No need to calibrate: characterization and compilation for high-fidelity circuit execution using imperfect gates

TL;DR

The study tackles the bottleneck of calibrating two-qubit entangling gates by proposing a characterize-and-compile pipeline that learns a small set of gate parameters from fast tomography and encodes their nonlocal content in Weyl chamber coordinates (c1,c2,c3). A hardware-agnostic workflow then synthesizes target unitaries from an extended gate set using the Cartan decomposition, supported by a coverage-set/invariant framework and accelerated routines for controlled and single-axis pulses. Key contributions include an end-to-end pipeline for extended gate-set construction and synthesis, practical acceleration strategies, and IBM-device demonstrations showing up to 7x improvements in QFT success probability and up to 9x reductions in MSE for TFIM simulations. The approach reduces calibration overhead and enables scalable, high-fidelity circuit execution across architectures, with potential extensions to drift-robust characterization and multi-axis controls for broader platforms.

Abstract

We propose and validate on real quantum computing hardware a new method for extended two-qubit gate set design, replacing iterative, fine calibration with fast characterization of a small number of gate parameters which are then tracked and corrected in circuit compilation. Coherent contributions to the pulse unitary that would traditionally be considered sources of error are treated as part of the gate definition, and compensated in software via single-qubit rotations. This approach enables rapid device-wide generation of high-fidelity two-qubit entangling gates, which are combined with standard calibrated gates to produce an expanded gate set. We show how these gates are directly usable as part of a quantum compiler, synthesizing generic two-qubit circuit blocks into minimal-duration sequences of the characterized gates interleaved with compensating single-qubit rotations. Benchmarking against circuits compiled using the default $CX$ gate alone on 127-qubit IBM hardware shows up to 7X improvement in success probability for Quantum Fourier Transform circuits up to 26 qubits, and up to 9X lower mean-square error in Trotter simulations of the one-dimensional transverse-field Ising model. Our hardware-agnostic characterization and compilation methodology makes it practical to scale up expressive gate sets on quantum computing architectures while minimizing the need for onerous fine-tuning of low-level control waveforms.

Figures (9)

• Figure 1: The pulse-efficient compilation procedure. (a) An extended two-qubit gate set $\mathcal{B}$ for a specific pair of qubits. Each gate is represented by a point in the Weyl chamber (the space of all two-qubit unitaries up to local equivalence). Here, $\mathcal{B}$ contains a calibrated $CX$ gate $U_\text{pulse}^{(2)}$, along with two shorter gates that were characterized as $U_\text{pulse}^{(0)}$ and $U_\text{pulse}^{(1)}$. (b) The corresponding coverage set $\mathcal{S}_\mathcal{B}$, representing the regions of the Weyl chamber reachable by different combinations of these gates interleaved with single-qubit gates. We organize the coverage set by the number of applications of each two-qubit entangling gate, encoded as a vector $\bm{k} = (k_{0}, \ldots, k_{n-1})$ having entries ordered right-to-left with respect to circuit ordering, where $k_i \in \{0,1,2\}$ indicates which of the three entangling gates is used. For example, $\mathbf{k}=(1, 0, 0)$ denotes the space of unitaries reachable by two applications of $U_\text{pulse}^{(0)}$ followed by $U_\text{pulse}^{(1)}$, interleaved with single-qubit gates. Although three applications of $CX$ ($\mathbf{k}=(2, 2, 2)$) is universal, many unitaries can be synthesized with other lower-duration combinations of the gate set. (c) During compilation, when presented with a two-qubit block, the lowest-duration pulse sequence containing the coordinates of that block is chosen. Inner single-qubit gates are computed via numerical optimization and outer gates are computed by the Cartan decomposition, producing a synthesis of $U_\text{block}$ using gates drawn from $\mathcal{B}$.
• Figure 2: The robustness and efficiency of the numerical invariant-matching synthesis scheme for randomly generated targets, using the Q-CTRL Boulder Opal optimizer Ball2021. (a) Distributions of the minimum converged cost for different numbers of initial points, from 10,000 random instances. (b) Box plot of the total number of optimization steps required to reach different target thresholds using a retry-if-fail approach, over 10,000 random instances.
• Figure 3: A protocol for complete, robust and efficient characterization of a controlled pulse unitary $U_\text{pulse}$. (a) A controlled pulse unitary consists of two $SU(2)$ matrices on the block diagonal, separated by an unknown phase. (b) The process for characterizing a target $U_\text{pulse}$. Single-qubit gate tomography, via any appropriate preparation and measurement sets, e.g. $R_\text{prep}, R_\text{meas} \in \{I, H, S.H\}$, is used to characterize the blocks $U_0$ and $U_1$ "simultaneously", i.e. using the same experimental configuration for each. This enables the construction of the state $\ket{\psi}$, which is applied to estimate the phase difference $2\phi$ between the two blocks via measurement of the first qubit in the $X$ and $Y$ bases ($R_\text{meas} \in \{H, S.H\}$). Curve fitting the sinusoidal relationships of different pulse iteration counts $n$ against the unitary parameters enables more accurate estimation via robustness to noise and amplification of small parameters.
• Figure 4: Numerical simulations of the controlled-pulse characterization process in the presence of shot noise and $10^{-2}$ depolarizing probability per application of $U_\text{pulse}$, and using a fixed iteration set of $\{1, 2, 4, 8\}$. (a)-(f) Example of the single-qubit tomography and fit data for one of two Haar-random $SU(2)$ matrices forming a block-diagonal unitary, using a shot count of 128 per experiment. The notation $\langle A B\rangle$ denotes state preparation in the $A$ basis and measurement in the $B$ basis. This example had a $U_\text{pulse}$ reconstruction infidelity of $1.4\times 10^{-4}$. (g) Box plot of the reconstruction infidelity (compared to the true unitary $U_\text{pulse}$) from the full fitting process to characterize $U_0$, $U_1$, and $\phi$. For each shot count, data was generated over 50 Haar-random controlled unitaries.
• Figure 5: Illustration of the waveform for a calibrated echoed cross-resonance gate. The full waveform implements a gate locally equivalent to $CX$. The raw cross-resonance waveform in the dashed box can be extracted and characterized to obtain a gate locally equivalent to $\mathop{\mathrm{\mathcal{C}}}\nolimits(\pi/4 + \delta, 0, 0)$, for some (likely small) value $\delta$ which is determined through the characterization process. The area of the raw CR waveform can also be scaled by a factor $\alpha$ to obtain, through characterization, a gate locally equivalent to $\mathop{\mathrm{\mathcal{C}}}\nolimits(\theta_\alpha, 0, 0)$ with $\theta_\alpha = \alpha \pi/4 + \delta_\alpha$. This allows approximate targeting of desirable additional angles for an extended gate set.

Theorems & Definitions (1)

• proof