# EWL

A simple Python library to simulate and execute EWL quantum circuits on IBM Q with symbolic calculations using SymPy.

## Installation

`pip install ewl`

## Examples

- Prisoner's dilemma

## Usage

### Initialization

This library uses SymPy to perform symbolic calculations. It's convenient to import it as `sp`

and define frequently used constants for future use.

```
import sympy as sp
i = sp.I
pi = sp.pi
sqrt2 = sp.sqrt(2)
```

When using this library in Jupyter Notebook, call `init_printing`

to enable pretty printing.

`sp.init_printing()`

### EWL instance

First, you need to define the initial (preferably entangled) quantum state:

```
from sympy.physics.quantum.qubit import Qubit
psi = (Qubit('00') + i * Qubit('11')) / sqrt2
```

It is also necessary to define two unitary strategies that represent the classical strategies:

```
C = sp.Matrix([
[1, 0],
[0, 1],
])
D = sp.Matrix([
[0, i],
[i, 0],
])
```

Then you need to define the players' strategies. Each strategy must be a unitary matrix as it represents a single-qubit quantum gate.

```
alice = sp.Matrix([
[1, 0],
[0, 1],
])
```

The library comes with a series of built-in parametrizations, including the original one from EWL paper as well as other 2- and 3 degrees of freedom parametrizations (see here).

```
from ewl.parametrizations import *
bob = U_Eisert_Wilkens_Lewenstein(theta=pi / 2, phi=0)
```

At this point you can also use arbitrary symbols and compound expressions to generalize the analysis.

```
theta, gamma = sp.symbols('theta gamma', real=True)
charlie = U_Eisert_Wilkens_Lewenstein(theta=theta, phi=gamma / 2)
```

You also need to define the payoff matrix, possibly with symbols, for arbitrary number of players.

```
payoff_matrix = sp.Array([
[
[3, 0],
[5, 1],
],
[
[3, 5],
[0, 1],
],
])
```

Finally, you can make an instance of quantum game in the EWL protocol by providing the initial quantum state, a list of players' strategies and the payoff matrix with corresponding shape. The library supports arbitrary number of players, although it works best for 2-player games.

```
from ewl import EWL
ewl = EWL(psi=psi, C=C, D=D, players=[alice, bob], payoff_matrix=payoff_matrix)
```

### Calculations

Based on the provided initial quantum state, the library automatically calculates the corresponding matrix of *J* and *J*^{†} gates.

```
ewl.J
ewl.J_H
```

Based on the players' strategies, the library also calculates the amplitudes of the result quantum state in the computational basis.

```
ewl.amplitudes()
ewl.amplitudes(simplify=False)
```

From the amplitudes one can easily derive the probabilities of possible game results. By default, the expressions are simplified using trigonometric identities. Make sure to enable `real=True`

flag when defining real-valued symbols to allow for further simplification.

```
ewl.probs()
ewl.probs(simplify=False)
```

Finally, based on the payoff matrix and previously mentioned probabilities, the library calculates the payoff functions as symbolic expressions (possibly with parameters from the initial state and strategies).

```
ewl.payoff_function(player=0) # first player
ewl.payoff_function(player=1, simplify=False) # second player
ewl.payoff_function(player=None) # payoff sum
```

You can also obtain a tuple containing the payoffs for each respective player.

```
ewl.payoffs()
ewl.payoffs(simplify=False)
```

For quantum games parametrized with exactly two symbols, it is possible to plot a three-dimensional graph of the payoff function.

```
from ewl.plotting import plot_payoff_function
plot_payoff_function(
ewl, player=0,
x=alpha, x_min=0, x_max=pi / 2,
y=beta, y_min=0, y_max=pi / 2)
```

### Parameters

Here's how you can list all symbols used either in the initial quantum state or in the players' strategies:

`ewl.params`

You can also substitute the symbols with specific values to obtain a non-parametrized instance of quantum game as new EWL instance:

`ewl_fixed = ewl.fix(theta=0, gamma=pi / 2)`

It is also possible to substitute specific players' strategies for further analysis of a certain case of the game.

`ewl_CD = ewl.play(C, D)`

### Example games

The library comes with a series of built-in example games, in particular a few variants of Quantum Prisoner's Dilemma with different parametrizations from various articles (see here).

### Qiskit integration

This library also integrates with Qiskit, allowing arbitrary quantum games in the EWL protocol to be executed on IBM Q devices. First, you need to load your credentials:

```
from qiskit import IBMQ
IBMQ.load_account()
```

When running locally, make sure to save the access token to disk first using `IBMQ.save_account`

.

In order to access backend-specific features of EWL instance, first you need to convert it to `EWL_IBMQ`

instance. Note that the input quantum game must be non-parametrized (cannot have any symbols).

```
from ewl.ibmq import EWL_IBMQ
ewl_ibmq = EWL_IBMQ(ewl_fixed)
```

You can also specify and apply noise model used in quantum simulation.

```
from qiskit.providers.aer.noise import NoiseModel, pauli_error
p_error = 0.05
bit_flip = pauli_error([('X', p_error), ('I', 1 - p_error)])
phase_flip = pauli_error([('Z', p_error), ('I', 1 - p_error)])
noise_model = NoiseModel()
noise_model.add_all_qubit_quantum_error(bit_flip, ['u1', 'u2', 'u3'])
noise_model.add_all_qubit_quantum_error(phase_flip, ['x'], [0])
ewl_ibmq = EWL_IBMQ(ewl_fixed, noise_model=noise_model)
```

You can draw the original quantum circuit of quantum game in the EWL protocol.

`ewl_ibmq.draw()`

It is also possible to draw the quantum circuit transpiled for a specific backend.

`ewl_ibmq.draw_transpiled(backend_name='ibmq_quito', optimization_level=3)`

Here's how you can execute the quantum game on a specific statevector simulator:

`ewl_ibmq.simulate_probs(backend_name='statevector_simulator')`

You may also run the quantum circuit on QASM simulator and get histogram data of the experiment.

`ewl_ibmq.simulate_counts(backend_name='qasm_simulator')`

Finally, you can run the quantum game on a real quantum device:

`ewl_ibmq.run(backend_name='ibmq_quito', optimization_level=3)`

## Citation

```
@software{PythonEWL2022,
author = {Tomasz Zawadzki and Piotr Kotara},
title = {A Python tool for symbolic analysis of quantum games in EWL protocol with IBM Q integration},
howpublished = {\url{https://github.com/tomekzaw/ewl}},
}
```