Template Struct UnitaryEvolver¶

Defined in File UnitaryEvolver.hpp

Struct Documentation¶

template<int n_ctrl = Dynamic, int dim = Dynamic, typename Matrix = DMatrix<dim, dim>> struct UnitaryEvolver¶

A struct to store the diagonalised drift and control Hamiltonians. On initialisation the Hamiltonians are diagonalised and the eigenvectors and values stored. This initial diagonalisation may be slow and takes $O(\textrm{dim}^3)$ time for a $\textrm{dim}\times \textrm{dim}$ Hamiltonian. However, it allows each step of the Suzuki-Trotter expansion to be implimented in $O(\textrm{dim}^2)$ time with matrix multiplication and only scalar exponentiation opposed to matrix exponentiation which takes $O(\textrm{dim}^3)$ time.

Template Parameters:

n_ctrl – The number of control Hamiltonians.
dim – The dimension of the vector space the Hamiltonians act upon.
Matrix – The type of matrix to use. Matrix must take the value DMatrix<dim, dim> or SMatrix for dense or sparse matrices, respectively.

Public Functions

template<typename T = Matrix> inline UnitaryEvolver(std::enable_if_t<std::is_same<T, DMatrix<dim, dim>>::value, DMatrix<dim, dim>> drift_hamiltonian, DMatrix<dim_x_n_ctrl, dim> control_hamiltonians)¶

Initialises a new unitary evolver with the Hamiltonian

\[ H(t)=H_0+\sum_{j=1}^{\textrm{length}}a_j(t)H_j, \]

where $H_0$ is the drift Hamiltonian and $H_j$ are the control Hamiltonians modulated by control amplitudes $a_j(t)$ which need not be specified during initialisation.

Parameters:

drift_hamiltonian – The drift Hamiltonian.
control_hamiltonians – The control Hamiltonians.

template<typename T = Matrix> inline UnitaryEvolver(std::enable_if_t<std::is_same<T, SMatrix>::value, DMatrix<dim, dim>> drift_hamiltonian, DMatrix<dim_x_n_ctrl, dim> control_hamiltonians)¶

Initialises a new unitary evolver with a sparse Hamiltonian.

\[ H(t)=H_0+\sum_{j=1}^{\textrm{length}}a_j(t)H_j, \]

where $H_0$ is the drift Hamiltonian and $H_j$ are the control Hamiltonians modulated by control amplitudes $a_j(t)$ which need not be specified during initialisation.

Parameters:

drift_hamiltonian – The drift Hamiltonian.
control_hamiltonians – The control Hamiltonians.

inline UnitaryEvolver(size_t l, Eigen::Array<complex<double>, dim, 1> d0, vector<Eigen::Array<complex<double>, dim, 1>> ds, Matrix u0, Matrix u0_inverse, vector<Matrix> us, vector<Matrix> us_individual, vector<Matrix> us_inverse_individual, vector<Matrix> hs, Matrix u0_inverse_u_last)¶

Initialises a new unitary evolver using the struct attributes.

Parameters:

l – The number of control Hamiltonians. Initialises UnitaryEvolver::length.
d0 – The eigenvalues, $\operatorname{diag}(D_0)$, of the drift Hamiltonian: $H_0=U_0D_0U_0^\dagger$. Initialises UnitaryEvolver::d0.
ds – The eigenvalues, $\left(\operatorname{diag}(D_i)\right)_{i=1}^{\textrm{length}}$, of the control Hamiltonians: $H_i=U_iD_iU_i^\dagger$ for all $i\in\left[\textrm{length}\right]$. Initialises UnitaryEvolver::ds.
u0 – The unitary transformation, $U_0$, that diagonalises the drift Hamiltonian: $H_0=U_0D_0U_0^\dagger$. Initialises UnitaryEvolver::u0.
u0_inverse – The inverse of the unitary transformation, $U_0^\dagger$, that diagonalises the drift Hamiltonian: $H_0=U_0D_0U_0^\dagger$. Initialises UnitaryEvolver::u0_inverse.
us – The unitary transformations, $(U_i^\dagger U_{i-1})_{i=1}^{\textrm{length}}$, from the eigen basis of $H_{i-1}$ to the eigen basis of $H_i$. Initialises UnitaryEvolver::us.
us_individual – The unitary transformations, $\left(U_i\right)_{i=1}^{\textrm{length}}$, that diagonalise the control Hamiltonians: $H_i=U_iD_iU_i^\dagger$ for all $i\in\left[\textrm{length}\right]$. Initialises UnitaryEvolver::us_individual.
us_inverse_individual – The inverse of the unitary transformations, $(U_i^\dagger)_{i=1}^{\textrm{length}}$, that diagonalise the control Hamiltonians: $H_i=U_iD_iU_i^\dagger$ for all $i\in\left[\textrm{length}\right]$. Initialises UnitaryEvolver::us_inverse_individual.
control_hamiltonians – The control Hamiltonians: $H_i$ for all $i\in\left[\textrm{length}\right]$. Initialises UnitaryEvolver::hs.
u0_inverse_u_last – The unitary transformation, $U_0^\dagger U_{\textrm{length}}$, from the eigen basis of $H_{\textrm{length}}$ to the eigen basis of $H_0$. Initialises UnitaryEvolver::u0_inverse_u_last.

inline DMatrix<dim, 1> propagate(DMatrix<Dynamic, n_ctrl> ctrl_amp, DMatrix<dim, 1> state, double dt)¶

Propagates the state vector using the first-order Suzuki-Trotter expansion. More precisely, a state vector, $\psi(0)$, is evolved under the differential equation

\[ \dot\psi=-iH\psi \]

using the first-order Suzuki-Trotter expansion:

\[\begin{split} \begin{align} \psi(N\Delta t)&=\prod_{i=1}^N\prod_{j=0}^{\textrm{length}} e^{-ia_{ij}H_j\Delta t}\psi(0)+\mathcal E\\ &=\prod_{i=1}^N\prod_{j=0}^{\textrm{length}} U_je^{-ia_{ij}D_j\Delta t}U_j^\dagger\psi(0)+\mathcal E. \end{align} \end{split}\]

where $a_{nj}\coloneqq a(n\Delta t)$, we set $a_{n0}=1$ for notational ease, and the additive error $\mathcal E$ is

\[\begin{split} \begin{align} \mathcal E&=\mathcal O\left( \Delta t^2\left[\sum_{i=1}^N\sum_{j=1}^{\textrm{length}}\dot a_{ij} \norm{H_j} +\sum_{i=1}^N\sum_{j,k=0}^{\textrm{length}}a_{ij}a_{ik} \norm{[H_j,H_k]}\right] \right)\\ &=\mathcal O\left( N\Delta t^2\textrm{length}\left[\omega E+\alpha^2+E^2\right] \right) \end{align} \end{split}\]

where $\dot a_{nj}\coloneqq\dot a_j(n\Delta t)$ and

\[\begin{split} \begin{align} \omega&\coloneqq\max_{\substack{i\in\left[1,N\right]\\ j\in\left[1,\textrm{length}\right]}}\left|\dot a_{ij}\right|,\\ \alpha&\coloneqq\max_{\substack{i\in\left[1,N\right]\\ j\in\left[0,\textrm{length}\right]}}\left|a_{ij}\right|,\\ E&\coloneqq\max_{j\in\left[0,\textrm{length}\right]}\norm{H_j}. \end{align} \end{split}\]

Note the error is quadratic in $\Delta t$ but linear in $N$. We can also view this as being linear in $\Delta t$ and linear in total evolution time $N\Delta t$. Additionally, by Nyquist’s theorem this asymptotic error scaling will not be achieved until the time step $\Delta t$ is smaller than $\frac{1}{2\Omega}$ where $\Omega$ is the largest energy or frequency in the system.

Parameters:

ctrl_amp – $\left(a_{ij}\right)$ The control amplitudes at each time step expressed as an $N\times\textrm{length}$ matrix where the element $a_{ij}$ corresponds to the control amplitude of the $j$th control Hamiltonian at the $i$th time step.
state – $\left[\psi(0)\right]$ The state vector to propagate.
dt – ( $\Delta t$) The time step to propagate by.

Returns:

The propagated state vector: $\psi(N\Delta t)$.

template<int l = Dynamic> inline DMatrix<dim, l> propagate_collection(DMatrix<Dynamic, n_ctrl> ctrl_amp, DMatrix<dim, l> states, double dt)¶

Propagates a collection of state vectors using the first-order Suzuki-Trotter expansion. More precisely, a collection of state vectors, $\left(\psi_k(0)\right)_{k}$, are evolved under the differential equation

\[ \dot\psi_k=-iH\psi_k \]

using the first-order Suzuki-Trotter expansion:

\[\begin{split} \begin{align} \psi_k(N\Delta t)&=\prod_{i=1}^N\prod_{j=0}^{\textrm{length}} e^{-ia_{ij}H_j\Delta t}\psi_k(0)+\mathcal E\\ &=\prod_{i=1}^N\prod_{j=0}^{\textrm{length}} U_je^{-ia_{ij}D_j\Delta t}U_j^\dagger\psi_k(0)+\mathcal E. \end{align} \end{split}\]

where $a_{nj}\coloneqq a(n\Delta t)$, we set $a_{n0}=1$ for notational ease, and the addative error $\mathcal E$ is

where $\dot a_{nj}\coloneqq\dot a_j(n\Delta t)$ and

Parameters:

ctrl_amp – $\left(a_{ij}\right)$ The control amplitudes at each time step expressed as an $N\times\textrm{length}$ matrix where the element $a_{ij}$ corresponds to the control amplitude of the $j$th control Hamiltonian at the $i$th time step.
states – $\left[\left(\psi(0)\right)_{k}\right]$ A collection of state vectors to propagate expressed as a matrix with each column corresponding to a state vector.
dt – ( $\Delta t$) The time step to propagate by.

Template Parameters:

l – The number of state vectors to propagate.

Returns:

The propagated state vectors: $\left(\psi_k(N\Delta t)\right)_k$.

inline DMatrix<dim, Dynamic> propagate_all(DMatrix<Dynamic, n_ctrl> ctrl_amp, DMatrix<dim, 1> state, double dt)¶

Propagates the state vector using the first-order Suzuki-Trotter expansion and returns the resulting state vector at every time step. More precisely, a state vector, $\psi(0)$, is evolved under the differential equation

\[ \dot\psi=-iH\psi \]

using the first-order Suzuki-Trotter expansion:

\[\begin{split} \begin{align} \psi(n\Delta t)&=\prod_{i=1}^n\prod_{j=0}^{\textrm{length}} e^{-ia_{ij}H_j\Delta t}\psi(0)+\mathcal E \quad\forall n\in\left[0, N\right]\\ &=\prod_{i=1}^n\prod_{j=0}^{\textrm{length}} U_je^{-ia_{ij}D_j\Delta t}U_j^\dagger\psi(0)+\mathcal E \quad\forall n\in\left[0, N\right]. \end{align} \end{split}\]

where $a_{nj}\coloneqq a(n\Delta t)$, we set $a_{n0}=1$ for notational ease, and the additive error $\mathcal E$ is

where $\dot a_{nj}\coloneqq\dot a_j(n\Delta t)$ and

Parameters:

ctrl_amp – $\left(a_{ij}\right)$ The control amplitudes at each time step expressed as an $N\times\textrm{length}$ matrix where the element $a_{ij}$ corresponds to the control amplitude of the $j$th control Hamiltonian at the $i$th time step.
state – $\left[\psi(0)\right]$ The state vector to propagate.
dt – ( $\Delta t$) The time step to propagate by.

Returns:

The propagated state vector at each time step: $\left(\psi(n\Delta t)\right)_{n=0}^N$.

inline complex<double> evolved_expectation_value(DMatrix<Dynamic, n_ctrl> ctrl_amp, DMatrix<dim, 1> state, double dt, DMatrix<dim, dim> observable)¶

Calculates the expectation value with respect to an observable of an evolved state vector evolved under a control Hamiltonian modulated by the control amplitudes. The integration is performed using propagate().

Parameters:

ctrl_amp – $\left(a_{ij}\right)$ The control amplitudes at each time step expressed as an $N\times\textrm{length}$ matrix where the element $a_{ij}$ corresponds to the control amplitude of the $j$th control Hamiltonian at the $i$th time step.
state – $\left[\psi(0)\right]$ The state vector to propagate.
dt – ( $\Delta t$) The time step to propagate by.
observable – $(\hat O)$ The observable to calculate the expectation value of.

Returns:

The expectation value of the observable: $\langle\hat O\rangle \equiv\psi^\dagger(N\Delta t)\hat O\psi(N\Delta t)$.

inline DMatrix<Dynamic, 1> evolved_expectation_value_all(DMatrix<Dynamic, n_ctrl> ctrl_amp, DMatrix<dim, 1> state, double dt, DMatrix<dim, dim> observable)¶

Calculates the expectation values with respect to an observable of a time series of state vectors evolved under a control Hamiltonian modulated by the control amplitudes. The integration is performed using propagate_all().

Parameters:

ctrl_amp – $\left(a_{ij}\right)$ The control amplitudes at each time step expressed as an $N\times\textrm{length}$ matrix where the element $a_{ij}$ corresponds to the control amplitude of the $j$th control Hamiltonian at the $i$th time step.
state – $\left[\psi(0)\right]$ The state vector to propagate.
dt – ( $\Delta t$) The time step to propagate by.
observable – $(\hat O)$ The observable to calculate the expectation value of.

Returns:

The expectation value of the observable: $\left(\psi^\dagger(n\Delta t)\hat O\psi(N\Delta t)\right)_{n=0}^N$.

inline std::tuple<complex<double>, Eigen::Matrix<double, Dynamic, n_ctrl>> switching_function(DMatrix<Dynamic, n_ctrl> ctrl_amp, DMatrix<dim, 1> state, double dt, DMatrix<dim, dim> cost)¶

Calculates the switching function for a Mayer problem with an expectation value as the cost function. More precisely if the cost function is

\[ J\left[\vec a(t)\right]\coloneqq\langle\hat O\rangle \equiv\psi^\dagger[\vec a(t);T] \hat O\psi[\vec a(t);T], \]

where $T=N\Delta t$, then the switching function is

\[ \phi_j(t)\coloneqq\frac{\delta J}{\delta a_j(t)} =2\operatorname{Im}\left(\psi^\dagger[\vec a(t);T] \hat OU(t\to T)H_j\psi[\vec a(t);t]\right). \]

using the first-order Suzuki-Trotter expansion we can express the switching function as

\[\begin{split} \begin{align} &\phi_j(n\Delta t)=\frac{1}{\Delta t}\pdv{J}{a_{nj}}\\ &=\!2\operatorname{Im}\!\left(\psi^\dagger(T) \hat O\!\!\left[\prod_{i>n}^N\prod_{k=1}^{\textrm{length}} e^{-ia_{ik}H_k\Delta t}\right]\!\!\! \left[\prod_{k=j}^{\textrm{length}} e^{-ia_{nk}H_k\Delta t}\right]\!H_j\!\! \left[\prod_{k=0}^{j-1} e^{-ia_{nk}H_k\Delta t}\right] \!\psi(\left[n-1\right]\Delta t)\right), \end{align} \end{split}\]

where for numerical efficiency we replace $e^{-ia_{ik}H_k\Delta t}$ with $U_ke^{-ia_{ik}D_k\Delta t}U_k^\dagger$ as in propagate().

Parameters:

ctrl_amp – $\left(a_{ij}\right)$ The control amplitudes at each time step expressed as an $N\times\textrm{length}$ matrix where the element $a_{ij}$ corresponds to the control amplitude of the $j$th control Hamiltonian at the $i$th time step.
state – $\left[\psi(0)\right]$ The initial state vector.
dt – ( $\Delta t$) The time step.
cost – $(\hat O)$ The observable to calculate the expectation value of.

Returns:

The expectation value, $\psi^\dagger(T)\hat O\psi(T)$, and the switching function, $\phi_j(n\Delta t)$ for all $j\in\left[1,\textrm{length}\right]$ and $n\in\left[1,N\right]$.

Public Members

size_t length¶: The number of control Hamiltonians

Eigen::Array<complex<double>, dim, 1> d0¶: The eigenvalues, $\operatorname{diag}(D_0)$, of the drift Hamiltonian: $H_0=U_0D_0U_0^\dagger$.

vector<Eigen::Array<complex<double>, dim, 1>> ds¶: The eigenvalues, $\left(\operatorname{diag}(D_i)\right)_{i=1}^{\textrm{length}}$, of the control Hamiltonians: $H_i=U_iD_iU_i^\dagger$ for all $i\in\left[\textrm{length}\right]$.

Matrix u0¶: The unitary transformation, $U_0$, that diagonalises the drift Hamiltonian: $H_0=U_0D_0U_0^\dagger$.

Matrix u0_inverse¶: The inverse of the unitary transformation, $U_0^\dagger$, that diagonalises the drift Hamiltonian: $H_0=U_0D_0U_0^\dagger$.

vector<Matrix> us¶: The unitary transformations, $(U_i^\dagger U_{i-1})_{i=1}^{\textrm{length}}$, from the eigen basis of $H_{i-1}$ to the eigen basis of $H_i$.

vector<Matrix> us_individual¶: The unitary transformations, $\left(U_i\right)_{i=1}^{\textrm{length}}$, that diagonalise the control Hamiltonians: $H_i=U_iD_iU_i^\dagger$ for all $i\in\left[\textrm{length}\right]$.

vector<Matrix> us_inverse_individual¶: The inverse of the unitary transformations, $(U_i^\dagger)_{i=1}^{\textrm{length}}$, that diagonalise the control Hamiltonians: $H_i=U_iD_iU_i^\dagger$ for all $i\in\left[\textrm{length}\right]$.

vector<Matrix> hs¶: The control Hamiltonians: $H_i$ for all $i\in\left[\textrm{length}\right]$.

Matrix u0_inverse_u_last¶: The unitary transformation, $U_0^\dagger U_{\textrm{length}}$, from the eigen basis of $H_{\textrm{length}}$ to the eigen basis of $H_0$.

Public Static Attributes

static const int dim_x_n_ctrl = (dim == Dynamic || n_ctrl == Dynamic) ? Dynamic : dim * n_ctrl ¶: The dimension of rows in each control Hamiltonian multiplied by the number of control Hamiltonians. This is the number of rows in for the control_hamiltonians argument for UnitaryEvolver::UnitaryEvolver().