Dimensionality reduction for closed-loop quantum gate calibration

arXiv:2412.05230 · quant-ph · Submitted 2024-12-06 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Dimensionality reduction for closed-loop quantum gate calibration".

Mira: Here's the gist of the paper. The gist: This work presents a systematic method based on SVD stacking for reducing the dimensionality of the parameter space traversed in gate calibration, starting from an arbitrary high-dimensional pulse representation.

Kai: How it works The core idea is to interface numerical gate design with calibration via a dimensionality reduction procedure designed to significantly reduce the number of pulse parameters that need to be calibrated experimentally Starting from an initial point in parameter space p0, the experimental calibration process involves selecting modified pulse parameters p0 -> p0 + δp to further minimize some experimental objective characterizing gate performance.

Mira: To analyze how a local shift in pulse parameters by δp from p0 moves the final operation in unitary space, the authors consider the Jacobian JU˜ (p0) of U˜, which provides a local linear approximation to U˜(p0 + δp).

Kai: The singular value decomposition (SVD) of JU˜ (p0), written as JU˜ (p0) = WΣV† =Xr i=1 σiwiv⊤ i, provides a geometrically motivated approach to pulse-dimensionality reduction by yielding orthogonal directions v1,..., vr in parameter space ordered according to the strength with which they impact the final unitary locally at p0.

Title and authors: Kai: So let's talk about who wrote this. Emma Berger, Vivek Maurya, Z. M. McIntyre, Ken Xuan Wei, Holger Haas, and Daniel Puzzuoli are the authors of "Dimensionality reduction for closed-loop quantum gate calibration".

Mira: They seem to be coming from a mix of places—IBM Quantum in Yorktown Heights and universities like Berkeley and Southern California—which makes sense because this is clearly a problem bridging numerical design with real hardware.

Lev: I'm interested in how they handle the disconnect between the theoretical, high-dimensional parameterizations used in design, and what actually happens when you run that pulse on a real superconducting qubit.

Kai: The title itself points to the core idea: using dimensionality reduction for closed-loop calibration of quantum gates. It’s about finding a smarter way to map the control space onto the experimental tuning space.

Mira: Think of it like this: you have a giant room full of dials, and you need to find the right settings for one specific machine, but instead of turning every single dial at once, you figure out which few dials actually matter most for that specific machine's performance.

Lev: That sounds like a necessary step before we even consider running the actual calibration sequence on the cryogenic setup.

The paper's summary: Kai: The summary of "Dimensionality reduction for closed-loop quantum gate calibration" explains that they use SVD stacking to create a geometrically motivated approach to pulse-dimensionality reduction.

Mira: Basically, they look at the Jacobian JU˜ at the optimized parameter point p0, and the singular value decomposition gives them orthogonal directions in parameter space ordered by how much they affect the final unitary locally.

Lev: So if we find these directions, v1 through vr, we only need to tune along those directions to get close to what works experimentally without having to search the whole parameter space M.

Kai: The paper points out that because the dimension of SU(d) is d squared minus one at most d squared minus one parameters are relevant for tuning up the final unitary locally at a point p0, regardless of how many original pulse parameters M there are <ref:2412.05230#pg1>.

Mira: They then take this idea further by stacking Jacobians from multiple model realizations to get an Ld2 times M matrix, and SVD-stacking that entire structure to find the most important reduced parameterization.

Lev: That addresses the issue of being too dependent on a single model realization—it makes it more robust against those uncertainties we always worry about in simulations.

The paper's improvements: Kai: One major improvement they highlight is that this method allows for generating a single-qubit Xpi/two gate that’s robust against variations in pulse amplitude and detuning <ref:2412.05230#pg1>.

Mira: They achieve this by using a three-phase calibration process: first, numerical optimization to find p0, then dimensionality reduction to get the reduced parameterization, and finally, experimental tuning of those reduced parameters.

Lev: That phase structure is key for hardware; it means we can use the simulation to guide the experiment much more effectively than just blindly sweeping parameters.

Kai: They also showed a second application for an Xpi/two gate that is robust against offresonant spectator interactions, which eliminates a specific kind of coherent error in phase-sweep spectroscopy data <ref:2412.05230#pg1>.

Mira: For that second pulse, they used objective functions constructed from first-order perturbation expansions of the unitary evolution, specifically including terms for amplitude uncertainty and cross-resonance.

Lev: When I look at the results for that spectator-robust pulse, they got an EPC of roughly seven times ten to the minus five in randomized benchmarking. That's a specific number we need to track when we design our own control sequences on the qubit.

Conclusion: Kai: So, summarizing "Dimensionality reduction for closed-loop quantum gate calibration," this paper gives us a systematic way to interface numerical gate design and calibration by systematically reducing the dimensionality of an arbitrary pulse parameterization.

Mira: The main implication is that even when you start with a very high-dimensional pulse representation, you can identify a subspace whose dimension is equal to the output space SU(d), which simplifies calibration significantly.

Lev: For us on hardware, this means we don't have to worry about tuning thousands of parameters; we just tune the few directions identified by that SVD stacking procedure.

Kai: The work shows that this approach works for two different gate designs, demonstrating robustness against amplitude errors and spectator interactions using these reduced parameterizations.

Mira: It really provides a much-needed interface between the theoretical design space and the device-based calibration reality, moving past just relying on brute-force search methods.

Lev: I think what's important is that this method gives us a concrete, low-dimensional set of parameters to tune experimentally, which is exactly what we need for practical, scalable quantum control.

Kai: So that’s the main point on "Dimensionality reduction for closed-loop quantum gate calibration". We've seen how SVD stacking helps cut down the tuning space from potentially huge dimensions to something related to SU(d).

Emma Berger, Vivek Maurya, Z. M. McIntyre, Ken Xuan Wei, Holger Haas, Daniel Puzzuoli, PhD, IBM Quantum, T.J. Watson Research Center, University of California Berkeley Department of Physics, Center for Quantum Information Science and Technology University of Southern California Department of Physics & Astronomy McGill University IBM Quantum IBM Canada Vancouver BC

IBM Quantum · University of California Berkeley · Center for Quantum Information Science and Technology · University of Southern California · McGill University · IBM Canada

quant-ph

Submitted: 2024-12-06

Updated: 2026-08-24

Comments: 15 pages, 7 figures

DOI: 10.1103/w2v4-7pvc

Code: https://github.com/danpuzzuoli/calibration-directions

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist: This work presents a systematic method based on SVD stacking for reducing the dimensionality of the parameter space traversed in gate calibration, starting from an arbitrary

Key concepts

Jacobian JU˜ (p0): This matrix approximates how a small change in pulse parameters ($\delta p$) locally affects the final unitary operation $U(\tilde{p})$. Its SVD reveals orthogonal directions in parameter space that most strongly influence the gate's performance at a specific point $p_0$. This guides which parameters are most critical for calibration.
SVD-stack: This is a method to perform dimensionality reduction across multiple numerical simulations. It involves stacking Jacobians derived from different model parameter realizations into a large matrix and then applying SVD to find the dominant, low-dimensional subspace. This helps create a robust reduced parameterization.
SU(d) Dimension: The dimension of the special unitary group SU(d) is $d^2 - 1$. The paper leverages this mathematical property to argue that at most $d^2 - 1$ parameters are locally relevant for tuning the final unitary, regardless of how many initial pulse parameters ($M$) were used in the design.
Reduced Parameterization: This is a simplified set of four control parameters ($x_i$) derived from the original high-dimensional pulse waveform. These reduced parameters control a modification $\delta p(x)$ that is applied to $p_0$. Tuning this small vector $x$ allows for efficient on-device calibration.

Terminology

Calibration — 5× in this paper · explained in 9 episodes
Parameterization — 1× in this paper · explained in 2 episodes

Summary

The gist: This work presents a systematic method based on SVD stacking for reducing the dimensionality of the parameter space traversed in gate calibration, starting from an arbitrary high-dimensional pulse representation.

How it works

The core idea is to interface numerical gate design with calibration via a dimensionality reduction procedure designed to significantly reduce the number of pulse parameters that need to be calibrated experimentally Starting from an initial point in parameter space p0, the experimental calibration process involves selecting modified pulse parameters p0 -> p0 + δp to further minimize some experimental objective characterizing gate performance. To analyze how a local shift in pulse parameters by δp from p0 moves the final operation in unitary space, the authors consider the Jacobian JU˜ (p0) of U˜, which provides a local linear approximation to U˜(p0 + δp).

The singular value decomposition (SVD) of JU˜ (p0), written as JU˜ (p0) = WΣV† =Xr i=1 σiwiv⊤ i, provides a geometrically motivated approach to pulse-dimensionality reduction by yielding orthogonal directions v1,..., vr in parameter space ordered according to the strength with which they impact the final unitary locally at p0. The dimension of SU(d) is d2 − 1, and for this reason, at most d2 − 1 parameters are relevant for tuning up the final unitary locally at a point p0, regardless of the number of pulse parameters M.

To avoid over-dependence on specific model parameters used in optimization, the authors perform dimensionality reduction on a collection of Jacobians, L = 1,..., L Jacobians, where Uci denotes the final unitary obtained from a numerical simulation performed with a given realization ci ∈ R K of K model parameters. They aggregate information by computing the SVD of the Ld2 × M matrix constructed by stacking the L Jacobians vertically, referred to as “SVD-stack” in [36]. The choice of L can be guided by the degree of confidence in the model parameters or knowledge of their range.

The reduced parameterization is then used for on-device pulse calibration by tuning a vector x = (x1,..., x4) of four parameters xi, which control the modification δp(x) to the original waveform according to δp(x) = X 4 i=1 xivi, x ∈ R 4.

Application and Results

The method was demonstrated in two applications:

Generation of a single-qubit Xπ/2 gate robust against variations in pulse amplitude and detuning.
An Xπ/2 gate robust against offresonant spectator interactions.

For the amplitude- and detuning-robust Xπ/2 pulse, the calibration process involved three phases: (1) a numerical optimization to determine pulse parameters p0, (2) dimensionality reduction of the pulse parameterization at p0 yielding the reduced parameterization, and (3) experimental calibration via tune-up of the reduced parameters. The post-calibration data, which achieves ⟨Z⟩ ≈ 0 for a large number of ϕ and N, demonstrates that the reduced parameterization was effective in tuning the experimental data.

For the spectator-robust Xπ/2 pulse, the authors showed that it eliminates the signature of spectator interactions in phase-sweep spectroscopy data. The randomized benchmarking (RB) for this pulse achieved an EPC of roughly 7 × 10−5.

Technical Details

In the amplitude- and detuning-robust Xπ/2 pulse, the numerical optimization cost function included several metrics: (i) gate infidelity F(G, U(T)), (ii) first-order perturbation-theory terms to minimize model uncertainties in Eq. (5), (iii) leakage outside the computational subspace L(U(T)) at the final time, and (iv) leakage averaged over the entire control sequence. The calibration was performed by tuning the four reduced parameters using iterated one-dimensional line searches along each of the four calibration directions.

For the spectator-robust pulse, the objective functions were constructed based on first-order perturbation expansion of U(t) in powers of H(t) and incorporated terms corresponding to amplitude uncertainty, cross-resonance, reverse cross-resonance, classical crosstalk, ZZ coupling, and frequency uncertainty. The spectator-robust pulse achieved objective values in Table I where comparable values were attained for gate fidelity and robustness properties compared to a 32-ns DRAG pulse.

Conclusion

This work introduced a method for systematically reducing the dimensionality of an arbitrary pulse parameterization, while preserving the ability to locally search the space of gates in calibration. The approach identifies a subspace with a dimension equal to that of the output space SU(d), providing a significant simplification to the calibration process when the initial pulse parameterization used for numerical gate design has a dimensionality far exceeding that of SU(d). This provides a much-needed interface between numerical gate design and device-based calibration.

Acknowledgments

The authors thank David C. McKay, Lev S. Bishop, and Emily Pritchett for helpful discussions and comments. The paper also references numerous foundational works in optimal control theory, quantum dynamics simulation tools like Qiskit Dynamics [47], and specific literature on superconducting qubit characterization.

References

[1] V. F. Krotov and I. Feldman, An iterative method for solving optimal-control problems, Eng. Cybern. 21, 123 (1983)

[2] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨uggen, and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005)

[3] P. Doria, T. Calarco, and S. Montangero, Optimal control technique for many-body quantum dynamics, Phys. Rev. Lett. 106, 190501 (2011)

[4] N. Rach, M. M¨uller, T. Calarco, and S. Montangero, Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape, Phys. Rev. A 92, 062343 (2015)

[5] S. Machnes, E. Ass´emat, D. Tannor, and F. K. Wilhelm, Tunable, flexible, and efficient optimization of control pulses for practical qubits, Phys. Rev. Lett. 120, 150401 (2412)

[6] A. Trowbridge, A. Bhardwaj, K. He, D. I. Schuster, and Z. Manchester, Direct Collocation for Quantum Optimal Control, in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), Vol. 1 (IEEE, 2023) pp. 1278–1285

[7] J. D. Chadwick and F. T. Chong, Efficient control pulses for continuous quantum gate families through coordinated re-optimization, in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), Vol. 1 (IEEE, 2023) pp. 1286–1294

[8] D. J. Egger and F. K. Wilhelm, Adaptive hybrid optimal quantum control for imprecisely characterized systems, Phys. Rev. Lett.

Improvements for AI systems

Improved on-device calibration efficiency by reducing search space dimension: "The SVD of this map, evaluated at the vector p0 of optimized pulse parameters, permits the identification of a three-dimensional subspace of R M (spanned by v1, v2, v3) to which we may restrict the calibration vector δp without restricting the corresponding change in U(p0 + δp). This allows for calibration by tuning only a reduced set of parameters where the dimension of SU(d) is d squared – 1."
Robust gate design against system drift: The method enables designing gates robust against amplitude and detuning errors, which can be achieved by minimizing objective functions including first-order perturbation-theory terms intended to minimize the impact of the model uncertainties given in Eq. (5). This allows for maintaining high fidelity even when qubit frequencies and coupling strengths drift over time.
Robust gate design against spectator interactions: The dimensionality reduction procedure allows for designing gates robust against coherent errors due to a spectator qubit, by minimizing objective functions that penalize terms like the cross-resonance term, which is quantified as s(t)U†(t)Z1U(t)X˜2(t). This results in an EPC of roughly 7 × 10−5 for the spectator-robust pulse.