Quantum State Engineering Under Multiple Expectation-Value Constraints

Abstract

This work introduces a formulation of quantum state engineering termed expectation-value targeting: the task of preparing a pure state whose expectation values with respect to a prescribed set of observables attain specified targets. This formulation subsumes standard ground-state preparation problems in quantum chemistry and many-body physics, while extending beyond variational energy minimization to multi-constraint state synthesis. The problem amounts to solving a system of nonlinear constraints on an exponentially large state space, for which no general efficient classical approaches are known. Variational quantum algorithms tackle this problem by restricting the search to a low-dimensional parameter space, and relying on classical optimization techniques for solutions. However, these approaches can become extremely ineffective for the present problem, where competing constraints can induce rugged landscapes and vanishing gradients (barren plateaus). Adaptive variational methods, in which the ansatz is constructed iteratively from a pool of candidate operators rather than fixed in advance, have been developed primarily for ground-state preparation. However, we show that the present problem also admits a similar construction. We introduce QUEST (Quantum Unitary Engineering of States to Target), a framework purpose-built for expectation-value targeting, in which the engineered state is constructed as a depth-adaptive sequence of Pauli rotations, with each rotation chosen to descend a sum-of-squared-residuals cost. QUEST provides a constructive route to expectation-value targeting, building the engineered state one Pauli rotation at a time, and establishes adaptive synthesis as a primitive for state preparation under multiple, potentially inconsistent target constraints.