Topology-Aware Block Coordinate Descent for Qubit Frequency Calibration of Superconducting Quantum Processors

TL;DR

Calibration of qubit frequencies in superconducting processors is bottlenecked by crosstalk and exponential search space. The authors prove the Snake optimizer is equivalent to Block Coordinate Descent (BCD) and cast the block ordering as a Sequence-Dependent Traveling Salesman Problem (SD-TSP) solved by a nearest-neighbor algorithm, using a reduced local objective $G_{B_j}$ defined via a crosstalk footprint. They demonstrate $O(N)$ per-epoch scaling under local crosstalk and provide convergence analyses and robustness claims supported by simulations, culminating in an implementation-ready workflow for NISQ-era calibration. The work offers scalable, topology-aware calibration that maintains accuracy while dramatically reducing runtime, enabling larger superconducting quantum processors to be calibrated more efficiently.

Abstract

Pre-execution calibration is a major bottleneck for operating superconducting quantum processors, and qubit frequency allocation is especially challenging due to crosstalk-coupled objectives. We establish that the widely-used Snake optimizer is mathematically equivalent to Block Coordinate Descent (BCD), providing a rigorous theoretical foundation for this calibration strategy. Building on this formalization, we present a topology-aware block ordering obtained by casting order selection as a Sequence-Dependent Traveling Salesman Problem (SD-TSP) and solving it efficiently with a nearest-neighbor heuristic. The SD-TSP cost reflects how a given block choice expands the reduced-circuit footprint required to evaluate the block-local objective, enabling orders that minimize per-epoch evaluation time. Under local crosstalk/bounded-degree assumptions, the method achieves linear complexity in qubit count per epoch, while retaining calibration quality. We formalize the calibration objective, clarify when reduced experiments are equivalent or approximate to the full objective, and analyze convergence of the resulting inexact BCD with noisy measurements. Simulations on multi-qubit models show that the proposed BCD-NNA ordering attains the same optimization accuracy at markedly lower runtime than graph-based heuristics (BFS, DFS) and random orders, and is robust to measurement noise and tolerant to moderate non-local crosstalk. These results provide a scalable, implementation-ready workflow for frequency calibration on NISQ-era processors.

Figures (6)

• Figure 1: Flowchart of algorithm and results. Arrangement of qubits and crosstalk hypothesis are sent to the Block coordinate descent(BCD) algorithm. How to block parameters and how to determine the reduced cost function are defined in this algorithm, then the complexity of BCD algorithm can be determined. With nearest-neighbor algorithm(NNA) for sequence-dependent traveling salesman problem(TSP), a optimized strategy with less complexity is found. Parameters leading to higher fidelity will be chosen by BCD algorithm. Due to the different relationship between crosstalk hypothesis and actual crosstalk, the fidelity may increase strictly or non-strictly.
• Figure 2: Optimization performance. (a)We employ probability density functions to describe the distribution of our optimization metric(the average gate error rate) before and after optimization with BCD-NNA, starting from randomly chosen initial points under different error models. (b,c) For a fixed error model, distributions of the average gate error rate before and after optimization when starting from different randomly selected initial points. During optimization in panel (c), the evaluations exhibit a relative standard deviation of 0.2, whereas no such variability is present in panel (b). (d,e,f) Demonstration of the optimization process showing the actual average gate error rate varies with the number of optimization steps starting from different initial points in panel (a). The curves show the actual average gate error rate, while the optimization itself uses evaluations with different relative standard deviations.
• Figure 3: Impact of nonlocal crosstalk errors on the optimization process In the numerical simulation, we altered the maximum range of random selection for nonlocal crosstalk parameters in the error simulator, incrementally increasing it from (a) to (d), while maintaining the random selection ranges for other local crosstalk parameters unchanged. By repeatedly selecting parameters within their respective ranges to determine the error model, we optimized it using different optimization algorithms. The probability density plots show the distribution of differences in average gate error rates, calculated as the results from crosstalk-unaware optimization minus those from crosstalk-aware optimization.
• Figure 4: NNA for BCD (a) Probability density plot of the difference in average gate error rate, calculated as BCD-random minus BCD-NNA. (b) Probability density plot of the algorithmic complexity ratio, calculated as BCD-random divided by BCD-NNA, demonstrating markedly lower complexity for BCD-NNA. In both (a) and (b), the optimization algorithms start from random initial points of random error model parameters. The difference between the two optimization algorithms stems solely from the distinct optimization sequences resulting from their parameter block strategies. For each optimization under the same error model parameter, BCD-NNA selects a path derived from the NNA within the strategy subspace, while BCD-random makes random selections within the strategy subspace each time.
• Figure 5: Algorithm scaling with increasing number of qubits. (a) Optimization of quantum chips with varying numbers of qubits using the BCD-NNA algorithm: comparison of circuit error rates before and after optimization. (b) Algorithmic complexity comparison under the search-space complexity model ($S_i = k^{|B_i|}$ with $k=100$), following the theoretical framework of Ref. Klimov2020. This model represents the worst-case complexity for derivative-free optimization over all frequency configurations. Under this model, BCD-NNA achieves substantial complexity reduction compared to graph-based heuristics (BFS, DFS) employed in the original Snake implementation Google2024, demonstrating the significant benefit of the SD-TSP formulation for optimizing block order.

Theorems & Definitions (1)

• Theorem 1