Probing Qubit Memory Errors at the Part-per-Million Level

Robust qubit memory is essential for quantum computing, both for near-term devices operating without error correction, and for the long-term goal of a fault-tolerant processor. We directly measure the memory error ${\epsilon }_m$ for a ${}^{43}{\mathrm{Ca}}^{+}$ trapped-ion qubit in the small-error regime and find ${\epsilon }_m<{10}^{-4}$ for storage times $t\lesssim 50\mathrm{ms}$. This exceeds gate or measurement times by three orders of magnitude. Using randomized benchmarking, at $t=1\mathrm{ms}$ we measure ${\epsilon }_m=1.2(7)\times {10}^{-6}$, around ten times smaller than that extrapolated from the $T_2^{*}$ time, and limited by instability of the atomic clock reference used to benchmark the qubit.

Figure 1

Two-point Ramsey experiments for measuring the qubit memory error. (a) Sequences used to measure the maximum and minimum of the Ramsey fringe. The phase difference between the final $\pi /2$ pulses remains fixed at $\tilde{\varphi }-\varphi =180°$ throughout the experiments. (b) SPAM-corrected loss of Ramsey fringe contrast. The line represents exponential contrast decay (fixed at 0 for ${\tau }_R=0$) fitted to the data with ${\tau }_R>1\mathrm{s}$, where the SPAM error is negligible compared with the contrast error; the fit gives $T_2^{*}=22(3)\mathrm{s}$. The right-hand ordinate gives the memory error, averaged over the Bloch sphere, associated with a given Ramsey contrast loss.

Figure 2

“Standard” randomized benchmarking of single-qubit gates. (a) Gate diagram of the sequence used, including an example decomposition of a Clifford gate $C_i$ into $\pi /2$ pulses. The qubit is prepared in $|\uparrow ⟩$ and then subjected to $m$ random gates from the Clifford set, followed by an ($m+1$)th gate that is chosen to rotate the qubit to one of its two basis states, selected with equal probability. Each RB sequence is alternated with a measurement of ${\epsilon }_{\mathrm{SPAM}}$, for which the delay between state preparation and measurement is equal to the duration of the RB sequence; this allows us to check for any systematic dependence of ${\epsilon }_{\mathrm{SPAM}}$ on sequence duration. (b) Measured sequence fidelities (blue circles) and SPAM fidelities (grey diamonds), as a function of sequence length. The fit to a decay $\frac{1}{2}(a{p}^m+1)$ (blue line) gives an average Clifford gate error of ${\epsilon }_g=1.7(2)\times {10}^{-6}$. The fit intercept is consistent with the mean measured SPAM error of $2.7(4)\times {10}^{-3}$ (grey line). Error bars on each point are the standard error of the mean over $k=50$ sequence randomizations. Shaded regions represent the $1\sigma$ uncertainties of the fits.

Figure 3

Single-qubit memory errors ${\epsilon }_m$ measured by interleaved randomized benchmarking. (a) IRB sequence, with delays $\tau$ inserted between each Clifford gate $C_i$. An example decomposition of a Clifford gate $C_i$ into $+X_{\pi /2}$ and $+Y_{\pi /2}$ gates is shown. Also shown are the associated BB1 decompositions: each $X_{\pi /2}$ gate is followed by a sequence of rotations $\{{\varphi }_{\pi },(3\varphi {)}_{2\pi },{\varphi }_{\pi }\}$; similarly for each $Y_{\pi /2}$ gate, but around an axis $\tilde{\varphi }$. For the reference sequence (ref), an additional delay $m\tau$ is inserted after the last Clifford gate to keep the time between the qubit initialization and readout equal to that in the IRB sequence, thus minimizing any systematic differences in ${\epsilon }_{\mathrm{SPAM}}$. (b) Qubit frequency offset $\mathrm{\Delta }f$ versus offset $\mathrm{\Delta }B$ of the magnetic quantization field, relative to the field-independent “clock” point. (c) Average memory error measured for interleaved delays $1\mathrm{ms}<\tau <10\mathrm{s}$. At each $\tau$, we choose $m$ to give a total sequence infidelity of $\sim 0.1$, which is much larger than the SPAM error. The IRB data (blue circles) is consistent with the Ramsey results (grey squares, from Fig. 1). We fit a decoherence model to the IRB data (blue line—see Supplemental Material [21], §A3 for details), which contains white noise and $1/f$ noise with a low-frequency cutoff (this cutoff is consistent with the duration of the longest sequences). IRB sequences including dynamical decoupling (red diamonds) show that the error due to $1/f$ noise can be suppressed at long times using this technique. The dashed black line shows the calculated error contribution from the local oscillator (atomic clock) to which the qubit phase is compared: this accounts for the measured memory error in the ${\epsilon }_m\lesssim {10}^{-4}$ regime. Finally the yellow, green, and purple data points show the error measured when the magnetic field is deliberately offset from the field-independent point by $\mathrm{\Delta }B=(10,25,50)\mathrm{mG}$; note that the local oscillator was left at $\mathrm{\Delta }f=0$, so that the observed error arises mostly from the corresponding qubit detuning $\mathrm{\Delta }f\approx (0.12,0.76,3.0)\mathrm{Hz}$ (dotted lines).