Thermofield double states

Quantum simulation of materials at finite temperatures requires the generation of thermal quantum states.

Thermal quantum states are given by density matrices in which the eigenstate occupation probability follows the Boltzmann distribution.

Such density matrices cannot be generated from pure unitary quantum evolution. They either require the quantum system to interact with some environment, or to trace over some components of an entangled quantum state.

The latter approach is the most promising for the generation of thermal states of arbitrary quantum Hamiltonians. One approach based on thermofield double states, enables the generation of an N qubit thermal state using a 2N qubit pure state, i.e. two copies of the system of interest.

How to prepare thermofield double states?

One approach recently implemented in ion trap and superconducting qubit experiments employs the quantum approximate optimization algorithm (QAOA). The idea is that the infinite temperature (fully mixed) state is easy to prepare and the ground state of a simple (mixing) Hamiltonian that entangles pairs of qubits, one from the system of interest, and the other from the subsystem to be traced out. One can also identify a Hamiltonian that whose ground state describes the system of interest at zero temperature.

In the limit of a large number of steps, QAOA effectively performs an adiabatic transformation from the infinite temperature double state (easy to prepare) to the zero temperature one (hard to prepare). This suggests it should also be able to well-approximate intermediate temperature states using comparatively shallow circuits by solving a variational optimization problem; the cost function used is some measure of fidelity of the obtained density matrix with respect to a thermal state at the desired temperature.

Other approaches are based on deterministic algorithms, but require deeper circuits not really suitable for near-term quantum processors, for example relying on the phase estimation algorithm to obtain the desired thermal state.

Last year researchers from China proposed an alternate scheme which uses a continuous variable quantum state (qmode) to assist with the preparation of the thermofield double state. In this case, the challenges appear to be preparation of the required qmode resource state, decomposing the target Hamiltonian into a series of unitaries controlled by the qmode, and finite precision in measuring the quadrature of the qmode.

Since future materials science applications of quantum processors will primarily be concerned with simulating finite temperature properties, developing more efficient and robust schemes for the preparation of thermal states will be a topic of growing interest in the coming years.