Quantization of Blackjack: Quantum Basic Strategy and Advantage (2011.12342v3)

Abstract: Quantum computers that process information by harnessing the remarkable power of quantum mechanics are increasingly being put to practical use. In the future, their impact will be felt in numerous fields, including in online casino games. This is one of the reasons why quantum gambling theory has garnered considerable attention. Studies have shown that the quantum gambling theory often yields nontrivial consequences that classical theory cannot interpret. We devised a quantum circuit reproducing classical blackjack and found possible quantum entanglement between strategies. This circuit can be realized in the near future when quantum computers are commonplace. Furthermore, we showed that the player's expectation increases compared to the classical game using quantum basic strategy, which is a quantum version of the popular basic strategy of blackjack.

• The paper introduces a quantum formalism for Snackjack by mapping player and dealer actions to qubit operations and applying the EWL protocol.
• It shows that optimal quantum strategies shift the player's expected payoff from -1.7% in classical play to as high as +10.2% under maximal entanglement.
• A detailed quantum circuit is presented to simulate probabilistic card drawing and conditional game actions, highlighting practical implementation challenges.

This paper explores the application of quantum game theory to Blackjack, specifically using a simplified version called "Snackjack" (2011.12342). The core idea is to investigate whether quantum phenomena, particularly entanglement between player and dealer strategies, can provide an advantage to the player compared to classical strategies.

Classical Snackjack Formalism:

The authors first define the rules for Snackjack, a simplified Blackjack using only eight cards (two Aces, two 2s, four 3s) with a target score of 7. Aces count as 1 or 4. The dealer must stand on soft 6 or more, and the player can only hit once.

They then frame this classical game using quantum notation:

• The player's strategy (Hit/Stand) and the dealer's state are represented by qubits. The initial state is typically $\ket{0} \otimes \ket{1}$ (Player Stand, Dealer Active).
• Player strategies are unitary operators: $\hat{I}$ for Stand and $\hat{X}$ (Pauli-X) for Hit.
• The dealer's strategy is fixed ($\hat{I}$), as their actions are deterministic based on their hand total.
• The player's expected payoff (utility) $P_1$ depends on the initial cards $(p, d)$ and their chosen strategy $\hat{S}_1$:

  $$P_1(p,d,\hat{S}_1) = E_{\mathrm{std}}\bra{0}\otimes\bra{1}(\hat{S}_1\otimes\hat{I})\rho + E_{\mathrm{hit}}\bra{1}\otimes\bra{1}(\hat{S}_1\otimes\hat{I})\rho$$

  where $E_{\mathrm{std}}$ and $E_{\mathrm{hit}}$ are the payoffs for standing and hitting, respectively, given the initial deal.

Introducing Quantum Entanglement:

The authors apply the Eisert-Wilkens-Lewenstein (EWL) protocol to introduce entanglement between the player's and dealer's strategy qubits before the player chooses their action.

• An entangling operator $$\mathcal{J} = \mathrm{exp}(-i\frac{\gamma}{2} \hat{X} \otimes \hat{U})$$ is applied, where $\gamma$ controls the "entanglement intensity" and $\hat{U}$ is a unitary operator parametrized by $\theta$ (specifically $\hat{U} = \sin\theta \hat{X} + \cos\theta \hat{Z}$).
• A disentangling operator $\mathcal{J}^\dagger$ is applied after the player's strategy operator $\hat{S}_1$.
• Crucially, this setup reproduces the classical game if the player chooses classical strategies ($\hat{I}$ or $\hat{X}$).
• However, it allows the player to choose other quantum strategies, like the Pauli operators $\hat{Y}$ and $\hat{Z}$.
• These quantum strategies, combined with entanglement, lead to final states involving components like $\ket{00}$ and $\ket{10}$, which don't occur classically. These correspond to new payoff possibilities, $E_{00}$ and $E_{10}$.
• The player's expected payoff now depends on $\gamma$, $\theta$, and the chosen strategy from $\{\hat{I}, \hat{X}, \hat{Y}, \hat{Z}\}$, incorporating $E_{\mathrm{std}}, E_{\mathrm{hit}}, E_{00}, E_{10}$. For instance, the payoff for strategy $\hat{Y}$ is $$\sin^2\gamma(\cos^2\theta E_\mathrm{std} +\sin^2\theta E_{00}) +\cos^2\gamma E_\mathrm{hit}$$.

Quantum Circuit Implementation:

A significant part of the paper details a quantum circuit designed to simulate Snackjack, including the entanglement protocol.

• Hilbert Space: The circuit uses several registers: $\mathcal{H}_{\rm deck}$ (state of cards in the deck), $\mathcal{H}_{\rm deck\,copy}$ (a copy for technical reasons), $\mathcal{H}_{\rm p-hand}$ (player's hand), $\mathcal{H}_{\rm d-hand}$ (dealer's hand), $\mathcal{H}_{\rm p-strategy}$ (player strategy qubit), $\mathcal{H}_{\rm d-strategy}$ (dealer strategy qubit), and $\mathcal{H}_{\rm control}$ (ancilla qubits for card drawing).
• Card Representation: Each potential card position (e.g., first Ace, second Ace, first 2, etc.) is a qubit. $\ket{1}$ means the card exists/is held, $\ket{0}$ means it doesn't/isn't.
• Hit Operator ($\hat{\mathcal{H}}$): This is the core mechanism for drawing cards. It uses the control qubits $\ket{\Psi}$ (initially in an equal superposition of all basis states) to probabilistically select a card sector. Controlled-SWAP operations are used: if the control qubit state corresponds to card $i$ and card $i$ exists in the deck ($\ket{D_i} = \ket{1}$), then the state of the deck qubit $\ket{D_i}$ is swapped with the corresponding player/dealer hand qubit $\ket{p_i}$ or $\ket{d_i}$. Measuring the control qubits reveals which card was drawn without collapsing the strategy or hand superposition if quantum strategies are used. A check ensures a valid card is drawn.
• Classical Circuit (Fig. 1 & 2): Shows the flow for classical Snackjack using this quantum representation.
• Quantum Circuit (Fig. 3): Inserts the $\mathcal{J}$ and $\mathcal{J}^\dagger$ gates around the player's strategy operation $\hat{S}_1$ to implement the EWL protocol.

graph TD A[Initialize: Deck, Hands, Strategy Qubits (p=0, d=1)] --> B(Apply J Gate); B --> C{Player Chooses Strategy S1}; C -- S1 = I --> D(Apply J_dagger Gate); C -- S1 = X --> E(Apply X on Player Strategy Qubit); E --> F(Player Hit Operation: Apply H Gate); F --> D; C -- S1 = Y --> G(Apply Y on Player Strategy Qubit); G --> D; C -- S1 = Z --> H(Apply Z on Player Strategy Qubit); H --> D; D --> I(Dealer Reveals Card & Hits if Needed: Apply H Gate conditionally); I --> J[Measure Strategy Qubits]; J --> K[Calculate Payoff based on final state (00, 01, 10, 11) and Hands]; style A fill:#f9f,stroke:#333,stroke-width:2px style B fill:#ccf,stroke:#333,stroke-width:2px style C fill:#ff9,stroke:#333,stroke-width:2px style D fill:#ccf,stroke:#333,stroke-width:2px style E fill:#f9f,stroke:#333,stroke-width:2px style F fill:#9cf,stroke:#333,stroke-width:2px style G fill:#f9f,stroke:#333,stroke-width:2px style H fill:#f9f,stroke:#333,stroke-width:2px style I fill:#9cf,stroke:#333,stroke-width:2px style J fill:#f9f,stroke:#333,stroke-width:2px style K fill:#9f9,stroke:#333,stroke-width:2px

Diagram: Simplified flow of the quantum game circuit.

Results and Quantum Basic Strategy (QBS):

• Classical: The calculated expected value for the player using classical basic strategy (CBS) in this Snackjack version is -1.7% (dealer advantage).
• Quantum: By calculating the expected payoffs for all four strategies $\{\hat{I}, \hat{X}, \hat{Y}, \hat{Z}\}$ under entanglement, the authors derive a Quantum Basic Strategy (QBS).
• Advantage: With maximal entanglement ($\gamma = \pi/2$) and optimal parameter choice ($\theta = \pi/2$), using QBS yields a player expectation of +10.2%. This significant shift occurs because quantum strategies allow access to scenarios (like $E_{00}, E_{10}$) that can be more favorable than classical Hit/Stand outcomes depending on the initial deal.
• Parameter Dependence: The player's advantage depends strongly on the entanglement parameters $\gamma$ and $\theta$. For $\theta=0$, no advantage is gained regardless of $\gamma$. The maximum advantage occurs at $\gamma=\pi/2, \theta=\pi/2$. A half-entangled game ($\gamma=\pi/4, \theta=\pi/2$) still provides a +1.8% advantage.

Practical Implications:

• Demonstrates Quantum Advantage: Shows concretely how quantum effects (entanglement) applied to strategies (not just the game physics) can alter optimal play and expected outcomes in a game setting.
• Circuit Blueprint: Provides a practical, albeit complex, quantum circuit design for simulating card games, including probabilistic card drawing and conditional actions within a quantum framework. This could be adapted for simulating other card games or probabilistic processes on quantum computers.
• Implementation Considerations: Realizing this requires a quantum computer capable of maintaining coherence across multiple qubits representing cards, hands, and strategies, and implementing multi-qubit controlled gates (for $\mathcal{J}$ and $\hat{\mathcal{H}}$). The number of qubits scales with the deck size. NISQ devices would likely face significant noise challenges.
• Future Applications: Suggests potential impacts on online gambling if quantum computers become accessible, potentially requiring casinos to adapt rules to account for quantum strategies. The card-drawing circuit element might be reusable in other quantum simulations involving sampling or probabilistic state changes.