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arXiv:2608.22489 · Physics

Time reversal of complex evolution on a quantum computer

Mahdi Kourehpaz, Dima L. Shepelyansky

cond-mat.stat-mechquant-phSubmitted Aug 23, 2026• Announced Aug 25, 2026• 0 citations
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Abstract

The Boltzmann-Loschmidt dispute in 1876-1877 discussed the problem of time reversal of ther- malization from time reversible classical equations of motion. Here, 150 years later, we highlight this problem in the frame of quantum computing. A quantum protocol is proposed that allows to perform a time reversal of complex evolution in the regime of many-body quantum chaos with many qubits. The system represents an evolution of qubits on a square lattice with inter-qubit next-nearest static couplings with a driven pulsed magnetic field. The system entropy grows rapidly to maximal values but returns to initial 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 the butterfly effect of qubit when inversion of only one qubit breaks time reversibility of the whole system. It is argued that this protocol is accessible to nowadays quantum computers and annealers with hundreds of qubits.

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The Boltzmann-Loschmidt dispute in 1876-1877 discussed the problem of time reversal of ther- malization from time reversible classical equations of motion. Here, 150 years later, we highlight this problem in the frame of quantum computing. A quantum protocol is proposed that allows to perform a time reversal of complex evolution in the regime of many-body quantum chaos with many qubits. The system represents an evolution of qubits on a square lattice with inter-qubit next-nearest static couplings with a driven pulsed magnetic field. The system entropy grows rapidly to maximal values but returns to initial 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 the butterfly effect of qubit when inversion of only one qubit breaks time reversibility of the whole system. It is argued that this protocol is accessible to nowadays quantum computers and annealers with hundreds of qubits.
A legend tells that once Loschmidt asked Boltzmann on what happens to his statistical theory if one inverts the velocities of all particles, so that, due to the reversibility of Newton's equations, they return from the equilibrium to a nonequilibrium initial state. Boltzmann only replied ``then go and invert them''. This problem of the relationship between the microscopic and macroscopic descriptions of the physical world and time-reversibility has been hotly debated from the XIXth century up to nowadays. At present, no modern computer is able to perform Boltzmann's demand for a macroscopic number of particles. In addition, dynamical chaos implies exponential growth of any imprecision in the inversion that leads to practical irreversibility. Here we show that a quantum computer composed of a few tens of qubits, and operating even with moderate precision, can perform Boltzmann's demand for a macroscopic number of classical particles. Thus, even in the regime of dynamical chaos, a realistic quantum computer allows to rebuild a specific initial distribution from a macroscopic state given by thermodynamic laws.
Echoes are ubiquitous phenomena in several branches of physics, ranging from acoustics, optics, condensed matter and cold atoms to geophysics. They are at the base of a number of very useful experimental techniques, such as nuclear magnetic resonance, photon echo and time-reversal mirrors. Particularly interesting physical effects are obtained when the echo studies are performed on complex systems, either classically chaotic, disordered or many-body. Consequently, the term Loschmidt echo has been coined to designate and quantify the revival occurring when an imperfect time-reversal procedure is applied to a complex quantum system, or equivalently to characterize the stability of quantum evolution in the presence of perturbations. Here, we present the articles which discuss the work that has shaped the field in the past few years.
The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n . As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.
The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n. As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.
Time reversal in a macroscopic system contradicts daily experience. It is practically impossible to restore a shattered cup to its original state by just time reversing the microscopic dynamics that led to its breakage. Yet, with the precise control capabilities provided by modern quantum technology, the unitary evolution of a quantum system can be reversed in time. Here, we implement a time-reversal protocol in a dipolar interacting, isolated many-body spin system represented by Rydberg states in an atomic gas. By changing the states encoding the spin, we flip the sign of the interaction Hamiltonian, and demonstrate the reversal of the relaxation dynamics of the magnetization by letting a demagnetized many-body state evolve back in time into a magnetized state. We elucidate the role of atomic motion using the concept of a Loschmidt echo. Finally, by combining the approach with Floquet engineering, we demonstrate time reversal for a large family of spin models with different symmetries. Our method of state transfer is applicable across a wide range of quantum simulation platforms and has applications far beyond quantum many-body physics, reaching from quantum-enhanced sensing to quantum information scrambling. Published by the American Physical Society 2024