QuantumGateDesign.jl

Table of Contents

• Control Functions
• • Rabi Oscillator
• Forward Simulation
• Gradient Evaluation
• QuantumGateDesign.jl
• Table of Contents
• Problem Description
• Workflow
• Real-Valued Representation
• State Storage
• Optimization
• Problem Setup
• Visualization

Problem Description

QuantumGateDesign.jl solves optimization problems of the form

\[\min_{\boldsymbol{\theta}} \mathcal{J}_1(U(t_f)) + \int_0^{t_f} \mathcal{J}_2 U(t) dt,\]

where the dynamics of $U$ are governed by Schrodinger's equation:

\[\frac{dU}{dt} = -iH(t;\boldsymbol{\theta}) U, \quad 0 \leq t \leq t_f,\quad U(0) = U_0 \in \mathbb{C}^{N_s},\]

and the Hamiltonian can be decomposed into a constant 'system'/'drift' component, and several 'control' hamiltonians which are modulated by scalar functions:

\[H(t;\boldsymbol\theta) = H_d + \sum_{j=1}^{N_c} c_j(t; \boldsymbol\theta) \cdot H_j.\]

Workflow

The basic workflow is:

1. Set up the physics of the problem.
2. Set up the control functions.
3. Choose an initial guess for the control vector.
4. Choose a target gate.
5. Optimize the control vector using ipopt.

The user can also directly simulate the time evolution of the state vector and compute gradients of the gate-infidelity, outside of the optimization loop.

Real-Valued Representation

We perform all our computations using real-valued arithmetic. When the real and imaginary part of a state vector are not handled in separate data structures, we 'stack' them into one vector. For example, the complex-valued state $\boldsymbol\psi = [1+2i, 3+4i]^T$ becomes the real-valued state $[1, 2, 3, 4]$.

In several parts of the source code, we use u to indicate the real part of a state, and v to indicate the imaginary part.

For control pulses, the real part is indicated by p, and the imaginary part by q.

Eventually we would like to replace the u/v and p/q notation with something more descriptive, like real/imag.

State Storage

Because we solve the initial value problem stated above using Hermite interpolation, after solving the problem we have computed the state vector and its derivatives at many points in time. For a problem with multiple initial conditions, the history is stored as a four-dimensional array. The four indices correspond respectively to

1. the component of the state vector or derivative,
2. the derivative order (with a $1/j!$ factor),
3. the timestep index,
4. and the initial condition.

Therefore, given a 4D array history, history[i,j,k,l] gives the $i$-th component of the real part (for $i \leq N_s$) of the $j$-th derivative (divided by $j!$) of the state vector corresponding to the $l$-th initial condition after $k-1$ timesteps.