Elliptic Curves in Continuous-Variable Quantum Systems

TL;DR

An algorithm for computing elliptic curve group addition using a single continuous-variable mode, based on weak measurements of a system with a cubic potential energy is given, which could lead to improvements in the efficiency of elliptic curve discrete logarithms using a quantum device.

Abstract

Elliptic curves are planar curves which can be used to define an abelian group. The efficient computation of discrete logarithms over this group is a longstanding problem relevant to cryptography. It may be possible to efficiently compute these logarithms using a quantum computer, assuming that the group addition operation can be computed efficiently on a quantum device. Currently, however, thousands of logical qubits are required for elliptic curve group addition, putting this application out of reach for near-term quantum hardware. Here we give an algorithm for computing elliptic curve group addition using a single continuous-variable mode, based on weak measurements of a system with a cubic potential energy. This result could lead to improvements in the efficiency of elliptic curve discrete logarithms using a quantum device.

Figures (4)

• Figure 1: (a) The elliptic curve corresponding to the classical Hamiltonian \ref{['eq:classical-hamiltonian']} with parameters $a = 0.075$ and $b=-1$, and $m=1$, and $E=1.62$ (arbitrary units). (b) The Wigner function derived from the Eigenvector of the quantum Hamiltonian \ref{['eq:Schrodinger-Eq']} with the same parameters and $\hbar = 0.1$. In some places, there were values of the Wigner function as large as $W=1.53$, but to make the form of the function easily visible, values of the Wigner function are cut off at $\pm 0.75$.
• Figure 2: (a) The post-measurement state after a weak measurement of a quadrature operator $\hat{X}_\theta$, with $\theta = \pi/12$ and $r = 1.5$. Similarly to Fig.\ref{['fig:Wigner-function']}, the range of values for the Wigner function has been restricted, in this case to the interval $\pm 1.42$. In some places, there were values of the Wigner function as large as $W=2.84$. (b) Marginal distribution for the second quadrature measurement (of the operator $\hat{X}_{\theta+\pi/2}$). The blue curve $f(x)$ is the probability density function for the marginal distribution, and the orange curve $F(x)$ is the cumulative distribution function, where the three peaks correspond to the three blue regions of concentration in phase space of the Wigner function in Fig. 2a. In Fig. 2a we also see two regions of concentration with wavelike patterns in between these, which are not present in the marginal due to destructive interference.
• Figure 3: (a) The post-measurement state after a weak measurement of a quadrature operator $\hat{X}_\theta$, with $\theta = \pi/12$ and $r = 0.07$. Similarly to Fig.\ref{['fig:Wigner-function']}, the range of values for the Wigner function has been restricted, in this case to the interval $\pm 1.31$. In some places, there were values of the Wigner function as large as $W=2.61$. (b) Marginal distribution for the second quadrature measurement (of the operator $\hat{X}_{\theta+\pi/2}$). The blue curve $f(x)$ is the probability density function for the marginal distribution, and the orange curve $F(x)$ is the cumulative distribution function, where the peaks correspond to the blue regions of concentration in phase space of the Wigner function in Fig. 3a.
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