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arxiv: 2411.15294 · v2 · pith:GLQWUYD2new · submitted 2024-11-22 · 🪐 quant-ph · cs.ET· cs.GT

Imperfect-Information Games on Quantum Computers: A Case Study in Skat

Ulrich Armbr\"uster , Stefan Edelkamp , Gabriel Maresch , Erik Schulze This is my paper

Pith reviewed 2026-05-23 08:26 UTC · model grok-4.3

classification 🪐 quant-ph cs.ETcs.GT
keywords Skatquantum computingimperfect information gamesquantum countingquantum algorithmscard gamespayoff maximizationgame theory
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The pith

Skat rules and hidden cards can be encoded in quantum registers so a score operator and quantum counting identify the move with highest expected payoff.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors translate the imperfect-information card game Skat into a quantum circuit by placing known and hidden card assignments into quantum registers and building gates that enforce legal play. A score operator then projects each possible first move onto the subspace of outcomes where the player wins, allowing quantum counting to estimate the fraction of winning paths for each choice. Because the full decision tree grows too rapidly for classical enumeration once the number of hidden cards is large, the method is presented as a route to payoff maximization that becomes feasible only on quantum hardware. A reader would care if the approach scales, since many practical decisions under uncertainty share the same structure of hidden information and branching outcomes.

Core claim

By representing the current Skat position with quantum registers that hold the visible cards and the superposition over possible hidden assignments, constructing unitary gates that advance the game according to the rules, and applying a score operator whose eigenvalues mark winning terminal states, the winning probability for each legal action is obtained via quantum counting. The resulting payoff function is maximized to recommend the action that yields the highest long-run success rate, an evaluation the authors state cannot be performed classically for sufficiently large instances because of the size of the underlying search tree.

What carries the argument

Quantum registers holding the game state together with a score operator that projects onto the winning subspace, used inside a quantum counting routine to estimate the number of paths leading to victory.

If this is right

  • Selecting the legal move whose measured winning fraction is largest gives the action that maximizes the payoff function.
  • Quantum advantage is expected once the number of possible hidden-card assignments exceeds the scale at which classical exhaustive search remains practical.
  • The method supplies a statistical recommendation rather than a guaranteed winning line, consistent with the imperfect-information setting.
  • Game rules are enforced exactly through the choice of unitary operators that implement legal card play and scoring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same register-and-score-operator construction could be applied to other imperfect-information card games such as bridge once their rule sets are expressed as quantum gates.
  • Running the circuit on near-term hardware with a reduced deck would expose whether decoherence or measurement overhead prevents useful tree sizes from being reached.
  • Hybrid schemes that pre-filter the superposition with classical sampling might lower the qubit and gate resources required for the initial state.

Load-bearing premise

The rules of Skat together with the distribution of hidden cards can be faithfully realized as a quantum circuit and score operator whose measurement statistics match the true winning probabilities without being overwhelmed by decoherence or state-preparation cost.

What would settle it

Exact classical enumeration of winning probabilities for a concrete Skat position with a small but non-trivial number of hidden cards, followed by direct comparison against the output of the corresponding quantum counting circuit executed on a simulator.

Figures

Figures reproduced from arXiv: 2411.15294 by Erik Schulze, Gabriel Maresch, Stefan Edelkamp, Ulrich Armbr\"uster.

Figure 1
Figure 1. Figure 1: FIG. 1. Skat deal with Diamond 8 and Ace being taken, and [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Skat, classified among other games in terms of strate [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Trick taking partial order for a spades game. Totally [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Payoff per game for different choices of the declarer, [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Measurement of Φ [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Superposition of the twelve possible card distributions [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Superposition of the 24 possible card distributions [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Superposition of the 24 possible card distributions [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Superposition of the eight possible card distributions [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Scheme of a star-topology QPU. Qubits can only [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Benchmarking our quantum approach against a clas [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
read the original abstract

For decades it is known that Quantum Computers might serve as a tool to solve a very specific kind of problems that have long thought to be incalculable. Some of those problems are of a combinatorial nature, with the quantum advantage arising from the exploding size of a huge decision tree. Although this is of high interest as well, there are more opportunities to make use of the quantum advantage among non-perfect information games with a limited amount of steps within the game. Even though it is not possible to answer the question for the winning move in a specific situation, people are rather interested in what choice gives the best outcome in the long run. This leads us to the search for the highest number of paths within the game's decision tree despite the lack of information and, thus, to a maximum of the payoff-function. We want to illustrate on how Quantum Computers can play a significant role in solving these kind of games, using an example of the most popular German card game Skat. Therefore we use quantum registers to encode the game's information properly and construct the corresponding quantum gates in order to model the game progress and obey the rules. Finally, we use a score operator to project the quantum state onto the winning subspace and therefore evaluate the winning probability for each alternative decision by the player to be made by using quantum algorithms, such as quantum counting of the winning paths to gain a possible advantage in computation speed over classical approaches. Thus, we get a reasonable recommendation of how to act at the table due to the payoff-function maximization. This approach is clearly not doable on a classical computer due to the huge tree-search problem and we discuss peculiarities of the problem that may lead to a quantum advantage when exceeding a certain problem size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes encoding Skat (an imperfect-information card game) on a quantum computer by mapping game states to quantum registers, implementing rules via quantum gates, and applying a score operator whose measurement via quantum counting yields winning probabilities for payoff-maximizing decisions. It asserts that the exponential size of the decision tree renders the problem intractable classically but potentially amenable to quantum advantage at large scale.

Significance. If the encoding, gate constructions, and resource scaling can be shown to work, the approach would extend quantum algorithms from perfect-information games to a new class of imperfect-information combinatorial problems, with possible practical implications for decision-making under uncertainty. The emphasis on payoff maximization rather than single-instance winning moves is a constructive framing.

major comments (3)
  1. [Abstract] Abstract: the claim that quantum registers 'encode the game's information properly' and that a score operator 'project[s] the quantum state onto the winning subspace' is not supported by any explicit mapping from card configurations (including hidden hands) to basis states, nor by any derivation showing that measurement statistics recover the correct payoff function.
  2. [Abstract] Abstract and main text: no circuit construction, gate count, depth estimate, or scaling analysis is supplied to justify the assertion of quantum advantage 'when exceeding a certain problem size,' despite this being the central contrast with classical tree search.
  3. [Abstract] The manuscript provides neither a toy-model verification on a reduced game instance nor any resource estimate (qubits, gates, shots) that would allow assessment of whether decoherence or the continuous distribution of hidden-card assignments remains tractable.
minor comments (2)
  1. [Abstract] Abstract contains minor grammatical issues (e.g., 'For decades it is known' should read 'For decades it has been known').
  2. [Abstract] Several long sentences in the abstract could be split for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below, clarifying the conceptual scope of the proposal while indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that quantum registers 'encode the game's information properly' and that a score operator 'project[s] the quantum state onto the winning subspace' is not supported by any explicit mapping from card configurations (including hidden hands) to basis states, nor by any derivation showing that measurement statistics recover the correct payoff function.

    Authors: The manuscript is a conceptual outline of a quantum-register encoding for Skat states (including hidden hands) and a score operator whose measurement yields payoff estimates via quantum counting. We agree that the abstract and main text do not supply explicit basis-state mappings or a derivation of the payoff function. We will revise the abstract to qualify the claims as high-level and add a dedicated subsection with a concrete example mapping for a reduced card configuration together with the corresponding measurement statistics. revision: yes

  2. Referee: [Abstract] Abstract and main text: no circuit construction, gate count, depth estimate, or scaling analysis is supplied to justify the assertion of quantum advantage 'when exceeding a certain problem size,' despite this being the central contrast with classical tree search.

    Authors: The central contrast drawn in the manuscript is the exponential growth of the decision tree under imperfect information, which quantum counting can in principle address by estimating the measure of winning paths. We acknowledge that no explicit circuit constructions, gate counts, or quantitative scaling analysis are provided. We will add a new section that supplies qualitative scaling arguments based on the qubit requirements for encoding card distributions and the known complexity of quantum counting, while making clear that a full resource analysis lies beyond the scope of the present case study. revision: partial

  3. Referee: [Abstract] The manuscript provides neither a toy-model verification on a reduced game instance nor any resource estimate (qubits, gates, shots) that would allow assessment of whether decoherence or the continuous distribution of hidden-card assignments remains tractable.

    Authors: We agree that the absence of a toy-model verification and concrete resource estimates limits the ability to assess practical tractability, including decoherence and the handling of hidden-card distributions. The work is framed as a proposal for the overall approach rather than an implementation study. We will revise the manuscript to include a qualitative discussion of qubit counts needed to encode a Skat hand and to note that detailed toy-model simulations and decoherence analysis are reserved for follow-up work. revision: partial

Circularity Check

0 steps flagged

No circularity; high-level proposal lacks explicit derivations or self-referential reductions.

full rationale

The manuscript outlines a conceptual mapping of Skat rules to quantum registers, gates, and a score operator followed by quantum counting, but supplies no equations, circuit constructions, or parameter fits. No step reduces a claimed prediction or advantage to an input by definition, self-citation chain, or renaming. The discussion of potential quantum advantage at large problem size remains an unelaborated assertion rather than a derivation that collapses onto its premises. The paper is therefore self-contained at the level of a proposal and exhibits no load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal rests on standard quantum-computing assumptions applied to an unformalized game model. No free parameters, new physical entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Quantum superposition and entanglement can represent distributions over hidden information without loss of correctness for payoff estimation.
    Implicit in the statement that quantum registers encode the game's information properly.
  • domain assumption Quantum gates exist that simulate every legal sequence of moves while obeying Skat rules.
    Stated directly in the abstract as 'construct the corresponding quantum gates in order to model the game progress and obey the rules.'

pith-pipeline@v0.9.0 · 5855 in / 1435 out tokens · 59080 ms · 2026-05-23T08:26:47.432670+00:00 · methodology

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