AQCEL Quantum Circuit Optimization

• AQCEL Optimization Protocol is an initial-state-dependent quantum circuit optimization technique that leverages control qubits' states to eliminate unnecessary operations.
• It introduces a state label manager to track qubit states dynamically, significantly reducing measurement overhead during circuit execution.
• The CX-pair removal mechanism cuts two-qubit gate counts, thereby enhancing circuit fidelity and overall efficiency on near-term quantum hardware.

The AQCEL Optimization Protocol is an initial-state-dependent quantum circuit optimization technique designed to eliminate unnecessary controlled operations and redundant two-qubit gates by leveraging knowledge of the control qubits’ computational basis states throughout circuit execution. The methodology focuses on reducing circuit depth and gate counts, thereby significantly enhancing the fidelity and efficiency of quantum circuits running on near-term quantum hardware. The protocol's major advancements include the introduction of a state label manager that systematically tracks and utilizes qubit state information to minimize measurements, and a CX-pair removal process that eliminates redundant pairs of CNOT (CX) gates arising from standard decompositions of multi-controlled operations (Kaji et al., 5 Sep 2025).

1. Principle and Structure of AQCEL Optimization

AQCEL (Advancing Quantum Circuit by icEpp and Lbnl) is predicated on the observation that for known initial states, many controlled gates (such as CNOTs and Toffolis) are superfluous because their ^^^^0^^^^ conditions are never met during the actual circuit evolution. Whereas general-purpose optimizers maintain gate-by-gate unitary equivalence for arbitrary initial states, AQCEL preserves only the final state for the specific initial state(s) under consideration. The core workflow:

• Decompose multi-qubit controlled gates into RCCX (relative-phase Toffoli) and CX gates.
• At every controlled gate, examine the computational basis amplitudes of the control qubits immediately before the gate (via classical simulation or quantum measurement/tracking).
• If the "^^^^0^^^^" state (e.g., all control qubits in |1⟩ for a Toffoli) is absent or only present, gates can be deleted or reduced to unconditioned operations.
• Aggregate state information to minimize the number of measurements/tracking operations as the circuit advances.

This approach ensures the resulting circuit implements the same transformation on the designated initial states, with considerably reduced complexity (Jang et al., 2022).

2. State Label Management and Minimization of Measurements

A key improvement in the most recent protocol iteration is the "^^^^1^^^^" which assigns a label to each qubit reflecting its known state or logical/statistical grouping. Labels include: "0", "1", "Bell", "0/1" (superposition not flagged as Bell), and "unknown." The manager propagates these labels as the circuit evolves:

• If a controlled gate is reached and any control qubit's label is "unknown," a measurement occurs; otherwise, previous information is used.
• When a label is deducible from the circuit structure or earlier computation, further measurements are avoided.
• If a qubit's label transitions due to gate action (e.g., entanglement), the manager updates appropriately, ensuring correct inference for subsequent gates.

This systematic management reduces the need for repeated quantum measurements, thus lowering overhead and the risk of errors introduced by noisy readout. In typical circuits with substantial regularity (e.g., quantum parton shower circuits), this technique enables a large fraction of controlled operations to be eliminated or simplified without extra measurement cost (Kaji et al., 5 Sep 2025).

3. CX-Pair Removal Mechanism

During the decomposition of multi-controlled gates, paired CX (CNOT) gates acting in reverse order often appear, particularly when ancilla qubits initialized to |0⟩ act as temporary targets:

• If the target qubit remains |0⟩ between the two CXs, the CX pair constitutes an identity and is redundant.
• The protocol identifies such pairs and removes them, shifting any subsequent control dependencies back to the original controls.
• This process yields additional reductions in two-qubit gate count and circuit depth beyond what is achievable with sole state-dependent pruning.

A quantikz diagrammatic representation clarifies the simplification:

\begin{quantikz}[column sep=0.7em] \lstick{\ket{\phi}} & \ctrl{3} & \gate{\cdots} & \ctrl{3} & \qw \ \lstick{\ket{0}} & \qw & \qw & \qw & \qw \ \lstick{\ket{0}} & \qw & \qw & \qw & \qw \ \lstick{\ket{\psi}} & \gate{U} & \qw & \qw & \qw \end{quantikz}

is reduced to

\begin{quantikz}[column sep=0.7em] \lstick{\ket{\phi}} & \ctrl{3} & \qw \ \lstick{\ket{0}} & \qw & \qw \ \lstick{\ket{0}} & \qw & \qw \ \lstick{\ket{\psi}} & \gate{U} & \qw \end{quantikz}

With this mechanism, the CX count is substantially lowered, especially in circuits built with many Toffolis and sizable control-ancilla structures (Kaji et al., 5 Sep 2025).

4. Effects on Gate Counts and Fidelity

The AQCEL protocol, with both state label tracking and CX-pair removal, leads to significant reductions in two-qubit gate counts for standard quantum algorithms. For the one-step QPS (Quantum Parton Shower) circuit:

• An unoptimized circuit contained 115 CX gates.
• Standard AQCEL optimization reduced this to 23 CX gates.
• With the new CX-pair removal, the CX count dropped further to 11.

This reduction directly translates to higher fidelity on hardware. The paper reports Hellinger fidelity

$$F = \left( \sum_k \sqrt{p_k^{\rm ideal}\, q_k^{\rm opt}} \right)^2$$

where $p_k^{\rm ideal}$ and $q_k^{\rm opt}$ are output probabilities from the noiseless simulator and hardware experiments, respectively. Optimized circuits using full AQCEL achieve fidelities up to 0.83, compared to conventional techniques which saturate at significantly lower values under similar noise and threshold conditions (Kaji et al., 5 Sep 2025).

5. Application to Quantum Parton Shower Circuits

Experimental validation used the QPS algorithm, a quantum simulation of parton radiation processes in high energy physics:

• Optimized QPS circuits, especially after 2-step evolution, exhibited two-qubit gate counts at 54% of those in initial AQCEL or Qiskit-only routines with full all-to-all connectivity.
• On IBM’s ibm_fez quantum computer, the state label manager and CX-pair removal combined to produce circuits that not only ran with lower error rates (reflected in output distribution fidelity) but also resisted additional mis-optimization events even as noise thresholds were increased.

These results underline the protocol’s effectiveness for real-world quantum computing tasks where circuit depth and error rates are paramount (Kaji et al., 5 Sep 2025).

6. Broader Implications and Outlook

AQCEL’s initial-state-dependent optimization framework, especially with its recent extensions, is impactful in several ways:

• For noisy intermediate-scale quantum (NISQ) computers, the primary bottleneck is the accumulation of errors from two-qubit gates. AQCEL’s ability to suppress unnecessary CNOTs and re-order control dependencies mitigates this bottleneck.
• The protocol’s state label manager provides a basis for dynamic or real-time optimization pipelines where state knowledge can be propagated and updated throughout complex hybrid classical-quantum workflows.
• As quantum algorithms for application domains (e.g., quantum chemistry, optimization, machine learning) are increasingly tailored to stationary or structured initial states, the rationale for initial-state-dependent circuit pruning grows stronger, suggesting wide deployability for AQCEL’s techniques.

A plausible implication is that future quantum compilers will incorporate such initial-state-aware optimizations as native routines, especially as hardware characterization and measurement tracking become more robust. The AQCEL protocol thus points to a paradigm shift where circuit synthesis is not strictly input-agnostic but can exploit problem and state structure to minimize resources and maximize algorithmic fidelity.