REVIEW 2 major objections 7 minor 12 references
Reviewed by Pith at T0; open to challenge.
T0 means a machine referee read the full paper against a public rubric. The mark states how deep the mechanical check went, never who wrote it. the ladder, T0–T4 →
T0 review · glm-5.2
Quantum moves can't beat perfect classical play in board games
2026-07-08 01:57 UTC pith:6HPSGGE4
load-bearing objection Quantum moves do not help against perfect classical play in combinatorial games; clean negative result with a Byzantine consensus corollary. the 2 major comments →
Quantum combinatorial games
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper's central discovery is that the constraint binding quantum moves to classical reachability is strong enough to preserve the entire classical outcome-guarantee structure. The key objects are the sets Sigma_T and Pi_T, which record, for any target outcome T, whether the current player or the opponent can force that outcome. Theorem 21 shows by induction that if every classical position in the support of a quantum state lies in Sigma_T (or Pi_T), then the quantum state itself lies in the corresponding quantum guarantee set. The induction step for the opponent's turn is the load-bearing one: because a valid quantum move U on a state |phi> must send each component g in the support to a
What carries the argument
quantizable combinatorial game
Load-bearing premise
The definition of a valid quantum move is permissive: a unitary counts as valid if its output is a superposition of positions each reachable by some classical move from some position in the input support, without requiring that each individual classical component map to a superposition of its own legal successors. This permissiveness is what enables the flaw-amplification trick, and the authors acknowledge that a stricter 'completely valid' definition would block it. The main
What would settle it
Construct a quantizable combinatorial game and a position where a quantum player can guarantee an outcome that no classical player can guarantee from the same position, under any reasonable definition of valid quantum move. This would directly contradict Theorem 21.
If this is right
- For any finite deterministic perfect-information game, adding quantum moves to one or both players does not change which outcomes can be guaranteed under perfect play — the classical minimax value of the game is preserved.
- Classical consensus algorithms for distributed systems with adversarial nodes remain correct when adversaries gain quantum communication capabilities, provided honest nodes play classically.
- The flaw-amplification result suggests that in practical quantum-vs-classical game play, the classical player must play perfectly or risk exploitation — but this advantage depends on the permissive definition of valid quantum moves and may vanish under stricter definitions.
- The framework extends the reach of Zermelo's theorem to quantum games in a way that eliminates probabilistic uncertainty: the quantum guarantee is not just over distributions but over deterministic outcomes matching the classical ones.
Where Pith is reading between the lines
- If the result extends to games with chance (by modeling chance as a third player, as the authors suggest), then quantum moves would also fail to improve outcomes in backgammon-like games against perfect classical play, narrowing the space where quantum game theory offers genuine advantage to games with hidden information or simultaneous moves.
- The distinction between 'valid' and 'completely valid' quantum moves suggests a spectrum of quantum game formulations: the main no-advantage theorem is robust across this spectrum, but the practical advantage against imperfect play is sensitive to where on the spectrum the rules sit, which could inform the design of quantum game implementations on actual hardware.
- The preservation of classical guarantee structure through quantum moves resembles a conservation law: the classical game tree's win/loss/draw labeling is an invariant under quantum extensions, which raises the question of whether other game-theoretic invariants (e.g., Sprague-Grundy values for impartial games) might also be preserved.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper proposes a general framework for quantizing finite two-player combinatorial games with perfect information. The authors define a quantum variant G^Q of a classical game G in which positions are superpositions of classical positions and moves are unitaries subject to a permissive validity condition. The central result (Theorem 21, Corollary 22) is a quantum Zermelo theorem: against a perfect classical opponent, quantum moves confer no advantage—the set of guaranteeable outcomes is identical (up to distributions) in the classical and quantum variants. The proof proceeds by strong induction on N(g), the maximum remaining moves, using a bisimulation argument to reduce arbitrary combinatorial games to a quantizable form (injective and fixed-length). A secondary result (Lemma 24) shows that against an imperfect classical opponent, a quantum player can exploit the permissive validity definition to amplify mistakes. The framework is applied to the Byzantine Generals problem, yielding a corollary that classical consensus algorithms tolerating classical adversaries also tolerate quantum adversaries.
Significance. The paper addresses a natural and well-motivated question: whether quantum moves can improve a player's guaranteed outcome in a combinatorial game against a perfect classical opponent. The main theorem is a clean, parameter-free result—the proof uses only the definitions of Sigma_T, Pi_T, valid quantum moves, and induction on N(g), with no fitted constants or normalization choices. The bisimulation machinery (Proposition 12, Corollary 35) ensuring that the transcribed/extended variants preserve Sigma_T and Pi_T is standard but correctly applied. The framework's applicability to the Byzantine consensus problem (Corollary 38) demonstrates relevance beyond board games. The flaw-amplification result (Lemma 24) is correctly scoped to the permissive validity definition and provides a concrete, falsifiable contrast with the main theorem. The contrast with Dorbec and Mhalla's framework, where quantum moves do affect outcomes, is clearly attributed to the difference in how moves are restricted (Section 1, Related Work).
major comments (2)
- Section 4 / Definition 37: The Byzantine consensus application is presented as a sketch with details deferred to Appendix D, but the modeling raises a question about the game's structure. Player X commits to an algorithm A before player O chooses the initial setup (d, B), and then player X's subsequent moves are fixed (applying sys then A). This means player X has no strategic choices after initialization, yet the game is defined as a combinatorial game where players alternate moves. The bisimulation and Theorem 21 require that the game satisfies Definition 1, including condition 2 (taking turns) and condition 3 (no moves implies finished). It is not immediately clear that G_BF satisfies these conditions as stated, particularly whether player X's 'fixed' moves constitute valid moves in the sense of Definition 1. The authors should clarify how the fixed-strategy phase fits the combinatori
- Definition 13, item 4 (valid quantum move): The validity condition requires that for all g' in Supp(U|phi>), there exists g in Supp(|phi>) and sigma in Sigma^C with g ->sigma g'. This is the permissive notion that enables the flaw-amplification trick (Lemma 24). The authors note (Remark 25) that the main theorem holds under the stricter 'completely valid' notion as well. However, the paper does not formally state or prove Theorem 21 under the completely valid notion; Remark 25 only asserts it. Since the completely valid notion is arguably more natural (it requires U|g> to be a superposition of positions reachable from g specifically, not just from some g in the support), a brief formal verification— even a one-paragraph argument—would strengthen the claim and clarify the relationship between the two notions. This is load-bearing because the practical-advantage claim (Section 3.2) depends
minor comments (7)
- Definition 1, condition 4: The condition that moves on finished positions do not change the outcome is motivated by the quantum setting (superpositions of finished and unfinished positions). The paper could note more explicitly that this is a standard 'pass' or 'skip' convention in combinatorial game theory, to reassure readers familiar with the BCG convention.
- Remark 15: The verification that G^Q satisfies condition 3 (no moves implies finished) relies on Lemma 18, which is proven later. A forward reference or a brief note that this is deferred would improve readability.
- Section 2: The notation N(g) for maximum remaining moves is introduced in Definition 2, but the term 'birthday' is mentioned parenthetically. A brief citation to the standard combinatorial game theory literature where this term appears would help readers cross-referencing.
- Lemma 24 proof: The construction extends |phi> and |h> to orthonormal bases and defines U by U|e_n> = |f_n>. The proof states this U is valid because Supp(U|phi>) = {h} and g ->sigma h. The validity condition (Definition 13, item 4) requires that for all g' in Supp(U|phi>), there exists g in Supp(|phi>) with g ->sigma g'. This is satisfied since h is reachable from g. However, the proof could note that U need not be completely valid (in the sense of Remark 25), which is why the trick fails under the stricter notion. Stating this explicitly would connect the result to the discussion in Remark 25.
- The paper would benefit from a concrete small example (e.g., a simplified tic-tac-toe position) illustrating the flaw-amplification trick of Lemma 24. The current description is abstract; a worked example with specific positions and a specific unitary would make the Section 3.2 contribution more accessible.
- Reference [RVDNMB+25] is cited as a 2025 IEEE Conference on Games paper. If this is a forthcoming publication, the authors should verify the citation details against the published version.
- Page 3, 'Kyle Burke et al. [BFT20] study several properties... most notably (for our research) showing that quantum moves haveimpact on the game's outcome': missing space in 'haveimpact' (should be 'have an impact').
Simulated Author's Rebuttal
We thank the referee for a careful and constructive report. Both major comments are well-taken and will be addressed in the revised manuscript. The first concerns a gap in clarifying how the Byzantine consensus game G_BF satisfies the combinatorial game definition (Definition 1), particularly regarding the fixed-strategy phase. The second requests a formal verification that Theorem 21 holds under the stricter 'completely valid' notion of quantum move, rather than just the assertion in Remark 25. We agree with both points and will revise accordingly.
read point-by-point responses
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Referee: Section 4 / Definition 37: The Byzantine consensus application is presented as a sketch with details deferred to Appendix D, but the modeling raises a question about the game's structure. Player X commits to an algorithm A before player O chooses the initial setup (d, B), and then player X's subsequent moves are fixed (applying sys then A). This means player X has no strategic choices after initialization, yet the game is defined as a combinatorial game where players alternate moves. The bisimulation and Theorem 21 require that the game satisfies Definition 1, including condition 2 (taking turns) and condition 3 (no moves implies finished). It is not immediately clear that G_BF satisfies these conditions as stated, particularly whether player X's 'fixed' moves constitute valid moves in the sense of Definition 1.
Authors: The referee raises a valid concern. The issue is that Definition 1 requires alternating turns (condition 2) and that a position with no valid moves is finished (condition 3), but in G_BF as described, player X's moves during the simulation phase are not strategic choices—they are determined by the pre-committed algorithm A. We see two ways to address this, and will adopt the cleaner one in the revision. The most natural fix is to reformulate G_BF so that player X's 'move' during each simulation round is the unique deterministic action prescribed by (sys, A). This is a valid move in the sense of Definition 1: it is an element of Sigma_move (specifically, the composition of sys with A applied to non-Byzantine nodes), it is the only valid move for player X in that position, and it alternates the turn to player O. Condition 2 is satisfied because the transition changes the player-to-move. Condition 3 is satisfied because if no move exists for the current player, the position is finished. The key observation is that a 'move' in our framework need not involve a strategic choice—it need only be an element of Sigma that is valid on the current position. A position where the current player has exactly one valid move is perfectly consistent with Definition 1. We will clarify this explicitly in the revision by stating that during the simulation phase, player X's move set is a singleton {sigma_A} for each round, where sigma_A is the deterministic action prescribed by the committed algorithm. We will also verify that all six conditions of Definition 1 are satisfied and state this explicitly, rather than leaving it as 'pretty straightforward' as we do for other constructions. Additionally, we will note that the initialization phase—where X chooses A and O chooses (d, B)—fits the comb revision: no
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Referee: Definition 13, item 4 (valid quantum move): The validity condition requires that for all g' in Supp(U|phi>), there exists g in Supp(|phi>) and sigma in Sigma^C with g ->sigma g'. This is the permissive notion that enables the flaw-amplification trick (Lemma 24). The authors note (Remark 25) that the main theorem holds under the stricter 'completely valid' notion as well. However, the paper does not formally state or prove Theorem 21 under the completely valid notion; Remark 25 only asserts it. Since the completely valid notion is arguably more natural (it requires U|g> to be a superposition of positions reachable from g specifically, not just from some g in the support), a brief formal verification—even a one-paragraph argument—would strengthen the claim and clarify the relationship between the two notions.
Authors: The referee is correct that Remark 25 asserts but does not formally verify that Theorem 21 holds under the completely valid notion. We will add a formal argument in the revision. The key observation is straightforward: in the proof of Theorem 21, the only quantum moves that appear are (a) the classical moves U_omega constructed via Lemma 18, which are completely valid by construction (each U_omega|g> = |Gamma(g, omega(g))> is a single classical position reachable from g), and (b) the opponent's arbitrary valid moves U, which in statement (5) are universally quantified. If we restrict to completely valid moves, the universal quantification in (5) ranges over a smaller set, so the implication 'Supp|phi> subset Pi^C_T implies |phi> in Pi^Q|P_D(T)' becomes easier to satisfy, not harder. For statement (4), the move we construct is classical and hence completely valid. Thus both directions of the induction go through unchanged. We will write this up as a formal remark or proposition following Theorem 21, stating that Theorem 21 holds verbatim when 'valid' is replaced by 'completely valid', with the one-paragraph proof sketched above. This also makes explicit why the permissive notion is the right default for the main theorem: it makes the theorem stronger, and the flaw-amplification result (Lemma 24) shows that the permissive notion is not vacuous—it has real consequences for imperfect play. revision: yes
Circularity Check
No significant circularity: the main theorem is a parameter-free inductive proof with no fitted inputs or self-citation chains.
full rationale
The paper's central result (Theorem 21) is derived by strong induction on N(g), the maximum number of remaining moves. The proof uses only the definitions of Σ_T, Π_T (Definition 3), valid quantum moves (Definition 13, item 4), and Lemma 18 (which constructs a unitary from an injective game's transition map). No parameters are fitted to data, no normalization is chosen to force the result, and no self-citation chain bears the load of the argument. The bisimulation arguments (Proposition 12, Corollary 35) are standard and self-contained. Corollary 22 follows directly from Theorem 21 and the classical Zermelo theorem (Corollary 6, which cites Zermelo [Zer13] — an external, century-old result). The Byzantine consensus corollary (Corollary 38) is a direct application of Corollary 22 to a game-theoretic encoding of the problem. The flaw-amplification result (Lemma 24) is correctly scoped and does not feed back into the main theorem. The permissive validity definition (footnote 4, Remark 25) is acknowledged by the authors as strengthening the theorem rather than weakening it, since the theorem must hold against a more powerful quantum player. No step in the derivation chain reduces to its own inputs by construction.
Axiom & Free-Parameter Ledger
axioms (4)
- domain assumption Positions of a combinatorial game can be encoded as length-L bitstrings (G ⊆ {0,1}^L), enabling unitary operations on the Hilbert space.
- domain assumption The game is finite (finite positions, moves, outcomes) and short (ends after finitely many moves).
- ad hoc to paper A valid quantum move U on |φ⟩ requires only that U|φ⟩ is a superposition of positions reachable by some classical move from some position in Supp|φ⟩.
- standard math Bisimulation preserves the Σ_T and Π_T sets of a game.
read the original abstract
A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games. We investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo's classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. In a more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond the mere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.
Reference graph
Works this paper leans on
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[1]
SinceN(g ′ 1)< N(g 1), we know that equation (2) already holds forN(g ′ 1). In particular, g′ 2 ∈Π T . This means that fromg 2 we can move intoΠ T , sog 2 ∈Σ T . Suppose instead thatg 2 ∈Σ T ; we must show thatg 1 ∈Σ T . Sinceg 2 is not finished, we know there is some moveg 2 σ2 − →g′ 2 ∈Π T . Sinceg 1 ∼g 2, andg 2 is not finished, there is some move g1 σ...
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[2]
Thusg ′ 1 ∈Π T since (2) already holds forg ′
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[3]
Thusg 1 ∈Σ T . □ B Making a game injective and of fixed length To make a game injective, we employ the following construction. Definition 27.Thetranscribed variantG tr of a combinatorial gameGis defined as follows. –Aspositions,G tr hastranscripts, pairs(g, R)∈ G ×Σ ∗, whereg∈ GandR≡(R 1, . . . , Rn) is a sequence of moves inGfrom0tog, so0 R1 − − →g1 R2 −...
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[4]
We haveg∈ Fiff (g, R)∈ F tr, and in that caseO(g) =O((g, R)), by definition
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[5]
(a) Ifg σ− →g′, then we have the corresponding move (g,(σ 1,
Supposeg̸∈ F. (a) Ifg σ− →g′, then we have the corresponding move (g,(σ 1, . . . , σn)) σ− →(g′,(σ 1, . . . , σn, σ)) with g′ ∼(g ′,(σ 1, . . . , σn, σ)). (b) If (g,(σ 1, . . . , σn)) σ− →(g′,(σ 1, . . . , σn, σ)) then we have the corresponding moveg σ− →g′ ∼ (g′,(σ 1, . . . , σn, σ)). □ Lemma 33.A combinatorial gameGand its extended variantG ext are bisi...
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[6]
We haveg∈ Fiff (g, P)∈ F ext, and in that case,O((g, P))≡ O(g), by definition
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[7]
Supposeg̸∈ F (a) Wheng σ− →g′ for someσ∈Σ, we have a corresponding move (g, P) σ− →(g′, P), with g′ ∼(g ′, P). (b) When (g, P) σ− →(g′, P ′), we knowσ̸=•(sinceg /∈ F), so we have a corresponding move g σ− →g′ inG, withg ′ ∼(g ′, P ′). □ WhenceG ∼ G ext ∼(G ext)tr. To conclude thatG ∼(G ext)tr, we need the fact that bisimilarity is a transitive relation on...
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[8]
Assumeg 1 ̸∈ F 1, then: (a) Ifg 1 σ1 − →g′ 1, theng 2 σ2 − →g′ 2 for someσ 2 andg ′ 2, withg ′ 1 ∼g ′
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[9]
SinceG 2 ∼ G 3 we also have thatg 3 σ3 − →g′ 3 for someσ 3 andg ′ 3 withg ′ 2 ∼g ′
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[10]
(b) This case is analogous to (a)
This gives thatg ′ 1 ∼g ′ 3. (b) This case is analogous to (a). So,G 1 ∼ G 3. □ Corollary 35.Any combinatorial gameGis bisimilar to its extended and then transcribed variant, G ∼(G ext)tr. Proof.This follows directly from Lemmas 32, 33, and 34. □ C Omitted theory and proofs (Quantum combinatorial games) Lemma 36.Given a moveUon a position|φ⟩in the quantum...
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[11]
Then|φ 2⟩=|g 1⟩by definition of∼
Letg 1 ∼ |φ 2⟩be given. Then|φ 2⟩=|g 1⟩by definition of∼. Now,g 1 ∈ Fiff|g 1⟩ ∈ F Q, by definition of the quantum variant. Ifg 1 ∈ F, thenO(|g 1⟩) =O(g 1), again by definition of the quantum variant. 18 Dieks Scholten, Bram Westerbaan, and Simona Samardjiska
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[12]
(a) Ifg σ− →g′, then by Lemma 18 we get that|g⟩ Uσ − − →Uσ |g⟩=|g ′⟩, andg ′ ∼ |g ′⟩. (b) If|g⟩ U− → |φ′⟩, then sinceUmust be classical,U|g⟩=|φ ′⟩ ≡ |g ′⟩for someg ′ ∈ G. By definition of a valid quantum move, there is aσ∈Σ C withg σ− →g′(∼ |φ ′⟩). Hence we have a bisimulation,G ∼ G C–C . □ D Quantum Byzantine consensus We model the network as a graphG= (...
discussion (0)
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