Forward and backward constrained bisimulations for quantum circuits using decision diagrams

Efficient methods for the simulation of quantum circuits on classical computers are crucial for their analysis due to the exponential growth of the problem size with the number of qubits. Here we study lumping methods based on bisimulation, an established class of techniques that has been proven successful for (classic) stochastic and deterministic systems such as Markov chains and ordinary differential equations. Forward constrained bisimulation yields a lower-dimensional model which exactly preserves quantum measurements projected on a linear subspace of interest. Backward constrained bisimulation gives a reduction that is valid on a subspace containing the circuit input, from which the circuit result can be fully recovered. We provide an algorithm to compute the constraint bisimulations yielding coarsest reductions in both cases, using a duality result relating the two notions. As applications, we provide theoretical bounds on the size of the reduced state space for well-known quantum algorithms for search, optimization, and factorization. Using a prototype implementation, we report significant reductions on a set of benchmarks. In particular, we show that constrained bisimulation can boost decision-diagram-based quantum circuit simulation by several orders of magnitude, allowing thus for substantial synergy effects.