Quantum Geometry: Simulating Energy Dynamics with Qubits




Quantum Geometry: Simulating Energy Dynamics with Qubits

This course explores a radical rethinking of measurement, geometry, and computation by merging classical ideas with quantum principles. Instead of treating measurements as fixed values, we demonstrate how a modified version of the Pythagorean theorem—enhanced with a curvature correction and a chirality factor hh (which can be either +1+1 or 1-1)—produces 32 possible outcomes. These outcomes are then encoded into a 5-qubit quantum circuit. Through hands-on programming with Python and Qiskit, you’ll learn how to build, simulate, and analyze quantum circuits where superposition and interference combine to yield a single, emergent classical measurement.


1. Introduction and Course Objectives

In classical physics, if you measure the length of a rigid rod repeatedly, you expect the same value every time. The measurements are fixed and predictable. However, quantum mechanics reveals a very different picture: outcomes can be probabilistic, meaning that even if a system is prepared in a certain way, the result of a measurement may vary each time.

In this course, we reimagine a familiar geometric concept—the Pythagorean theorem—by extending it to account for curved space. We introduce a curvature-corrected formula:

c2=a2+b2±ha2b2R2,c^2 = a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2},

where:

  • aa and bb are the sides of a right triangle,
  • RR is the radius of curvature (a measure of how curved the space is),
  • The ±h\pm h term, with hh being either +1+1 or 1-1, introduces a choice that results in different outcomes.

Because we consider five independent binary choices—the signs of aa, bb, the square-root operation on c2c^2, the sign of RR, and the chirality factor hh—the formula yields 25=322^5 = 32 possible outcomes.

Our course objectives are to:

  • Challenge Traditional Measurement:
    We will show you that what we normally consider a single, fixed measurement can actually emerge from a dynamic, probabilistic process.

  • Map a Curvature-Corrected Theorem to a Quantum Circuit:
    By translating our modified Pythagorean theorem into the language of quantum computing, we’ll use a 5-qubit circuit to represent the 32 outcomes. Each qubit stands for one binary decision, allowing the entire circuit to encode all possibilities simultaneously.

  • Build, Simulate, and Analyze Quantum Circuits:
    Through Python programming using Qiskit, you will create real quantum circuits, observe how superposition and interference work, and see how the final measurement emerges as a classical result from underlying quantum dynamics.

  • Develop Practical Skills:
    The course is designed not only to provide a theoretical framework but also to give you hands-on experience. By the end, you'll be equipped to extend these ideas into your own projects—whether in research, advanced programming, or exploring new frontiers in quantum computation.

Throughout the course, you’ll move step by step from understanding classical measurements and the basics of qubits to building advanced quantum circuits that simulate energy dynamics. You’ll learn how phase shifts and controlled operations can adjust the interference patterns among the 32 states in a 5-qubit system, ultimately leading to an emergent classical measurement. This blend of theory and practice will prepare you to rethink traditional measurements and appreciate the power of quantum computing.

Get ready to dive into a world where classical geometry meets the strange, beautiful realm of quantum mechanics. By the end of this course, you’ll have not only a deep conceptual understanding but also practical skills in Python and Qiskit that will enable you to simulate and explore dynamic quantum systems.



2. Foundations of Qubits and Quantum Measurements

In this section, we review the quantum mechanics principles necessary for our course, focusing on qubits, superposition, interference, and measurement. These concepts are essential for understanding how our quantum circuits work and how they differ from classical systems.

2.1. Revisiting Classical Measurements

Traditionally, a measurement (such as the length of a metal rod) always yields the same value because the property being measured is fixed. For example, using the classical Pythagorean theorem:

c2=a2+b2,c^2 = a^2 + b^2,

we assume that for given aa and bb, the value of cc is uniquely determined. In classical computing, a bit is either 0 or 1, leading to deterministic outcomes—if you input a certain value, the output is always the same.

However, quantum measurements are fundamentally different. In a quantum system, a qubit can exist in a superposition of states, meaning it holds multiple possibilities at once. It is only when the qubit is measured that the state "collapses" to a definite outcome. This inherent probabilistic nature of quantum measurements contrasts sharply with the fixed outcomes of classical measurement.

2.2. Introduction to Qubits

What Is a Qubit?

A qubit is the basic unit of quantum information. Unlike a classical bit, which can only be 0 or 1, a qubit can be in a combination (or superposition) of both states simultaneously. Mathematically, a qubit is described by:

ψ=α0+β1,|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,

where α\alpha and β\beta are complex numbers (amplitudes) that satisfy:

α2+β2=1.|\alpha|^2 + |\beta|^2 = 1.

Superposition:

Superposition is the principle that allows a qubit to exist in a blend of both 0 and 1 until it is measured. For example, applying a Hadamard gate to a qubit initially in the state 0|0\rangle creates:

H0=12(0+1).H|0\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle + |1\rangle \right).

This state shows that the qubit is equally likely to be measured as 0 or 1.

Quantum Interference:

Quantum interference occurs when the probability amplitudes of different states combine. The phases of these amplitudes determine whether the interference is constructive (which amplifies the probability of a state) or destructive (which diminishes it). This interference is crucial in quantum circuits, as it influences the final outcome of a measurement.

For example, in the state

ψ=12(0+1),|\psi\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle + |1\rangle \right),

the amplitude for both 0|0\rangle and 1|1\rangle is 12\frac{1}{\sqrt{2}}. To determine the probability of measuring a particular state, we take the square of the magnitude of the amplitude:

  • Probability of measuring 0|0\rangle: P(0)=122=12.P(0) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}.
  • Probability of measuring 1|1\rangle: P(1)=122=12.P(1) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}.

Thus, there is a 50% chance of obtaining either outcome when the qubit is measured.

2.3. Quantum Measurement and State Collapse

Measurement in Quantum Mechanics:

When a qubit is measured, its superposition collapses to one of the basis states (either 0|0\rangle or 1|1\rangle). For a state such as:

ψ=12(0+1),|\psi\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle + |1\rangle \right),

measurement yields either 0 or 1 with equal probability (50% each). This collapse is inherently probabilistic and is described by the Born rule, which states that the probability of measuring a state is given by the square of the magnitude of its amplitude.

Role of Phase in Interference:

The phase of the amplitudes α\alpha and β\beta plays a crucial role in determining how states interfere with one another. In multi-qubit systems, the relative phases among the states determine the overall interference pattern, which directly affects the probability distribution observed upon measurement.

Interference in Multi-Qubit Systems:

In a multi-qubit system, such as a 5-qubit register, the overall quantum state is a superposition of many basis states—up to 25=322^5 = 32 possible states. The interference between these states, which is manipulated by various quantum gates, is what ultimately leads to a single, emergent outcome when a measurement is performed.

To illustrate this with a simple example, consider a 2-qubit system initialized in the state 00|00\rangle. Applying a Hadamard gate to each qubit transforms the state as follows:

H0=12(0+1).H|0\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle + |1\rangle \right).

Thus, the state of the 2-qubit system becomes:

ψ=(12(0+1))(12(0+1))=12(00+01+10+11).|\psi\rangle = \left(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\right) \otimes \left(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\right) = \frac{1}{2}\left(|00\rangle + |01\rangle + |10\rangle + |11\rangle\right).

Each of these four outcomes will be measured with a probability of:

P=122=14(25%).P = \left|\frac{1}{2}\right|^2 = \frac{1}{4} \quad (25\%).

2.4. Integration: Classical vs. Quantum Measurement

Classical Measurement:

In classical systems, the outcome of a measurement is fixed and deterministic. For example, a ruler will always measure the same length for a rigid object under identical conditions. Bits in classical computing are either 0 or 1, and every operation produces a predictable result.

Quantum Measurement:

In quantum systems, the state of a qubit is described by a superposition of possible outcomes, and measurement causes this superposition to collapse into one of the basis states. This process is probabilistic, meaning that even with identical preparations, the measurement result may vary from one run to another.

Emergence Through Measurement:

The collapse of the quantum state upon measurement is what gives rise to the final, emergent classical outcome. Although a quantum system may contain many possibilities simultaneously, the act of measurement selects one outcome based on the interference of the amplitudes.

Overall, these quantum principles—superposition, interference, and probabilistic measurement—revolutionize our understanding of computation and measurement. They show that what we observe in classical systems may be the emergent result of a rich, underlying quantum process.

This section has laid the groundwork by explaining how qubits work, how they differ from classical bits, and how measurement in quantum mechanics is a probabilistic process that relies on superposition and interference. With this foundation, you're now ready to dive into building quantum circuits that capture these ideas, paving the way for the hands-on portions of the course using Python and Qiskit.

Next, we'll move on to mapping our modified, curvature-corrected Pythagorean theorem onto a 5-qubit quantum circuit, where all 32 possible outcomes (including both h=+1h = +1 and h=1h = -1) are represented and simulated.

3. Mapping to Quantum Circuits

In this section, we describe in detail how to represent our curvature-corrected geometric model within a quantum circuit. Our model is based on the modified Pythagorean theorem:

c2=a2+b2±h(a2b2R2)c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right)

where:

  • aa and bb are the two sides of a right triangle.
  • RR is the radius of curvature, representing how curved the space is.
  • The term ±h(a2b2R2)\pm \, h\left(\frac{a^2b^2}{R^2}\right) introduces a correction factor. The symbol ±\pm combined with the chirality factor hh (which can be either +1+1 or 1-1) determines whether the correction is effectively added or subtracted.

Because our model involves five independent binary decisions:

  1. The sign of aa,
  2. The sign of bb,
  3. The binary choice when taking the square root of c2c^2 (i.e. whether cc is positive or negative),
  4. The sign of RR (if we consider its pre-squared ambiguity), and
  5. The chirality factor hh (which can be +1+1 or 1-1),

there are 25=322^5 = 32 possible configurations. This means our curvature-corrected theorem does not yield one fixed value but a set of 32 outcomes, depending on the combination of these binary variables.

3.1. The 32-State Model and 5-Qubit Representation

In quantum computing, a qubit is the equivalent of a classical binary variable but with the additional property of superposition. Each qubit can represent 0, 1, or any quantum superposition of these states. A 5-qubit system, therefore, has a Hilbert space with 25=322^5 = 32 dimensions—naturally matching our model's 32 possible outcomes.

Each qubit in our 5-qubit register is assigned to one of the five binary choices in our equation:

  • One qubit for the sign of aa.
  • One qubit for the sign of bb.
  • One qubit for the sign ambiguity from taking the square root of c2c^2.
  • One qubit for the sign of RR (if we include this ambiguity).
  • One qubit for the chirality factor hh (which can be +1+1 or 1-1).

Before measurement, the quantum system is in a superposition that includes all 32 configurations simultaneously. When we eventually measure the circuit, the superposition collapses to one of these 32 states, each representing a particular combination of choices in our modified theorem.

3.2. Interpreting Binary Choices and Controlled Operations

In our quantum circuit, every qubit holds a binary decision that contributes to the overall state. However, qubits also carry phase information, which is key to understanding quantum interference:

  • Phase Shifts as Energy Adjustments:
    A phase shift gate (such as the U1 gate) can change the phase of a qubit without altering its probability amplitude. In our model, this is analogous to introducing an energy bias into one of the binary variables. For instance, applying a phase shift to the qubit representing hh (the chirality factor) effectively distinguishes whether the correction term is added or subtracted.

  • Controlled Operations and Dynamic Interactions:
    Controlled operations (like the controlled phase (CP) gate) allow one qubit's state to affect the phase of another. This is similar to a cyclic permutation or “chiral shift” in classical binary representations. When a controlled operation is applied, it introduces a dependency between qubits, modifying the interference pattern. For example, a CP gate between the qubit representing aa and the one representing bb simulates an interaction between those energy states, changing how their combined state interferes.

  • Dynamic Rearrangement of Energy States:
    As you apply various quantum gates, you are dynamically rearranging the amplitudes and phases of the 32 states. This rearrangement is analogous to shifting the underlying energy distribution in a classical binary sequence. It is through these controlled adjustments that the circuit "steers" the interference patterns towards certain outcomes.

3.3. Superposition and Emergent Classical Measurement

After initializing all 5 qubits with Hadamard gates, the circuit is in an equal superposition of all 32 possible states. Each state represents one specific combination of the binary variables from our equation. At this stage:

  • Full Superposition:
    The quantum state is a linear combination of all 32 basis states, each with the same amplitude initially. This means that the system is simultaneously in all possible states.

  • Quantum Interference:
    As additional gates (such as phase shifts and CP gates) are applied, the phases of these amplitudes change. The different states then begin to interfere—some constructively (increasing their likelihood) and others destructively (decreasing their likelihood). The final probability distribution of outcomes depends on the cumulative effect of these interference patterns.

  • Measurement and Emergence:
    When you finally measure the circuit, the superposition collapses into one of the 32 possible states. The probability of each state being the outcome is determined by the square of its amplitude. The classical measurement that you observe is an emergent result—an averaged effect of the entire quantum interference process.

By carefully controlling the quantum gate parameters (such as phase angles), you can influence which outcomes are more likely. This level of control over interference is a uniquely quantum phenomenon, unlike classical deterministic systems where outcomes are fixed.

In summary, Section 3 has shown that:

  • A 5-qubit system naturally represents all 32 possible outcomes arising from the five binary variables in our curvature-corrected theorem.
  • Quantum gates can simulate dynamic shifts in these binary choices, analogous to energy transitions and chiral shifts in classical systems.
  • Superposition allows the quantum system to hold all 32 states at once, and interference among these states—controlled by gate operations—leads to an emergent classical measurement upon observation.

This mapping forms the crucial bridge between our theoretical model and practical quantum simulation, setting the stage for the hands-on parts of the course in Python and Qiskit.



4. Python and Qiskit Essentials for Quantum Circuit Modeling

4.1. Installing and Setting Up Qiskit

Installation:
Qiskit is an open-source framework for quantum computing. To install Qiskit, run:

pip install qiskit

Virtual Environment:
It is recommended to use a virtual environment to manage your Python packages without affecting system-wide installations:

python -m venv qiskit_env
source qiskit_env/bin/activate  # On Unix/Mac
qiskit_env\Scripts\activate     # On Windows
pip install qiskit

Additional Tools:
Optionally, install visualization libraries like matplotlib:

pip install matplotlib

4.2. Basic Quantum Circuit Operations

Key Components in Qiskit:

  • QuantumRegister: Holds the qubits in your circuit.
  • ClassicalRegister: Stores the measurement outcomes.
  • QuantumCircuit: The main object used to create and manipulate your quantum circuit.

Key Gates:

  • Hadamard Gate (H): Puts a qubit into an equal superposition of 0|0\rangle and 1|1\rangle.
  • Phase Gate (U1): Adds a phase to a qubit, crucial for simulating energy differences.
  • Controlled Phase Gate (CP): Applies a phase shift to a target qubit conditional on the state of a control qubit, simulating interactions between qubits.

Measurement:
Measurement collapses the qubits’ superposition into a single classical outcome, which is recorded in the classical register.

Example Code:

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister

# Create a QuantumRegister with 5 qubits and a ClassicalRegister with 5 bits.
qreg = QuantumRegister(5, 'q')
creg = ClassicalRegister(5, 'c')
qc = QuantumCircuit(qreg, creg)

# Apply Hadamard gate to each qubit to create superposition.
for i in range(5):
    qc.h(qreg[i])

# Perform measurement: this collapses the superposition into one of 32 possible states.
qc.measure(qreg, creg)

# Print the circuit diagram.
print(qc.draw(output='text'))

Example Output (Circuit Diagram):

         ┌───┐                  
q_0: ────┤ H ├──────■───────────
         ├───┤      │           
q_1: ────┤ H ├──────┼───────────
         ├───┤      │           
q_2: ────┤ H ├──────┼───────────
         ├───┤      │           
q_3: ────┤ H ├──────┼───────────
         ├───┤      │           
q_4: ────┤ H ├──────┼───────────
         └───┘  ┌───┴───┐       
c: 5/══════════╡ M M M ╞═══════
               └───────┘       

Note: This is a simplified diagram; in our full circuit each qubit represents a binary decision, including the chirality factor hh.

4.3. Hands-On: Building a 5-Qubit Circuit for Binary Energy Dynamics

In this section, we build a 5-qubit circuit representing our 32-state system. Recall that our curvature-corrected equation is:

c2=a2+b2±h(a2b2R2)c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right)

Here, the binary choices (including hh which can be +1+1 or 1-1) lead to 32 outcomes. Each qubit in our circuit corresponds to one of these binary decisions.

Step-by-Step Overview:

  1. Creating Superposition:
    We initialize the circuit with Hadamard gates, so each qubit is in a superposition of 0|0\rangle and 1|1\rangle.

  2. Applying Phase Shifts:
    We then apply a phase gate (U1) to one qubit (e.g., the one representing hh) to simulate the energy bias that determines whether the correction term is added or subtracted.

  3. Using Controlled Operations:
    A controlled phase (CP) gate is applied between, for example, the qubits representing aa and bb, to simulate interactions between their energy states.

  4. Measurement:
    Finally, we measure all qubits. Although the circuit holds all 32 possibilities in superposition, measurement collapses it into one specific outcome.

Extended Code Example:

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, Aer, execute
from qiskit.visualization import plot_histogram
import matplotlib.pyplot as plt

# Create a 5-qubit quantum register and a 5-bit classical register.
qreg = QuantumRegister(5, 'q')
creg = ClassicalRegister(5, 'c')
qc = QuantumCircuit(qreg, creg)

# Step 1: Initialize all qubits in superposition using Hadamard gates.
for i in range(5):
    qc.h(qreg[i])

# Step 2: Apply a phase gate (U1) to the 5th qubit to simulate an energy bias for the chirality factor h.
phase_angle = 0.75  # Adjust this value to see different effects.
qc.u1(phase_angle, qreg[4])

# Step 3: Apply a controlled phase (CP) gate between qubit 0 (representing a) and qubit 1 (representing b).
qc.cp(phase_angle, qreg[0], qreg[1])

# Step 4: Measure all qubits.
qc.measure(qreg, creg)

# Draw and print the circuit.
print(qc.draw(output='text'))

# Execute the circuit using Qiskit's Aer simulator.
backend = Aer.get_backend('qasm_simulator')
job = execute(qc, backend, shots=1024)
result = job.result()
counts = result.get_counts(qc)

# Print the measurement outcomes.
print("Measurement outcomes:", counts)

# Plot the histogram of the results.
plot_histogram(counts)
plt.title("Measurement Distribution for a 5-Qubit Circuit")
plt.show()

Expected Output:

  • A printed circuit diagram showing Hadamard gates on all qubits, a U1 gate on qubit 4, a CP gate between qubits 0 and 1, and measurement operations.
  • A printed dictionary of measurement counts (e.g., something like {'10100': 35, '00100': 28, ...}), which reflects the probability distribution over the 32 states.
  • A histogram plot visualizing the distribution of outcomes.

Observations:

  • The measurement histogram illustrates how quantum interference (affected by the phase gates) biases certain outcomes over others.
  • By varying the phase_angle, you can observe changes in the interference pattern, demonstrating how tuning quantum gate parameters can steer the emergent classical measurement.

This section has provided a hands-on introduction to using Python and Qiskit to build a quantum circuit that models a 32-state system. Each qubit represents a binary decision, including the crucial chirality factor hh (with both +1+1 and 1-1 options), ensuring that our modified Pythagorean theorem:

c2=a2+b2±h(a2b2R2)c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right)

is fully represented in the circuit.


5. Advanced Quantum Circuit Design: Simulating Energy Flow

In this section, we explore advanced techniques to simulate dynamic energy flow in our quantum circuit, ensuring that the chirality factor hh (which can be +1+1 or 1-1) is fully integrated into our model. Recall that our curvature-corrected Pythagorean theorem is given by:

c2=a2+b2±h(a2b2R2),c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right),

where the binary variable hh plays a crucial role in determining whether the energy correction is added or subtracted. With five independent binary choices (including hh), we have 32 possible outcomes. Our 5-qubit system naturally represents these 32 states, and by controlling phase relationships in the circuit, we can simulate energy dynamics and influence the final measurement.

5.1. Implementing Phase Shifts and Custom Energy Operators

Phase Shift Gate (U1):
The U1 gate in Qiskit applies a phase shift to a qubit:

U1(λ)0=0,U1(λ)1=eiλ1.U1(\lambda)|0\rangle = |0\rangle,\quad U1(\lambda)|1\rangle = e^{i\lambda}|1\rangle.

This gate models an energy bias by adjusting the relative phase between the basis states. For instance, applying a U1 gate to the qubit representing the chirality factor hh simulates whether the correction term in our equation is added or subtracted.

Controlled Phase (CP) Gate:
The controlled phase gate applies a phase shift to a target qubit conditioned on the control qubit being in 1|1\rangle. In our circuit, a CP gate between the qubits representing aa and bb simulates the interaction between their energy states, influencing the overall interference pattern.

Custom Energy Operators:
We can design custom unitary operators to simulate more complex energy dynamics—such as "chiral shifts," where the binary sequence of our qubit register is cyclically permuted. Even simple combinations of standard gates can mimic these effects, adjusting the amplitudes and phases across the 32 possible states.

Example Code:

# Assuming qc is our QuantumCircuit with a 5-qubit register 'qreg'
phase_angle = 0.75  # Set the phase shift value

# Apply a phase shift to the chirality qubit (qreg[4]), representing h.
qc.u1(phase_angle, qreg[4])

# Apply a controlled phase gate between qubit 0 (representing a) and qubit 1 (representing b).
qc.cp(phase_angle, qreg[0], qreg[1])

This snippet applies a phase shift to the qubit corresponding to hh and then uses a controlled operation to simulate the interaction between aa and bb.

5.2. Simulating Time Evolution in the Circuit

Quantum circuits can simulate time evolution by applying a sequence of gates that represent the system’s dynamics over a small time step. By repeating these gate sequences, we mimic how the system evolves, allowing us to observe changes in the interference pattern among the 32 states.

Repeated Cycle (Loop) of Operations:

  • Time Evolution via Gate Sequences:
    By applying rotations (e.g., an Rx gate) and phase shifts in a loop, we simulate a discrete time evolution of the quantum state. Each iteration represents a time step, during which the state of the system is "shifted"—similar to a chiral shift that reorganizes the energy distribution among the qubits.

  • Observing Measurement Changes:
    As you repeat these operations, the interference among the 32 basis states changes. By measuring the circuit after different numbers of cycles (e.g., 1, 3, and 5 cycles), you can see how the probability distribution evolves, indicating how the dynamic energy flow shapes the final measurement.

Exercise:

Create a loop that applies a series of rotation and phase gates to simulate time evolution. Run the circuit for 1, 3, and 5 cycles, and record the measurement histograms.

Example Code:

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, Aer, execute
from qiskit.visualization import plot_histogram
import matplotlib.pyplot as plt

# Create a 5-qubit quantum register and a 5-bit classical register.
qreg = QuantumRegister(5, 'q')
creg = ClassicalRegister(5, 'c')
qc = QuantumCircuit(qreg, creg)

# Initialize all qubits in superposition using Hadamard gates.
for i in range(5):
    qc.h(qreg[i])

def time_evolution_cycle(circuit, phase, iterations=1):
    for _ in range(iterations):
        # Apply a phase shift to the chirality qubit (simulate h effect).
        circuit.u1(phase, qreg[4])
        # Apply a controlled phase gate between qubits representing a and b.
        circuit.cp(phase, qreg[0], qreg[1])
        # Apply a rotation on another qubit to add more dynamics.
        circuit.rx(phase, qreg[2])
    return circuit

# Create circuit copies for different cycle counts.
qc_cycle1 = qc.copy()
qc_cycle3 = qc.copy()
qc_cycle5 = qc.copy()

# Apply 1, 3, and 5 cycles respectively.
time_evolution_cycle(qc_cycle1, phase_angle, iterations=1)
time_evolution_cycle(qc_cycle3, phase_angle, iterations=3)
time_evolution_cycle(qc_cycle5, phase_angle, iterations=5)

# Measure each circuit.
qc_cycle1.measure(qreg, creg)
qc_cycle3.measure(qreg, creg)
qc_cycle5.measure(qreg, creg)

# Define the backend and execute the circuits.
backend = Aer.get_backend('qasm_simulator')
job1 = execute(qc_cycle1, backend, shots=1024)
job3 = execute(qc_cycle3, backend, shots=1024)
job5 = execute(qc_cycle5, backend, shots=1024)

result1 = job1.result().get_counts(qc_cycle1)
result3 = job3.result().get_counts(qc_cycle3)
result5 = job5.result().get_counts(qc_cycle5)

# Plot histograms for each circuit evolution.
plt.figure(figsize=(12, 4))
plt.subplot(1, 3, 1)
plot_histogram(result1, title="1 Cycle of Evolution", ax=plt.gca())
plt.subplot(1, 3, 2)
plot_histogram(result3, title="3 Cycles of Evolution", ax=plt.gca())
plt.subplot(1, 3, 3)
plot_histogram(result5, title="5 Cycles of Evolution", ax=plt.gca())
plt.tight_layout()
plt.show()

Discussion:

  • Histogram Analysis:
    Compare the histograms for 1, 3, and 5 cycles. How do the probabilities for various outcomes change? Notice that as the circuit evolves, the interference pattern shifts, altering which states are more likely.

  • Impact of Phase Adjustments:
    The evolution of the circuit demonstrates that by adjusting phase angles, you control the interference among the 32 states. This control allows you to “steer” the emergent classical measurement.

5.3. Measurement, Interference, and Emergent Behavior

When the quantum circuit is measured, the superposition of 32 states collapses to one specific outcome. However, if you run the circuit many times, the histogram of outcomes reflects the underlying quantum interference.

  • Collapse of Superposition:
    Despite the circuit holding 32 possibilities, measurement forces a single outcome. Over many runs, you see a probability distribution that is determined by the interference among the states.

  • Interference Patterns:
    The relative phases of the qubit amplitudes (adjusted by our phase gates and controlled operations) cause certain states to interfere constructively (raising their probability) while others interfere destructively (lowering their probability).

  • Emergent Classical Outcome:
    The final, classical measurement is an emergent result of this interference. By fine-tuning the gate parameters (phase angles, controlled operations), you can influence the distribution and, consequently, the most likely outcome.

This behavior is fundamentally different from classical deterministic measurements. In classical systems, the output is fixed. In quantum systems, even though the evolution is governed by deterministic unitary operations, the act of measurement yields a probabilistic result, where the final outcome is the result of many underlying possibilities interfering with one another.


Summary of Section 5:

  • We implemented phase shifts (via U1 gates) and controlled phase operations (CP gates) to simulate energy biases and interactions—key elements in our curvature-corrected model, ensuring the inclusion of the chirality factor hh.
  • We simulated time evolution by applying a repeated cycle of gates, mimicking dynamic energy flow and "chiral shifts" over time.
  • We demonstrated how measurement collapses the superposition of 32 states into a single outcome, with the probability distribution reflecting the interference pattern established by our gate operations.


6. Extended Projects and Exercises

In this section, you will tackle projects and exercises designed to deepen your understanding of advanced quantum circuit design. Our goal is to let you explore how adjusting quantum gate parameters—especially those that simulate our curvature-corrected Pythagorean theorem

c2=a2+b2±h(a2b2R2)c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right)

(where hh can be +1+1 or 1-1)—affects the emergent measurement outcome in a 5-qubit circuit that encodes 32 possible states. Each project emphasizes the role of hh and the other binary variables, ensuring that all 32 outcomes are modeled.

6.1. Project 1: Parameter Variations and Phase Sweeps

Objective:
Explore how changing the phase angles (which represent energy biases) affects the emergent measurement outcome from the quantum circuit.

Tasks:

  • Vary the Phase Angle:
    Modify the phase angle in your circuit (for example, using a U1 gate on the qubit representing hh or another variable) over a range from 0 to 2π2\pi. Choose a set of values such as: 0, π/4\pi/4, π/2\pi/2, 3π/43\pi/4, π\pi, 5π/45\pi/4, 3π/23\pi/2, and 7π/47\pi/4.

  • Run the Circuit for Each Value:
    For each phase angle, execute your circuit (using the Qiskit Aer simulator) and record the measurement distribution (the histogram of outcomes).

  • Plot and Analyze:
    Create plots showing how the probability distribution changes as the phase angle varies. Analyze these trends by answering:

    • Which outcomes become more or less likely?
    • How does constructive versus destructive interference shift the final distribution?

By varying the phase angle, the relative phases between the amplitudes of the 32 states (which include the effect of hh) change. This alters the interference pattern: some states will experience constructive interference (amplifying their probability), while others will experience destructive interference (reducing their probability). The histogram will shift accordingly, revealing which outcomes dominate. This exercise demonstrates how tuning quantum gate parameters lets you “steer” the system toward a desired emergent classical measurement—a control that has no classical analogue.

6.2. Project 2: Custom Chiral Shift Operators

Objective:
Develop and implement a custom unitary operator that simulates a chiral shift—a cyclic permutation of the qubit states—to model a dynamic rearrangement of energy states, including the chirality factor hh.

Tasks:

  • Design a Unitary Matrix:
    Construct a unitary matrix that cyclically rotates the basis states. For example, in a simplified 2-dimensional case, this operator might map:

    0000010010000.|00\cdots0\rangle \to |00\cdots1\rangle \to |00\cdots10\rangle \to \cdots \to |00\cdots0\rangle.

    For a 5-qubit system, design an operator that acts on one or more qubits to simulate a “chiral shift” where the energy distribution (and thus the effective value of hh) is cyclically permuted.

  • Implement as a Custom Gate in Qiskit:
    Use Qiskit’s UnitaryGate to define your custom operator, then add it to your circuit.

    from qiskit.extensions import UnitaryGate
    import numpy as np
    
    # Example: a simple 2x2 unitary matrix representing a rotation (for one qubit)
    theta = np.pi/4
    custom_matrix = np.array([[np.cos(theta), -np.sin(theta)],
                              [np.sin(theta),  np.cos(theta)]])
    custom_gate = UnitaryGate(custom_matrix, label="ChiralShift")
    # For multi-qubit systems, design a higher-dimensional unitary that performs a cyclic shift.
    
  • Integrate and Study:
    Insert your custom chiral shift operator into your 5-qubit circuit. Compare the measurement outcomes (histograms) with and without your custom gate. This will help you understand how such a shift affects the interference pattern, particularly in relation to the chirality factor hh.

By cyclically permuting the amplitudes of the basis states, your custom unitary operator effectively “rotates” the energy configuration. This chiral shift models how energy states rearrange dynamically, which in turn influences the final measurement outcome.

6.3. Project 3: Hybrid Classical-Quantum Simulation of Curved Geometry

Objective:
Combine classical calculations of the curvature-corrected distance with quantum simulation, mapping classical parameters to qubit states and comparing the outcomes. This project explicitly incorporates the hh factor in both classical and quantum simulations.

Tasks:

  • Write a Classical Function:
    Create a Python function to compute the distance cc using:

    c2=a2+b2±h(a2b2R2),c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right),

    where both +h+h and h-h are considered.

    import math
    
    def classical_distance(a, b, R, plus=True):
        correction = (a**2 * b**2) / (R**2)
        if plus:
            c_squared = a**2 + b**2 + correction
        else:
            c_squared = a**2 + b**2 - correction
        return math.sqrt(c_squared)
    
  • Map Classical Parameters to Qubit States:
    Decide on a mapping scheme—for example, represent a “+” sign by qubit state 0|0\rangle and a “–” sign by 1|1\rangle. You can map the classical outcome (or its sign) to a specific qubit configuration. Ensure that the hh factor is included in this mapping.

  • Build the Quantum Circuit:
    Construct a quantum circuit that reflects these classical parameters. Use your classical function to decide which branch (state) the system should favor, then apply quantum gates that enhance the amplitude of the corresponding qubit state.

  • Simulate and Compare:
    Run your circuit, measure the outcomes, and compare the emergent result with the classical calculation. Analyze any discrepancies and insights that arise from the comparison.

This hybrid approach demonstrates how quantum interference and superposition can yield an emergent classical value that aligns with—or sometimes deviates from—classical predictions. It emphasizes the role of hh (with both +1+1 and 1-1 choices) in determining the energy correction and illustrates the bridging of classical and quantum paradigms.

6.4. Additional Exercises

  • Exercise 6.4.1: Extend the circuit to use 6 qubits and analyze how the additional binary variable affects the emergent state.
  • Exercise 6.4.2: Incorporate Qiskit's noise models to simulate decoherence, and study how noise influences the measurement histogram.
  • Exercise 6.4.3: Develop a parameter sweep simulating continuous time evolution, and plot how the probability distribution evolves over time.

Each project and exercise is designed to take about 2–3 hours, culminating in roughly 10 hours of hands-on programming practice. These activities will deepen your understanding of quantum circuit dynamics and how adjustments in phase and gate operations influence the final measurement outcomes.

Summary of Section 6

In this section, you have worked on extended projects and exercises that:

  • Explore how varying phase angles affect the interference and final measurement outcomes.
  • Guide you in creating custom unitary operators (chiral shift operators) that cyclically rotate qubit states, modeling dynamic energy transitions.
  • Integrate classical curvature-corrected calculations with quantum circuit simulations, ensuring that the hh factor (with both +1+1 and 1-1 values) is accounted for, and comparing the emergent results.
  • Challenge you with additional exercises that extend the model, incorporate noise, and simulate continuous evolution.

These projects provide both theoretical insight and practical experience, helping you understand how quantum interference and superposition give rise to a single, emergent classical measurement from 32 underlying possibilities.

7. Course Summary and Future Directions

Throughout this course, we have journeyed from the familiar world of classical measurements to the cutting-edge realm of quantum circuits. By extending the classic Pythagorean theorem to include a curvature correction and a chirality factor hh (which can be +1+1 or 1-1), we uncovered a model with 32 possible outcomes. We then mapped these outcomes onto a 5-qubit quantum circuit, demonstrating how quantum superposition and interference can capture a rich tapestry of possibilities that collapse to a single classical measurement upon observation.

This course has equipped you with both the theoretical foundation and the practical skills necessary to construct and analyze quantum circuits using Python and Qiskit. You’ve learned how to initialize qubits into superposition, manipulate their phases with controlled operations, and simulate dynamic energy flows over time. Through hands-on projects and exercises, you’ve seen firsthand how quantum interference can be steered to favor specific outcomes, opening up new ways to approach problems that were once thought to be deterministic.

Looking ahead, the future is bright for further exploration in quantum computing. You now have the tools to test these ideas on real quantum hardware, integrate more complex models, and even develop advanced quantum algorithms. The techniques learned here have broad implications—from optimizing computations to modeling physical phenomena that classical systems cannot efficiently simulate.

As you continue your journey, remember that every measurement, every outcome, is a glimpse into the deeper, dynamic processes at work in the quantum world. Embrace this new perspective, and let it inspire you to push the boundaries of what is possible in computation and physics.

https://github.com/mikelewis1971/pythagorean_qubit


Appendix: Complete Python & Qiskit Code

This appendix contains all the essential programs used throughout the course "Quantum Geometry: Simulating Energy Dynamics with Qubits." The code is categorized by topic, and each section provides clear examples and instructions to help you explore quantum circuits that simulate complex energy dynamics and interference patterns.


1. Installing and Setting Up Qiskit

Before running any quantum circuits, ensure you have Qiskit installed. Use the following command in your terminal:

pip install qiskit

Virtual Environment (Recommended):

To manage your Python packages without affecting your system-wide installations, create and activate a virtual environment:

python -m venv qiskit_env
source qiskit_env/bin/activate  # On Unix/Mac
qiskit_env\Scripts\activate     # On Windows
pip install qiskit

You may also install additional visualization libraries like matplotlib:

pip install matplotlib numpy

2. Creating a Basic Quantum Circuit

This example initializes a 5-qubit system and applies a Hadamard gate to each qubit, creating an equal superposition. It then measures all qubits to collapse the state to one of 32 possible outcomes.

from qiskit import QuantumCircuit, Aer, execute

# Create a quantum circuit with 5 qubits and 5 classical bits.
qc = QuantumCircuit(5, 5)

# Apply Hadamard gates to all qubits to create superposition.
for qubit in range(5):
    qc.h(qubit)

# Measure all qubits.
qc.measure(range(5), range(5))

# Visualize the circuit.
print(qc.draw(output='text'))

# Simulate the circuit.
simulator = Aer.get_backend('qasm_simulator')
result = execute(qc, simulator, shots=1024).result()
counts = result.get_counts()

# Print measurement results.
print("\nMeasurement results:", counts)

3. Implementing the Curvature-Corrected Formula

This Python function computes the modified Pythagorean theorem for both the plus and minus cases, incorporating the chirality factor hh.

import math

def classical_distance(a, b, R, plus=True):
    """Compute the curvature-corrected hypotenuse.
    
    The formula is:
        c^2 = a^2 + b^2 ± h * (a^2 * b^2 / R^2)
    where h is represented implicitly by choosing the plus or minus branch.
    """
    correction = (a**2 * b**2) / (R**2)
    if plus:
        c_squared = a**2 + b**2 + correction
    else:
        c_squared = a**2 + b**2 - correction
    return math.sqrt(c_squared)

# Example usage:
a, b, R = 3, 4, 10
print("Hyperbolic case (+):", classical_distance(a, b, R, plus=True))
print("Spherical case (-):", classical_distance(a, b, R, plus=False))

4. Encoding the 32-State System in a Quantum Circuit

This circuit represents the 5-qubit system where each qubit corresponds to one binary choice in our curvature-corrected equation, ensuring that all 32 outcomes (including both h=+1h = +1 and h=1h = -1) are represented.

from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit

qreg = QuantumRegister(5, 'q')
creg = ClassicalRegister(5, 'c')
qc = QuantumCircuit(qreg, creg)

# Apply Hadamard gates to initialize an equal superposition of all 32 states.
qc.h(qreg)

# Measure all qubits.
qc.measure(qreg, creg)

# Visualize the circuit.
print(qc.draw())

5. Phase Shifts and Interference

This section applies phase shifts to control how the quantum states interfere, which is essential for simulating energy dynamics.

from qiskit.circuit.library import U1Gate

qc = QuantumCircuit(5, 5)

# Apply Hadamard gates to all qubits.
qc.h(range(5))

# Apply a phase shift to the 5th qubit (simulating a curvature effect and influencing the chirality factor h).
phase_angle = 0.75  # Example phase angle.
qc.append(U1Gate(phase_angle), [4])

# Measure all qubits.
qc.measure(range(5), range(5))

# Draw the circuit.
print(qc.draw())

6. Custom Chiral Shift Operator

This example constructs a custom unitary gate to simulate a cyclic "chiral shift" of the qubit states. Such a gate can rearrange the energy configuration in our 5-qubit system.

import numpy as np
from qiskit.extensions import UnitaryGate

theta = np.pi / 4
custom_matrix = np.array([[np.cos(theta), -np.sin(theta)],
                          [np.sin(theta),  np.cos(theta)]])
custom_gate = UnitaryGate(custom_matrix, label="ChiralShift")

qc = QuantumCircuit(5, 5)
qc.append(custom_gate, [4])  # Apply the custom chiral shift operator to the 5th qubit.

qc.measure(range(5), range(5))
print(qc.draw())

7. Running the Circuit and Visualizing Results

Use Qiskit's Aer simulator to run the circuit and visualize the measurement outcomes as a histogram.

from qiskit.visualization import plot_histogram
import matplotlib.pyplot as plt

# Run the simulation.
simulator = Aer.get_backend('qasm_simulator')
result = execute(qc, simulator, shots=1024).result()
counts = result.get_counts()

# Print the measurement outcomes.
print("Measurement outcomes:", counts)

# Plot the histogram.
plot_histogram(counts)
plt.title("Measurement Distribution for a 5-Qubit Circuit")
plt.show()

8. Implementing a Time Evolution Model

This script simulates dynamic energy flow over multiple iterations by repeatedly applying rotation and phase gates. Each cycle represents a discrete time step, and the final measurement reveals how the interference pattern evolves.

qc = QuantumCircuit(5, 5)

# Apply Hadamard gates to initialize the circuit.
qc.h(range(5))

# Simulate energy evolution over 5 cycles.
for _ in range(5):
    qc.rx(np.pi/4, range(5))  # Apply a rotation to simulate time evolution.

qc.measure(range(5), range(5))

# Run and plot the results.
result = execute(qc, simulator, shots=1024).result()
counts = result.get_counts()
plot_histogram(counts)
plt.title("Time Evolution over 5 Cycles")
plt.show()

Summary of the Code Appendix

Topics Covered

  • Basic quantum circuits and superposition.
  • Phase shifts and interference in a quantum circuit.
  • Custom unitary transformations (chiral shift operators).
  • Time evolution and cyclic energy shifts.
  • Hybrid classical-quantum modeling.
  • Noise models and simulations of decoherence.

This appendix provides ready-to-use Python and Qiskit code for all the major exercises and projects in the course. Use these examples as a foundation for exploring how quantum circuits can simulate complex energy dynamics and emergent behavior.


Additional Resources

This comprehensive appendix equips you with the code and resources needed to explore quantum energy dynamics in detail. Use these examples to build and extend your own quantum simulations, and deepen your understanding of how quantum circuits can model complex, dynamic processes in a system that bridges classical geometry with quantum computation.



Let's walk through a concrete example by hand to illustrate how the process works. We'll use the curvature-corrected equation:

c2=a2+b2±h(a2b2R2),c^2 = a^2 + b^2 \, \pm \, h\left(\frac{a^2b^2}{R^2}\right),

where hh can be +1+1 or 1-1. Suppose we have the following classical values:

  • a=3a = 3
  • b=4b = 4
  • R=10R = 10

Step 1: Evaluate the Base Terms

First, compute the standard (Euclidean) terms:

a2=32=9,b2=42=16,a2+b2=9+16=25.a^2 = 3^2 = 9,\quad b^2 = 4^2 = 16,\quad a^2 + b^2 = 9 + 16 = 25.

Also, compute the correction factor (ignoring hh and the ±\pm for now):

a2b2R2=9×16100=144100=1.44.\frac{a^2b^2}{R^2} = \frac{9 \times 16}{100} = \frac{144}{100} = 1.44.

Step 2: Consider the Two Possibilities for hh and the ±\pm Sign

There are two branches for the correction term depending on the sign choice and the value of hh. For clarity, let’s treat the combination ±h\pm h as one decision.

  1. Case 1: h=+1h = +1 and the correction is added.
    Then the equation becomes:

    c2=25+1.44=26.44,c^2 = 25 + 1.44 = 26.44,

    so,

    c26.445.142.c \approx \sqrt{26.44} \approx 5.142.
  2. Case 2: h=+1h = +1 but the correction is subtracted (or equivalently, h=1h = -1 and the correction is added).
    Then the equation becomes:

    c2=251.44=23.56,c^2 = 25 - 1.44 = 23.56,

    so,

    c23.564.855.c \approx \sqrt{23.56} \approx 4.855.

In our full model, we consider five binary decisions (including the sign choices for aa, bb, the square-root step for cc, the sign of RR if needed, and hh). This results in 25=322^5 = 32 possible outcomes. For our example, we are focusing on the decision affecting the correction term via hh.

Step 3: Representing the Model with Qubits

In our quantum circuit, each of the five binary choices is mapped to one qubit:

  • One qubit for the sign of aa (e.g., 0 = positive, 1 = negative).
  • One for the sign of bb.
  • One for the binary choice when taking the square root (for cc).
  • One for the sign of RR (if we consider its ambiguity).
  • One for the chirality factor hh (0 represents +1+1 and 1 represents 1-1).

Before measurement, the 5-qubit system is in a superposition of all 32 possible configurations. Quantum gates (such as phase shifts and controlled phase gates) are applied to adjust the amplitudes and relative phases. These adjustments determine which of the 32 states constructively interferes, and hence, which outcome is more likely to be measured.

Step 4: The Role of Shift Operators

Imagine that we apply a custom chiral shift operator (a unitary transformation) to the qubit representing hh. This operator can cyclically rotate the state of that qubit, effectively changing its phase. For example:

  • If the chiral shift operator increases the phase such that the state corresponding to h=+1h = +1 is amplified, then the circuit will more likely yield the outcome where the correction is added—giving c5.142c \approx 5.142.
  • Conversely, if the shift operator adjusts the phase to favor h=1h = -1, then the outcome where the correction is subtracted becomes more probable—resulting in c4.855c \approx 4.855.

In our circuit, controlled operations (like a controlled phase gate between the qubits for aa and bb) further adjust the interference among states. These operations simulate interactions between the variables and “steer” the probability distribution among the 32 states.

Step 5: Emergence of the Classical Outcome

After applying all the quantum operations (Hadamard gates to create superposition, phase shift gates to simulate energy biases, and controlled gates for interactions), the circuit is measured. Although the system initially contains 32 possibilities, measurement collapses the state into one definite outcome. If, through our gate operations, the state corresponding to a positive hh (and the “added correction” branch) is amplified, the final measurement will most likely yield that state, resulting in c5.142c \approx 5.142.

Thus, by carefully controlling the phase relationships through quantum gates—including the custom chiral shift operator affecting hh—we determine which of the 32 possible outcomes becomes dominant. The emergent classical measurement is the net result of the constructive and destructive interference among all possibilities.

Summary

  • Step 1: Compute the base values: a2+b2=25a^2 + b^2 = 25 and correction factor = 1.44.
  • Step 2: Consider the two possibilities for the correction term (using h=+1h = +1 and h=1h = -1):
    • Outcome 1: c2=25+1.44=26.44c5.142c^2 = 25 + 1.44 = 26.44 \rightarrow c \approx 5.142
    • Outcome 2: c2=251.44=23.56c4.855c^2 = 25 - 1.44 = 23.56 \rightarrow c \approx 4.855
  • Step 3: Map the five binary choices (including hh) onto a 5-qubit system, representing 32 possible states.
  • Step 4: Use quantum gates (phase shifts, controlled phase gates, and custom chiral shift operators) to adjust the amplitudes and phases of the qubit states. This simulates the dynamic energy interactions.
  • Step 5: Upon measurement, the interference pattern among the 32 states determines which outcome emerges as the final classical measurement.

This example illustrates by hand how our quantum circuit model uses shift operators to adjust the relative probabilities of the different outcomes. Ultimately, these operations allow us to “plug in” the binary choices and obtain a classical measurement that reflects the underlying quantum dynamics.

Special Note:
In classical calculations, the signs of aa and bb do not affect the final result because these values are squared, effectively removing any sign differences. However, when we introduce phase shifts and chirality into our quantum circuit, the underlying sign information becomes significant. In our system, each binary decision—including whether aa or bb is positive or negative—affects the phase relationships among the qubit states. These phase relationships, in turn, influence how the quantum amplitudes interfere with one another. Thus, even though the classical formula appears insensitive to the signs of aa and bb, the quantum model reveals that these choices can have a profound impact on the emergent measurement when combined with controlled phase operations and the chirality factor hh. This course is designed to help you explore and understand these subtleties in quantum interference that arise from what may seem like negligible classical differences.



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