Linear and nonlinear quantum algorithms made explicit

Abstract

The promise of technological revolution drives the race to implement quantum algorithms on a real-world quantum computer, but the literature is vast and often-times unwieldy. Where there was little thought before, we concern ourselves with the explicit physical cost of implementing quantum algorithms. Linear quantum algorithms are the standard quantum algorithms that use qubits obeying the linear Schrödinger equation, and comprise optimisation routines faster than their classical counterparts. Nonlinear quantum algorithms are a speculative branch of quantum computing, and assume control of a qubit undergoing nonlinear quantum dynamics. In our analyses of linear quantum algorithms, we introduce a quantum search rubric that standardises quantum circuit notation and makes fundamental gate costs transparent. We give a thorough breakdown of common quantum algorithms into their component parts, and give the explicit cost of each component in terms of fundamental quantum gates. We discuss various methods for producing a quantum superposition state representing all elements of the permutation group on n elements, Sn. Our first method involves amplitude amplification. Combining features of two other methods, by Abrams and Lloyd [1] and Berry et al. [2], we devise six di↵erent algorithms for preparing the quantum superposition state and contrast their resource requirements in terms of qubits and fundamental quantum gates. Through this comparison we determine a new state-of-the-art algorithm for producing a superposition of all permutations. Throughout, we derive explicit gate counts and total qubit count in anticipation that real-world quantum computers might implement some of these methods in the near future. Despite their power, linear quantum algorithms are still insu cient for solving NP–complete and harder problems. Abrams and Lloyd’s nonlinear quantum algorithm purports to solve NP– complete decision problems, but it is not explicit [3]. We expand on their comments and use subsequent work by Childs and Young [4] to design a nonlinear quantum algorithm that solves NP–complete decision problems, exploiting the nonlinearity of the Gross-Pitaevskii equation. Our algorithm is a non-trivial extension of both algorithms, and requires a prolonged mathematical proof of correctness. We describe the explicit gate cost and nonlinear evolution time of each step of the algorithm. Finally, we use our algorithm to solve the NP–complete problem of graph comparison.