Et eksempel på et trekk i [[Kvantesjakk]] som bryter med intuisjonen om at vi flytter kvantepartikler rundt i et rom. Oppkalt etter Discordbrukeren som oppdaget denne elegante metoden for merging. Merge ved å "splitte" en brikke i en 50/50-superposisjon, hvor den éne brikken har fase +/-i i forhold til den andre (altså $\pi/4$ eller $3 \pi / 4$). For å demonstrere: Startposisjon: ![[Pasted image 20240924210350.png|200]] Etter en vanlig split move har vi følgende: ![[Pasted image 20240924210423.png|200]] Vi ser at hver del av superposisjonen har samme fase (og at de ikke er sammenfiltret, som man ser ved at de har forskjellige farger): ![[Pasted image 20240924210435.png|200]] Ved å flytte én av brikkene én rute, blir den relative fasen skiftet med $\pi/4$ (som er en gameplayregel: å flytte en brikke ganger fasen dens med $i$, men dette er ikke synlig for klassiske moves fordi globale faser er uobserverbare). Her er elementet øverst til høyre markert, og vi ser at det andre har fase $3 \pi / 4$ i forhold til det (eller ekvivalent $-\pi / 4$): ![[Pasted image 20240924210735.png|200]] Dersom man nå gjør et split move med elementet øverst til høyre, til den tomme ruten like nedenfor *og* ruten diagonalt nedenfor med det andre elementet i, så kanselleres fasen nede til høyre (takket være at å bevege seg induserer et $\pi / 4$ skifte, til $\pi / 2$, som bare er minus, siden $e^{i \pi /2} = -1$), og vi ender opp med en merget brikke nede til venstre: ![[Pasted image 20240924211140.png|200]] (Hvorfor ender man ikke opp med $3/4$ i ruten til venstre og $1/4$ i ruten til høyre (med relativ fase $\pi/2$)? Etter splitting er jo amplituden redusert til (kvadratroten av) $1/4$!) Dersom vi istedet hadde markert den andre brikken i det nest siste steget, ville vi sett at fasen var $\pi / 4$ i forhold til den andre: ![[Pasted image 20240924211658.png|200]] Hvis man splitter dette elementet i stedet, til rutene nede til venstre og oppe til venstre, så vil de (etter $\pi / 4$ rotasjon av fasen) få samme fase i ruten oppe til venstre, og dermed sammenfalle i den ruten. ![[Pasted image 20240924211909.png|200]] Man kan se på det siste tilfellet som en omvei til hva man intuitivt "skulle" fått av å simpelthen flytte elementet nede til venstre opp til høyre for å merge kvantetilstanden. (Hvorfor funker ikke det? Usikker! Men merk at dersom de to elementene har relativ fase $+- \pi / 4$ så flippes fasen dersom man prøver på dette, mens dersom de har relativ fase $1$ eller $\pi / 2$ så skjer ingenting.) Som QuantumChris [skrev i Discorden](https://discord.com/channels/168032661376466944/664544527247343630/1047951910096863322): > I've been thinking about removing the Merge move from the rules. I want to hear what you think. When I introduced the Merge move, I wanted to give players a way to use quantum interference in the game. The Split move was a nice way to illustrate superposition. The logical counterpart to Split was Merge, which enabled me to demonstrate interference. It provided a way to “unsplit” pieces, which can be valuable from a strategic standpoint. Still, something always bugged me about it. It has been the source of a number of difficulties with the game, from UI to AI, and it has always been a source of confusion for new players. When @Autoskip discovered the AutoSkip Merge (Merge by splitting with the correct relative phases), I thought that was a much more elegant demonstration of interference than the Merge move. Going a step further, I might not have implemented Merge, if I'd come up with the AutoSkip Merge. I’m also considering a UX update with this rule change. If we don’t have Merge move, we can simplify the “drag across two squares” interface because we no longer need to distinguish between source squares and target squares. We only ever have a single source, so every move can now be 3 clicks: source-target-target. Clicking the same target twice indicates a standard move. This change also opens up the option of using drag for standard moves. Let me know what you think about the potential rule change, and the interface updates. Mitt spørsmål til Autoskip: > @autoskip I had a question about the "Autoskip Merge". I see how it makes sense that given how the relative phase shifts by pi/4 each time you move a part of a superposition, you'll be able to cancel one part and so "merge". However, I would still expect there to be a small amplitude remaining in the square that got cancelled. Let me explain my logic: > > Say you have the following: > > i/sqrt(2) K | (empty) > ----------|------------- > (empty) | 1/sqrt(2) K > > You want to cancel the amplitude of the bottom right square, by making a superposition move with the top left part to the two bottom squares. Then you'd get: > > (empty) | (empty) >----------|------------- >-1/2 K | -1/2 K + 1/sqrt(2) K > >which is the same as > >(empty) | (empty) >----------|------------- >-1/2 K | (-1 + sqrt(2))/2 K > >which roughly evaluates to the probabilities > >(empty) | (empty) >------------|--------- >P(K) = 0.25 | P(K) = 0.04 > >Now, these probabilities don't sum to one, so this needs to be renormalized. Assuming renormalization affects the whole wavefunction equally (if not, why not?), I would expect there to be a small amplitude remaining in the lower right square, given that it didn't fully cancel. I.e., some state yielding the following (here I just multiplied both sides by 1/0.29): > >(empty) | (empty) >-------------|--------------- >P(K) = 0.86 | P(K) = 0.14 > >Yet in the game you of course get > >(empty) | (empty) >----------|----------- >K | (empty) > >Where's my mistake? Eksakt renormalisering: Sum av sannsynligheter er $\sum_i (x\cdot\psi(x_i))^2 = x^2(1 + (\sqrt{2}-1)^2)/4 = 1$ så vi får at renormaliseringsfaktoren er $x = 2/\sqrt{(1 + (\sqrt{2}-1)^2)} = \sqrt{2 + \sqrt{2}} \approx 1.85$. Seems about right. (Sanity check: $1/0.29 \approx 3.45 \approx 1.85^2$.) #### Autoskip's response > I may have found it, and gotten it named after me, but the Autoskip Merge is the result of semi-random messing around, and you seem to have a better grasp of what's happening than me (not helped by the fact that it's been ages since I last played Quantum Chess or read the paper, and my entire knowledge of quantum mechanics is from that and Schrödinger's cat level explanations for the James and Jo public). > > That said, I have found a quantum logic simulator website, and back when it was fresh in my mind (and I knew where my copy of the paper was), I added the iSwap and √iSwap to my local version, and bookmarked it, so I did manage to recreate it here. > > ![[Pasted image 20240927234043.png|500]] > > …the url is massive though, so I'm hesitant to paste it in discord. > Each √iSwap-iSwap pair is a split move, and the central iSwap that doesn't have a √iSwap to pair with is the normal move that offsets the phase rotation. > > Going through the steps (and sub steps), this is how things are after each move - the first √iSwap is what turns it into a superposition, then an iSwap aligns the phases, completing the Quantum Chess "split" move, the second iSwap is the normal Quantum Chess move that offsets the phase (I could've done that to either of the superpositions, but I arbitrarily chose the bottom one), then the √iSwap that starts off the second "split" move (which, due to interference, merges, making the Autoskip Merge) is actually the last move that does anything quantum - I only included the third iSwap because that's how Quantum Chess's "split" move works. > > ![[Pasted image 20240927234243.png|500]] > ![[Pasted image 20240927234304.png|500]] > ![[Pasted image 20240927234319.png|500]] > ![[Pasted image 20240927234332.png|500]] > ![[Pasted image 20240927234345.png|500]] #### Min forklaring > @Autoskip I spent some more time thinking about this, and my original understanding of the iSwap and √iSwap operations were actually sound, so that's not where my error was. The difference between my understanding and what the game does is actually that when splitting a piece, the extra factor 1/√2 is applied to the _square_—and anything it might already contain!—rather than just the part of the wavefunction being split. This is due to the fact that, although we usually think of the pieces as "quantum particles", it is actually the _squares_ that are implemented as qudits in the quantum simulation (presumably with d = 13, one basis stase for each kind of piece plus one for "Empty"). In a way it is a shame that the physical intuition of viewing the piece as a quantum particle fails in this instance. On the other hand it is actually quite interesting that this gives a concrete separation between the two interpretations, as I had up until now assumed they would be equivalent. (More accurately, while you might expect that the wavefunction of the invading piece simply gets added to whatever was already in the target square, the √iSwap rather splits the content of each square in two and then swaps one half for each other (after multiplying the contents of the first square with i). This also explains something that always confused me, namely why you couldn't merge a split piece simply by moving one half into the square of the other (while avoiding phase cancellations).) > > ![[Pasted image 20240929233352.png|400]]