dbac.hbac

Heat-bath algorithmic cooling (HBAC), the standard entropy-based protocol, implemented minimally so DBAC can be compared with it. Each round resets the last qubit to a thermal state and sorts the populations of the joint state; hbac runs several rounds and returns the first qubit.

See HBAC baseline for how HBAC differs from DBAC and a side-by-side comparison.

hbac(rho[, epsilon_b, step])	Run step rounds of heat-bath algorithmic cooling (reset + compress) on rho, then trace out every qubit except the first (the target qubit, which the sorting step cools).
refresh_reset_qubit(rho, epsilon_b)	Reset the last qubit of a joint density matrix to a thermal state with polarization epsilon_b, by rescaling each 2x2 block of rho that acts on that qubit.
compress_density_matrix(rho)	Sort-and-compress step of HBAC: reorder the populations of rho in descending order, discarding coherences, to approximate the state achievable by the optimal (sorting) unitary.

Running HBAC

hbac(rho, epsilon_b=0.8, step=1)

Run step rounds of heat-bath algorithmic cooling (reset + compress) on rho, then trace out every qubit except the first (the target qubit, which the sorting step cools).

Parameters:

• rho (ndarray) – 2^n x 2^n density matrix, last qubit is the reset qubit.

• epsilon_b (float) – Polarization of the fresh reset qubit each round, i.e. populations (1 +- epsilon_b) / 2.

• step (int) – Number of reset-and-compress rounds.

Returns:

2x2 reduced density matrix of the first qubit.

Return type:

ndarray

Example

>>> import numpy as np
>>> from dbac.hbac import hbac
>>> mixed = np.eye(2) / 2                      # maximally mixed qubit
>>> rho = np.kron(np.kron(mixed, mixed), mixed)
>>> hbac(rho, epsilon_b=0.8, step=6).real.round(3)
array([[0.985, 0.   ],
       [0.   , 0.015]])

The two parts of a round

refresh_reset_qubit(rho, epsilon_b)

Reset the last qubit of a joint density matrix to a thermal state with polarization epsilon_b, by rescaling each 2x2 block of rho that acts on that qubit.

Parameters:

• rho (ndarray) – 2^n x 2^n density matrix of n qubits.

• epsilon_b (float) – Polarization of the reset qubit.

Returns:

New density matrix after resetting the last qubit.

Return type:

ndarray

compress_density_matrix(rho)

Sort-and-compress step of HBAC: reorder the populations of rho in descending order, discarding coherences, to approximate the state achievable by the optimal (sorting) unitary.

Parameters:

rho (ndarray) – Density matrix (Hermitian, positive semidefinite, trace 1).

Returns:

Diagonal density matrix with sorted probabilities.

Return type:

ndarray