Minimal Energy Cost to Initialize a Quantum Bit with Tolerable Error

TL;DR

This paper investigates the minimal energy required to initialize a quantum bit within a finite time and tolerable error, revealing a fundamental increase over Landauer's limit due to finite-time effects and proposing an optimal protocol.

Contribution

It derives a quantitative relation for the additional energy cost in finite-time qubit initialization and proposes an optimal protocol to minimize this energy expenditure.

Findings

Energy cost increases as error decreases.

Explicit formulas for additional energy cost at specific error rates.

An optimal finite-time isothermal protocol is proposed.

Abstract

Landauer's principle imposes a fundamental limit on the energy cost to perfectly initialize a classical bit, which is only reached under the ideal operation with infinite-long time. The question on the cost in the practical operation for a quantum bit (qubit) has been posted under the constraint by the finiteness of operation time. We discover a raise-up of energy cost by $\mathcal{L}^{2}(\epsilon)/\tau$ from the Landaeur's limit ($k_{B}T\ln2$) for a finite-time $\tau$ initialization with an error probability $\epsilon$. The thermodynamic length $\mathcal{L}(\epsilon)$ between the states before and after initializing in the parametric space increases monotonously as the error decreases. For example, in the constant dissipation coefficient ($\gamma_{0}$) case, the minimal additional cost is $0.997k_{B}T/(\gamma_{0}\tau)$ for $\epsilon=1\%$ and $1.288k_{B}T/(\gamma_{0}\tau)$ for $\epsilon=0.1\%$. Furthermore, the optimal protocol to reach the bound of minimal energy cost is proposed for the qubit initialization realized via a finite-time isothermal process.