Fast Bayesian Tomography of a Two-Qubit Gate Set in Silicon

Benchmarking and characterizing quantum states and logic gates is essential in the development of devices for quantum computing. We introduce a Bayesian approach to self-consistent process tomography, called fast Bayesian tomography (FBT), and experimentally demonstrate its performance in characterizing a two-qubit gate set on a silicon-based spin qubit device. FBT is built on an adaptive self-consistent linearization that is robust to model approximation errors. Our method offers several advantages over other self-consistent tomographic methods. Most notably, FBT can leverage prior information from randomized benchmarking (or other characterization measurements), and can be performed in real time, providing continuously updated estimates of full process matrices while data are acquired.

Figure 1

False-color scanning electron microscope image of the device. Two quantum dots are formed underneath gates G1 (blue) and G2 (red). The gates CB, G3, and G4 form confinement barriers that laterally define the quantum dots. RG is the reservoir gate that supplies electrons to the quantum dots. The gate electrodes ST, SLB, and SRB define a single-electron transistor , designed to sense charge movement in the quantum dot region. An alternating current running through the ESR line generates an oscillating magnetic field $B_1$ (light blue) to manipulate the electron spins. The direction of the external magnetic field $B_0$ is indicated by the white arrow.

Figure 2

High-level schematic of the FBT protocol and experiment, with the $U_{1,\downarrow }^{\pi /2}$ gate used as an illustrative example. (a) The estimated noise channel $\overline{\mathrm{\Lambda }}$, presented as a Hinton diagram of the noise residual $(I-\overline{\mathrm{\Lambda }})$, for the $U_{1,\downarrow }^{\pi /2}$ gate at three key stages during data acquisition:① the initial estimate based only on the fidelity; ② the estimate after half of the total number of sequences; and ③ the final estimate after all sequences have been measured. The noise residual $(I-\overline{\mathrm{\Lambda }})$ reveals greater, clearer detail than the Pauli transfer matrix itself. (b) The estimates for three parameters [indicated in (a)], with mean values and uncertainties, as they are updated throughout the experiment. The three parameters were chosen only for clarity of presentation, having nonzero, nonoverlapping mean values. (c) The expected reconstruction error, as determined by the trace of the covariance for the gate, throughout the experiment. Inset: as in the main plot, but using a logarithmic scale. (d) The magnitude of the two error processes (the approximation error and shot noise) throughout the experiment, characterized by the trace of their respective covariances. The approximation error is calculated as a moving average with window size of 100 (the darker line shows the moving average, the lighter lines the raw values). A threshold for approximation error, at two orders of magnitude less than the shot noise, indicates when it can be neglected. For ease of reference this point is indicated by a vertical dotted line in all graphs (b)–(e). (e) The speed of the FBT protocol compared to the time to perform each sequence in the experiment. The light blue lines show the raw time of execution of each sequence (which varies with the length of any particular sequence), the darker line the moving average.

Figure 3

Device benchmarks. (a) Gate infidelity and incoherence for each individual primitive gate, extracted from the final posterior distribution. All error bars are two standard deviations, which corresponds to a $95\mathrm{\%}$ credible interval. (b) Fidelity and concurrence for each of the four Bell states ${\mathrm{\Phi }}^{+},{\mathrm{\Phi }}^{-},{\mathrm{\Psi }}^{+},{\mathrm{\Psi }}^{-}$. Having tomographic estimates for the primitive gates allows us to infer SPAM-free estimates of these benchmarks, compared to the conservative estimates in Ref. [30] via state tomography. (c) Readout assignment matrix indicating assignment error rates for different preparations and measurements. The $x$ axis represents the prepared two-qubit state and the $y$ axis represents the corresponding measurement basis.

Figure 4

Repurposing randomized benchmarking data for FBT. (a) The RB sequences can be repurposed as tomographic data for FBT by decomposing Clifford gates $C_i$ into their primitive components $G_i$. (b) The RB decay curve for the projected state probability as a function of the number of Clifford gates in each sequence. (c) The iterative estimates for two parameters [selected as examples and indicated in (d)] throughout the RB experiment. (d) The final estimate for the $U_{1,\downarrow }^{\pi /2}$ gate, presented as a Hinton diagram of the noise residual $(I-\overline{\mathrm{\Lambda }})$. (e) The magnitudes of the two error processes. The approximation error is significantly larger in the repurposed RB sequences than the tomography data due to the use of much longer sequence lengths in RB.

Figure 5

Gate-set noise estimates presented as noise channel residuals $(I-\overline{\mathrm{\Lambda }})$.

Figure 6

Gate-set estimates presented as full gate Pauli transfer matrices.

Figure 7

$\mathrm{RB\; U}\mathrm{PDATE}(\overline{x},{\mathrm{\Gamma }}_x,\overline{f},{\sigma }_f^2,N_{\mathrm{samples}})$ randomized Benchmarking prior update

Figure 8

$\mathrm{FBT}(m_k,\overline{x},{\mathrm{\Gamma }}_x,S_k)$ fast Bayesian tomography