Testing Quantum Mechanics with Quantum Computers: Qubit Information Capacity

Abstract: Motivated in part by John Wheeler's assertion that the continuum nature of Hilbert Space conceals the information-theoretic character of the quantum wavefunction, a specific discretisation of complex Hilbert Space is proposed. Although the Schrödinger equation is not modified, the bases in which the quantum state is mathematically defined must satisfy certain `rationality conditions'. This leads to the notion of 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 the $2^{N+1}-2$ degrees of freedom demanded by complex Hilbert Space (exponentially growing in $N$) and hence unitary quantum mechanics (QM, where $N_{\mathrm{max}}=\infty$). Here, state reduction to the classical limit corresponds to a steady coarsening of the discretisation with time. Using the Diósi-Penrose model for gravitised state reduction, it is estimated that $N_{\mathrm{max}}$ lies between about 200 and 400 for current qubit technologies, and will never exceed 1,000. It is therefore predicted that the exponential advantage of Shor's algorithm over classical algorithms, will have saturated at 1,000 qubits. Hence, insofar as a classical computer will never factor a 2048-bit RSA integer, neither will a quantum computer. This predicted breakdown of QM is potentially testable in a few years. Importantly, the proposed QIC is not itself due to gravitational or other types of state collapse.