some work on the presentation

presentation/main.tex

@@ -12,14 +12,10 @@
 \usepackage{listings}
 %\usepackage{struktex}
 \usepackage{qcircuit}
+\usepackage{adjustbox}
 
-\newtheorem{definition}{Definition}
-\newtheorem{postulate}{Postulate}
-\newtheorem{corrolary}{Corrolary}
-\newtheorem{lemma}{Lemma}
-\newtheorem{theorem}{Theorem}
 
-\usetheme[progressbar=frametitle]{metropolis}
+\usetheme{metropolis}
 
 \setbeamercolor{background canvas}{bg=white!20}
 
@@ -66,7 +62,9 @@
 
     \begin{itemize}
         \item Some (physical) problems are classically hard to solve.
+        \pause
         \item The quantum simulator: Mapping a hard problem to quantum hardware that can simulate this specific problem.
+        \pause
         \item The (universal) quantum computer: able to simulate any unitary transformation on the system.
     \end{itemize}
 
@@ -77,7 +75,9 @@
 \begin{frame}{Quantum Errors and Quantum Error Correction}
     \begin{itemize}
         \item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
+        \pause
         \item Fault tolerant QC needs a way to correct for those errors.
+        \pause
         \item Several strategies exist one important class of quantum error correction codes are \textbf{stabilizer codes}.
     \end{itemize}
 
@@ -87,22 +87,387 @@
 \section{Binary Quantum Computing}
 
 {
-
 \begin{frame}{Qbits}
 
-    \begin{definition}
+    \textbf{Definition}
+    {\itshape
         A qbit is a two-level quantum mechanical system $\ket{0}, \ket{1}$ with $\braket{0}{1} = 0$.
 
-        In the following $Z = \sigma_Z, X = \sigma_X, Y = \sigma_Y, I = \id$ will be used.
+        In the following $Z = \sigma_Z, X = \sigma_X, Y = \sigma_Y$ will be used. $I$ is the identity matrix.
-    \end{definition}
+    }
     
     Where $Z\ket{0} = +1\ket{0}$ and $Z\ket{1} = -1\ket{1}$.
 
-    \begin{postulate}
-        A $n$-qbit system is the tensor product of the single-qbit systems.
-    \end{postulate}
-    
 
 \end{frame}
 }
+{
+\begin{frame}{Qbits and Gates}
+    \begin{itemize}
+        \item{
+            \textbf{Postulate}
+            {\itshape
+                A $n$-qbit system is the tensor product of the single-qbit systems.
+            }
+        }
+    \pause
+        %\item{
+        %        For $n$ qbits define the integer state $\ket{j}$ as
+
+        %    \begin{equation}
+        %        \ket{j} := \ket{\mbox{0b}i_0i_1...i_{n-1}} := \ket{i_0}_s \otimes \ket{i_1}_s \otimes ... \otimes \ket{i_{n-1}}_s
+        %    \end{equation}
+        %}
+        \item{
+            A transformation $U \in SU(2^n)$ is called \textit{gate} acting on the system. 
+            For $\bar{U} \in SU(2)$ the gate $U_i$ acting on qbit $i$ is given by
+            \begin{equation}
+                U_i := \left(\bigotimes\limits_{k < i} I\right) \otimes \bar{U} \otimes \left(\bigotimes\limits_{k > i} I\right)
+            \end{equation}
+        }
+    \pause
+        \item{
+            For $\bar{U} \in SU(2)$ and qbits $i \neq j$ 
+            \begin{equation}
+                CU_{i,j} := \ket{1}\bra{1}_j \otimes U_i + \ket{0}\bra{0}_j \otimes I
+            \end{equation}
+            is the controlled $\ket{U}$ gate acting on $i$ with control-qbit $j$.
+        }
+    \pause
+    \end{itemize}
+\end{frame}
+}
+
+
+{
+\begin{frame}{Gates}
+     Some notable gates are
+    \begin{itemize}
+        \item{
+            the Hadamard gate
+            \begin{equation}
+                H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)
+            \end{equation}
+             which transforms from the $Z$ to the $X$ basis}
+        \pause
+        \item{the rotation gate
+             \begin{equation}
+                R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)
+             \end{equation}
+             that performs a rotation around the $Z$ axis.}
+            \pause
+
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Gates}
+    \begin{itemize}
+        \item{The $Z$ and $S$ gates are given by:
+             \begin{equation}
+                 Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) = R_{\pi}
+             \end{equation}
+             \begin{equation}
+                S := \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right) = R_{\frac{\pi}{2}}
+             \end{equation}
+
+             The $S$ gate transforms from $X$ to $Y$ basis.
+
+            }
+        \pause
+        \item{
+            \textbf{Theorem}
+            {\itshape
+                Any gate $U \in SU(2^n)$ can be approximated arbitrarely good using the $H$, $R_\phi$
+                and $CZ_{i,j}$ gate.
+            }
+
+        }
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Integer States}
+    \begin{itemize}
+        \item{
+            The eigenstates of the $Z_i$ are called integer states. They have the form
+            \begin{equation}
+                \ket{j} = \ket{\mbox{0b}l_1...l_n} = \ket{l_1}_s \otimes ... \otimes \ket{l_n}s
+            \end{equation}
+            }
+        \pause
+        \item{
+            For $n$ qbits there exist $2^n$ such states and they form a basis
+
+            \begin{equation}
+                \ket{\psi} = \sum\limits_{i=0}^{n-1} \ket{i}\braket{i}{\psi} = \sum\limits_{i=0}^{n-1} c_i\ket{i}
+            \end{equation}
+
+            with the condition $\sum\limits_{i=0}^{n-1} c_i^2 = 1$.
+        }
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Measurement}
+    \begin{itemize}
+        \item{Measurements are performed in $Z$ basis, i.e. for a qbit $i$ 
+            $Z_i$ is measured. 
+            }
+        \item{
+            The results of the measurements are associated with a classical result $s \in \{0, 1\}$
+            using
+
+            \begin{equation}
+                Z_i\ket{\psi'} = (-1)^s \ket{\psi'}
+            \end{equation}
+
+            after the measurement.
+        }
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Quantum Circuits}
+    \begin{itemize}
+        \item{
+            Writing a unitary transformation as a product of the generator gates is unreadable.
+            To fix this problem quantum circuits have been introduced.
+            }
+        \pause
+        \item{
+            Qbits are represented by horizontal lines.
+        }
+    \pause
+        \item{
+            Gates acting on a qbit are boxes on the lines.
+        }
+    \pause
+    \item{
+            Control-qbits are connected to the gate via a vertical line.
+        }
+    \pause
+    \item{
+            Circuits are read left to right.
+        }
+    \pause
+    \item{
+            Example:
+            \Qcircuit @C=1em @R=.7em {
+            & \gate{H} & \ctrl{1} & \qw &\qw \\
+            & \gate{H} & \gate{Z} & \gate{H} &\qw \\
+            }
+        }
+    \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
+    \begin{itemize}
+        \item{
+            For a set of $n$ spins in a magnetic field one can rescale the Hamiltonian
+            of the system to
+            \begin{equation}
+                H = -\sum\limits_{i=1}^{n-1} Z_i Z_{i-1} + g\sum\limits_{i=0}^{n-1} X_i
+            \end{equation}
+            }
+        \pause
+        \item{
+            The time evolution of such a system is given by the transfer matrix
+            \begin{equation}
+                T := \exp(-itH) \in SU(2^n)
+            \end{equation}
+        }
+    \pause
+    \item{
+            By associating every qbit with one spin (both are two-level systems)
+            one should be able to simulate the behaviour of the spin chain using 
+            a quantum computer.
+        }
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
+    \begin{itemize}
+        \item{
+            Trotterizing the matrix exponential
+            \begin{equation}
+                \exp(t(A + B)) = \left(\exp(\frac{t}{N}A)\exp(\frac{t}{N}B)\right)^N + \mathcal{O}\left(\frac{t^2}{N^2}\right)
+            \end{equation}
+            }
+        \item{
+            For $n=3$ spins one gets a circuit
+            {\centering\adjustbox{max width=\textwidth}{
+\Qcircuit @C=1em @R=.7em {
+& \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
+& \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} &\qw \\
+& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} &  \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} &\qw \\
+}
+
+            }}
+            }
+        \item{Applying this circuit $N$ times gives an approximation for the time evolution of a state.}
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=\linewidth]{spin_chain/time_evo_6spin_g3.png}
+\end{center}
+\end{figure}
+\end{frame}
+}
+
+\section{Stabilizers}
+
+{
+\begin{frame}{The multilocal Pauli Group and the Clifford Group}
+    \begin{itemize}
+        \item{\textbf{Definition}
+            {\itshape
+            $P := \{\pm1, \pm i\} \cdot \{X, Y, Z, I\}$ is called the Pauli group.\\
+            $P_n := \left\{p_1 \otimes ... \otimes p_n \middle| p_i \in P \right\}$ is called the multilocal Pauli group.
+            }}
+            \pause
+        \item{
+            \textbf{Definition}
+            {\itshape
+            $C_n := \left\{U \in SU(2^n) \middle| \forall p \in P_n: U^\dagger p U \in P_n\right\}$
+            is called the Clifford group
+            }
+            }
+            \pause
+        \item{
+                One can show that $C_n$ is generated by $H, S, CZ_{i,j}$.
+            }
+    \end{itemize}
+
+\end{frame}
+}
+
+{
+\begin{frame}{Generators of a Group}
+    \begin{itemize}
+        \item{
+            \textbf{Definition}
+            {\itshape
+            For a finite group $G$ one says $G$ is generated by $g_1, ..., g_N$ iff any $g \in G$ can be expressed
+            as a product of $g_1, ..., g_N$. These generators are chosen to be the minimal set for which this
+            condition holds.
+
+            One also writes
+            \begin{equation}
+                G = \langle g_1,...,g_N \rangle \equiv \langle g_i\rangle_i
+            \end{equation}
+            }
+            }
+        \item{
+            The generators are not unique. For instance $C_1$ can be generated using $H, S$ or $\sqrt{-iX}, \sqrt{iZ}$.
+            }
+        \item{
+            The generators of a group have some kind of independence property.
+            }
+    \end{itemize}
+
+\end{frame}
+}
+
+{
+\begin{frame}{Stabilizers and Stabilizer Spaces}
+    \begin{itemize}
+        \item{
+            \textbf{Definition}
+            {\itshape
+            A finite abelian subgroup $S$ of $P_n$ is called a set of stabilizers iff $-I \notin S$ and
+            all elements of $S$ commute.
+            }
+            }
+        \item{
+            From $-I \notin S$ follows that all elements of $S$ are hermitian.
+            }
+        \item{
+            From the definition as tensor products of Pauli matrices follows that the
+            elements of $S$ have eigenvalues $\pm1$.
+            }
+        \item{
+            These properties together yield that all elements of $S$ can be diagonalized simultaneously.
+            Further there exists a vector space $V_S$ with $s \ket{\psi} = +1 \ket{\psi}$ for all $s \in S$
+            and all $\ket{\psi} \in V_S$. This space is called the stabilizer space of $S$
+            and all $\ket{\psi}$ are called stabilizer states.
+            }
+    \end{itemize}
+
+\end{frame}
+}
+
+{
+\begin{frame}{Stabilizer States}
+    \begin{itemize}
+        \item{
+            One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$  the stabilizer space $V_S$
+            has dimension $1$. 
+            }
+        \item{
+            Therefore the state $\ket{\psi}$ that is the +1 eigenstate of all stabilizers is (up to a 
+            global phase) unique.
+            }
+        \item{Notable stabilizer states are:
+            \begin{itemize}
+                \item{ $\ket{0b0..0} = \ket{0}\otimes ... \otimes \ket{0}$ and $\ket{0b1..1} = \ket{1}\otimes ... \otimes \ket{1}$
+                    which are stabilized by $\langle Z_i \rangle_i$ and $\langle -Z_i \rangle_i$ respectively.
+                }
+                \item{
+                    $\ket{+}$ stabilized by $\langle X_i \rangle_i$ (and $\ket{-}$ analogously).
+                }
+                \item{
+                    $\ket{0b00} + \ket{0b11}$ the Bell/EPR state which is stabilized by $\langle X_1X_2, Z_1Z_2\rangle$.
+                }
+            \end{itemize}
+            }
+    \end{itemize}
+
+\end{frame}
+}
+{
+\begin{frame}{Dynamics of Stabilizer States}
+    \begin{itemize}
+        \item{
+            Under a transformation $U \in SU(2^n)$ the state changes to
+            \begin{equation}
+            \ket{\psi'} = U \ket{\psi}
+            \end{equation}
+            }
+        \item{
+            The stabilizers of $\ket{\psi}$ change to 
+            \begin{equation}
+                \ket{\psi'} = U\ket{\psi} = US^{(i)}\ket{\psi} = US^{(i)}U^\dagger U\ket{\psi} = US^{(i)}U^\dagger\ket{\psi'}
+            \end{equation}
+            this gives that if $U \in C_n$ $\ket{\psi'}$ is a stabilizer state again with the stabilizers
+            $S' = \langle US^{(i)}U^\dagger\rangle_{i=1,...,n}$
+            }
+        \item{
+            Under the transformations $C_n$ one can describe the dynamics of the stabilizer states by their
+            stabilizers.
+            }
+        \item{Because the stabilizers are given by $n$ matrices which are the tensor product of $n$ Pauli matrices
+            this can be simulated in $n^2$ time instead of $2^n$.}
+    \end{itemize}
+
+\end{frame}
+}
+
+{
+\begin{frame}{Measurements on Stabilizer States}
+    \begin{itemize}
+        \item{
+            }
+    \end{itemize}
+
+\end{frame}
+}
+
 \end{document}

presentation/spin_chain/time_evolution.py

@@ -1,64 +1,87 @@
 import numpy as np
 import matplotlib.pyplot as plt
-
+import matplotlib
-from pyqcs import State, sample
-from transfer_matrix import T_time_slice
-from hamiltonian import H
 
 from scipy.linalg import expm
+from pyqcs import State, sample
 
-nqbits = 2
+from transfer_matrix import T_time_slice
-g = 0.20
+from hamiltonian import H
-N = 50
+from bootstrap import bootstrap
+
+
+np.random.seed(0xdeadbeef)
+
+matplotlib.rcParams.update(
+        {'errorbar.capsize': 2
+        , 'figure.figsize': (16, 9)}
+        )
+
+nqbits = 6
+g = 3
+N_trot = 80
 t_stop = 9
-delta_t = 0.05
+delta_t = 0.09
 qbits = list(range(nqbits))
 
-n_sample = 400
+n_sample = 2200
 measure = 0b10
 
+measure_coefficient_mask = [False if (i & measure) else True for i in range(2**nqbits)]
+
 
 results_qc = []
 results_np = []
+errors_sampling = []
+
 print()
 for t in np.arange(0, t_stop, delta_t):
     # QC simulation
     state = State.new_zero_state(nqbits)
 
-    for _ in range(N):
+    T_dt = T_time_slice(qbits, t, g, N_trot)
-        state = T_time_slice(qbits, t, g, N) * state
+    for _ in range(N_trot):
+        state = T_dt * state
 
-    #result = sample(state, measure, n_sample)
+    result = sample(state, measure, n_sample)
+    results_qc.append(result[0] / n_sample)
 
-    #results_qc.append(result[0] / n_sample)
+    errors_sampling.append(bootstrap(result[0], n_sample, n_sample // 4, n_sample // 10, np.average))
 
-    amplitude = np.sqrt(np.sum(np.abs(state._qm_state[[False if (i & measure) else True for i in range(2**nqbits)]])**2))
+    #amplitude = np.sqrt(np.sum(np.abs(state._qm_state[measure_coefficient_mask])**2))
-    results_qc.append(amplitude)
+    #results_qc.append(amplitude)
 
     # Simulation using matrices
     np_zero_state = np.zeros(2**nqbits)
     np_zero_state[0] = 1
 
-    itH = np.matrix(-1j * t * H(nqbits, g))
+    itH = np.matrix(-0.5j * t * H(nqbits, g))
     T = expm(itH)
 
     np_state = T.dot(np_zero_state)
-    amplitude = np.sqrt(np.sum(np.abs(np_state[[False if (i & measure) else True for i in range(2**nqbits)]])**2))
+    amplitude = (np.sum(np.abs(np_state[measure_coefficient_mask])**2))
     results_np.append(amplitude)
 
     print(f"simulating... {int(t/t_stop*100)} %  ", end="\r")
 print()
 print("done.")
 
-errors_trotter = np.arange(0, t_stop, delta_t)**2 / N**2
+results_qc = np.array(results_qc)
+
+errors_trotter = (np.arange(0, t_stop, delta_t) * g)**2 / N_trot**2
+errors_sampling = np.array(errors_sampling)
 
 
-h0 = plt.errorbar(np.arange(0, t_stop, delta_t), results_qc, yerr=errors_trotter, label=f"Quantum computing ({n_sample} samples, {N} trotterization steps)")
+h0 = plt.errorbar(np.arange(0, t_stop, delta_t)
+                , results_qc
+                , yerr=(errors_trotter + errors_sampling)
+                , label=f"Quantum computing ({n_sample} samples, {N_trot} trotterization steps)"
+                , )
 h1, = plt.plot(np.arange(0, t_stop, delta_t), results_np, label="Classical simulation using explicit transfer matrix")
 plt.xlabel("t")
 plt.ylabel(r"$|0\rangle$ probability amplitude for second spin")
 plt.title(f"{nqbits} site spin chain with g={g} coupling to external field")
 plt.legend(handles=[h0, h1])
 
+plt.savefig("time_evo_6spin_g3.png", dpi=400)
 plt.show()
-