Bayesian view of single-qubit clocks, and an energy versus accuracy tradeoff

We bring a Bayesian approach to the analysis of clocks. Using exponential distributions as priors for clocks, we analyze how well one can keep time with a single qubit freely precessing under a magnetic field. We find that, at least with a single qubit, quantum mechanics does not allow exact timekeeping, in contrast to classical mechanics, which does. We find the design of the single-qubit clock that leads to maximum accuracy. Further, we find an energy versus accuracy tradeoff—the energy cost is at least $k_{B}T$ times the improvement in accuracy as measured by the entropy reduction in going from the prior distribution to the posterior distribution. We propose a physical realization of the single-qubit clock using charge transport across a capacitively coupled quantum dot.

Figure 1

A spin precesses with angular velocity $\omega$ on the equator of the Bloch sphere. Arrival of the exponential random variable with rate $\lambda$ triggers a projective measurement in the $\left\{\right|+\rangle ,|-\rangle \}$ basis. The time reported depends on the measurement outcome.

Figure 2

Probability density functions for the prior $T$ and the two posteriors $T_{+}$ and $T_{-}$ at $\omega /\lambda =0.80654$ and $\varphi =0.246576$. The time axis has the units of $\frac{1}{\lambda }$, and the distributions are plotted by setting $\lambda$ to unity. The declared time for a measurement $``{-}^{\prime \prime }$ is the mean of $T_{-}$, which is significantly different from the mean of $T$.

Figure 3

Intensity plot of expected quality factor $Q\left[T\right|S\left(\omega T\right)]$ shows the maximum at $\omega /\lambda =0.81,\varphi =0.24$.

Figure 4

A spin precesses on a circle of latitude making angle $\theta$ with the north pole. Arrival of the rate-$\lambda$ exponential random variable triggers a projective measurement in direction $({\theta }_m,{\varphi }_m)$. We maximize the quality factor against $\varphi ,{\varphi }_m,\theta ,{\theta }_m,\omega ,\lambda$ and find $\theta ={\theta }_m=\pi /2$.

Figure 5

Setup for a quantum dot single-qubit clock based on a capacitively coupled quantum dot pair QD1 and QD2. As per our proposal, the exponential prior is the tunneling event in QD1 and the precessing spin is housed inside QD2. Measurement is enabled when the freely precessing spin inside QD2 is “disturbed” by the tunneling event in QD1 via the mutual Coulomb repulsion $U_{12}$. This leads to the spin in QD2 tunneling into either ferromagnetic contact, respectively, kept at a chosen quantization axis, leading to the measurement step.