Quantum Future-Trial Computing as a Geometric Primitive Resource Model: Gate-Reducible Equivalence, Hamiltonian Rollback, and Resource-Separation Hypotheses

This paper reformulates Quantum Future-Trial Computing (QFTC) as a two-regime theory separating gate-reducible quantum computation from geometric-primitive physical resource models.

The central correction is that if all QFTC components, including branch-conditioned trial unitaries, reversible predicate circuits, phase-commit operations, and rollback transformations, are uniformly decomposable into polynomial-size quantum circuits, then QFTC does not define a strict extension beyond BQP. Under this gate-reducible assumption, the paper proves that QFTC_gate = BQP.

Rather than treating this as a limitation, the paper uses it as a normalization theorem and rebuilds QFTC as a physical resource model. The geometric-primitive regime, QFTC_geom, treats reversible Hamiltonian rollback, Fubini–Study geometric action, phase-memory retention, decoherence leakage, predicate construction cost, calibration overhead, and rollback capacity as explicit computational resources.

The resulting theory does not claim automatic superiority over standard quantum computation. Instead, it proposes a resource-separation hypothesis: physically realizable rollback-induced branch transformations may, in some architectures, be implemented with lower geometric action, coherence cost, or control overhead than their compiled universal-gate equivalents.

The paper also establishes Grover-compatible constraints, avoiding false claims of unstructured-search speedup under identical oracle accounting. It provides rollback stability bounds under trace-distance error, an open-system feasibility condition involving coherence time and effective rollback error, thermodynamic consistency through reversible control cost, and experimentally testable signatures based on work-register restoration with branch-level phase retention.

In its strongest defensible form, QFTC is not presented as postselection, nonlinear quantum mechanics, time travel, or an unconditional complexity-class extension. It is presented as a geometric rollback resource theory for reversible quantum computation, where the central scientific question is whether physical rollback primitives can outperform compiled gate implementations under honest resource accounting.