unipolator

Unitary Interpolation, allows for the fast repeated exponentiation of parametric Hamiltonians [1]. Construct propagators (and their derivatives) of time dependent quantum systems of the form $H(t) = H_0 + \sum_{i=1} c_i(t) H_i$ (for example in optimal control problems) or quantum circuits with parametric gates. We utilize a grid based interpolation scheme to calculate propagators from cached matrix decompositions. The computation of a propagator for a time step is as fast as a single Trotter step, but with the ability to achieve machine precision.

Update: Now supports numpy $\geq$ 2.0.

To reproduce the plots from the publication (see the .ipynb scripts in the test/ folder) use python $\leq$ 3.10, as tikzplotlib currently does not support python 3.11.

Install and Import

Install via

pip install unipolator

and then simply import into a project via

from unipolator import *

Initialize the Unitary Interpolation Object:

Describe a system with a double complex array of Hamiltonians H_s, with H_s[i,...] = H_i ∆t, so that for $n$ control Hamiltonians H_s is a $(n+1) \times d \times d$ array for a $d$ dimensional Hilbert space. Define the bounds of the interpolation (hyper-) volume via $n$ dimensional double arrays c_mins and c_maxs and define the number of bins for every dimension via the $n$ dimensional int64 array c_bins, to initialize the unitary interpolation cache

ui = UI(H_s, c_mins, c_maxs, c_bins)  

Equivalently if we wish to propagate only wavevectors $\ket{\psi(t)} = U(t) \ket{\psi(0)}$ we initialize the unitary interpolation cache via

ui_vector = UI_vector(H_s, c_mins, c_maxs, c_bins, m)  

where m is the number of wavevectors that are calculated in parallel. The package contains further methods listed at the bottom of this document.

Automatic Binning

The method UI_bins automatically calculates the optimal binning for a target infidelity (default I_tar=1e-10). Use via

bins = UI_bins(H_s, c_mins, c_maxs, I_tar=1e-10)

By calling

ui = UI_auto(H_s, c_mins, c_maxs, I_tar=1e-10)

or

ui_vector = UI_vector_auto(H_s, c_mins, c_maxs, I_tar=1e-10, m)

this method is called automatically during the initialization of the unitary interpolation cache.

Calculate:

We can now use ui to calculate matrix exponentials, their derivatives, pulse sequences, and their gradients via the following methods:

1. expmH calculates the unitary $U = \exp(-i H(c) \Delta t)$ for a given set of coefficients c (double array of length $n$), pass U_ui to the method to store the result (this avoids allocating new memory for every call, and allows reusing the same arrays)

   ui.expmH( c, U_ui)  
   

   Similarly we pass two $d \times m$ arrays V_in and V_out, with the $m$ input wavevectors and for the propagated wavevectors, via

   ui_vector.expmH( c, V_in, V_out)
   

2. dexpmH also outputs the derivatives of the unitaris (wavevectors) with respect to the control paracters c. This requires the additional passing of a $n \times d \times d$ array dU to store the derivatives in

   ui.dexpmH( c, U, dU)
   

   During the initalization we can also select which dervatives we wish to compute, via the additional argument which_diffs which requires an int64 array with the indexes of the control parameters for which we wish to compute the derivatives.

   In the wavevector case, we replace the output variable dU with an additional $n \times m \times n$ arrays dV_out, so that

   ui_vector.dexpmH( c, V_in, V_out, dV_out)
   

3. expmH_pulse calculates the propagator of a piecewise constant pulse for a given set of coefficients c_s, now a 2d array of shape $N \times n$, where $N$ is the number of timesteps

   ui.expmH_pulse(cs, U)
   

4. grape calculates the infidelity of such a pulse with respect to a arget unitary U_target (using the indexes target_indexes of U_target), as well as the gradients of the control parameters along the pulse by using the GRAPE trick. We pass an array dI_dj of shape $n \times N$ to store the gradients at every time step for every control parameter

   ui.grape(cs, U_target, target_indexes, U, dU, dI_dj)
   

Other Methods:

The package also contains classes for eigenvalue based exponentiations, Krylov based exponentiations and (symmetric-) Trotterisations, namely

• Hamiltonian_System(H_s),

• Hamiltonian_System_vector(H_s, m), where m is the number of wavevectors that are calculated in parallel,

• Trotter_System=(H_s, m_times), where m_times is the number of doublings $2^\mathrm{m}$ Trotter steps,

• Symmetric_Trotter_System=(H_s, m_times),

• Trotter_System_vector(H_s, n_times) where n_times is the number of performed Trotter steps ,

• Symmetric_Trotter_System_vector(H_s, n_times).

• In the test Subdirectory we provide additional functions to generate Random Hamiltonians, construct infidelities and more.

Author:

Michael Schilling

References:

[1] Schilling, Michael, et al. Exponentiation of Parametric Hamiltonians via Unitary Interpolation arxiv.org/abs/2402.01498