5-Local Hamiltonian is QMA-complete

First use [F9] to amplify the source verifier to completeness at least $3\mathrm{/}4$ and soundness at most $1\mathrm{/}4$. Compile each copy's terminal predicate into a reversible circuit with one accept output, and include its input-copy CNOTs and clean predicate workspace in a fixed polynomial gate envelope $T_1(n)\ge n+1$. Fix a constant $C$ large enough that a second amplification with $r=O(\log (C(T_1(n)+1{)}^2))$ repetitions, together with its reversible majority predicate, has gate count at most $T_{max}(n):=C(T_1(n)+1{)}^2$; such a $C$ exists because $\log (C(T_1+1{)}^2)\le O(\log C+\log (T_1+1))$ and $T_1\log (T_1+1)\le O((T_1+1{)}^2)$. Put $N_{max}=T_{max}(n)+1$ and $\delta (n)=1\mathrm{/}(64N_{max}^3)$. This is a polynomial-time computable rational target, and the second amplification has completeness at least $1-\delta (n)$ and soundness at most $\delta (n)$. For each input $x$, form an input-specific circuit $V_x$ with fresh scratch input wires initially zero, and count the root input register, every copied-input register, all predicate workspace and every other initially zero work wire among the clean ancillas $A$. Its prefix loads $x$ into the root input register using $X=H{T}^{4}H$; the CNOT fanout in [F9] then prepares a separate $\mathrm{\mid }x⟩$ in each verifier-copy input register. Run the amplified verifier gates, and reversibly compute their complete acceptance predicate, specialized to this fixed $x$, into one output qubit. Thus $H_{in}$ enforces zero on each classical-input scratch register at time zero, and the prefix and fanout set every verifier input to this fixed $x$ before verification. The input-specific full gate list is generated in polynomial time from $x$, and the envelope $T_{max}$ includes these prefix, fanout and predicate gates. If the verifier has zero gates, include one idle identity time slot in the Hamiltonian list so its padded length $T\ge 1$ without changing acceptance. Construct the unary history Hamiltonian with $H=H_{in}+H_{prop}+H_{out}+J_{clock}{H}_{clock}$. Every propagation term uses at most two work qubits and a three-bit clock window by [F3]; input, output and clock penalties are at most 2-local.

For a YES input, [F12] lets the twice-amplified verifier use a pure witness accepted with probability at least $1-\delta (n)$; let $\mathrm{\mid }\eta ⟩$ be its normalized history. Input, propagation and clock energies vanish; [F5] gives output energy at most $\delta (n)\le 1\mathrm{/}(64N^3)$ because $T\le T_{max}(n)$. Thus ${\lambda }_{min}(H)\le a:=\delta (n)$. For a NO input, [F6] gives legal-clock ground energy at least $1\mathrm{/}(2N^3)$. The norm of the non-clock sum is at most $K=T+\mathrm{\mid }A\mathrm{\mid }+N$ because each propagation edge has norm at most one, the input sum has norm at most the number $\mathrm{\mid }A\mathrm{\mid }$ of clean ancillas, and $\mathrm{\parallel }{H}_{out}\mathrm{\parallel }=N$. Choose the explicit integer $J_{clock}=2K+8N^3{K}^2+1$; then [F7] gives illegal-clock projection error less than $1\mathrm{/}(8N^3)$, so ${\lambda }_{min}(H)\ge 3\mathrm{/}(8N^3)>b:=1\mathrm{/}(4N^3)$. Since $a\le 1\mathrm{/}(64N^3)$, the promise gap is at least $15\mathrm{/}(64N^3)$, inverse polynomial. Each matrix term and its norm bound is computed from the finite gate list and polynomial weights.

The verifier gate alphabet has fixed algebraic matrix entries, so the generated local terms have effectively encoded algebraic entries of polynomial description length. Apply [F8] with $\tau \le (b-a)\mathrm{/}8$ and output rationalized terms with thresholds $a+\tau$ and $b-\tau$. The resulting promise gap is at least $3(b-a)\mathrm{/}4$, and the number, bit length and norm of the rational terms remain polynomial. This is a deterministic polynomial-time map preserving YES and NO by [F11]; [F2] makes the problem QMA-hard. Membership is [F1], so it is QMA-complete.