We propose a compound quantum architecture for broadband axion dark matter detection combining three independently validated components: quantum phase estimation (QPE) for sub-SQL photon counting (~20× per-step sensitivity), Grover amplitude amplification (~212× search speedup over 180,000 tuning steps), and entangled multi-cavity networks for further scaling. The single-cavity compound yields ~4,200× scan speed advantage over classical ADMX pipelines. All claims are supported by 10 executable companion scripts validated against published results. SSPE circuit: 32 qubits, 525 gates.