adding codes and lecture slides

Author: mhjensen

doc/src/week11/prograns/qpe.py

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+import numpy as np
+
+# Define basic quantum gates
+def hadamard(n):
+   """Creates an n-qubit Hadamard gate."""
+   H = np.array([[1, 1], [1, -1]]) / np.sqrt(2)
+   return np.linalg.matrix_power(np.kron(H, np.eye(2 ** (n - 1))), 1)
+
+def phase_shift(theta):
+   """Single-qubit phase shift gate U = e^(2πiθ)."""
+   return np.array([[1, 0], [0, np.exp(2j * np.pi * theta)]])
+
+def controlled_U(U, n):
+   """Controlled-U gate for n qubits."""
+   # Identity for control 0
+   dim = 2 ** (n + 1)
+   CU = np.eye(dim, dtype=complex)
+   # Apply U when control is 1
+   for i in range(2 ** n):
+       if (i >> (n - 1)) & 1:  # Check if the last qubit is 1
+           CU[i, i] = U[1, 1]
+   return CU
+
+def inverse_qft(n):
+   """Inverse Quantum Fourier Transform (QFT†)."""
+   dim = 2 ** n
+   QFT = np.zeros((dim, dim), dtype=complex)
+   omega = np.exp(-2j * np.pi / dim)
+   for i in range(dim):
+       for j in range(dim):
+           QFT[i, j] = omega ** (i * j) / np.sqrt(dim)
+   return QFT
+
+# Initialize parameters
+t = 4  # Number of counting qubits (precision)
+theta = 0.3125  # Phase we are trying to estimate (0.3125 = 5/16)
+n_qubits = t + 1  # Total qubits (t counting + 1 target)
+
+# Create the initial state |0...0⟩|1⟩
+state_dim = 2 ** n_qubits
+state = np.zeros(state_dim, dtype=complex)
+state[-1] = 1  # |0...01⟩
+
+# Create Hadamard on counting qubits
+for i in range(t):
+   H_i = np.eye(2 ** i) if i != 0 else 1
+   H = np.kron(H_i, np.array([[1, 1], [1, -1]]) / np.sqrt(2))
+   H = np.kron(H, np.eye(2 ** (t - i - 1)))
+   state = H @ state
+
+# Apply controlled U gates with increasing powers
+U = phase_shift(theta)
+for i in range(t):
+   CU = controlled_U(np.linalg.matrix_power(U, 2 ** i), t)
+   state = CU @ state
+
+# Apply inverse QFT on counting qubits
+QFT_inv = inverse_qft(t)
+QFT_inv = np.kron(QFT_inv, np.eye(2))  # Do not apply on target qubit
+state = QFT_inv @ state
+
+# Measure probabilities
+probabilities = np.abs(state) ** 2
+most_likely_state = np.argmax(probabilities)
+binary_result = bin(most_likely_state >> 1)[2:].zfill(t)
+estimated_phase = int(binary_result, 2) / (2 ** t)
+
+# Output results
+print(f"Estimated Phase (binary): {binary_result}")
+print(f"Estimated Phase (decimal): {estimated_phase}")
+print(f"Actual Phase: {theta}")

doc/src/week13/Latexfiles/qrbm.pdf

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+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath, amsfonts, amssymb, bm}
+\usepackage{graphicx}
+\usepackage{hyperref}
+\usepackage{physics}
+\usepackage{mathtools}
+\usepackage{braket}
+
+\title{Advanced Topics in Quantum Boltzmann Machines}
+\author{Quantum Computing Lecture Series}
+\date{\today}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+\newpage
+
+%-----------------------------------------------------------
+\section{Introduction to Quantum Boltzmann Machines (QBMs)}
+Quantum Boltzmann Machines (QBMs) are a quantum generalization of classical Boltzmann machines. They leverage quantum effects such as superposition and entanglement to model complex probability distributions. QBMs are well-suited for quantum machine learning tasks, particularly for generative modeling. 
+
+\subsection{Motivation}
+Classical Boltzmann Machines suffer from high-dimensional sampling complexity. Quantum mechanics offers exponential state space and quantum tunneling effects that can alleviate these issues.
+
+\subsection{Key Concepts}
+\begin{itemize}
+    \item Quantum States as Probability Distributions
+    \item Quantum Tunneling for Escaping Local Minima
+    \item Exponential State Space in Quantum Systems
+\end{itemize}
+
+%-----------------------------------------------------------
+\section{Mathematical Framework}
+\subsection{Hamiltonian of a Quantum Boltzmann Machine}
+The quantum analog of the classical energy-based model is expressed by the Hamiltonian \( H \):
+
+\begin{equation}
+    H = H_Z + H_X,
+\end{equation}
+
+where:
+\begin{align}
+    H_Z &= -\sum_{i} b_i Z_i - \sum_{i<j} w_{ij} Z_i Z_j, \\[5pt]
+    H_X &= -\sum_i \Gamma_i X_i.
+\end{align}
+
+\begin{itemize}
+    \item \( Z_i \) and \( X_i \) are Pauli operators acting on the \( i \)-th qubit.
+    \item \( b_i \) represents the bias terms.
+    \item \( w_{ij} \) represents the interaction between qubits.
+    \item \( \Gamma_i \) is the transverse field strength.
+\end{itemize}
+
+%-----------------------------------------------------------
+\subsection{Density Matrix and Boltzmann Distribution}
+The quantum Boltzmann distribution is defined by the density matrix:
+
+\begin{equation}
+    \rho = \frac{e^{-\beta H}}{Z},
+\end{equation}
+
+where:
+\begin{itemize}
+    \item \( \beta = 1/k_B T \) is the inverse temperature.
+    \item \( Z = \Tr(e^{-\beta H}) \) is the partition function.
+\end{itemize}
+
+In the classical case, this reduces to a Gibbs distribution.
+
+%-----------------------------------------------------------
+\section{Training Quantum Boltzmann Machines}
+\subsection{Objective Function}
+The goal of training a QBM is to minimize the Kullback–Leibler (KL) divergence between the data distribution \( p_{\text{data}} \) and the model distribution \( p_{\theta} \):
+
+\begin{equation}
+    \mathcal{L}(\theta) = \text{KL}(p_{\text{data}} || p_{\theta}) = \sum_x p_{\text{data}}(x) \log \frac{p_{\text{data}}(x)}{p_{\theta}(x)}.
+\end{equation}
+
+\subsection{Gradient-Based Optimization}
+The gradient of the loss function is computed using:
+
+\begin{equation}
+    \nabla_\theta \mathcal{L}(\theta) = \mathbb{E}_{p_{\text{data}}}[\nabla_\theta E(x)] - \mathbb{E}_{p_{\theta}}[\nabla_\theta E(x)],
+\end{equation}
+
+where \( E(x) \) is the energy function derived from the Hamiltonian.
+
+%-----------------------------------------------------------
+\section{Quantum Sampling Techniques}
+\subsection{Quantum Monte Carlo Methods}
+Quantum Monte Carlo (QMC) simulates quantum systems by sampling from the quantum density matrix using classical resources. However, it faces limitations due to the **sign problem**.
+
+\subsection{Quantum Annealing}
+Quantum annealers leverage adiabatic evolution to reach low-energy states efficiently:
+
+\begin{equation}
+    H(t) = (1 - t/T) H_B + (t/T) H_P,
+\end{equation}
+
+where:
+\begin{itemize}
+    \item \( H_B \) is the mixing Hamiltonian.
+    \item \( H_P \) is the problem Hamiltonian.
+\end{itemize}
+
+%-----------------------------------------------------------
+\section{Advantages and Challenges}
+\subsection{Advantages}
+\begin{itemize}
+    \item Quantum Parallelism
+    \item Efficient Sampling in Complex Systems
+    \item Potential for Exponential Speedups
+\end{itemize}
+
+\subsection{Challenges}
+\begin{itemize}
+    \item Noisy Quantum Hardware
+    \item High Cost of Quantum Simulation
+    \item Quantum Decoherence
+\end{itemize}
+
+%-----------------------------------------------------------
+\section{Applications}
+\subsection{Generative Modeling}
+Quantum Boltzmann Machines can generate complex probability distributions with applications in:
+\begin{itemize}
+    \item Image and Text Generation
+    \item Quantum Chemistry
+    \item Financial Modeling
+\end{itemize}
+
+\subsection{Optimization Problems}
+QBMs are suitable for solving optimization problems where classical approaches suffer from local minima.
+
+%-----------------------------------------------------------
+\section{Conclusion}
+Quantum Boltzmann Machines offer a promising path for leveraging quantum resources in machine learning. While hardware limitations currently restrict scalability, ongoing research in quantum algorithms and quantum hardware is likely to overcome these obstacles.
+
+%-----------------------------------------------------------
+\section{References}
+\begin{thebibliography}{9}
+
+\bibitem{Amin2018}
+M. Amin et al., "Quantum Boltzmann Machines", \textit{Physical Review X}, 2018.
+
+\bibitem{Hinton1985}
+G. Hinton, "Boltzmann Machines: Constraints and Learning", \textit{Cognitive Science}, 1985.
+
+\bibitem{Nielsen2000}
+M. Nielsen and I. Chuang, "Quantum Computation and Quantum Information", Cambridge University Press, 2000.
+
+\end{thebibliography}
+
+\end{document}