198)  So which rule takes precedence? Or is it neither? Apache reverse proxy error page when proxied server is down I am using Apache2 as a reverse proxy for tomcat, my configuration being something like:  ProxyRequests Off  ProxyPass        / http://localhost:8080/ ProxyPassReverse / http://localhost:8080/   My question is: can I configure Apache to show up a courtesy page (an "under construction" HTML static page) whenever Tomcat is down? Generic Parse Method without Boxing I am trying to write a generic Parse method that converts and returns a strongly typed value from a NamedValueCollection.  I tried two methods but both of these methods are going through boxing and unboxing to get the value.  Does anyone know a way to avoid the boxing?  If you saw this in production would you not like it, how bad is it for performance?  Usuage:  var id = Request.QueryString.Parse&lt;int&gt;("id");   Attempt #1:  public static T Parse&lt;T&gt;(this NameValueCollection col, string key) {     string value = col[key];      if (string.IsNullOrEmpty(value))         return default(T);      if (typeof(T) == typeof(int))     {         //return int.Parse(value); // cannot convert int to T         //return (T)int.Parse(value); // cannot convert int to T         return (T)(object)int.Parse(value); // works but boxes     }     if (typeof(T) == typeof(long))     {         return (T)(object)long.Parse(value); // works but boxes     }     ...      return default(T); }   Attempt #2 (using reflection):  public static T Parse&lt;T&gt;(this NameValueCollection col, string key) {     string value = col[key];      if (string.IsNullOrEmpty(value))         return default(T);      try     {         var parseMethod = typeof(T).GetMethod("Parse", new Type[] { typeof(string) });          if (parseMethod == null)             return default(T);          // still boxing because invoke returns an object         var parsedVal = parseMethod.Invoke(null, new object[] { value });         return (T)parsedVal;     }     // No Proper Parse Method found     catch(AmbiguousMatchException)      {     }      return default(T); }