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 t _c ~&tgr; ₀ /(n log ₂ n), where &tgr; ₀ is the qubit ‘lifetime’, n is the number of qubits, S(0) = 0 and S(t _c )=1. At t _ t _c the entropy is small: S ~nt ² J ² log ₂ (1/t ² J² ), where J is the inter-qubit interaction strength. At t > t _c the number of ‘wrong’ states increases exponentially as 2 ^S(t) . Therefore, t _c may be interpreted as a maximal time for operation of a quantum computer. At t >>t _c the system entropy approaches that for chaotic eigenstates.