Nonexponential fidelity decay in randomized benchmarking with low-frequency noise

We show that nonexponential fidelity decays in randomized benchmarking experiments on quantum-dot qubits are consistent with numerical simulations that incorporate low-frequency noise and correspond to a control fidelity that varies slowly with time. By expanding standard randomized benchmarking analysis to this experimental regime, we find that such nonexponential decays are better modeled by multiple exponential decay rates, leading to an instantaneous control fidelity for isotopically purified silicon metal-oxide-semiconductor quantum-dot qubits which is 98.9% when the low-frequency noise causes large detuning but can be as high as 99.9% when the qubit is driven on resonance and system calibrations are favorable. These advances in qubit characterization and validation methods underpin the considerable prospects for silicon as a qubit platform for fault-tolerant quantum computation.

Figure 1

(a) Randomized benchmarking consists of applying multiple sequences of random Clifford gates, a final recovery Clifford gate to ensure that each sequence ends with the qubit in an eigenstate, and reading out the qubit state. In interleaved randomized benchmarking, an additional test Clifford gate is inserted in between the random Clifford gates. (b) Bloch sphere representation for the breakdown of a general noisy operation ${\mathcal{C}}_N$ into an ideal ${\mathcal{C}}_I$ rotation followed by a noise operation $\mathcal{D}$.

Figure 2

(a) Sequence fidelity as a function of sequence length $m$, with the qubit subject to Gaussian distributed $T_2^{*}$ associated noise. Each black line represents a fidelity decay for one particular value of detuning $\mathrm{\Delta }\omega$ (only 10% of traces shown). The linear decay on the logarithmic scale illustrates that these individual traces are indeed exponential, while the ensemble average (thick blue line) is nonexponential. The thick red line is the ensemble average for a Lorentzian distributed noise. (b) Gaussian distributed detuning frequencies and (c) Lorentzian distributed detuning frequencies associated with individual traces.

Figure 3

Semilog plot of $F_m^{\uparrow }-(1-F_m^{\downarrow })$ for the reference sequence of randomized benchmarking on a silicon quantum-dot qubit [12]. Both the two-fidelity model and a single-fidelity model including residual SPAM can fit the data, but for the single-fidelity model an unreasonably large SPAM has to be included.

Figure 4

(a) Reference sequence of randomized benchmarking on a silicon quantum-dot qubit [12]. The separate fidelities from the two-fidelity model have been plotted to show how the initial decay is dominated by the low $q$ value, whereas the higher value of $p$ is indicative of the average decay in the longer-lived high-fidelity regime. Histogram of spin up $|\uparrow \rangle$ and spin down $|\downarrow \rangle$ corresponding to data point (b) $m=2$ and (c) $m=150$. Results with expected spin-up outcome are shown in red, while blue represents data with expected spin-down result. The gray regions illustrate the overlapping areas.