Time Reversal of Complex Evolution on a Quantum Computer

The Boltzmann–Loschmidt dispute in 1876–1877 concerned the problem of the time reversal of thermalization in time-reversible classical equations of motion. Here, 150 years later, we revisit this problem in the context of quantum computing. We propose a quantum protocol that allows us to perform a time reversal of complex evolution in the regime of many-body quantum chaos with many qubits. The system consists of qubits evolving on a square lattice with inter-qubit next-nearest-neighbor static couplings and a driven pulsed magnetic field. The system entropy grows quite rapidly to maximal values but returns to its initial comparably small values after time reversal. This time reversal is shown to be stable with respect to quantum gate imperfections. However, similar to the Lorenz butterfly effect, there is a butterfly effect of a qubit in which the inversion of only one qubit breaks the time reversibility of the whole system. We argue that this protocol is accessible to current quantum computers and annealers with hundreds of qubits.