<p>The integrity of quantum gates (QG) under environmental noise and non-equilibrium conditions is a crucial challenge for scalable quantum computing. Moving beyond traditional metrics, we introduce the Wigner-algebra parity operator as a powerful, model-independent tool to directly measure quantum coherence degradation. We develop this by constructing a parity-deformed Lindblad master equation for a squeezed harmonic oscillator (SHO), representing a dissipative qubit. The oscillator is limited to its two lowest energy levels, and its non-equilibrium thermal state is described within the framework of Tsallis statistics. We then evaluate the performance of a Hadamard gate affected by this combined decoherence. Our main finding is that the parity deformation parameter, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\upsilon$</EquationSource> <EquationSource Format="MATHML"><math> <mi>υ</mi> </math></EquationSource> </InlineEquation>, acts as an algebraic reference and a fine-tuning parameter for system-environment entanglement. We show that the range <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0\le \upsilon <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>υ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is a critical regime where the system evolves toward a pure state, while outside this range, purity is lost. Importantly, we find that the Wigner negativity, conditioned on parity, provides a direct measure of non-classicality and coherence loss during gate operation. Numerical simulations indicate that increasing <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\upsilon$</EquationSource> <EquationSource Format="MATHML"><math> <mi>υ</mi> </math></EquationSource> </InlineEquation> not only slows down entanglement dissipation but also broadens the duration of entanglement peaks, improving gate stability. Additionally, the post-gate entanglement landscape displays a distinct, parity-dependent noise pattern. Our analysis reveals a direct trade-off between gate fidelity and environmental entanglement, which can be effectively controlled by tuning the parity deformation parameter. This work introduces a new approach for diagnosing and reducing decoherence in quantum processors by using algebraic deformation to identify critical thresholds where quantum interference is eliminated by thermal and squeezing noise. In other words, our model allows us to dissect and understand decoherence channels in a large class of oscillator-based qubits. By tuning <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\upupsilon$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">υ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$q$</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation>, we can theoretically predict how changes in qubit design or environmental engineering (which effectively alter these parameters) would impact gate fidelity.</p>

Hadamard Gate in Non-equilibrium Conditions Under Parity Defined in the Wigner Algebra

期刊论文
发表日期：

2026年2月20日

查看全文

E. Eilkhas,
B. Lari,
H. Hassanabadi

摘要

The integrity of quantum gates (QG) under environmental noise and non-equilibrium conditions is a crucial challenge for scalable quantum computing. Moving beyond traditional metrics, we introduce the Wigner-algebra parity operator as a powerful, model-independent tool to directly measure quantum coherence degradation. We develop this by constructing a parity-deformed Lindblad master equation for a squeezed harmonic oscillator (SHO), representing a dissipative qubit. The oscillator is limited to its two lowest energy levels, and its non-equilibrium thermal state is described within the framework of Tsallis statistics. We then evaluate the performance of a Hadamard gate affected by this combined decoherence. Our main finding is that the parity deformation parameter, $\upsilon$ $υ$ , acts as an algebraic reference and a fine-tuning parameter for system-environment entanglement. We show that the range $0\le \upsilon <1$ $0 \leq υ < 1$ is a critical regime where the system evolves toward a pure state, while outside this range, purity is lost. Importantly, we find that the Wigner negativity, conditioned on parity, provides a direct measure of non-classicality and coherence loss during gate operation. Numerical simulations indicate that increasing $\upsilon$ $υ$ not only slows down entanglement dissipation but also broadens the duration of entanglement peaks, improving gate stability. Additionally, the post-gate entanglement landscape displays a distinct, parity-dependent noise pattern. Our analysis reveals a direct trade-off between gate fidelity and environmental entanglement, which can be effectively controlled by tuning the parity deformation parameter. This work introduces a new approach for diagnosing and reducing decoherence in quantum processors by using algebraic deformation to identify critical thresholds where quantum interference is eliminated by thermal and squeezing noise. In other words, our model allows us to dissect and understand decoherence channels in a large class of oscillator-based qubits. By tuning $\upupsilon$ $υ$ and $q$ $q$ , we can theoretically predict how changes in qubit design or environmental engineering (which effectively alter these parameters) would impact gate fidelity.