Quantum Computing Spacetime

Abstract

A causal set C can describe a discrete spacetime, but this discrete spacetime is not quantum, because C is endowed with Boolean logic, as it does not allow cycles. In a quasi-ordered set Q, cycles are allowed. In this paper, we consider a subset QC of a quasi-ordered set Q, whose elements are all the cycles. In QC, which is endowed with quantum logic, each cycle of maximal outdegree N in a node, is associated with N entangled qubits. Then QC describes a quantum computing spacetime. This structure, which is non-local and non-casual, can be understood as a proto-spacetime. Micro-causality and locality can be restored in the subset U of Q whose elements are unentangled qubits which we interpret as the states of quantum spacetime. The mapping of quantum spacetime into proto-spacetime is given by the action of the XOR gate. Moreover, a mapping is possible from the Boolean causal set into U by the action of the Hadamard gate. In particular, the causal order defined on the elements of U induces the causal evolution of spin networks.