⚛️ quantum physics

Quantum Fourier Transform using Dynamic Circuits

This paper demonstrates the resource efficiency and practical advantages of using dynamic quantum circuits to implement the Quantum Fourier Transform on IBM hardware, achieving record-breaking process fidelities on up to 37 qubits through a novel "feed-forward-compensated dynamical decoupling" protocol and an efficient fidelity certification method.

Original authors: Elisa Bäumer, Vinay Tripathi, Alireza Seif, Daniel Lidar, Derek S. Wang

Published 2026-07-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Elisa Bäumer, Vinay Tripathi, Alireza Seif, Daniel Lidar, Derek S. Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive puzzle, but instead of using a single, super-smart brain that tries to hold every piece in its mind at once, you have a team of helpers who can shout out answers to each other as they go. This is the world of quantum computing, a field where machines use the strange rules of the subatomic world to solve problems that would take normal computers thousands of years. The key players here are qubits, the tiny building blocks of these computers. Unlike regular computer bits that are either a 0 or a 1, qubits can be both at the same time, like a spinning coin that is heads and tails until you catch it.

However, these spinning coins are incredibly fragile. If you try to do too many things with them at once, or if they have to wait around too long, they get confused and the answer falls apart. This is where dynamic circuits come in. Think of a standard quantum program like a recipe written on a single piece of paper that you must follow from start to finish without looking up. A dynamic circuit is more like a cooking show where the chef tastes the sauce halfway through, decides if it needs more salt, and then immediately adds it before moving to the next step. This ability to measure a qubit, read the result, and instantly use that information to change what happens next is a game-changer. It allows the computer to be smarter and faster, but only if the team can coordinate perfectly without dropping the ball.

This paper is about a specific, famous recipe called the Quantum Fourier Transform (QFT). In the old way of doing things (called the "unitary" method), the QFT is like a giant dance where every dancer has to hold hands with every other dancer to get the right rhythm. As you add more dancers (qubits), the number of hand-holds explodes, making the dance incredibly hard to keep up with and prone to mistakes. The researchers in this paper asked: "What if we could skip the hand-holding and just have the dancers shout their positions to the next person?" They found that by using dynamic circuits—measuring the dancers and feeding that information forward—the complexity drops dramatically. Instead of needing a chaotic web of connections, they only need a simple line of communication.

The team, working on IBM's superconducting quantum hardware, put this idea to the test. They built two versions of the QFT: the old, complicated hand-holding dance and the new, shouting dynamic version. To make sure the shouting didn't get drowned out by noise, they invented a special technique they call "feed-forward-compensated dynamical decoupling" (FC-DD). You can think of this as a noise-canceling headset for the qubits. While the computer is waiting for the "shout" (the measurement result) to come back, the qubits usually just sit there and get jostled by the environment. The FC-DD protocol gently taps the qubits with specific pulses to keep them steady and focused during this waiting time, ensuring they don't lose their place.

The results were a clear victory for the new method. Without these noise-canceling tricks, the performance of both methods crashed as the number of qubits grew, dropping below 1% accuracy after just 9 qubits. But with the new FC-DD technique, the dynamic circuit version soared. The researchers achieved a process fidelity (a measure of how close the result is to the perfect answer) of over 50% on up to 16 qubits, and still managed to stay above 1% on up to 37 qubits. In contrast, the old unitary method struggled to get past 11 qubits with similar accuracy.

To visualize this, the team ran a test with 10 qubits. They prepared a specific pattern and asked the computer to transform it. The old method produced a blurry, flat mess where the answer was hard to spot. The new dynamic method, however, produced a sharp, clear peak that looked almost exactly like the perfect, theoretical answer. The paper suggests that this approach doesn't just work for this one specific dance; it opens the door to compiling much larger and more complex quantum algorithms efficiently. By proving that dynamic circuits can drastically reduce the resources needed and improve accuracy, the researchers have shown a promising path toward making quantum computers truly powerful tools for the future.

Technical Summary: Quantum Fourier Transform using Dynamic Circuits

Problem Statement
The Quantum Fourier Transform (QFT) followed immediately by measurement (QFT+M) is a fundamental subroutine in critical algorithms such as Shor's algorithm and quantum phase estimation. In the standard unitary formulation, an $n$ -qubit QFT requires $O(n^2)$ two-qubit gates with all-to-all connectivity. On current hardware with limited connectivity, transpiling this circuit necessitates additional SWAP gates, increasing circuit depth and resource overhead, which exacerbates noise and decoherence. While dynamic quantum circuits—capable of mid-circuit measurement and classical feed-forward—offer a theoretical pathway to reduce these requirements, their practical advantage on noisy intermediate-scale quantum (NISQ) hardware has been limited by error rates and a lack of robust certification methods for processes involving measurements.

Methodology
The authors experimentally implement the QFT+M subroutine on IBM's superconducting quantum hardware (specifically the ibm_kyiv processor) using two approaches:

Unitary Implementation: The standard circuit requiring $O(n^2)$ two-qubit gates.
Dynamic Implementation: A circuit utilizing the "semi-classical fan-out" capability. By applying the principle of deferred measurement, conditional quantum operations are replaced with classically controlled operations based on mid-circuit measurement outcomes. This formulation eliminates the need for all-to-all connectivity and reduces the scaling of two-qubit gates to $O(n)$ mid-circuit measurements.

To address the noise inherent in these operations, the authors employ Dynamical Decoupling (DD):

For the unitary circuit, they use the universally robust (UR10) pulse sequence during idle periods to suppress pulse errors.
For the dynamic circuit, they introduce a novel protocol called Feed-Forward-Compensated Dynamical Decoupling (FC-DD). This protocol accounts for the specific timing constraints of dynamic circuits, where pulses cannot be applied during the feed-forward latency ( $\tau_{FF} \approx 653$ ns). The FC-DD sequence inserts an $X_2$ sequence (two uniformly spaced $X$ pulses) at the boundaries of the feed-forward window and utilizes a uniformly spaced $XY_4$ sequence in the remaining idle time ( $\tau \approx 591$ ns) between the measurement pulse and the feed-forward operation.

Key Contributions

Efficient Process Fidelity Certification: The authors derive a general expression for the process fidelity of a noisy unitary followed by a projective measurement. They propose an efficient sampling scheme that avoids the exponential cost of full state tomography. This method allows for the estimation of fidelity by preparing specific initial states ( $\sigma^*_k = \text{QFT}^\dagger |k\rangle$ ) and measuring output probabilities, leveraging virtual $R_Z$ gates to prepare these states noiselessly.
FC-DD Protocol: The development of a context-aware error suppression strategy specifically designed for the latency constraints of mid-circuit measurements and feed-forward operations in dynamic circuits.
Experimental Demonstration: The first experimental demonstration of the QFT+M advantage on dynamic circuits across a wide range of qubit counts (up to 40 qubits) on superconducting hardware.

Results
The experiments demonstrate a significant performance gap between the dynamic and unitary implementations, particularly when error suppression is applied:

Without Error Suppression: Both implementations suffer rapid fidelity decay, dropping below 1% for $n > 9$ qubits, rendering the dynamic advantage negligible in this regime.
With Error Suppression (FC-DD vs. UR10):
- The dynamic circuit implementation with FC-DD achieves certified process fidelities > 50% for up to 16 qubits and > 1% for up to 37 qubits.
- In contrast, the unitary implementation with standard DD achieves comparable performance only up to approximately 11 qubits.
- For a 10-qubit test case, the dynamic circuit achieved an 88% fidelity compared to 6% for the unitary circuit.
Visual Verification: When applied to a periodic state, the dynamic circuit produced a sharply peaked distribution closely resembling the ideal case, whereas the unitary circuit resulted in a flattened distribution with broad peaks, consistent with the fidelity metrics.

Significance and Claims
The paper claims that leveraging dynamic circuits allows for a drastic reduction in resource requirements for the QFT+M primitive, scaling from $O(n^2)$ two-qubit gates to $O(n)$ mid-circuit measurements without connectivity constraints. The authors assert that their results set new benchmarks for process fidelity across all quantum computing platforms to date.

The significance of this work lies in demonstrating that dynamic circuits can optimize the compilation of core quantum algorithmic primitives. The authors emphasize that while the assumption of low crosstalk (due to decoupled qubits in this specific QFT implementation) is crucial for the success of FC-DD, useful cases exist where this holds, such as when using ancilla qubits or multiple registers in algorithms like Shor's. The work suggests that refining error suppression and mitigation techniques for mid-circuit operations is essential for unlocking the full potential of dynamic circuits in compiling larger-scale quantum algorithms.

Technical Summary: Quantum Fourier Transform using Dynamic Circuits

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