On Quantum Walks and Iterated Quantum Games

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On Quantum Walks and Iterated Quantum Games

• G. Abal, R. Donangelo, H. Fort
• Universidad de la República, Montevideo, Uruguay
• UFRJ, RJ, Brazil

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0. THE MAIN IDEA
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1.QUANTUM WALKS AND QUANTUM GAMES
1.1 Quantum Walks

• Quantum walks (QWs) are expected to have
  potential for the development of new quantum
  algorithms.
• When two quantum walks are considered, the joint
  state of both walkers may be entangled in several
  ways this opens new possibilities for quantum
  information manipulation.
• Quantum walks have been realized using
  technologies ranging from NMR to linear optics.

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1.2 Quantum Games

• Classical Game Theory constitutes a powerful tool
  for strategic analysis and optimization.
• Bipartite quantum games (QGs), in which players
  can resort to quantum operations, open new
  possibilities for information processing.
• It was shown that in QG, given a sufficient
  amount of entanglement, the players can achieve
  results not available to classical players

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2. FROM QW TO ITERATED QG
2.1 Discrete-time QW on the Line

• The Hilbert space of a quantum walk on the line
  is composed of two parts, H Hx ? Hc .
• Hx ? x? ? integer x 0, 1,2 . . .
  associated to discrete positions on the line.
• Hc is spanned by the 2 orthonormal kets 0 ?, 1
  ?.
• The quantum walk with two walkers A,B takes place
  in a Hilbert space HAB HA ? HB .
• The evolution operator

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The well known case of a Hadamard walk is
obtained if Uc H ? H, where H is the one-qubit
Hadamard gate defined by H0gt (0gt1gt)/v2
and H1gt (0gt-1gt)v2 .
Here we shall be concerned with more general coin
operations Uc which cannot be written as products
of local operations. The conditional shift
operation can be represented as
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Lets see how works W, or
2.2 QW as a QG
R1, S -2, T -S 2 P - R -1
But, wait a minute,
exchanging 1 by C 0 by D, this is eq. to a well
known game
The PRISONERS DILEMMA GAME !
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T gt R gt P gt S
Non Optimal Situation !
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2.3 Escape from the Prisoners Dilemma Repeated
Games

• Repeated games differ from one-shot games
  because the actions of the agents can produce
  retaliation or reward.
• Agents need a strategy (that is, a rule to update
  their behavior), and, some strategies favor
  cooperation.

• Lets specify a strategy by a 4-tuple pR, pS,
  pT, pP
• where pX is the conditional probability of
  cooperation of an agent after he got the payoff X
  in the previous round.
• Examples 1/2, 1/2, 1/2, 1/2
  RANDOM
• 1, 0, 0, 1 win-stay, lose-shift or PAVLOV

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2.4 Implementation of Iterated QG
Consider 2 agents A (Alice) and B (Bob), players
in an iterated QG.
Connection with the QW is made by 3 simple rules
1. The coin states of the QW are interpreted as
0 gt ? C (cooperation) 1 gt ? D
(defection) 2. Each agent can alter his/her own
coin qubit by applying a unitary operation (a
strategy) UA or UB in Hc ?Hc 3. The position
corresponds to the accumulated payoff. If XA is
the position operator for Alice, XAxAgt xA ,
XAxAgt her average payoff is ltXAgt trace (rXA)
(idem for Bob).
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The first qubit from the left is Alices and the
second is Bobs , ?
The possible strategies available to Alice are
represented by the set of unitary 2-qubit
operations that dont alter the second qubit
The coeff. ai are expressed in terms of the
conditional prob pX as
pRpTpSpP 1
And similar expresions for Bob.
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For instance, the quantum version of Pavlovs 1,
0, 0, 1, played by A may be implemented through
an operator
If the 3 phases are chosen 0, a CNOT operation
results in which Bobs coin is the control
qubit 00 ? 00 01 ? 11 10 ? 10 11 ? 01
The joint coin operation is constructed as UC
UB UA, assuming Alice moves 1st or UC UA UB,
otherwise.
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2.4-A Example Pavlov vs. Random
Alice plays randomly and Bob responds with
Pavlov. The operation transforms a product state
into a maximally entangled (Bell) state.
Alice plays Pavlov and Bob plays random, an
operation which disentangles a Bell state.
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2.4-B Pavlov vs. Random Results
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3. PARAMETERIZED QUANTUM STRATEGIES

• Lets consider now the results for strategies
  that interpolate between Random and Pavlov. For
  both pR pS 1,
• ?
• neglecting phases, each players strategy depends
  on a single real parameter

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Assuming Alice plays first, the joint coin
operation is
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Or more illuminating perhaps
Optimal situation for both players.
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4. CONCLUSIONS

• A connection between iterated bipartite quantum
  games and discrete-time quantum walk on the line
  was established.

• In particular, conditional strategies, depending
  on the previous state of both players, are
  naturally formulated within this scheme.

• As a by-product of this correspondence, popular
  strategies in Game Theory can be mapped into
  elementary quantum gates.

Examples of this Pavlov ? CNOT
Random ? Hadamard
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• An example of a QG in which both agents are
  allowed to choose a strategy that interpolates
  continuously between Pavlov and Random has been
  analyzed in detail using two unbiased initial
  coin states.

• Within this limited strategic choice, in the case
  of initial
• coin state (00gt 11gt)/v2 there is a Pareto
  optimal Nash equilibrium when Alice plays Pavlov,
  ? 0, and Bob responds using ? p/20.

• In one-shot quantum games, the initial state
  must include a minimum amount of entanglement so
  that truly quantum features emerge. In the
  iterated QG based on the QW, entanglement is
  dynamically generated, so that entangled initial
  states are not a requirement.

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• Obviously, this scheme for quantizing the
  iterated PD game also works for 22 games with
  arbitrary payoff matrix. There are several
  popular games that seem interesting to analyze
  within this framework.

• This connection introduces an entire new set of
  coins and shift operators that may be useful for
  quantum information processing tasks and opens
  the possibility to experimental tests using the
  facilities that are being developed for the QW.