Quantum-Adaptive KS($\varphi$): A Parameterized Three-Qubit Gate Family Embedding Toffoli with Measurement-Free Phase Kickback and Intrinsic Error Non-Amplification

Abstract

We introduce Quantum-Adaptive KS($\varphi$) ($K$ = kickback, $S$ = sandwich), a parameterized three-qubit gate family that structurally embeds the Toffoli (CCX) gate within two additional components: (1)a palindromic Hadamard sandwich on the first control qubit $q_0$ that conjugates $Z$-type errors to $X$-type in the CCX frame, providing simultaneous sensitivity to both error types without ancilla overhead; and (2)a controlled-phase (CP) gate whose quantum phase kickback propagates post-CCX target-state information into the control-qubit phase without measurement. The term Quantum- Adaptive refers to amplitude steering conditioned by the compile-time parameter $\varphi$ via a Quantum Neural Cellular Automaton (QNCA) majority-inspired bias rule; the gate does not self-modify at runtime. Two QA-KS($\pi$) gates chained on a shared control qubit $q_0$ produce outputs completely orthogonal to two sequential CCX gates on $q_0$=1 inputs (output fidelity F=0.000), while agreeing exactly on $q_0$=0 inputs (F=1.000). This subspace-dependent divergence is the direct computational signature of coherent phase retention across gate boundaries -- impossible for CCX-only circuits. On the $q_1$ = 0 subspace the gate acts deterministically (up to a relative phase), providing intrinsic error non-amplification. On the $q_1$ = 1 subspace it produces four-component entangled superpositions, making it a strictly distinct quantum-native primitive from CCX. We present the complete $8 \times 8$ unitary matrix, confirmed exact to $||U^{\dagger}U-I||_{\infty} < 10^{-15}$, and define two canonical variants: QA-KS$_{\pi/2}$ ($\varphi = \pi/2$, $S$ gate) and QA-KS$_{\pi}$ ($\varphi = \pi$, $Z$ gate). Qiskit depolarizing-noise simulation demonstrates near-unit fidelity at $p \leq 10^{-2}$ with an honest depth cost at higher error rates. The gate preserves the three-qubit footprint of CCX with no qubit overhead.