Integration of spectator qubits into quantum computer architectures for hardware tune-up and calibration

Performing efficient quantum computer tune-up and calibration is essential for growth in system complexity. In this work we explore the link between facilitating such capabilities and the underlying architecture of the physical hardware. We focus on the specific challenge of measuring (“mapping”) spatially inhomogeneous quasistatic calibration errors using spectator qubits dedicated to the task of sensing and calibration. We introduce an architectural concept for such spectator qubits: arranging them spatially according to prescriptions from optimal two-dimensional approximation theory. We show that this insight allows for efficient reconstruction of inhomogeneities in qubit calibration, focusing on the specific example of frequency errors which may arise from fabrication variances or ambient magnetic fields. Our results demonstrate that optimal interpolation techniques display near optimal error scaling in cases where the measured characteristic (here the qubit frequency) varies smoothly, and we probe the limits of these benefits as a function of measurement uncertainty. For more complex spatial variations, we demonstrate that the noise mapping for quantum architectures formalism for adaptive measurement and noise filtering outperforms optimal interpolation techniques in isolation and, crucially, can be combined with insights from optimal interpolation theory to produce a general purpose protocol.

Figure 1

Impact of sensor-qubit layout on field mapping quality in a 2D device. (a) Fixed $5\times 5$ data qubits (gray dots) used in all analyses. Left to right: sensor qubits (red dots) are arranged in regularly spaced $d\times d$ grid for $d=4$ (left) or according to Padua points of order $\kappa =1,4$ (middle, right). Generating curve for Padua point sets illustrated by red dashed line. (b) Mapping error in radians (rad) vs polynomial order $n$ for three different interpolation strategies. Top row of insets depicts an example of true polynomial field of degree $n$ on a fixed $5\times 5$ regular grid of data qubits. Main panel: each data marker corresponds to an average over 50 trials of randomly generated true fields. In a single trial, a true field is generated as a polynomial of degree $n$ with random coefficients, and the output of the polynomial is scaled and shifted to lie between $[0,\pi ]$ radians. Interpolation at Padua points of order $\kappa$ is performed using Lagrange polynomials with $\kappa =n$ (black dashed triangles). Interpolation on a $d\times d$ regular grid is performed using RBF, where the total number of sensors is within $\pm 3$ qubits on both regular and Padua grids, with $d=2,3,4,5,9$ (blue squares). For $d=9$, sensor qubits that overlap with data qubits are removed, leading to a total of 54 sensor qubits. In the extreme case where the number of sensor qubits on regular grid are much greater than Padua grid for all orders $n<9$, RBF interpolation is performed on a fixed, regular grid using $d=9$ (“Max regular” in green dashed markers). Error bars represent one standard deviation within the distribution of 50 random trial polynomials. For all interpolation schemes $m=50$ measurements per sensor qubit is used.

Figure 2

Performance of interpolation strategies and NMQA for reconstructing different true fields for both regular and Padua geometries. True fields are depicted as a color scale on a fixed $5\times 5$ grid of data qubits in insets. Main panels (a)–(c) show expected mapping error in radians (rad) vs number of sensors for (a) linear, (b) Franke function, and (c) nonpolynomial true field, with visible error bars depicting 1 std. error over 50 repeated simulations. For Padua sensor locations, we plot NMQA (open red circles) vs Lagrange polynomial interpolation (black dashed triangles). For a regular $d\times d$ grid, we plot NMQA (filled circles) vs radial basis functions (blue squares). Standard assignment approach is shown as a threshold using a fixed arrangement of nine sensor qubits and averaged over four nonunique orientations (gray solid line). Data shown for $m=50$ measurements per sensor qubit, with $d=2,3,4,6,9$ and $\kappa =1,2,3,4,5,10$.

Figure 3

Expected mapping error in radians (rad) vs measurements per sensor qubit $m$ for linear field ($n=1$) in top row (a)–(c) and nonpolynomial field in bottom row (d)–(f). Columns depict increasing order of the Padua grid from left to right $\kappa =1,4,10$. In each panel, mapping error vs $m$ number of measurements per sensor qubit is plotted for NMQA (open red circles) vs Lagrange polynomial interpolation (black dashed triangles) over 50 trials with visible error bars depicting 1 std. error. Padua sensor arrangements (red dots) and associated generating curve (red dashed) for each column is shown as bottom left insets in (d)–(f). For a linear field in (a), $\kappa =n=1$ is optimal Lagrange-Padua interpolation for $m\to \infty$ and (b),(c) represent an unnecessary increase in the number of sensor qubits. In (d)–(f), a nonpolynomial true field permits no finite order polynomial representation.

Figure 4

Placement of sensor and data qubits on the unit square ${[-1,1]}^2$ for standard local neighborhood value-assignment (top), regular (middle), and Padua (bottom) configurations. A $5\times 5$ grid of data qubits (gray circles) is fixed in all panels, while sensor qubit (red circles) are varied. Top: four possible orientations of local neighborhood assignment of data qubits to a single sensor. Shaded regions depict fixed neighborhood groups. Middle: regular nested grid of $d\times d$ sensors, with $d=2,3,4,6$. For $d=9$, sensor qubits that overlap exactly with data qubits are removed, leading to a total of 54 sensor qubits. Bottom: Padua locations for order $\kappa =1,2,3,4,5,10$ and associated generating curve (red dashed).