Quantum von Neumann Machine and the Cost of Coherent Programmes

A quantum von Neumann machine that retains its stored programme in superposition must execute the selected operations without destroying coherence between programme branches, creating a resource requirement that cannot be assessed by examining each programme separately. I establish this distinction for programmes that select controlled-Z operations on the edges of a graph. Every classically selected programme uses only Clifford operations, yet coherent execution requires a number of injected non-Clifford T states proportional to the programme length for every fixed normalised diamond-distance tolerance below one half. For a graph with m edges and n active data vertices, the lower bound for exact execution is m + n states, whilst an explicit implementation uses at most 4m. The lower bound permits arbitrary programme-dependent phases, stabilizer workspace, measurements and adaptive control. Its extension to imperfect execution bounds both the probability of substantial resource consumption and the expected consumption, without imposing a separate limit on workspace or measurement count. The proof derives a uniform bound on Pauli expectations of the joint programme-data output and uses its stabilizer subspaces to constrain every low-resource trajectory. A supporting criterion identifies precisely when a Clifford instruction library admits an exact Clifford interpreter under the chosen binary programme encoding. These results separate the cost of interpreting coherent instructions from the cost of the operations selected by a classical programme, providing a quantitative constraint on compilation and resource estimation for stored-programme quantum computation.