The Efficiency of Feynman's Quantum Computer

• URL: http://arxiv.org/abs/2309.09331v1
• Date: Sun, 17 Sep 2023 17:33:30 GMT
• Title: The Efficiency of Feynman's Quantum Computer
• Authors: Ralph Jason Costales, Ali Gunning, Tony Dorlas
• Abstract summary: Feynman's circuit-to-Hamiltonian construction enables the mapping of a quantum circuit to a time-independent Hamiltonian. We investigate the efficiency of Feynman's quantum computer by analysing the time evolution operator $e-ihatHt$ for Feynman's clock Hamiltonian $hatH$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Feynman's circuit-to-Hamiltonian construction enables the mapping of a quantum circuit to a time-independent Hamiltonian. Here we investigate the efficiency of Feynman's quantum computer by analysing the time evolution operator $e^{-i\hat{H}t}$ for Feynman's clock Hamiltonian $\hat{H}$. A general formula is established for the probability, $P_k(t)$, that the desired computation is complete at time $t$ for a quantum computer which executes an arbitrary number $k$ of operations. The optimal stopping time, denoted by $\tau$, is defined as the time of the first local maximum of this probability. We find numerically that there is a linear relationship between this optimal stopping time and the number of operations, $\tau = 0.50 k + 2.37$. Theoretically, we corroborate this linear behaviour by showing that at $\tau = \frac{1}{2} k + 1$, $P_k(\tau)$ is approximately maximal. We also establish a relationship between $\tau$ and $P_k(\tau)$ in the limit of a large number $k$ of operations. We show analytically that at the maximum, $P_k(\tau)$ behaves like $k^{-2/3}$. This is further proven numerically where we find the inverse cubic root relationship $P_k(\tau) = 6.76 \; k^{-2/3}$. This is significantly more efficient than paradigmatic models of quantum computation.

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