Spacetime at the Planck Scale: The Quantum Computer View

Paola Zizzi
Dipartimento di Astronomia dell' Università di Padova
Vicolo dell' Osservatorio, 2
35122 Padova, Italy
zizzi@pd.astro.it

Abstract What is "spacetime" at the Planck scale? Once we understand that, we will be able to formulate the theory of Quantum Gravity, the theory which should reconcile General Relativity and Quantum Mechanics. In fact, it is widely believed that at the Planck scale, the quantum aspects of gravity become relevant. Moreover, it is generally assumed that at the Planck scale, spacetime is not any longer a smooth manifold, but has a discrete structure. There are two main approaches to quantum gravity that assume quantum spacetime to be discrete: Loop Quantum Gravity (and spin foams), and String (and M) Theory. Other interesting approaches are non-commutative geometry, Causal Set Theory and kinds of discrete models of spacetime at the Planck scale, like lattice versions of loop quantum gravity, and Cellular Networks. In our particular approach to quantum gravity, we assume discreteness of spacetime at the Planck scale, and we also include the issue of information, (more precisely quantum information ). In fact, as it was suggested by Wheeler (the "It from bit" proposal), information theory must play a relevant role in understanding the foundations of Quantum Mechanics. Wheeler's view is shared, in particular, by Zeilinger (who associates bits with elementary systems, i.e. two-level systems, and claims that the world appears quantised because information is quantised) and by Deutsch (who says that "quantum computing is quantum mechanics"). As it was first realized by Feynmann, a quantum computer can be exponentially more powerful than a classical one in simulating a quantum system. This line of thought is what we call here the "Quantum Computer View" (QCV). We believe that the QCV is universal, and thus can be extended to the "description" of quantum spacetime itself.
Approaches similars to ours, still encompassing the QCV, are those of Lloyd, Zimmermann , Hitchcock and Jaroszkiewicz. Our approach, (as well as the one of Zimmermann) is closely related to Loop Quantum gravity and spin networks. Spin networks are relevant for quantum geometry. They were invented by Penrose in order to approach a drastic change in the concept of spacetime, going from that of a smooth manifold to that of a discrete, purely combinatorial structure. Then, spin networks were re-discovered by Rovelli and Smolin in the context of loop quantum gravity. Basically, spin networks are graphs embedded in 3-space, with edges labeled by spins and vertices labeled by intertwining operators. In loop quantum gravity, spin networks are eigenstates of the area and volume-operators. We interprete spin networks as qubits when their edges are labeled by the spin-1/2 representation of SU(2). In this context, we use the quantum version of the Holographic Principle. In our model, quantum spacetime is discrete, quantised in Planck units, and each pixel of Planckian area, encodes a qubit (qubitisation of quantum spacetime). This is a quantum memory register. To process the quantum information stored in the memory, it is necessary to dispose of a network of quantum logic gates (which are unitary operators). The network must be part of quantum spacetime itself, as it describes its dynamical evolution. The quantum memory plus the quantum network form a quantum computer (quantum computer view of quantum spacetime). In the QCV, some new features of quantum spacetime emerge:
i) The dynamical evolution of quantum spacetime is a reversible process, as it is described by a network of unitary operators. This leads to some weird consequences…
ii) During a quantum computational process, quantum spacetime can be in a entangled state, which leads to non-locality of spacetime itself at the Planck scale (all pixels are in a non separable state, and each pixel loses its own identity).
iii) As entanglement is a particular case of superposition, quantum spacetime is in a superposed state, which is reminiscent of the Many-Worlds interpretation of Quantum Mechanics.
iv) Due to superposition and entanglement, quantum spacetime can compute a Boolean function for all inputs simultaneously (massive quantum parallelism). We argue that the functions which are quantum-evaluated by quantum spacetime are the laws of Physics in their most fundamental, discrete and abstract form. Moreover, as the laws are the outputs of quantum measurements, their origin is probabilistic.
v) By scratch space management, we find that at the Planck scale it is possible to compute composed functions of maximal depth. vi) The quantum information stored and processed by quantum spacetime prevents direct tests of the Planck scale.