Quantum Go

Is this nonsense?

If not, I wonder what it would be like to play a game where there was, say, a 50/50 chance of playing a stone of the opposite colour each turn…


The title is nonsense.

They created a game in which you play 2 stones each move and assign a probability to each of the stones. When a stone is played next to one of them, you can use any random source to decide which stone remains on the board and which stone is removed. You don’t need any quantum mechanics or entangled photons to play the game, any source of randomness will do.

The article would be much better without the use of the terms and concepts of quantum, entangled and photon.


Ah, the old polarity reversal of the quantum flow.


Invert the mobius strip and find the eigenvalue and all that.

I mean there’s also ‘quantum’ chess, which they have on steam.


I think the desire to use quantum entanglement effects as a source of randomness is to get better “quality” randomness, as getting truly random numbers is a surprisingly difficult feat. Cloudflare, for instance, has a wall of lava lamps for this purpose.

I assume the concern is that a sufficiently advanced AI could certainly start to cue in on small variations in the PRNG to sufficiently to affect game play, especially given the small set of possible outcomes. I don’t think any such AI is even close to existing today (the cryptography community would be reasonably terrified of any such AI advancement at this point), but I can see the argument for the future.


Here is another suggestion named “Quantum Go”

I have not checked either article, so I’m not sure if they are related, the same thing, or completely different.


Seems to be a different paper, the one mentioned in physics world is

but they reference the one you’ve linked.


I read quite carefully both articles. Quantum Go has been written in 2016 and is a precursor of the idea published later in July 2020 in the paper Quantum Go Machine.

In the original article, the idea was quite different. You can still play two stones per turn leaving them in an entangled state, but - when the opponent (or you) place a stone in an intersection adjacent to one of the entangled stones, is the player that can decide which entangled stone remains on the board and which one must be removed. Then the same decision can be taken by the opponent.

This has obviously nothing to do with the quantum theory (taking apart the idea that two stones can be on the board for a certain period of time in an indefinite “state”). But this variant of classic Go is quite interesting. No random decision, but a mere strategical 4th dimension that leaves you some time to define the real placement of a delocalized stone on the board. You can define it at a later stage.

In my opinion, this variation of the classic Go can be explored further especially in the proposed variant in which only one player is allowed to play entangled stones. The opponent (I suppose the strongest, but I’m not sure) should play a classic Go.

In both articles, however, there is something missing. The idea that two stones of different colors (both belongings to entangled pairs) could be placed on the same intersection. At the end of the day, their state is not yet defined. Why not? Obviously - following the rules - as soon as a player puts an entangled stone in a spot already occupied by one of the opponent’s entangled stone, a quick decision must be taken (yes, go stones are fermions and cannot stay in the same state in the same place.). Who first decides to maintain the stone in that spot would cause the automatic removal of the opponent one and consequently to maintain the other paired stone on the board.

Following the rules of the original article, in the case an interaction occurs, the players alternatively chose which stones should remain on the board starting from the player who is causing the interaction. He must decide the state of any of its stones affected by the interaction (if any) but he cannot yet decide the state of the stone causing the interaction. Afterward, it is the turn of the opponent to decide the state of its entangled stones. Finally, it is again the first player turn who can decide the state of the entangled stones that caused the interaction.

The sequence of the stones settlement during an interaction is important because it can affect the game rules (just think to a ko for example).

Here there is a big difference between the two articles even if the sequence for stone settlement during an interaction is the same in both articles. In the original article, if a stone collapses in a certain state (i.e. becomes real on the board), then all adjacent entangled stones collapse in the same state (becomes real on the board) meanwhile their paired sisters must be removed from the board. No decision is possible by the opponent unless each of his two entangled stones is in contact with other stones that just collapsed at the same turn.

In the second article instead, each stone is somehow independent and can collapses on a random state regardless of the final state of the surrounding stones.

This rule in the first article appears to be more coherent and leads to a game where the stones follow - at least locally - a logical sequence, like in the classic Go. The same cannot be said for the rule of the second article. Since the materialization of the stone is random, there is no connection between the place where the stone finally appears and any underlying strategy. The result is a confusing board where the stones have no relationships one with another until the end of the game.

Indeed, there is a simulation at the end of the second article (Quantum Go Machine) where the final status of a board is shown for a 19x19 game on a tty terminal. It is clear that even the endgame sequence becomes a nightmare with this rule. The board appears almost full of stones and the points of territory (Japanese rules) are really really few for both players because you are forced to play stones until each territory boundary is cleared to allow final counting.

I agree with @flovo that the essence of the second article doesn’t change if we assume a theoretical pure random sequence of “1” and “0”. But probably this wouldn’t have helped the researchers to spend money and time to realize a real working “Quantum Go Machine” that demonstrates nothing apart from the fact that the well known entangled polarization of photons phenomenon can be used as a “real” random source.

I find always interesting things here when I have time to read…


If they have different color, so they can stay in the same place. Pauli exclusion principle applies only to identical particles.

Btw. the concept of entanglement requires at least 2 particles, while the pair of stones played by a player is only one particle with a space wave function not vanishing in exactly 2 locations.



Thank you for the detailed summaries, @ayaros.

What about representing the state of the board with qubits? And they apply quantum computing gates to make plays and apply the capture rule. Eventually the

There are two fundamental approaches, I think:

  1. Use 1 qubit to represent each point. Since the board is ternary, perhaps the initial empty state of each point could be represented as the superposition of |0> and |1>, i.e., (2^-0.5) ( |0> + |1> ), while the basis states |0> and |1> would correspond the black and white.
  2. Use 2 qubits to represent each point. The initial empty state could be represented as either |01> and |10>, while |00> and |11> would represent black and white, respectively.

Then, to play the game, the players take turns choosing and applying quantum gates to the board states. For example, a turn might look like:

  1. Selecting a point on the board and choosing to apply a quantum gate that aims to rotate that bit(s) toward your desired state.
  2. Then automatically applying another set of quantum gate(s) that aims to resolve something roughly akin to a capturing/influence rule.

The first step represents the strategic action of the player, so the player has the freedom to choose which gate (out of a predefined set) and where to apply it, while the second step represents the resolution of the game mechanics that happens after each stone is placed (hopefully something very roughly analogous to resolving captures). The idea is that choosing to rotate a particular bit has impact on the bits around it, and the gates to realize the game mechanics could introduce entanglements, etc.

Eventually, if the game is properly designed, but players should want to pass, whereupon the state of the board is materialized with a measurement of the bits, which finally settles which points are black, white, or empty (although empty is only possible with using two bits). Then, the scores would be counted as how many points wound up for each player’s color, and maybe the fully surrounded empties (in the two bit case) should also be counted as territory.

I have no idea how to exactly design these gates to make for an interesting game that vaguely resembles Go in a quantum sense, but indeed the crux of the game design would be figuring this out.

Maybe the second article already tried to do something similar?


Uhm… interesting extension @yebellz.

They use 2-qubit correlated photon pairs to describe the quantum stone.

What you are proposing is essentially a sort of quantum field (the board) where the absence or presence of a stone in one of the intersections is somehow similar to the existence of a particle that is resulting from the excitation of the field in that point. It would be a discrete field rather than a continuous one (no matter for us).

I don’t want to push too much the parallelism with physics, but if it helps to have a more elegant picture of the concept, why not. The goal is not to use any existing theory as it is forcing the parallelism but to reuse some concepts that can help to simplify the approach to the idea we are discussing here.

If we assume that the color of one stone (black or white) is a quantum number as suggested by @flovo, we can have a compact description of the field in a specific intersection as a superimposition of eigenvalues of the stone’s color (black and white, as you suggested: |b> and |w>).

Let see if conceptually it works:

  • the concept to have two stones of different colors in the same place is now bypassed. Each state of an intersection can be ‘void’, ‘black’, or ‘white’. where: with ‘void’ we are referring to an even superimposition of states |b> and |w> that leave the intersection empty in a classical sense.
  • to have two stones of the same color in the same spot is forbidden. Pauli’s principle here applies. We defined only one quantum number, the color. Hence we cannot have two stones having the same state (black, or |b>) in the same place at the same time. But this is possible at different times (after capturing for example).
  • the empty board is not excited at the beginning. So, adding a couple of entangled stones is the result of the excitation of the field. Let see later what practically means excitation of the field in our context… now what is important is that we are creating an entangled pair, so we need another quantum number to describe the entanglement. the candidate cannot be the color because in our model we need an entangled pair having the same color at each turn. Let find a name for the second quantum number we are looking for. For example “presence”. This can be represented by the eigenvalues “not-present” and “present” well represented by |0> and |1> in the bra-ket notation.
  • when a player places an entangled pair of stones on the board is like it is making a measure of the first quantum number (the color) and two stones pop-up in different places. The wave function describing the color collapses in the selected points of the field (intersections). We have the first problem here since the color is not needed as a quantum number from a practical point of view, because the normal game foresees alternate black and white turns. We cannot manage the idea to add at each turn a stone of not predefined color. The same theoretical problem (if we want to maintain the parallelism with physics) arises for the chosen intersections. In other terms, color alternation is a predefined state that both players know in advance (no need to measure, whatever it means in this context). In a similar way, for the positions of the newly placed stones, it is not the result of casual quantum fluctuations of the field but the result of an inducted excitement of the same field in certain points that force the stones to appear in that specific locations. We can survive with this incoherence.
  • Adding a second quantum number make it impossible to have stones of the same color in the same place at the same time. Pauli’s principle works fine because one of the quantum numbers is the “presence” so if a stone placed in a certain position has an eigenvalue |1> then the other (with the same color) can be in the same location if and only if its “presence” status is |0> (i.e. not present). So, no conflicts with Pauli’s principle. I know… it is a sort of trick, but it works very well.
  • Black and White are mutually exclusive being eigenvalues of the goban-field. If the field in an intersection is measured to be |b>, it cannot be |w>. The important thing is to have one field only describing the goban-field and not superimposition of different fields having the same quantum numbers.
  • The previous point suggests finally that we can play a stone in a spot already occupied by another stone of any color. This would solve the idea (missing in the articles) to play two stones in the same location that I mentioned in my previous post. Both color and presence are quantum numbers each of which is a doublet. Each of them has binary mutually exclusive possible values in a specific space-time location (a specific intersection at a specific turn). If such a move is played, the conflict must be suddenly removed just following the rules described in the articles as occur in normal interaction. Only one stone will survive in that spot.

It would work fine from a “theoretical” (OMG) point of view. :sweat_smile:

What is missing now is the exciting part. The game.

  1. I like the rules proposed in the original article. The player can choose the surviving stones in an entangled pair affected by the interaction caused by a newly placed stone respecting the sequence of decisions:
    {player 1} - {player 2} - {player 1 on the last-placed stone}. No random choices.

  2. I also like the idea to play entangled stones in a place already occupied by another entangled stone.

Both points above would add a sort of strategical dimension to the game (as if it were necessary and Go wasn’t deep enough as it is :hot_face:).

But let me speculate on it. It is just for fun.


Today a server for playing Quantum Go was launched as a beta.


I tried it and it was hard, interesting and fun. Worth a look.


What are the rules then? Do you play two moves at once, or some normal moves and some “quantum moves” like the quantum chess mentioned earlier?

How do positions get resolved, what’s the mechanism/trigger for collapse essentially? :slight_smile:

I haven’t decided if I want to sign up just to find this out :slight_smile:

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All that information is on the website.

Here are the rules: https://quantumgo.net/quantum_go_v1.6.pdf

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This is all I get, and all the privacy policy etc is in Dutch (it might be German) I guess?

Anyway, maybe a brief summary of the ideas from the pdf @le_4TC linked:

  • play 2 or spots simultaneously, or pass. Each move essentially forks the game, giving a number of possible realities/board states.
  • all board states are equally likely to be chosen.
  • if a move is played that results in a capture in some reality then a reality/board state is picked at random, and the result is that realities where that capture is opposite to the chosen reality are discarded.
  • game ends with one intentional pass and another possibly unintentional pass, or when the last four moves are passes regardless of intention. The scoring uses stone scoring, a point per stone and 4 Komi to white.
  • Moves are illegal if they are illegal in all realities (stone exists, ko, suicide) or if the move already was played, that is pair or triple of moves with the same color. If an illegal move is attempted it becomes an unintentional pass.
  1. Practicalities
    The server only allows a certain maximum number of realities. When this number is exceeded the game gets determined down to one reality.

This actually makes a lot of sense given how massive the branching must be.

Apparently both players can see the total number of realities, the number of relative realities for a given color stone to be in a spot, which is a proportion and not a probability it stresses. They can if a measurement occurred and with which probability stones could have been captured, and if a capture occurred which locations the captured stones are etc.

Both players can request counting it says when there is only one reality left, and can then mark dead stones and count.

It stresses that

It should be noted, that this is not random Go. Where a player plays in random Go is determined at the time of the move, whereas in QGo it is determined at specific points in the game, that means (in most cases) not at the time of the move.

but that’s just a minor subtlety really. I get that it’s saying it’s not like whether stone chosen to be A or B is determined at the move the pair was played, instead it’s triggered by captures. Yet, it’s still the “Quantum” in quotes because it’s basically just a probabilistic game, but the name Quantum sounds cooler, when people are naming their games like Quantum tic-tac-toe and Quantum chess.


There are a few useful & interesting videos on youtube which explain the rules and some strategic ideas.

There is even some analysis software:

it seems allowing to choose more than 2 moves at once is bad idea. Unnecessary gameplay complication that makes it too hard to comprehend.

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