Rational Quantum Mechanics: Testing Quantum Theory with Quantum Computers

Abstract

Motivated in part by John Wheeler's assertion that the continuum nature of Hilbert Space conceals the `it-from-bit' information-theoretic character of the quantum wavefunction, a theory of quantum physics (Rational Quantum Mechanics - RaQM) is proposed based on a specific discretisation of complex Hilbert Space. The Schr\"{o}dinger equation is not modified in RaQM, even during measurement. However, the bases in which the quantum state is defined must satisfy certain rational-number constraints. These constraints lead to the notion of finite qubit information capacity $N_{\mathrm{max}}$: for any $N > N_{\mathrm{max}}$ qubit state, there is insufficient information in the $N$ qubits (linearly growing in $N$) to allocate even one bit to each of all $2^{N+1}-2$ continuum degrees of freedom (exponentially growing in $N$) associated with quantum mechanics/theory (QM, where $N_{\mathrm{max}}=\infty$). It is proposed that the discretisation of Hilbert Space in RaQM is due to gravity, hence QM is the (singular) continuum limit of RaQM at $G=0$. On this basis, it is estimated that $N_{\mathrm{max}}$ lies between about 200 and 400 for current qubit technologies, and will never exceed 1,000. Whilst QM and RaQM are experimentally indistinguishable for small numbers of qubits, RaQM predicts that the exponential advantage of quantum algorithms which, like Shor's, require bases with maximal $N$-qubit superposition/entanglement, will have saturated at 1,000 perfect qubits. Hence, insofar as a classical computer will never factor a 2048-bit RSA integer, RaQM predicts that a quantum computer won't either. This predicted breakdown of QM could be testable in less than 5 years.