Time Dynamics in Chaotic Many-body Systems: Can Chaos Destroy a Quantum Computer?

Abstract

Highly excited many-particle states in quantum systems (nuclei, atoms, quantum dots, spin systems, quantum computers) can be 'chaotic' superpositions of mean-field basis states (Slater determinants products of spin or qubit states). This is a result of the very high energy level density of many-body states which can be easily mixed by a residual interaction between particles. We consider the time dynamics of wave functions and increase of entropy in such chaotic systems. As an example, we present the time evolution in a closed quantum computer. A time scale for the entropy S(t) increase is tc ∼ τ0/ (n log2n), where τ0 is the qubit 'lifetime', n is the number of qubits, S(0) = 0 and S(tc) = 1. At t ≪ tc the entropy is small: S ∼ nt2J2log2(1/t2J2), where J is the inter-qubit interaction strength. At t > tc the number of 'wrong' states increases exponentially as 2S(t). Therefore, tc may be interpreted as a maximal time for operation of a quantum computer. At t ≫ tc the system entropy approaches that for chaotic eigenstates.