Universal Variational Quantum Computation

Abstract: Variational quantum algorithms dominate contemporary gate-based quantum enhanced optimisation, eigenvalue estimation and machine learning. Here we establish the quantum computational universality of variational quantum computation by developing two objective functions which minimise to prepare outputs of arbitrary quantum circuits. The fleeting resource of variational quantum computation is the number of expected values which must be iteratively minimised using classical-to-quantum outer loop optimisation. An efficient solution to this optimisation problem is given by the quantum circuit being simulated itself. The first construction is efficient in the number of expected values for $n$-qubit circuits containing $\mathcal{O}({poly} \ln n)$ non-Clifford gates -- the number of expected values has no dependence on Clifford gates appearing in the simulated circuit. The second approach yields $\mathcal{O}(L^2)$ expected values while introducing not more than $\mathcal{O}(\ln L)$ slack qubits, for a quantum circuit partitioned into $L$ gates. Hence, the utilitarian variational quantum programming procedure -- based on the classical evaluation of objective functions and iterated feedback -- is in principle as powerful as any other model of quantum computation. This result elevates the formal standing of the variational approach while establishing a new universal model of quantum computation.