Distilling Unitary Operations: A No-Go Theorem and Minimal Realization

• URL: http://arxiv.org/abs/2604.01048v1
• Date: Wed, 01 Apr 2026 15:53:45 GMT
• Title: Distilling Unitary Operations: A No-Go Theorem and Minimal Realization
• Abstract summary: Quantum gates executed on physical hardware are inevitably degraded by environmental noise.<n>We show that a 3-slot architecture provides the minimal realization for non-trivial universal purification.<n>Our results establish the strict theoretical boundaries of distilling clean operations from noisy gates.
• Score: 9.169822148498062
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum gates executed on physical hardware are inevitably degraded by environmental noise. While state purification effectively distills static quantum resources, the dynamic execution of quantum algorithms requires a higher-order approach to mitigate errors on the operations themselves. In this work, we investigate unitary purification: the task of utilizing a quantum higher-order operation to partially restore the ideal action of an unknown unitary corrupted by a known noise model. Focusing on canonical depolarizing noise, we first reveal a fundamental operational obstruction. We prove that within the indefinite causal order framework, no nontrivial 2-slot higher-order operation can universally purify the set of single-qubit unitaries. Overcoming this strict limitation, we establish that a 3-slot architecture provides the minimal realization for non-trivial universal purification. We analytically derive the optimal average fidelity for the 3-slot regime, demonstrating that it strictly surpasses trivial strategies by systematically utilizing ancillary qubits as a quantum memory to absorb errors. Furthermore, we provide a concrete quantum circuit construction for this optimal higher-order operation. Our results establish the strict theoretical boundaries of distilling clean operations from noisy gates, offering immediate architectural insights for robust gate design.

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