Quantum evolution is time-reversible and yet little advantage is gained from this in the circuit model of quantum computing. Indeed, most quantum algorithms expressed in the circuit model compute strictly from the present to the future, preparing initial states and proceeding forward with unitary transformations and measurements. We may call this predictive computation. In contrast, retrodictive quantum theory, retrocausality, and the time-symmetry of physical laws all suggest that quantum computation embodies richer --untapped-- modes of computation, which could exploit knowledge about the future for a computational advantage.
We demonstrate that by using symbolic partial evaluation, retrodictive reasoning can indeed be used as a computational resource that exhibits richer modes of computation at the boundary of the classical/quantum divide. Specifically, instead of fully specifying the initial conditions of a quantum circuit and computing forward, it is possible to compute, classically, in both the forward and backward directions starting from partially-specified initial and final conditions. Furthermore, this mixed mode of computation (i) can solve problems with fewer resources than the conventional forward mode of execution, sometimes even purely classically (de-quantization), (ii) can be expressed in a symbolic representation that immediately exposes global relational properties of the wavefunction that are needed for quantum algorithms and (iii) reveals that the entanglement patterns inherent in genuine quantum algorithms with no known classical counterparts are artifacts of the chosen symbolic representation.