How Alice is winning a game of coin flips with a quantum coin?

A simple coin toss

import qiskit

from qiskit import *

# Create circuit with 1 quantum and 1 classical bit
circuit = QuantumCircuit(1, 1)

# Superposition
circuit.h(0)

<qiskit.circuit.instructionset.InstructionSet at 0x121246320>

# Measure quantum bit and store result in classical bit
circuit.measure(0, 0)

<qiskit.circuit.instructionset.InstructionSet at 0x121245270>

qiskit.__qiskit_version__

{'qiskit-terra': '0.24.1', 'qiskit-aer': '0.12.1', 'qiskit-ignis': None, 'qiskit-ibmq-provider': '0.20.2', 'qiskit': '0.43.2', 'qiskit-nature': None, 'qiskit-finance': None, 'qiskit-optimization': None, 'qiskit-machine-learning': None}

from qiskit import IBMQ

IBMQ.save_account('Put your own unique Token here', overwrite=True)

IBMQ.load_account()

<AccountProvider for IBMQ(hub='ibm-q', group='open', project='main')>

circuit.draw('mpl')

!pip install matplotlib

Requirement already satisfied: matplotlib in /Users/prashantmudgal/env_q/lib/python3.11/site-packages (3.7.2)
Requirement already satisfied: contourpy>=1.0.1 in /Users/prashantmudgal/env_q/lib/python3.11/site-packages (from matplotlib) (1.1.0)
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provider = IBMQ.get_provider('ibm-q')

provider.backends()

[<IBMQSimulator('ibmq_qasm_simulator') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_lima') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_belem') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_quito') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_statevector') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_mps') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_extended_stabilizer') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_stabilizer') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_jakarta') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_manila') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibm_lagos') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibm_nairobi') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibm_perth') from IBMQ(hub='ibm-q', group='open', project='main')>]

qcomp = provider.get_backend('ibmq_quito')

job = execute(circuit, qcomp, shots=2)

from qiskit.tools.monitor import job_monitor

job_monitor(job)

Job Status: job has successfully run

import matplotlib
%matplotlib inline

from qiskit.tools.visualization import plot_histogram

result = job.result()

result

Result(backend_name='ibmq_quito', backend_version='1.1.42', qobj_id='0b1fa52f-66d7-4e93-9382-7526a2643f56', job_id='circuit-runner_ciqmqjgs80m10ues9450_fc83_0', success=True, results=[ExperimentResult(shots=2, success=True, meas_level=2, data=ExperimentResultData(counts={'0x0': 1, '0x1': 1}), header=QobjExperimentHeader(clbit_labels=[['c', 0]], creg_sizes=[['c', 1]], global_phase=0.7853981633974483, memory_slots=1, metadata={}, n_qubits=5, name='circuit-133', qreg_sizes=[['q', 5]], qubit_labels=[['q', 0], ['q', 1], ['q', 2], ['q', 3], ['q', 4]]), status=done)], date=2023-07-17 22:36:17+05:30, status=Successful completion, header=QobjHeader(backend_name='ibmq_quito', backend_version='1.1.42', _ibm_tracing_={'uber-trace-id': 'bc3bab6050ad75d80a8c8274aa83050d:7fadaf45e434d3e0:e4c09d084ec71b6:1'}), execution_id='009ddb12-24c4-11ee-86b0-b02628f7f59e', time_taken=1.6188080459999998, error=None, client_version={'qiskit': '0.43.2'})

counts = job.result().get_counts()
result = "heads" if  next(iter(counts.keys())) == "0" else "tails"

The momemnt of truth: Measure the result of your coin toss on a quantum computer

print(f"Your quantum coin landed on: {result}")

Your quantum coin landed on: heads

In a game of coin toss, how can Alice make sure that she wins all the time?

The following algorithm is based on the following paper - https://arxiv.org/pdf/2108.06271.pdf

How to win a quantum coin flip?

1. Alice prepares her penny in a state in some box where no one can see it

2. Bob decides whether he wants to flip the penny, applies X gate or not

3. Alice applies any gate (Hadamard)

4. We measure that outcome was always the same as the initial state

import numpy as np
from qiskit import QuantumCircuit
from qiskit_ibm_runtime import QiskitRuntimeService, Estimator, Session

circuit = QuantumCircuit(1, 1)

circuit.draw('mpl')

Alice decides what the initial state of her qubit will be, by default it is 0

# If she wants the initial state to be 1 then she applies x gate
init = 1 # make it 0 for state 0

if init == 1:
    circuit.x(0)

Alice applies Hadamard gate to create a superposition

circuit.h(0)

<qiskit.circuit.instructionset.InstructionSet at 0x12081ead0>

Bob decides whether to flip the state of penny that Alice gave to him or leave it as it is

# If he flips then flip = True, otherwise set it to false
flip = True

if flip == True:
    circuit.x(0)

Alice applies Hadamard gate

circuit.h(0)

<qiskit.circuit.instructionset.InstructionSet at 0x12081e5f0>

circuit.measure(0,0)

<qiskit.circuit.instructionset.InstructionSet at 0x12081efe0>

circuit.draw('mpl')

provider.backends()

[<IBMQSimulator('ibmq_qasm_simulator') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_lima') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_belem') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_quito') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_statevector') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_mps') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_extended_stabilizer') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQSimulator('simulator_stabilizer') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_jakarta') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibmq_manila') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibm_lagos') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibm_nairobi') from IBMQ(hub='ibm-q', group='open', project='main')>,
 <IBMQBackend('ibm_perth') from IBMQ(hub='ibm-q', group='open', project='main')>]

qcomp = provider.get_backend('ibmq_qasm_simulator')

job = execute(circuit, qcomp, shots=1)

from qiskit.tools.monitor import job_monitor

job_monitor(job)

Job Status: job has successfully run

result = job.result()

counts = job.result().get_counts()
counts

{'1': 1}

fin_result = 0 if  next(iter(counts.keys())) == "0" else 1

Moment of truth, no matter what Bob does, Alice wins!

print(f"Alice started with {init} and at the end she got {fin_result}. Alice wins!" )

Alice started with 1 and at the end she got 1. Alice wins!

Tossing Quantum Coins and Winning

A simple coin toss

The momemnt of truth: Measure the result of your coin toss on a quantum computer

In a game of coin toss, how can Alice make sure that she wins all the time?

How to win a quantum coin flip?

1. Alice prepares her penny in a state in some box where no one can see it

2. Bob decides whether he wants to flip the penny, applies X gate or not

3. Alice applies any gate (Hadamard)

4. We measure that outcome was always the same as the initial state

Alice decides what the initial state of her qubit will be, by default it is 0

Alice applies Hadamard gate to create a superposition

Bob decides whether to flip the state of penny that Alice gave to him or leave it as it is

Alice applies Hadamard gate

Moment of truth, no matter what Bob does, Alice wins!

Maths behind it

Alice started with state 1 and the end result is also 1