## Geological Sample density

Quantum sensing is a rapidly advancing field that leverages the principles of quantum mechanics to achieve ultra-precise measurements of various physical properties. One of the most promising use cases for quantum sensing is in the field of medical imaging.

Traditional medical imaging techniques like X-rays, CT scans, and MRI scans are limited in their ability to precisely localize and characterize small lesions or tumors. Quantum sensing-based techniques, on the other hand, have the potential to achieve much higher levels of sensitivity and accuracy.

For example, researchers have been exploring the use of diamond-based quantum sensors for imaging of biological samples. These sensors are capable of detecting the magnetic fields generated by the tiny electrical currents that flow through the body’s cells, providing a highly detailed view of the sample’s internal structure.

In addition to medical imaging, quantum sensing has applications in fields such as geology, metrology, and environmental monitoring. The ability to precisely measure physical properties is critical for making accurate predictions and diagnoses.

Here is a general outline of the steps involved in building such a model:

**Define the problem and the physical system:**In this case, we want to use quantum sensing to measure the physical properties of geological samples, such as their magnetic or electrical fields. The system could be a collection of qubits or other quantum devices that can interact with the sample.**Choose the appropriate quantum algorithm**: The choice of algorithm will depend on the specifics of the problem and the available quantum resources. For example, one possible algorithm for measuring magnetic fields is the gradient-based method, which uses the interaction between the sample and the quantum device to measure the gradient of the magnetic field. This can be implemented using PennyLane’s built-in gradient-based optimization functions.**Define the cost function**: The cost function is the function that we want to optimize in order to obtain the best estimate of the physical property we are trying to measure. This could be the difference between the measured and expected values of the property or a more complex function that takes into account noise and other factors.**Run the optimization:**Once the cost function is defined, we can run an optimization algorithm using PennyLane’s built-in optimization functions, such as gradient descent or Adam. The optimization will adjust the parameters of the quantum algorithm to minimize the cost function and obtain the best estimate of the physical property.

**Analyze the results:**After the optimization is complete, we can analyze the results and evaluate the accuracy and precision of our measurements. This may involve comparing the measured values to known values or using statistical methods to estimate uncertainty.

Suppose we are interested in estimating the **Density of a Geological Sample** using a quantum sensing model. In this case, the target value we are trying to estimate is the density of the sample, which is a physical property that describes how much mass is present per unit volume of the sample. Density is an important property in geology because it can provide information about the composition and structure of the sample, which in turn can help with geological exploration and resource management.

So, in the context of the quantum sensing model, the target value might be something like “the density of the geological sample is 2.5 g/cm^{3}.” The goal of the model would be to use the measured data and the quantum circuit to estimate this target value as accurately as possible.

Implementing a quantum sensing model in geology involves a combination of domain expertise, knowledge of quantum algorithms and optimization methods, and programming skills. Here’s some pseudo-code for a simple quantum sensing use case in geology, as discussed above:

` ````
```
# Import the necessary libraries
import pennylane as qml
import numpy as np
# Define the quantum device
dev = qml.device("default.qubit", wires=2)
# Define the quantum circuit
@qml.qnode(dev)
def circuit(params):
qml.RX(params[0], wires=0)
qml.RY(params[1], wires=1)
qml.CNOT(wires=[0, 1])
return qml.expval(qml.PauliZ(0))
# Define the cost function
def cost(params, target_value):
estimated_value = circuit(params)
return (estimated_value - target_value)**2
# Generate some mock data
target_value = 0.5
params_true = np.array([0.2, 0.4])
noise_std = 0.01
measured_value = target_value + noise_std*np.random.randn()
# Optimize the circuit to estimate the target value
params = np.random.randn(2)
opt = qml.AdamOptimizer(0.1)
for i in range(100):
params = opt.step(lambda p: cost(p, measured_value), params)
# Evaluate the results
estimated_value = circuit(params)
print("Target value: ", target_value)
print("Measured value: ", measured_value)
print("Estimated value: ", estimated_value)
print("True parameters: ", params_true)
print("Estimated parameters: ", params)

In this example, we’re trying to estimate a target value using a quantum circuit with two qubits and two parameters. The cost function is defined as the squared difference between the estimated and target values. We generate some mock data by adding noise to the target value and then use an optimizer to adjust the parameters of the quantum circuit to minimize the cost function. Finally, we print out the target, measured, and estimated values, as well as the true and estimated parameters. This is a very simple example, but it illustrates a quantum sensing model’s basic structure in geology.

The code I provided earlier is a simple example of a quantum sensing model for estimating a physical property of a geological sample. Here are the step-by-step explanations of the code:

**Import the necessary libraries:** The first step is to import the required libraries, including PennyLane and NumPy. PennyLane is used to define and simulate quantum circuits, while NumPy is used for numerical operations and generating random data.

` ````
```import pennylane as qml
import numpy as np

**Define the quantum device:** In this step, we define the quantum device on which we will simulate our quantum circuit. In this case, we are using the default.qubit simulator and a two-qubit circuit.

` ````
```
dev = qml.device("default.qubit", wires=2)

**Define the quantum circuit:** We define a simple quantum circuit that applies two rotations (RX and RY gates) to the two qubits and applies a controlled-NOT (CNOT) gate between them. The expectation value of a PauliZ operator on the first qubit is returned as the circuit output.

` ````
```@qml.qnode(dev)
def circuit(params):
qml.RX(params[0], wires=0)
qml.RY(params[1], wires=1)
qml.CNOT(wires=[0, 1])
return qml.expval(qml.PauliZ(0))

**Define the cost function: **We define a cost function that takes in the circuit parameters and a target value as input and returns the squared difference between the output of the circuit and the target value. This cost function is what we will minimize using an optimizer in order to estimate the physical property we are interested in.

` ````
```def cost(params, target_value):
estimated_value = circuit(params)
return (estimated_value - target_value)**2

**Generate mock data: **We generate some mock data by setting a target value for the physical property we are interested in, generating some random noise, and adding the noise to the target value to obtain a measured value. We also set the true circuit parameters for comparison later.

` ````
```target_value = 0.5
params_true = np.array([0.2, 0.4])
noise_std = 0.01
measured_value = target_value + noise_std*np.random.randn()

**Optimize the circuit parameters:** We use an optimizer (in this case, the Adam optimizer with a learning rate of 0.1) to adjust the circuit parameters in order to minimize the cost function and obtain an estimate of the physical property we are interested in.

` ````
```params = np.random.randn(2)
opt = qml.AdamOptimizer(0.1)
for i in range(100):
params = opt.step(lambda p: cost(p, measured_value), params)

**Evaluate the results:** Finally, we evaluate the results of our quantum sensing model by printing out the target value, the measured value, the estimated value (obtained from the circuit with the optimized parameters), and the true and estimated circuit parameters for comparison.

` ````
```estimated_value = circuit(params)
print("Target value: ", target_value)
print("Measured value: ", measured_value)
print("Estimated value: ", estimated_value)
print("True parameters: ", params_true)
print("Estimated parameters: ", params)

In the example code I provided for the quantum sensing model in geology, the target value we are trying to estimate is specified by the target_value variable, which is set to 0.5 in the code. This value represents the physical property of interest that we are trying to estimate using the quantum circuit.

I also generated mock data in the code by adding some random noise to the target value to obtain a measured value. The goal of the quantum sensing model is to use the measured value and the quantum circuit to estimate the target value. The optimization algorithm adjusts the circuit parameters to minimize the cost function, which is defined as the squared difference between the estimated and the measured value. The estimated value obtained from the circuit with the optimized parameters represents our best estimate for the target value.