Blind calibration of a quantum computer

TL;DR

Blind calibration addresses the quantum measurement calibration problem by enabling simultaneous recovery of multiple measurement errors from a single tomographic data set without assuming perfect state preparation. The authors formalize a bilinear calibration model and solve it via alternating minimization using projective Pauli measurements, validated on a trapped-ion quantum computer. They demonstrate recovery of native measurement errors and compare to direct calibration, achieving similar accuracy with fewer experimental shots and robustness to state-preparation noise. The approach enables post hoc, data-efficient calibration that scales to larger qubit systems by leveraging structured probe states and hierarchical calibration strategies.

Abstract

The calibration of quantum measurements is limited by the ability to accurately prepare quantum states under unknown device errors. We develop an accurate calibration protocol for the measurement apparatus of a quantum computer that is `blind' to the state preparation. Blind calibration quantifies and corrects measurement errors from simple tomographic data on a noisy quantum state. Importantly, it calibrates multiple error mechanisms in a single experiment, eliminating the need for bespoke, separate calibration experiments. Using a trapped-ion quantum computer, we systematically demonstrate the accuracy of the method. We use blind calibration to estimate the native calibration parameters of the experimental system. The recovered calibrations are consistent with directly measured values and perform similarly in predicting the state properties.

Figures (13)

• Figure 1: Blind calibration of a trapped ion quantum computer. (a) Since errors happen throughout a quantum computation, it is difficult to distinguish measurement errors from state preparation errors. The blind calibration protocol uses prior knowledge about the structure of the data in order to distinguish those errors in noisy data and thus allows us to accurately calibrate the device. (b) Experimental setup. Ions are held in a linear Paul trap, probed using individual laser beams, and detected using individual detector channels. (c) Ion fluorescence on the transition between the computational state $\ket 1$ and an excited state $\ket e$ is used for readout. A bright error occurs when an ion in the bright $\ket 1$ state off-resonantly decays to the dark $\ket 0$ state with probability $p_1$ or vice versa a dark error with probability $p_0$. (d) In a detector spillover error, a signal on one detector generates spurious photon counts on adjacent detectors due to electrical or optical crosstalk. This causes a false detection on the left detector with probability $p_{\text{left}}$, and on the right detector with probability $p_{\text{right}}$. (e) A crosstalk error occurs, when the laser beam targeting a particular ion causes unintended rotation by an angle $\xi_{r}(\xi_l) \cdot \frac{\pi}{2}$ on the neighboring right (left) ion. (f) A target $\frac{\pi}{2}$ operation over- or under-rotates by an angle $\xi_{\text{\rm OR}}\cdot \frac{\pi}{2}$.
• Figure 2: Numerical benchmarks of blind calibration using a three-qubit GHZ state and a randomly-drawn three-qubit product state. We simulate the four error types discussed in the main text, resulting in six calibration parameters. We set the parameter values to $\xi_{\text{actual}}$ given in \ref{['eq:xi actual']} of Appendix \ref{['app:error-models']}. (a) Evidence for validity of trace distance to the target state as a measure of success. Comparison of the state estimate obtained from calibrated tomography with calibration parameters $(1-c) \xi_{\text{ideal}} + c \xi_{\text{actual}}$ for $c \in [0,2]$ with $\xi_{\text{ideal}} = (1,0, \ldots)$ and $\xi_{\text{actual}}$ given in \ref{['eq:xi actual']} using $10000$ simulated shots per measurement basis from noisy state preparations using a Pauli noise channel with strength 1% per gate for the GHZ state (left) and the product state (right). Shown are the TD to the noisy (simulated) state preparation (diamonds) and the target state (circles). Error bars are one standard deviation. (b) Calibration error (left) for the seven recovered parameters. The recovered parameters are used to perform calibrated state tomography, with the resulting trace distances to the target (right). Both calibration error and trace distance are shown as a function of the number of experimental shots per basis with one standard deviation error bars, decaying as an inverse square root (dashed line). (c) State sensitivity for the product state. The two outer qubits are held at fixed angles, while the middle qubit is scanned through all possible states. The product state is shown by the red circle. The trace distance to the target and the calibration error remain constant to within a percent. (d) Calibration parameter recovery with increasing local depolarizing noise (left). Local depolarizing strength is chosen based on available noise models of similar hardware ionq_noise_model. The recovered parameters are used to perform calibrated state tomography (right). Calibration error is insensitive to depolarizing noise to within a tenth of a percent, which translates to insensitivity in trace distance to within half a percent.
• Figure 3: Recovery of intentional miscalibrations. In order to demonstrate the experimental viability of blind calibration, we intentionally inject overrotation and nearest-neighbour beam crosstalk errors (with $\xi_r = \xi_l$) of varying magnitude $[0,1,2.5,5]\cdot \%\frac{\pi}{2}$ by performing appropriate single-qubit rotations, when taking tomographic data of a randomly drawn 3-qubit product state in the $X$-$Z$ plane with polar angles given by $\theta_1=1.237\pi,\theta_2=0.670\pi,\theta_3=1.823\pi$. (a) Blind-calibration estimates of the overrotation (green circles) and crosstalk errors (yellow diamonds) compared to the injected errors (green/yellow dashed lines) as a function of the injected overrotation for different values of injected beam crosstalk. (b) Assessment of the quality of different calibrations using the trace distance (TD) to the target state of the state estimate obtained from standard tomography, direct-calibration tomography, and blind-calibration tomography.
• Figure 4: Blind calibration of the native errors of the trapped-ion quantum computer using six optimal probe states labeled OS1 to OS6, see Appendix \ref{['app:optimal states']} for details. Blind-calibration estimates of the calibration parameters are represented by green circles. Error bars are composed of the systematic error (dark green with caps) and one standard deviation statistical error (light green), see Appendix \ref{['app:error-bars']} for details. The average of the estimated calibration parameters over the six probe states and its standard deviation are represented by a solid green line and shaded area, respectively, and the estimates from the direct measurement by dashed yellow lines. (a) Estimates of dark error $p_0$ and right detector spillover error $p_{\text{right}}$. (b) Estimates of right beam crosstalk magnitude $\xi_{r}$ and right beam crosstalk phase $\phi_{r}$. (c) Calibration error $E(\xi, \delta)$ of the blind calibration ($\xi$) compared to direct measurements ($\delta$). (d,e) To assess the quality of different calibrations, we compute the distance (TD) to the target state of the state estimates obtained from standard tomography (triangles), direct-calibration tomography (diamonds), and blind-calibration tomography (circles). For blind-calibration tomography, we use the averages obtained from the six optimal probe states as our calibration parameter estimates. In (d), we show the quality of the estimates for tomographic data from the six probe states OS1--OS6. (e) To demonstrate that the blind calibration generalizes beyond the probe states, we also obtain tomographic data from several independent test states: the basis state $\ket{000}$, the Hadamard basis state $\ket{+++}$, four random product states labeled RP1 to RP4, the GHZ state, and two states generated by random deep circuits, labeled RD1 and RD2, see Appendix \ref{['app:optimal states']} for details. We show the difference $d_{\tr}(\rho^{*}, \psi) - d_{\tr}(\rho, \psi)$ of the trace distances between the standard tomography estimate $\rho$ and the direct/blind calibration estimate $\rho^{*}$ to the target state $\psi$. In this representation, standard tomography achieves a value of $0$ (solid line), and the more negative the values of direct/blind-calibration tomography are, the closer the corresponding state estimate is to the target state.
• Figure 5: Calibration parameter optimization. We find an optimal set of parameters to include in the blind calibration model. Each set contains decreasing numbers of parameters. Set 1 includes the full set of 9 calibration parameters: overrotations, beam crosstalk magnitude, beam crosstalk phase, detector spillover, and dark and bright errors. Set 2 removes overrotations. Set 3 removes beam crosstalk phase from Set 2. Set 4 removes beam crosstalk magnitude from Set 3. Set 5 removes detector spillover from Set 4, and thus only dark and bright errors remain. We perform rank-1 blind calibration for each set and show the calibration error in (b). We then perform rank-8 blind tomography of the test states and compare to direct tomography. The differences in trace distances are shown in (a). We conclude that removing overrotation from the model leads to an optimal recovery.