Update docs: change control variable from 'a' to 'u' everywhere

- Updated all manual pages and examples to use u, du, ddu naming
- Changed .trajectory.a → .trajectory.u
- Updated function parameters: a_bound → u_bound, da_bound → du_bound, etc.
- Updated regularizers: R_a → R_u, R_da → R_du, R_dda → R_ddu
- Updated math notation in index.md: a_t → u_t
- Consistent with source code refactoring

Author: aarontrowbridge

docs/literate/examples/multilevel_transmon.jl

@@ -44,7 +44,7 @@ levels = 5
 δ = 0.2
 
 ## add a bound to the controls
-a_bound = 0.2
+u_bound = 0.2
 
 ## create the system
 sys = TransmonSystem(levels=levels, δ=δ)
@@ -75,7 +75,7 @@ get_subspace_identity(op) |> sparse
 # We can then pass this embedded operator to the `UnitarySmoothPulseProblem` template to create the problem
 
 ## create the problem
-prob = UnitarySmoothPulseProblem(sys, op, T, Δt; a_bound=a_bound)
+prob = UnitarySmoothPulseProblem(sys, op, T, Δt; u_bound=u_bound)
 
 ## solve the problem
 solve!(prob; max_iter=50)
@@ -98,8 +98,8 @@ plot_unitary_populations(prob.trajectory; fig_size=(900, 700))
 ## create the a leakage suppression problem, initializing with the previous solution
 
 prob_leakage = UnitarySmoothPulseProblem(sys, op, T, Δt;
-    a_bound=a_bound,
-    a_guess=prob.trajectory.a[:, :],
+    u_bound=u_bound,
+    u_guess=prob.trajectory.u[:, :],
     piccolo_options=PiccoloOptions(
         leakage_constraint=true,
         leakage_constraint_value=1e-2,

docs/literate/examples/two_qubit_gates.jl

@@ -13,15 +13,15 @@
 # Specifically, for a simple two-qubit system in a rotating frame, we have
 
 # ```math
-# H = J_{12} \sigma_1^x \sigma_2^x + \sum_{i \in {1,2}} a_i^R(t) {\sigma^x_i \over 2} + a_i^I(t) {\sigma^y_i \over 2}.
+# H = J_{12} \sigma_1^x \sigma_2^x + \sum_{i \in {1,2}} u_i^R(t) {\sigma^x_i \over 2} + u_i^I(t) {\sigma^y_i \over 2}.
 # ```
 
 # where
 
 # ```math
 # \begin{align*}
 # J_{12} &= 0.001 \text{ GHz}, \\
-# |a_i^R(t)| &\leq 0.1 \text{ GHz} \\
+# |u_i^R(t)| &\leq 0.1 \text{ GHz} \\
 # \end{align*}
 # ```
 
@@ -53,7 +53,7 @@ Id = GATES[:I]
 
 ## Define the parameters of the Hamiltonian
 J_12 = 0.001 # GHz
-a_bound = 0.100 # GHz
+u_bound = 0.100 # GHz
 
 ## Define the drift (coupling) Hamiltonian
 H_drift = J_12 * (σx ⊗ σx)
@@ -62,9 +62,9 @@ H_drift = J_12 * (σx ⊗ σx)
 H_drives = [σx_1 / 2, σy_1 / 2, σx_2 / 2, σy_2 / 2]
 
 ## Define control (and higher derivative) bounds
-a_bound = 0.1
-da_bound = 0.0005
-dda_bound = 0.0025
+u_bound = 0.1
+du_bound = 0.0005
+ddu_bound = 0.0025
 
 ## Scale the Hamiltonians by 2π
 H_drift *= 2π
@@ -95,11 +95,11 @@ prob = UnitarySmoothPulseProblem(
     U_goal,
     T,
     Δt;
-    a_bound=a_bound,
-    da_bound=da_bound,
-    dda_bound=dda_bound,
-    R_da=0.01,
-    R_dda=0.01,
+    u_bound=u_bound,
+    du_bound=du_bound,
+    ddu_bound=ddu_bound,
+    R_du=0.01,
+    R_ddu=0.01,
     Δt_max=Δt_max,
     piccolo_options=PiccoloOptions(bound_state=true),
 )
@@ -166,11 +166,11 @@ prob = UnitarySmoothPulseProblem(
     U_goal,
     T,
     Δt;
-    a_bound=a_bound,
-    da_bound=da_bound,
-    dda_bound=dda_bound,
-    R_da=0.01,
-    R_dda=0.01,
+    u_bound=u_bound,
+    du_bound=du_bound,
+    ddu_bound=ddu_bound,
+    R_du=0.01,
+    R_ddu=0.01,
     Δt_max=Δt_max,
     piccolo_options=PiccoloOptions(bound_state=true),
 )

docs/literate/man/ket_problem_templates.jl

@@ -39,7 +39,7 @@ solve!(state_prob, max_iter=100, verbose=true, print_level=1);
 println("After: ", rollout_fidelity(state_prob.trajectory, system))
 
 # _extract the control pulses_
-state_prob.trajectory.a |> size
+state_prob.trajectory.u |> size
 
 # -----
 

docs/literate/man/unitary_problem_templates.jl

@@ -41,7 +41,7 @@ The `NamedTrajectory` object stores the control pulse, state variables, and the
 =#
 
 # _extract the control pulses_
-prob.trajectory.a |> size
+prob.trajectory.u |> size
 
 # -----
 

docs/src/index.md

@@ -36,26 +36,26 @@ Unitary Problem Templates:
 
 *Problem Templates* are reusable design patterns for setting up and solving common quantum control problems. 
 
-For example, a *UnitarySmoothPulseProblem* is tasked with generating a *pulse* sequence $a_{1:T-1}$ in orderd to minimize infidelity, subject to constraints from the Schroedinger equation,
+For example, a *UnitarySmoothPulseProblem* is tasked with generating a *pulse* sequence $u_{1:T-1}$ in order to minimize infidelity, subject to constraints from the Schroedinger equation,
 ```math
     \begin{aligned}
         \arg \min_{\mathbf{Z}}\quad & |1 - \mathcal{F}(U_T, U_\text{goal})|  \\
         \nonumber \text{s.t.}
-        \qquad & U_{t+1} = \exp\{- i H(a_t) \Delta t_t \} U_t, \quad \forall\, t \\
+        \qquad & U_{t+1} = \exp\{- i H(u_t) \Delta t_t \} U_t, \quad \forall\, t \\
     \end{aligned}
 ```
 while a *UnitaryMinimumTimeProblem* minimizes time and constrains fidelity,
 ```math
     \begin{aligned}
         \arg \min_{\mathbf{Z}}\quad & \sum_{t=1}^T \Delta t_t \\
-        \qquad & U_{t+1} = \exp\{- i H(a_t) \Delta t_t \} U_t, \quad \forall\, t \\
+        \qquad & U_{t+1} = \exp\{- i H(u_t) \Delta t_t \} U_t, \quad \forall\, t \\
         \nonumber & \mathcal{F}(U_T, U_\text{goal}) \ge 0.9999
     \end{aligned}
 ```
 
 -----
 
-In each case, the dynamics between *knot points* $(U_t, a_t)$ and $(U_{t+1}, a_{t+1})$ are enforced as constraints on the states, which are free variables in the solver; this optimization framework is called *direct trajectory optimization*. 
+In each case, the dynamics between *knot points* $(U_t, u_t)$ and $(U_{t+1}, u_{t+1})$ are enforced as constraints on the states, which are free variables in the solver; this optimization framework is called *direct trajectory optimization*. 
 
 -----