Checking Equivalence of States and Circuits

Equivalence checking is a basic task in the synthesis and verification of classical digital circuits. A hardware designer needs to know whether a circuit's implementation is functionally equivalent to its specification. In addition, the equivalence of different versions of the same (sub-)circuit must be checked throughout the complex computer-aided design process. Traditional combinational equivalence checking is solved in practice with high-performance solvers for Boolean Satisfiability, and its negative version (non-equivalence) is NP-complete.

Equivalence checking is likely to be just as important in quantum CAD, and the non-equivalence of quantum circuits is QMA-complete.¹ However, the equivalence of quantum states and operators can be subtle. Unlike their classical counterparts, qubits and quantum gates can differ by global and relative phase, and yet be equivalent upon measurement. Building upon the algorithmic blocks developed in Chapter 7, we present QuIDD algorithms to check quantum states and operators for equivalence. As we will see, the variety of algorithms available to solve this problem is surprising.