Stable quantum memories with limited measurement

We demonstrate the existence of a finite temperature threshold for a one-dimensional stabilizer code under an error correcting protocol that requires only a fraction of the syndrome measurements. Below the threshold temperature, encoded states have exponentially long lifetimes, as demonstrated by numerical and analytical arguments. We sketch how this algorithm generalizes to higher-dimensional stabilizer codes with stringlike excitations, such as the toric code.

Figure 1

This drawing illustrates the centering procedure for detected defects on a unit cell. Spin variables are in small, gray circles, and domain wall variables are in large blue (no defect present) and large red, hatched (defect present) circles. When a defect is detected on a measurement patch (blue, rectangular box), it is swapped to the center of the measurement patch via the $\mathrm{DSWAP}$ operator (black arrows). The defect immediately adjacent to it is also swapped onto the measurement patch so as not to pull apart defect pairs that would have otherwise dissipated. Measurement patch length scale $\lambda$ indicated by arrow on top, and unit cell length scale $\lambda$ indicated by arrow on bottom. The measurement fraction is defined as $m\equiv {\lambda }_m/\lambda$.

Figure 2

Lifetime enhancement of several system sizes $L$ as a function of temperature $T$ using a measurement fraction of $m=3/7$. Dotted line indicates example fit for threshold temperature $T_{th}$ extraction for $L=224$ data. Below $T\approx 0.16$, we find the lifetime to grow exponentially with $L$, indicative of a finite-temperature threshold. Inset: Finite-size scaling of $T_{th}$ with inverse length $1/L$. Extrapolation to the infinite system limit yields a threshold temperature of $T_{\mathrm{th}}=.155\left(6\right)$. Data collected via Monte Carlo simulation of Eq. (2)

Figure 3

(Top) Lifetime enhancement for temperatures both below and above the threshold as a function of system size $L$, using a measurement fraction $m=3/7$. Note the monotonic growth in lifetime at low temperatures, as well as the plateau in lifetime for moderately sized systems. A $3/7$ measurement fraction was used for this data. (Bottom) The same data rescaled by an additional factor ${\mathrm{\Gamma }}_0^{-1}$, to emphasize the origin and scaling of the plateau in system lifetimes.

Figure 4

(Top) The lifetime enhancement as a function of temperature for several measurement fractions $m$ of 32 (i.e., $L=32·3/m$. Note the threshold temperature decreases with $m$. For this data, $\mathrm{\Delta }=1.0$. (Bottom) Lifetime enhancement as a function of temperature for a variety of energy scales $\mathrm{\Delta }$ [see Eq. (1)] for an $L=224$ system with constant $\mathrm{m}=3/7$.

Figure 5

The threshold temperatures as a function of measurement fraction, $m$ (left), and the Hamiltonian energy scale $\mathrm{\Delta }$ [see Eq. (1)] (right). At zero measurement fraction, the critical temperature is 0, and at unit measurement fraction, the critical temperature is $O\left(1\right)$.

Figure 6

Here we sketch one possible $m=63/144$ (for a $12\times 12$ unit cell) geometry for the measurement rails for a realization of our protocol. More sparse geometries can be realized simply by moving the rails of measurement farther apart. Measured sites are in light blue, and vertex locations for the toric code are circles. Spin variables (not pictured) reside directly between neighboring vertices.

Figure 7

Comparison of three different functional forms used as proxies for $P(d_1:d_2)$. Protocol (i) is simply the error function expression given by Eq. (12). Protocol (ii) is the probability density $\frac{1}{2\pi D|t_2-t_1|}\exp (-\frac{|x_2-x_1{|}^2}{2D|t_2-t_1|})$. Protocol (iii) is the more complicated expression in the last line of Eq. (A2). In practice, each approximation for $P(d_1:d_2)$ is seen to perform approximately equally well.