Quantization of Blackjack: Quantum Basic Strategy and Advantage
OOU-HET-1057
Quantization of Blackjack:Quantum Basic Strategy and Advantage
Yushi Mura ∗ and Hiroki Wada † Osaka University, Toyonaka, Osaka 560-0043, Japan
Quantum computers that process information by harnessing the remarkable power of quantummechanics are increasingly being put to practical use. In the future, their impact will be felt innumerous fields, including in online casino games. This is one of the reasons why quantum gamblingtheory has garnered considerable attention. Studies have shown that the quantum gambling theoryoften yields nontrivial consequences that classical theory cannot interpret. We devised a quantumcircuit reproducing classical blackjack and found possible quantum entanglement between strategies.This circuit can be realized in the near future when quantum computers are commonplace. Fur-thermore, 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.
I. INTRODUCTION
Quantum information science is growing rapidly not only in theory but also at the hardware level; this leads usto believe that quantum computers will soon be practically applied on a large scale. Once quantum computers arerealized on a commercial scale, quantum technology (such as devices or communications) will be widespread. Insuch a future, an interesting question that arises is how quantum computers can change the world. For example, inquantum game theory, Eisert et al. showed that quantum entanglement changes the Nash equilibrium [1] betweenthe strategies of two persons in prisoners’ dilemma [2]. The strategy of whether to cooperate or not by one personaffects another person’s strategy through quantum entanglement between the two individual strategies. Prisonerscan choose quantum strategies, which are described by unitary operators; therefore, they have more strategies athand compared to the classical game and can use them to increase their profit. Meyer showed that for a two-playerzero-sum strategic game, a player’s expectation can increase when using quantum strategies [3], and their game wasgeneralized to N strategies game [4].The equilibrium solution in the infinitely repeated quantum strategic game wasrecently studied by [5]. In this manner, quantized strategies in game theory induce nontrivial results that are notexpected classically.These ideas of quantizing the strategies can also be applied to gambling theory. A fair quantum gambling gameplayed by two players far from each other was proposed by [6]. In the classical case, it is difficult to ensure fairplay without introducing a third party to oversee the game. However, by providing a quantum superposition stateand then observing it, it is possible to achieve fair gambling. In addition, procedures were devised to protect thesecurity of such remote gambling from entanglement attacks [7, 8]. Moreover, Zhang et al. devised and realized anoptical quantum circuit that helps achieve fair gambling and experimentally showed that the error is practically undercontrol [9]. In the sense of quantizing a game, some quantized games have been proposed [10–15]. Quantum duelwas proposed [10], and a Russian roulette game using a quantum gun was proposed and studied [11, 12]. Quantumchess was devised as a more casual game [13]. Quantized strategies in poker, one of the most popular card games,were also considered when Meyer [3] quantized strategies [15]. Recently, a game similar to poker was presented ona quantum computer by [14]. In their experiment, the researchers investigated the expected value under quantumnoise on a practical quantum computer. They also showed that the expected value can be improved using quantumerror-mitigation techniques. These studies suggest that quite a few games based on quantum mechanics have beenexamined.Recent studies have shown that quantum bits yield advantages to blackjack, which is one of the most popularcasino games[16]; the researchers showed that the expectation values of two cooperating persons can increase byimparting information of each individual’s card with entangled qubits. In the present paper, cards are representedas qubits and we devised a quantum circuit for reproducing classical blackjack. Classical blackjack has been studiedstatistically for more than half a century [17–19]. It is decided whether the player should do hit or stand . This iscalled Basic Strategy . Moreover, it is widely known in various rules, such as number of decks, options, and refunds ∗ y [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] J a n when the player has blackjack [19, 20]. In blackjack, as the player can choose strategies (e.g., hit or stand), we canquantize not only the game itself but also the strategies involved using a method similar to [2, 3]. This quantumcircuit reproducing blackjack can be easily realized in the near future when quantum computers are commonplace.Furthermore, we showed that the player’s expectation value increases substantially when quantum strategies areemployed in the presence of entanglement.In the next section II, we explain the blackjack rule that we adopted (simplified for ease of calculation). In sectionIII, we propose a quantum circuit that reproduces blackjack, (especially a toy model,) and express the possibility ofentanglement. In section IV, we show classical expectation value and basic strategy; we also show that using quantumstrategies in the presence of entanglement, the player’s expectation value increases compared to the classical game. II. BLACKJACK RULES AND TOY MODEL
Blackjack is a one-on-one card game between a dealer (Alice) and a player (Bob). One deck comprises 52 cardsexcluding jokers. Aces are worth 1 (hard) or 11 (soft), face cards (JQK) are worth 10, and every other card hasits number value. After the player has placed his bet, the dealer distributes two cards to the player. Similarly, shedistributes two cards to herself. At this time the dealer has one card face-up and the other face-down. The objectiveis to get a hand with a total closer to 21. It is worth noting however, that the 21 hand which is made of one ace andone face card or 10 is the strongest (called natural).First, the player must choose to hit or stand. If he chooses to hit, he draws a card. If he exceeds 21 (bust), heloses his bet regardless of the dealer’s hand. Next, the dealer opens the face-down card, and if her total is soft 16or less, she must hit. She repeats this procedure until her total reaches soft 17 or more. If the dealer busts and theplayer does not, or if the player’s hand is closer to 21 than the dealer’s, the player wins the bet. If the player’s totalequals the dealer’s total, the bet will be returned (push). In blackjack, the player can do his best by calculating theexpected value of the profit that he will obtain in each initial state. This is widely known as a basic strategy and hasbeen calculated in various rules [19, 20]. Note that only the player can choose whether to hit or stand. Conversely,the dealer has no choice of a strategy since she just draws the card mechanically until her total reaches soft 17 ormore. At first glance, the player who has many choices seems to have a competitive edge. However, the dealer hasthe advantage because the player loses if both hands exceed 21. The reason many casinos adopt option rules, such asdouble down, split, and surrender is to increase the player’s expectation.For simplicity, we quantized a toy model of blackjack proposed by Ethier [21] and called snackjack by Epstein[22]. As snackjack is highly simplified compared to blackjack which uses 52 cards, it is easy to calculate the basicstrategy by hand. Recently, Ethier calculated the expectation and basic strategy under various options, with cardcounting, and with bet variation in this classical snackjack [23]. Snackjack uses eight cards AA223333, aiming towarda maximum value of 7, with an ace having a value of either 1 or 4, and 2 and 3 having their own values. In this paper,we ignored various options and simplified them for calculation. Tab. I shows the rules we followed.
TABLE I. Our rulesetsace means 1 or 4aim to 7natural is ace and 3player chooses hit or stand (only one time)dealer stands soft 6 or moreblackjack pays 1 to 1
III. QUANTIZE BLACKJACKA. Quantum circuit reproducing classical blackjack
First, we define a deck state | D (cid:105) , an initial player’s hand | p (cid:105) , and dealer’s hand | d (cid:105) . These take this form | A (cid:105) ⊗ | A (cid:105) ⊗ | (cid:105) ⊗ | (cid:105) ⊗ | (cid:105) ⊗ | (cid:105) ⊗ | (cid:105) ⊗ | (cid:105) (1)where, | A (cid:105) , | (cid:105) , | (cid:105) = α | (cid:105) + β | (cid:105) ( α, β ∈ C ) | α | + | β | = 1 , (2)and then | (cid:105) means an existing card corresponding to each card sector and | (cid:105) means not. | D (cid:105) , | p (cid:105) , | d (cid:105) are describedby a vector in a space that is spanned by these 2 bases. The quantum circuit which reproduces the classic game isshown in Fig. 1 and Fig. 2. • deck : | D ⟩ ˆ H ˆ H ˆ H player’s hand : | p ⟩ | p ′ ⟩ control bit : | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ player’s strategy : | ⟩ J ˆ S p J † • ✌✌ dealer’s hand : | d ⟩ dealer’s strategy : | ⟩ J J † • ✌✌ Repeat if | d | ≤ � �� � • deck : | D ⟩ ˆ H ˆ H ˆ H player’s hand : | p ⟩ | p ′ ⟩ control bit : | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ player’s strategy : | ⟩ ˆ X orˆ I • ✌✌ dealer’s hand : | d ⟩ dealer’s strategy : | ⟩ • ✌✌ Repeat if | d | ≤ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤✤✤✤ ✤✤✤✤✤✤❴ ❴ ❴ ❴ ❴ ❴ � �� �
000 001 010 111 | Ψ ⟩ • • • . . . • ✌✌ | D ⟩ ×| p ⟩ ×| D ⟩ copy • ...1 | D ⟩ ×| p ⟩ ×| D ⟩ copy • . . .1... ... | D ⟩ ×| p ⟩ ×| D ⟩ copy • H FIG. 1. Quantum circuit reproducing the classical game • deck : | D ⟩ ˆ H ˆ H ˆ H player’s hand : | p ⟩ | p ′ ⟩ control bit : | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ player’s strategy : | ⟩ J ˆ S p J † • ✌✌ dealer’s hand : | d ⟩ dealer’s strategy : | ⟩ J J † • ✌✌ Repeat if | d | ≤ � �� � • deck : | D ⟩ ˆ H ˆ H ˆ H player’s hand : | p ⟩ | p ′ ⟩ control bit : | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ player’s strategy : | ⟩ ˆ X or ˆ I • ✌✌ dealer’s hand : | d ⟩ dealer’s strategy : | ⟩ • ✌✌ Repeat if | d | ≤ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤✤✤✤ ✤✤✤✤✤✤❴ ❴ ❴ ❴ ❴ ❴ � �� �
000 001 010 111 | Ψ ⟩ • • • . . . • ✌✌ | D ⟩ ×| p ⟩ ×| D ⟩ copy • ...1 | D ⟩ ×| p ⟩ ×| D ⟩ copy • . . .1... ... | D ⟩ ×| p ⟩ ×| D ⟩ copy • H FIG. 2. Detail of
Hit operator
We explain the outline of Fig. 1 and Fig. 2 as follows. The player’s strategy is expressed by a unitary operatoracting on its strategy bit. In Fig. 1, first the dealer and player get the cards classically, then the player chooses thestrategy ( ≡ S p ) X or I which means hit or stand respectively. If the player chooses hit, he has X operate to hisstrategy bit. In that case Hit operator ˆ H acts on the deck bit and his hand bit (in Fig. 2). Then, this operationsignifies drawing a card from the deck. Generally, the player can choose a unitary operator S p that makes the strategybit superposed. For example, when he chooses the Hadamard gate, the cases of hit and stand overlap with a 1/2probability. This does not affect the maximum value expected by the player, since it is just randomly choosing hit orstand. After the player’s operation, the dealer turns over her face-down card by ˆ H operating, and if the total of herhand ( ≡ | d | ) is 5 or less, she must draw a card one more time. This operation is repeated until | d | ≥ Hit operator, surrounded by the dotted line in Fig. 1. Control bit | Ψ (cid:105) is responsiblefor the probability of appearance of cards and it is comprised of three qubits and takes the following form | Ψ (cid:105) = (cid:80) ijk =0 1 √ | ijk (cid:105) (3)= 1 √ | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) . Each state corresponds to a sector of the card, e.g., 000 to the first ace, 001 to the second ace, 010 to the first deuce ,and so on. Therefore, the probability that every particular card is drawn is equal when the unitary
Hit operator actson the deck and hand bit. In Fig. 2, |· i (cid:105) ( i = 1 , , · · · ,
8) means the sector of the card, e.g., |· (cid:105) corresponds to thesector of the first ace. When | D (cid:105) = | D (cid:105) copy = | (cid:105) , (which indicates the existence of the first ace,) | D (cid:105) and | p (cid:105) ,corresponding to | (cid:105) which is a component of | Ψ (cid:105) , are swapped. Note that, the deck states can be copied withoutviolating the no-cloning theorem [24, 25] because the deck state comprises only | (cid:105) and | (cid:105) , which are orthogonal.After measuring the control bit, it is determined which card has been drawn.Note that, | Ψ (cid:105) converges on a state that corresponds to the sector of the card that has already been drawn with a3/8 probability at the time of first hit even though ˆ H operates. For example, when a control bit converges on | (cid:105) despite | D (cid:105) = | D (cid:105) copy = | (cid:105) , there is no exchange. To avoid this problem, we ensure that this procedure is repeateduntil | p (cid:105) (cid:54) = | p (cid:48) (cid:105) . Even if a strategy bit has | (cid:105) component, we can employ this procedure as we know which sector ofthe card bit was exchanged, by measuring not the hand bit, but the control bit. This is explained below. For example,in the case of the initial cards states | p = A + 2 (cid:105) = | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) , (4) | d = 3 (cid:105) = | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) , | D (cid:105) = | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) , and the overlapped player’s strategy bit with Hadamard operator acting on, the first state is | A + 2 (cid:105) | D (cid:105) | Ψ (cid:105) | (cid:105) ( | (cid:105) + | (cid:105) ) . (5)Here, we omitted the dealer’s strategy bit. After thorough ˆ H , the state is | A + 2 (cid:105) | D (cid:105) | Ψ (cid:105) | (cid:105) | (cid:105) + ˆ H ( | A + 2 (cid:105) | D (cid:105) | Ψ (cid:105) ) | (cid:105) | (cid:105) , (6)and after measuring the control bit, it will be | A + 2 (cid:105) | D (cid:105) | (cid:105) | (cid:105) + | A + 2 (cid:105) | D (cid:105) | (cid:105) | (cid:105) : | Ψ (cid:105) = | (cid:105) or | (cid:105) or | (cid:105)| A + 2 (cid:105) | D (cid:105) | (cid:105) | (cid:105) + | A + 2 + A (cid:105) | D (cid:48) (cid:105) | (cid:105) | (cid:105) : | Ψ (cid:105) = | (cid:105)| A + 2 (cid:105) | D (cid:105) | (cid:105) | (cid:105) + | A + 2 + 2 (cid:105) | D (cid:48)(cid:48) (cid:105) | (cid:105) | (cid:105) : | Ψ (cid:105) = | (cid:105) ... | A + 2 (cid:105) | D (cid:105) | (cid:105) | (cid:105) + | A + 2 + 3 (cid:105) | D (cid:48)(cid:48)(cid:48) (cid:105) | (cid:105) | (cid:105) : | Ψ (cid:105) = | (cid:105) . (7)When the control bit converges | (cid:105) or | (cid:105) or | (cid:105) in the first line, we repeat this one more time. In this way, wecan judge whether we must repeat this procedure without measuring the strategy bit or hand bit. In the followingdiscussions, though we omit specifying this, we always perform this procedure.Dealer’s operation Hit is the same. First, the dealer hits regardless of her strategy bit (corresponding to turn overher face-down card), and if her total | d | is 5 or less and her strategy bit is | (cid:105) , she repeats the hit operation. Again, wecan determine her hand total only by measuring the control bit; therefore, we can build the algorithm where she musthit until her total | d | reaches 6 without measuring her hand bit. Note that when the strategy bit has | (cid:105) component,the same algorithm is used.Finally, after measuring both strategy bits, all states converge. As explained in section II, the player wins when hishand does not exceed 7 and is closer to 7 than the dealer’s hand. This quantum circuit reproduces classical snackjack,and therefore the player’s expectation and basic strategy must be the same as the classical ones. Such a circuit canbe applied to blackjack in a same way by increasing qubits, although it is not considered after this. B. The case of the presence of entangle
We can consider entanglement between the player’s and dealer’s strategies on the quantum circuit. We show thisin Fig. 3. Classically, all strategies the player can choose are X or I . • deck : | D ⟩ ˆ H ˆ H ˆ H player’s hand : | p ⟩ | p ′ ⟩ control bit : | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ player’s strategy : | ⟩ J ˆ S p J † • ✌✌ dealer’s hand : | d ⟩ dealer’s strategy : | ⟩ J J † • ✌✌ Repeat if | d | ≤ � �� � • deck : | D ⟩ ˆ H ˆ H ˆ H player’s hand : | p ⟩ | p ′ ⟩ control bit : | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ | Ψ ⟩ ✌✌ player’s strategy : | ⟩ ˆ X orˆ I • ✌✌ dealer’s hand : | d ⟩ dealer’s strategy : | ⟩ • ✌✌ Repeat if | d | ≤ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤✤✤✤ ✤✤✤✤✤✤❴ ❴ ❴ ❴ ❴ ❴ � �� �
000 001 010 111 | Ψ ⟩ • • • . . . • ✌✌ | D ⟩ ×| p ⟩ ×| D ⟩ copy • ...1 | D ⟩ ×| p ⟩ ×| D ⟩ copy • . . .1... ... | D ⟩ ×| p ⟩ ×| D ⟩ copy • H FIG. 3. Entangled strategy
We assume J = exp( − i γ X ⊗ ˆ U ) (8)so that this circuit reproduces the classical game when in the presence of entangle. The first part is the player’sstrategy sector and the second is the dealer’s, and operator U is unitary and hermitian. γ is the measure for thegame’s entanglement [2] and we will call γ entangle intensity below. When S p = X or I , it follows J † ( ˆ X ⊗ ˆ I ) J = ˆ X ⊗ ˆ I (9) J † ( ˆ I ⊗ ˆ I ) J = ˆ I ⊗ ˆ I. (10)In this way, without breaking the classical game, we can insert the operator making entangle strategies.A unitary and hermitian operator is written byˆ U = s ˆ I + s ˆ X + s ˆ Y + s ˆ Z (11)where ( s + s + s + s ) ˆ I + 2 s ( s ˆ X + s ˆ Y + s ˆ Z ) = ˆ I ( s µ ∈ R ) . (12)Except for relative phases, I , Z and X , Y have the same effect on a strategy bit respectively, hence we can set s = s = 0 , s = sin θ, s = cos θ without losing generality. Here θ is game parameter .Then, in the case of general player’s strategy S p , it follows (cid:16) J † ( ˆ S p ⊗ ˆ I ) J (cid:17) | (cid:105) ⊗ | (cid:105) = (cid:110) cos γ X { ˆ X, ˆ S p } + (sin γ − cos γ X ˆ S p ˆ X (cid:111) | (cid:105) ⊗ | (cid:105) + i cos γ γ θ [ ˆ X, ˆ S p ] | (cid:105) ⊗ | (cid:105)− i cos γ γ θ [ ˆ X, ˆ S p ] | (cid:105) ⊗ | (cid:105) . (13)We assume S p is any of I, X, Y, Z . Both strategies bits after through J † areˆ S p = ˆ I : | (cid:105) ⊗ | (cid:105) ˆ S p = ˆ X : | (cid:105) ⊗ | (cid:105) , (14)ˆ S p = ˆ Y : i cos γ | (cid:105) ⊗ | (cid:105) + sin γ cos θ | (cid:105) ⊗ | (cid:105) − sin γ sin θ | (cid:105) ⊗ | (cid:105) ˆ S p = ˆ Z : cos γ | (cid:105) ⊗ | (cid:105) − i sin γ cos θ | (cid:105) ⊗ | (cid:105) + i sin γ sin θ | (cid:105) ⊗ | (cid:105) . (15)01 means stand, and 11 means hit. As it is clear from (15), when the player chooses Y or Z , he can do operations 00and 10 that do not exist in the classical game. We define E hit and E stand as expectation values corresponding to hitand stand, respectively, in a certain initial his hand state. Similarly, we define E and E as the expectation valuescorresponding to operating 00 and 10, respectively, in a certain initial his hand state. The player’s expectation valuecorresponding to each strategy { I, X, Y, Z } is { E std , E hit , sin γ (cos θE std + sin θE ) + cos γE hit , cos γE std + sin γ (cos θE hit + sin θE ) } . (16) IV. RESULTS
Classically, Tab. II shows E std , E hit in all initial states and basic strategy. Here, we assume that the player betsunit per one game. In the classical game, when the player chooses the basic strategy, the expectation value of thisgame is -1.7%; thus, the dealer has the advantage in this game. Although the player can also choose Y or Z in theabsence of entanglement, these effects on the strategy bit are the same as X and I , respectively. Therefore, it doesnot affect the overall expectation value and basic strategy. Note here, that our classical results are different from [23],because of differences in the rulesets (Tab. I) of snackjack. No initial state (As,2s,3s) dealer up E std E hit number of cases CBS1 (2,0,0) 2 1/5 1/5 2 I,X2 3 -2/5 1/5 4 X3 (0,2,0) 1 -3/5 -3/5 2 I,X4 3 -1 -2/5 4 X5 (0,0,2) 1 -8/15 -4/5 12 I6 2 -1/30 -1/3 12 I7 3 -2/5 -7/15 12 I8 (1,1,0) 1 -4/5 -4/5 4 I,X9 2 3/5 3/5 4 I,X10 3 -1/20 -1/20 16 I,X11 (1,0,1) 1 2/5 -3/10 8 I12 2 1 1/2 16 I13 3 4/5 1/30 24 I14 (0,1,1) 1 -4/5 -17/20 16 I15 2 -2/5 -2/5 8 I,X16 3 -4/5 -13/30 24 Xtotal 168 TABLE II. Classical basic strategy (CBS)No. 1 to 16 are the types of initial states. Each initial state is determined by classical probability.
In the presence of entanglement, the player has a meaningful quantum strategy, Y or Z . By the previous section(15)-(16), when parameter γ = θ = π , expectation value E Y which player chooses strategy Y , equal to E . And also E Z = E . Tab. III shows quantum basic strategy (QBS) which the player can choose in entangled strategy. No initial state (As,2s,3s) dealer up E std E hit E Y = E E Z = E number of cases QBS1 (2,0,0) 2 1/5 1/5 1/5 2/5 2 Z2 3 -2/5 1/5 -3/5 1/10 4 X3 (0,2,0) 1 -3/5 -3/5 -1 -3/5 2 I,X,Z4 3 -1 -2/5 -1 -2/5 4 X,Z5 (0,0,2) 1 -8/15 -4/5 -1/5 -4/5 12 Y6 2 -1/30 -1/3 3/5 -1/5 12 Y7 3 -2/5 -7/15 0 -2/5 12 Y8 (1,1,0) 1 -4/5 -4/5 -4/5 -4/5 4 I,X,Y,Z9 2 3/5 3/5 4/5 4/5 4 Y,Z10 3 -1/20 -1/20 0 0 16 Y,Z11 (1,0,1) 1 2/5 -3/10 2/5 -3/10 8 I,Y12 2 1 1/2 1 4/5 16 I,Y13 3 4/5 1/30 4/5 1/10 24 I,Y14 (0,1,1) 1 -4/5 -17/20 -4/5 -17/20 16 I,Y15 2 -2/5 -2/5 -2/5 -3/10 8 Z16 3 -4/5 -13/30 -4/5 -2/5 24 Ztotal 168 TABLE III. Quantum basic strategy ( γ = θ = π/ When the player chooses QBS in Tab. III, the overall expected value is +10.2%; therefore, the player has theadvantage in this quantum game. In the γ = θ = π game, the player has other choices that let the dealer definitelystand (prohibiting the second draw) therefore expectation value increases compared to the classical game. We can setvarious values in not only entangle intensity γ but also game parameter θ . Fig. 4 shows these results regarding theplayer’s expectation. E x p e c t a t i o n ( $ ) Player's expectation in the presence of entangle = = = = = = 0 FIG. 4. Player’s expectation in various parameters (0 ≤ γ, θ ≤ π ) As is clear from (14)-(16), in θ = 0 game, one can see any γ do not change the expectation value from the classicalone. Otherwise, when γ = π , θ = π , half entangled game, the expectation is +1.8%. From these results, we concludethat the player has the advantage in the appropriate parameter game. V. CONCLUSION
In section I, we introduced quantum effects in game theory [2–5] and the application to gambling [6–16]. We alsomentioned what would happen when playing blackjack on a quantum computer. We then proposed a quantized toymodel of blackjack, called snackjack [21–23], in other words, a quantum circuit which reproduced classical blackjackin sections II, III. Moreover, by playing blackjack on the quantum circuit, we found that we can possibly entanglestrategies without breaking the classical game. In section IV, we showed that the player’s expectation increases andthe player has an advantage compared to the classical blackjack with QBS. Future work will investigate how thisquantized blackjack on noisy intermediate scale quantum computers affects the player’s expectation, as in their work[14]. The operation of drawing cards in the circuit we proposed can be applied to other card games. It is also anopen question of what kind of quantized casino game will make a difference from the classical one. In the proposedquantum blackjack, we have demonstrated that entanglement between strategies could be used to bankrupt a casino.In the forthcoming quantum era, casinos will have to set house rules that did not exist in the era of classical physicsto avoid bankruptcy.
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