⚛️ quantum physics

Improving initial-state-dependent quantum circuit optimization by introducing state labels

This paper presents two key enhancements to the AQCEL quantum-state-dependent optimizer—the state label manager and the CX-pair removal process—which significantly reduce two-qubit gate counts and improve circuit fidelity on IBM quantum hardware, as demonstrated through the quantum parton shower algorithm.

Original authors: Toshiaki Kaji, Koji Terashi, Ryu Sawada

Published 2026-02-25

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CC BY 4.0

Original authors: Toshiaki Kaji, Koji Terashi, Ryu Sawada

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 send a very delicate message across a noisy, crowded room using a series of hand signals. In the world of quantum computing, these "hand signals" are quantum gates, and the "room" is the quantum computer hardware.

The problem is that the room is incredibly noisy. Every time you make a signal (perform a gate operation), there's a chance you'll mess up. The more signals you have to make, the higher the chance your final message will be garbled.

This paper introduces a smart new way to clean up the message before you even start sending it. It's called Aqcel, and the authors have just given it a major upgrade (from version 1.0 to 2.0) to make it even better at cutting out the noise.

Here is the breakdown of their invention using simple analogies:

1. The Core Problem: The "Over-Prepared" Chef

Imagine a chef (the quantum circuit) who has to cook a meal for a party. The chef's recipe says: "If the guest is wearing a red hat, add salt. If they are wearing a blue hat, add pepper."

But, the chef knows in advance that everyone at this specific party is wearing a red hat.

Old Way (Traditional Optimization): The chef still follows the recipe exactly, checking every guest's hat and adding salt or pepper, just in case. This takes time and uses up ingredients (resources).
Aqcel's Way (State-Dependent Optimization): The chef looks at the guest list, sees everyone has red hats, and realizes, "I don't need to check for blue hats or add pepper. I can just skip that whole step."

This is what Aqcel does. It looks at the starting state of the quantum computer (the "guest list") and removes any steps that are unnecessary for that specific starting point.

2. The Upgrade: The "Smart Note-Taker" (State Label Manager)

In the first version of Aqcel (v1), the chef was very thorough but a bit clumsy. Every time a step required checking a guest's hat, the chef would stop, walk over to the guest, and ask, "What color is your hat?"

Even if the chef had just asked the same guest five seconds ago and they hadn't moved, the chef asked again. This wasted time and energy.

The New Upgrade (v2) introduces a "Smart Note-Taker":

Instead of asking every time, the note-taker keeps a simple list of labels for every guest: "Red Hat," "Blue Hat," "Unknown," or "Entangled with the person next to them."
If the chef needs to know a guest's hat color, the note-taker checks the list first.
- If the list says "Red Hat," the chef skips the question entirely and just adds the salt.
- If the list says "Unknown," then the chef goes and asks.
Why this matters: In quantum computers, "asking" (measuring) is slow and introduces errors. By remembering what they already know, the new Aqcel saves time and reduces mistakes.

3. The Second Upgrade: The "Double-Back" Cleanup (CX-Pair Removal)

Sometimes, when the chef simplifies the recipe, they accidentally leave behind a weird pair of steps that cancel each other out.

Imagine the chef adds a pinch of salt, then immediately adds a pinch of salt again, then removes a pinch of salt. The net result is just one pinch of salt, but the chef did three actions.

In quantum circuits, these are called redundant CX pairs (a specific type of gate pair).

The Old Way: The chef left these extra steps in the recipe because they didn't realize they canceled out.
The New Way: The upgraded Aqcel has a "cleanup crew." It scans the recipe, spots these canceling pairs, and deletes them.
The Result: The recipe becomes much shorter. Fewer steps mean less time in the noisy room, which means the final message is much clearer.

4. The Real-World Test: The Particle Shower

To prove this works, the authors tested their new system on a simulation of Particle Physics (specifically, a "parton shower," which is like watching a billiard ball hit other balls and scatter).

They ran the experiment on a real IBM quantum computer (a very noisy machine).

The Result: The new Aqcel (v2) cut the number of necessary steps (gates) by nearly half compared to the old method.
The Fidelity: Because there were fewer steps, the final result was much closer to the "perfect" theoretical answer. It was like sending a message through a noisy room and having it arrive 99% clear instead of 80% clear.

Summary

Think of this paper as the difference between a clumsy tourist and a local guide:

The Tourist (Old Aqcel): Asks for directions at every single intersection, even if they just asked the person standing right next to them. They also take the long way around because they don't notice shortcuts.
The Local Guide (New Aqcel): Remembers where they've been, knows which streets are closed (redundant steps), and cuts out the unnecessary detours.

By being smarter about what they already know and cleaning up the extra steps, the new Aqcel allows quantum computers to do complex calculations with much higher accuracy, even while the hardware is still a bit "noisy."

1. Problem Statement

Quantum hardware is currently limited by noise and technical constraints, making the execution of quantum algorithms costly in terms of gate errors. Two-qubit entangling gates (such as CNOT/CX) are the primary source of these errors.

The Challenge: Traditional circuit optimization techniques aim to maintain equivalence for any input state, often missing opportunities to simplify circuits when specific initial states are known.
Limitations of Previous Work (Aqcel-v1): The authors previously introduced Aqcel, an initial-state-dependent optimizer that removes redundant control operations by measuring control qubits. However, Aqcel-v1 had two main inefficiencies:
1. Redundant Measurements: It performed state measurements every time it encountered a controlled operator, even if the state was already known from previous operations, wasting execution time and resources.
2. Residual Redundancy: After removing control operations, it often left behind redundant pairs of CX gates (specifically those arising from the decomposition of multi-controlled operators) that could not be removed by the original protocol.

2. Methodology: Aqcel-v2

The paper proposes Aqcel-v2, an enhanced version of the optimizer that introduces two key mechanisms to address the limitations above. The core philosophy remains: optimize the circuit based on the equivalence of final states for specific initial conditions rather than maintaining universal circuit equivalence.

A. State Label Manager

To eliminate unnecessary measurements, the authors developed a "state label manager" that tracks the quantum state of qubits using a minimal set of labels rather than full state vectors (which are classically intractable for large systems).

Labeling Scheme: Qubits are assigned one of five labels:
- 0 or 1: Definite computational basis states.
- Bell: Indicates entanglement with other qubits (e.g., $\alpha|00\rangle + \beta|11\rangle$ ), tracking parity groups.
- 0/1: A superposition of $|0\rangle$ and $|1\rangle$ without specific entanglement.
- Unknown: Requires measurement to determine the state.
Functionality:
- The manager updates labels based on gate operations (e.g., an X gate flips 0 to 1; a CX gate propagates entanglement).
- Measurement Avoidance: If a control qubit's label is known (e.g., 0, 1, or Bell), the optimizer skips the physical measurement on the quantum hardware.
- Reversion: It tracks the history of gates to revert labels to their original state (e.g., ancilla qubits returning to |0⟩), enabling further optimizations downstream.

B. Redundant CX-Pair Removal

When Aqcel decomposes a multi-controlled operator (e.g., Toffoli) into RCCX (Relative Phase Toffoli) gates and a controlled-U gate, it often creates pairs of RCCX gates surrounding the controlled-U.

The Mechanism: If the optimization process removes a control operation from the first RCCX gate, the second RCCX gate in the pair often becomes redundant.
Execution: The state label manager verifies if the target qubit of the CX pair remains in its initial state (typically |0⟩ for ancillas) throughout the sequence. If confirmed, the pair is removed, and the control of the subsequent U gate is "moved" to the original control qubit of the pair. This reduces circuit depth and gate count.

3. Key Contributions

State Label Manager: A novel classical tracking system that minimizes the number of mid-circuit measurements required during optimization, significantly reducing the classical overhead and quantum execution time.
CX-Pair Removal Algorithm: A specific procedure to identify and eliminate redundant CX gate pairs that persist after control operation removal, further reducing two-qubit gate counts.
Benchmarking on Real Hardware: The application of these improvements to the Quantum Parton Shower (QPS) algorithm, a high-energy physics simulation, demonstrating tangible gains in fidelity and gate reduction on real IBM quantum hardware.

4. Experimental Results

The authors tested Aqcel-v2 against Aqcel-v1 and standard Qiskit optimization on the IBM ibm_fez backend (156 qubits, Heron r2 processor) using 1-step and 2-step QPS circuits.

Measurement Efficiency:
- For the 1-step QPS circuit, the number of required measurements dropped from 22 (v1) to 3–4 (v2).
- For the 2-step circuit, measurements dropped from 61 to 15–22.
- Execution time for optimization was reduced to approximately 1/7th of the original time for the 1-step circuit.
Gate Reduction:
- 1-step Circuit: Two-qubit gates reduced from 115 (original) to 23 (v1) and further to 11 (v2) on ideal topology.
- 2-step Circuit: On ideal topology, the gate count was reduced to 54% of the Aqcel-v1 result when CX-pair removal was applied.
Fidelity Improvements:
- Stability: The state label manager prevented "mis-optimization" caused by measurement noise, resulting in much more stable fidelity scores, especially at low noise thresholds.
- Accuracy: On the 2-step QPS circuit, Aqcel-v2 achieved a Hellinger fidelity of 0.83 on real hardware (compared to lower scores for v1 and Qiskit-only).
- Physical Observables: The distribution of particle emissions simulated by Aqcel-v2 on real hardware closely matched the ideal simulator distribution, significantly outperforming Qiskit-only optimization (which showed 8% unphysical results).
- Quantum Interference: The experiment successfully captured quantum interference effects (by varying coupling constants $g_{12}$ ), proving that small-scale processors can simulate complex quantum dynamics when optimized effectively.

5. Significance and Conclusion

This paper demonstrates that initial-state-dependent optimization is a viable and powerful strategy for near-term quantum devices (NISQ).

Efficiency: By intelligently tracking qubit states via labels, the method drastically reduces the resource cost of the optimization process itself.
Performance: The removal of redundant CX pairs directly addresses the primary source of hardware errors, leading to higher fidelity results on real quantum computers.
Future Outlook: While Aqcel-v2 currently handles computational basis states ( $|0\rangle, |1\rangle$ ) and specific entanglement patterns, the authors note that extending the label system to handle X-basis states ( $|+\rangle, |-\rangle$ ) could unlock even greater optimization potential. The work provides a roadmap for enhancing quantum algorithm performance on current hardware without requiring full error correction.