Superconducting Qubit Control Pulses

• Superconducting qubit control pulses are specialized waveforms engineered to manipulate quantum states with high fidelity and minimal error using DRAG techniques.
• They integrate a VSM-based, nanosecond-routing architecture that enables independent, scalable broadcasting of tailored pulses to same-frequency qubits.
• Empirical results from randomized benchmarking demonstrate average fidelities over 0.998 per qubit, meeting fault-tolerance thresholds in quantum processors.

Superconducting qubit control pulses are specialized time- and frequency-domain waveforms designed to manipulate the quantum states of superconducting qubits with high fidelity and minimal error. The efficient generation, routing, and tailoring of these pulses—while minimizing hardware overhead and error rates—constitutes a cornerstone of scalable quantum processor engineering. Comprehensive strategies have been developed to achieve independent, simultaneous, and extensible control of multiple qubits sharing the same resonance frequency, with particular attention to minimizing leakage errors in weakly anharmonic devices such as transmons.

1. Hardware Architecture and Pulse Routing

A central architectural innovation for superconducting qubit control involves the reuse of a single microwave control chain across multiple same-frequency qubits. This is enabled by integrating a room-temperature vector switch matrix (VSM) that accepts multiple analog inputs and enables nanosecond-timescale, digitally triggered routing of control pulses to individual qubit drive lines. The VSM is controlled by digital marker signals that determine, for each qubit, which pulses are delivered and which are suppressed. Each control link (input-output channel) can be independently tuned in amplitude and phase, thereby providing the extensibility and individualization required for hardware-efficient designs in two-dimensional surface code lattices and other scalable architectures (Asaad et al., 2015).

Key configuration features:

• Analog inputs: Separate channels for Gaussian (in-phase) and derivative-of-Gaussian (quadrature) envelopes.
• Fast switching: Nanosecond digital enable lines per output.
• Individual control: Amplitude and phase tuning for any input–output combination.
• Scalability: Supports arbitrary groups of same-frequency qubits, critical for repeated unit cell motifs.

2. Pulse Engineering: Quadrature Tailoring and Leakage Suppression

High-fidelity gate operations in weakly anharmonic superconducting qubits (such as transmons) require suppression of population leakage to higher non-computational states—primarily the second excited state. The widely utilized derivative removal by adiabatic gate (DRAG) scheme constructs the control pulse as a sum of:

• A Gaussian envelope in the in-phase quadrature
• A derivative-of-Gaussian envelope in the quadrature component

The VSM enables these quadratures to be routed and tuned independently for each qubit—even when all qubits share the same carrier frequency. This enables the optimization of DRAG parameters (such as the weight of the derivative term) to minimize leakage and phase errors for each drive line individually, a requirement for practical deployment in large ensembles of identically fabricated (but individually varying) transmons.

3. Pulse Broadcasting Schemes

Three pulse broadcasting strategies are compared for executing single-qubit Clifford gate operations on multiple same-frequency qubits:

Scheme	Pulse Count Scaling	Control Individuality	Compilation Complexity
Sequential	Linear in number of qubits, $O(n)$	Full	Low (trivial)
Compiled	Variable, minimizes total pulses	Full	High (exponential in $n$)
5-Primitives (Selective Broadcasting)	Constant (exactly 5 per round; independent of $n$)	Full	Low (precompilation, fixed mapping)

• Sequential: Each qubit’s gate decomposition is broadcast in sequence—time overhead $\sim$ number of qubits.
• Compiled: Pulse primitives for all target Clifford operations are merged, with simultaneous application when shared pulses are found; minimum total pulses but combinatoric compilation cost.
• 5-Primitives Selective Broadcasting: A fixed sequence of five pulse primitives is broadcast every round. Digital markers enable or disable routing of each pulse to each qubit as needed to effect any of the 24 single-qubit Clifford gates. The overhead is strictly constant—five pulses per cycle—regardless of the number of qubits or gate diversity. This approach is hardware-optimal for scaling (Asaad et al., 2015).

4. Selective Broadcasting with Five Pulse Primitives

The five-primitives scheme utilizes a fixed, ordered set of five calibrated pulses. Each single-qubit Clifford gate is represented as a specific "pattern" (selection) of these primitives. For a given control round:

• Each qubit’s drive line receives only the pulses toggled "on" by its associated marker bits
• If a gate requires just the second and fourth primitives, markers for those pulses are set high for the corresponding qubit while others are held low
• All qubits can thus receive independent, simultaneous, and arbitrarily chosen Clifford gates with just five time-steps, without any scaling in pulse count or control hardware

This method leverages a precompiled mapping and is ideally suited for architectures requiring scalable, low-overhead control (e.g., surface-code fault-tolerant quantum computing). It allows control system resources—AWGs, microwave sources, etc.—to scale sublinearly with the number of qubits.

5. Empirical Performance and Error Characterization

Randomized benchmarking (RB) is employed to rigorously evaluate the gate fidelity and error rates for all three broadcasting approaches under single- and multi-qubit operation. Key findings:

• Average Clifford fidelities: $0.9982$–$0.9986$ per qubit; these values comfortably exceed the $0.99$ surface-code threshold.
• Fidelity in parallel operation: No significant degradation in error rates is observed when multiple qubits are driven simultaneously using selective broadcasting (including the five-primitives scheme). The limiting process is identified as qubit relaxation ($T_1$ decay), not control infidelity.
• Leakage to $|2\rangle$: Using a modified RB protocol (with final population-swapping $\pi$ pulse), leakage rates per Clifford are measured at $O(10^{-6})$—negligible in comparison to intrinsic gate errors.
• Cross-excitation (cross-driving): Actively monitored and found to be small; during simultaneous broadcasting, idling fidelities remain consistent with $T_1$ decay alone.
• Fidelity formula under $T_1$ decay:

$$F_{\mathrm{C}}^1 \simeq \left( \frac{1}{6}\left[3 + 2 e^{-t_{\mathrm{p}}/(2T_1)} + e^{-t_{\mathrm{p}}/T_1}\right] \right)^{\langle N_p \rangle}$$

where $t_{\mathrm{p}}$ is the pulse duration and $\langle N_p \rangle$ is the average primitive pulse count per Clifford.

6. Implementation and Scalability Considerations

Implementation relies on generating and synchronizing two analog quadrature waveforms for each independent drive (for the DRAG method), with digital marker synchronization via the VSM. The architecture:

• Streamlines hardware resource requirements by reusing microwave sources across multiple same-frequency qubits
• Supports extension to large units cell-based lattices (e.g., surface code), with the potential for further abstraction and automation in pulse pattern management as array size increases
• Exposes marker-control digital signals for external or software-driven synchronization, facilitating integration into classical control stacks for quantum hardware
• Allows for system-wide calibration and direct per-qubit pulse adjustment, critical for mitigating fabrication-induced parameter spreads

The five-primitives scheme in particular is compatible with quantum error correction thresholds and supports independent, simultaneous operation on arbitrary-sized ensembles without increased overhead.

7. Summary and Outlook

The selective broadcasting VSM-based architecture provides an efficient, independent, and extensible strategy for controlling large arrays of same-frequency superconducting qubits. By employing individually optimized DRAG pulse quadratures, and efficient pulse compilation and broadcasting schemes (with the five-primitives method yielding constant-time, hardware minimality), the approach achieves high single-qubit gate fidelities well within fault-tolerance requirements. Randomized benchmarking confirms that the scalability advantages do not sacrifice performance. This framework defines a viable path for scaling superconducting quantum processors, emphasizing hardware reuse, low-latency nano-second routing, and per-qubit pulse optimization (Asaad et al., 2015). The approach remains influential in the design of modern quantum information processors, where hardware constraints and noise mitigation must be balanced against the demands of algorithmic flexibility and fault-tolerant performance.