Linear Time & Storage Simulation of Non-Clifford Circuits via Symmetric Cartesian Collapse: A Trajectory-Based Solution to the Exponential Bottleneck

<title>Abstract</title> <p> The classical simulation of <bold>quantum computing</bold> is plagued by two distinct exponential problems: the complexity of tracking non-Clifford rotations ("Magic States") and the memory explosion required to track entanglement. Current "Dense Matrix" simulators attempt to calculate every possible probabilistic outcome simultaneously, leading to immediate memory failure on large systems. We propose that this approach fundamentally misunderstands the physical nature of quantum hardware. Real quantum systems can be modeled as single, discrete trajectories driven by symmetric coordinate equivalence and probability-locked state resolution. In this work, we present a solution based on <bold>Symmetric Cartesian Collapse</bold> to the " <bold>Cartesian Vertex</bold> " <italic>[x, y, z]</italic> or. We demonstrate that by forcing a simultaneous 3-axis collapse, we preserve the probability history of the state. We prove this via the "Double Hadamard" reversibility test, showing that our stochastic collapse preserves phase coherence even without global memory, allowing for linear-time simulation of universal circuits. Normal dense matrix simulators quantum information grows exponentially limiting to only 50 ~ 60 qubits but with these <italic>Symmetric Cartesian Collapse quantum simulators, we</italic> can simulate more than thousand qubits on an 8G RAM PC in less than ten seconds. V _cartesian = (± 1, ± 1, ±1) </p>