Ghost factors in Gauss-sum factorization with transmon qubits

A challenge in the Gauss-sum factorization scheme is the presence of ghost factors, nonfactors that behave similarly to actual factors of an integer, which might lead to the misidentification of nonfactors as factors or vice versa, especially in the presence of noise. We investigate type II ghost factors, which are the class of ghost factors that cannot be suppressed with techniques previously laid out in the literature. The presence of type II ghost factors and the coherence time of the qubit set an upper limit for the total experiment time, and hence the largest factorizable number with this scheme. Discernibility is a figure of merit introduced to characterize this behavior. We introduce preprocessing as a strategy to increase the discernibility of a system, and demonstrate the technique with a transmon qubit. This can bring the total experiment time of the system closer to its decoherence limit, and increase the largest factorizable number.

Figure 1

Block diagram of the experimental setup. The figure is divided into two parts: The room-temperature electronics at the top, and the cyrogenic electronics and the cryostat at the bottom. An arbitrary waveform generator (AWG) is used to synthesize the signals for qubit control and readout. The readout signals are upconverted and downconverted with an IQ mixer and a local oscillator. Both signals are combined and sent through the cryostat. The readout signal is then downconverted and parsed by an analog-to-digital converter (ADC).

Figure 2

Pulse sequence used in Gauss-sum factorization. Most operations are $\pi$ rotations around $cos\left({\varphi }_k\right)\widehat{x}+sin\left({\varphi }_k\right)\widehat{y}$, where ${\varphi }_k$ parametrizes the axis of rotation. The initialization pulses are $\pi /2$ rotations about ${\varphi }_i$ and ${\varphi }_f$, respectively. In our experiment, ${\varphi }_i={\varphi }_f=\pi /2$, and the measurement returns the probability of finding the system in the state $|1\rangle$. There is a wait time of $\tau$ between successive pulses.

Figure 3

Simulated example of Gauss sum with $T_2=500\mathrm{ns}$ and Gaussian random measurement noise added to Eq. (12). Factors are labeled above the corresponding bars. If the usual cutoff of $\frac{3}{4}$ is chosen, some of the factors, due to decoherence causing them to fall below the chosen cutoff, might be misidentified as nonfactors. We suggest a new cutoff based on the expected decay of the trial factors, and choosing it to be between the expected values of the factor and the worst ghost factor. This provides leeway for fluctuations without misidentification of the factors as nonfactors or vice versa.

Figure 4

Gauss-summand decay behavior, cutoffs, and discernibility. (a) Expected behavior of Gauss summands from theory. The signal from the factor ($l=21$, upper solid line) decays exponentially with a characteristic time $T_2$, while the signal from the ghost factor ($l=28$, lower solid line) oscillates between within the envelope formed by the factor. (b) Experimental measurements of signals from the factor (upper solid line) and ghost factor (lower solid line, oscillating). The dashed line is an exponential fit to yield $T_2$. (c) Decay behavior of the Gauss sum of the factor (upper solid line) and nonfactor (lower solid line), together with the cutoff (dashed line) calculated from the measured $T_2$. Our suggested cutoff always lies between the factor and the ghost factor, so misidentification is unlikely as long as the fluctuations are smaller than the separation between the two. (d) Experimental (black) and theoretical (gray) discernibility of the two factors. At very small $M$ the nonfactor is not sufficiently suppressed while at large $M$ decoherence closes the gap between the factor and the nonfactor, so both extremes result in a reduced discernibility.

Figure 5

Experimental results for the Gauss-sum factorization of $263193=3\times 7\times 83\times 151$ with the total number of pulses $M=17$. The factors are labeled above the corresponding bar without arrows, and the cutoff from Eq. (15) is marked out with a dashed line. The qubit drive is on resonance in (a) and (c) and with detuning noise in (b) and (d). The noise is introduced by randomly detuning the qubit drive to ${\omega }_d={\omega }_q-\delta$, where $\delta$ is sampled from a uniform distribution in the range $[0,2\pi \times 250\mathrm{MHz}]$. The results are shown without preprocessing [(a) and (b)] and with preprocessing [(c) and (d)]. In (a), ghost factors $l=28$ and 56, labeled with arrows, are close to the cutoff, although there is no misidentification of factors and nonfactors. In (b), the ghost factor $l=4$ exceeds some of the factors, so there will be misidentification regardless of how the cutoff is adjusted. In addition, the ghost factors $l=15$ and 36, labeled with arrows, exceed the chosen cutoff, so there is a risk of misidentifying these nonfactors depending on the choice of cutoff. In (c) and (d), there is a clear separation between factors and nonfactors, and there is no longer any misidentification of either. The worst nonfactors after preprocessing are $l=27$ and 91, labeled with arrows, which are the next worst ghost factors with $q=9$ and 13.

Figure 6

Experimental Gauss summand and discernibility for factor ($l=21$, upper lines) and nonfactor ($l=28$, lower lines) [similar to Figs. 4 and 4] with pulse delay $\tau =1\mathrm{ns}\ll t_{\pi }$. The dashed lines are the fits used to find the $T_2$ times. We found that in this regime, the $T_2$ time of the factor was much larger than the $T_2$ time of the nonfactor. This required an adjustment of the discernibility, which is shown in (b). The adjusted discernibility agrees better with the experimental results.

Figure 7

Filter functions of a factor [(a), (b)] and the worst ghost factor [(c), (d)], with constant ${\tau }_0=\tau +t_{\pi }$. For all four subfigures, the values of $t_{\pi }/{\tau }_0$ of each line are marked out on the color scale in the same order, with $t_{\pi }/{\tau }_0=1$ for the uppermost lines and $t_{\pi }/{\tau }_0=0$ for the lowermost lines. For the factor, assumption of instantaneous pulses (here $t_{\pi }=0$) in noise spectroscopy is lifted, resulting in some deviations to the peak height and location. For the ghost factor, the pulse sequence filters out a large range of frequencies of the noise spectrum.

Figure 8

Decoherence of factor and worst ghost factor in the filter function formalism in the presence of white noise (a) and $1/f$ noise (b). The dephasing behavior is found by using Eqs. (25) to (27) with $S\left(\omega \right)=\text{constant}$ for white noise and $S\left(\omega \right)\propto 1/\omega$ for $1/f$ noise. In both cases, the nonfactor (lower lines) decays more rapidly than the factor (upper lines). Note that only the envelope is plotted, so the rapid oscillation of the nonfactor is not shown.