Quantum Advantage for All

We show how to translate a subset of RISC-V machine code compiled from a subset of C to quadratic unconstrained binary optimization (QUBO) models that can be solved by a quantum annealing machine: given a bound $n$, there is input $I$ to a program $P$ such that $P$ runs into a given program state $E$ executing no more than $n$ machine instructions if and only if the QUBO model of $P$ for $n$ evaluates to 0 on $I$. Thus, with more qubits on the machine than variables in the QUBO model, quantum annealing the model reaches 0 (ground) energy in constant time with high probability on some input $I$ that is part of the ground state if and only if $P$ runs into $E$ on $I$ in no more than $n$ instructions. Translation takes $\mathcal{O}(n^2)$ time turning a quantum annealer into a polynomial-time symbolic execution engine and bounded model checker, eliminating their path and state explosion problems. Here, we take advantage of the fact that any machine instruction may only increase the size of the program state by $\mathcal{O}(w)$ bits where $w$ is machine word size. Translation time comes down to $\mathcal{O}(n)$ if memory consumption of $P$ is bounded by a constant, establishing a linear (quadratic) upper bound on quantum space, in number of qubits, in terms of algorithmic time (space) in classical computing. The generated QUBO models only have $\mathcal{O}(2^w\cdot n^2)$ solutions out of $\mathcal{O}(2^{n^2})$ choices and only require $\mathcal{O}(wn)$ attempts to find a solution on a quantum machine. The construction motivates a temporal and spatial metric of quantum advantage and provides a non-relativizing argument for $NP\subseteq BQP$ effectively utilizing the optimality of Grover's algorithm. Our prototypical open-source toolchain translates machine code that runs on real RISC-V hardware to models that can be solved by real quantum annealing hardware, as shown in our experiments.