Quantum ALife: Theory and Applications

Quantum computation can be combined with Artificial Life (ALife) models and agent-based systems leading to simultaneous contributions to (Quantum) Computer Science, Artificial Life, Complex Quantum Systems Science and different domains of application.

An example of an Artificial Life model following quantum rules is the quantum game of life model simulated in Netlogo:

http://ccl.northwestern.edu/netlogo/models/community/Quantum%20Life%20I

The model runs as follows:

The square lattice with periodic conditions at the borders is divided in cells, each cell can either be "on" (alive) or "off" (dead).

There are two transitions that can take place in a single round of the game:

- If the j-th cell is in a state at the beginning of a round |s(j)> it can remain on that state |s(j)> -> |s(j)>, or, alternatively, it can flip |s(j)> -> |1-s(j)>

In the first case the unit gate, 1=|0><0| + |1><1|, is applied so that we have:

1 |s(j)> = |s(j)>

In the second case the NOT gate is applied: |0><1| + |1><0|, so that we have:

NOT |s(j)> = |1-s(j)>

There are, thus, two alternative histories for the round, one in which the cell changes its state and another in which it does not. Assuming that the amplitudes associated with each history for the n-th round are, respectively, given by: ψ(s,j,1,n) and ψ(s,j,NOT,n) we have the following superposition of quantum circuits, for the :

ψ(s,j,1,n) 1 |s(j)> + ψ(s,j,NOT,n) NOT |s> = ψ(s,j,1,n) |s(j)> + ψ(s,j,NOT,n) |1-s(j)> = |ψ(j,n)>

The amplitudes can depend upon the initial cell configuration. In the most elementary case, we can have:

- If |s(j)> = |0>:

ψ(0,j,1,n) = sqrt(1 - p-alive)

ψ(0,NOT,n) = sqrt(p-alive)

- If |s(j)> = |1>:

ψ(1,j,1,n) = sqrt(p-alive)

ψ(1,j,NOT,n) = sqrt(1 - p-alive)

(Above, the p-alive corresponds to a global parameter)

In this case, the ket associated with the cell is always given by:

|ψ(j,n)> = sqrt(1 - p-alive) |0> + sqrt(p-alive) |1>

Thus, considering the projectors P0 = |0><0| and P1 = |1><1|, we have two virtual branches at the end of the round:

- Cell is dead: P0 |ψ(j,n)>

- Cell is alive: P1 |ψ(j,n)>

We are considering, in this example, an Everettian line which does not assume the collapse, but, instead, a simultaneous virtual branching with one actualized alternative, the actualized alternative corresponds to one of the virtual branches, the other branches are taken as non-actualized alternatives.

In the above case, the probability for the actualized alternatives are:

Prob(Dead) = 1 - p-alive

Prob(Alive) = p-alive

In the standard many worlds interpretation this would mean that for each alternative history a new quantum computation would be made so that we would have for the subsequent round two different quantum circuits leading to the same |ψ(j,n+1)> = sqrt(1-p-alive)|0> + sqrt(p-alive)|1>

A more complex dynamics takes place when the parameter defining the amplitudes over the round's histories depends upon the local lattice configuration of dead versus alive cells.

In the game simulated in Netlogo a round is divided in two stages each stage has its own local quantum circuits with a projection at the end marking the conditional quantum computation for the next stage:

- Birth stage

- Death stage

For the birth stage the cells that are dead can either become alive (NOT gate applied) or remain dead (unit gate applied) with amplitudes that depend upon the mean number of nearest neighbors (in a von Neumann neighborhood) that are alive at the beginning of the game's round. This means that a number of dead cells will become alive and a number of dead cells will remain dead at the end of the birth stage, depending upon the number of nearest neighbors. The actualized history will correspond to one of the local projected virtual branches with a probability measure depending upon the local amplitudes.

For the death stage a new quantum computation takes place: all of the alive cells can either remain alive or die with an amplitude that depends upon the weighted average of a global parameter and a the local mean number of alive neighbors (which allows for overcrowding to lead to death), the weight to which the global parameter or the local overcrowding rule holds is also a global parameter of the game.

At the end of the death stage another projection takes place defining the context for the next round's quantum circuits.

The resulting dynamics for this model, is a flocking pattern in the cellular automaton activation pattern, as can be seen in the following video, which shows two minutes of the Netlogo model, for an initially random configuration of alive and dead cells:

Quantum ALife approaches associated to agent-based modeling can be applied effectively to different areas of research, an example is econophysics. The following link is for an artificial financial market model where the number of agents buying and selling are modelled in terms of a two-level bosonic oscillator (this constitutes an example of quantum computation applied to agent-based modeling approaches to financial market modeling):

http://ccl.northwestern.edu/netlogo/models/community/Quantum_Financial_Market

Another more recent example, is that of an artificial economy that coevolves with a financial market introduced in the article "Financial Turbulence, Business Cycles and Intrinsic Time in an Artificial Economy" Gonçalves, C.P., Algorithmic Finance (2012), 1:2, 141-156, the following link contains the article and links to the Netlogo version of the model:

http://algorithmicfinance.org/1-2/pp141-156/

This last example includes quantum optimization in the modeling of an artificial economy and financial market that can be used to address financial risk modeling.

The research filed of Quantum Artificial Life, thus, provides for both a fundamental and applied framework for quantum computation with interdisciplinary extensions to other fileds like econophysics and quantum game theory.