📝 SummarXiv

Abstract

Motivated by John Wheeler's assertion that the continuum nature of Hilbert Space conceals the information-theoretic nature of the quantum wavefunction, a specific discretisation of complex Hilbert Space is proposed, leading to the notion of qubit information capacity $N_{\mathrm{max}}$: for any $N \ge N_{\mathrm{max}}$-qubit state, there is insufficient information in the $N$ qubits to allocate even one bit to each of the $2^{N+1}-2$ degrees of freedom demanded by complex Hilbert Space and hence unitary quantum mechanics. Using gravitised quantum mechanics, it is estimated that, for typical qubits in a quantum computer, $N_{\mathrm{max}} \approx 500-1,000$. By contrast, $N_{\mathrm{max}}=\infty$ in quantum mechanics. On this basis, it is predicted that the exponential speed up of algorithms such as Shor's will have saturated in quantum computers which use more than about 1,000 logical qubits. This predicted breakdown of quantum mechanics should be testable within the coming decade. If verified, factoring 2048-RSA integers using quantum computers will for all practical purposes be impossible. The existence of a finite qubit information capacity has profound implications for reimagining the foundations of quantum physics (including the measurement problem, complementarity and nonlocality) and for developing novel theories which synthesise quantum and gravitational physics.