Approximate Equivalence Checking
Checking the equivalence of quantum circuits has been considerably considered. However, in the NISQ (Noisy Intermediate-Scale Quantum) computing realm where quantum noise is present inevitably, equivalence checking of noisy quantum circuits is seldomly considered. We defined the approximate equivalence of (possibly noisy) quantum circuits based on the Jamiolkowski fidelity and proposed two algorithms for checking it. See our paper
Algorithm I
Our algorithm aims to check the equivalence of a noisy circuit and an ideal circuit, which relies on calculating the Jamiolkowski fidelity of two circuits. Suppose every noise is represented in an operator-sum form, our algorithm I choose an operation element of every noise every time in the noisy circuit and construct a miter of the two circuits. Viewing it as a tensor network and contracting it gives one item of the fidelity. When the number of noises in the circuit is not so many, there are only a few items to be calculated and this algorithm is very efficient.
Algorithm II
Algorithm II employs a computing all items in one run form. It utilize the matrix representation of the noises and extend the circuits with all the qubits doubled. Constructing a miter of the two circuits and contracting it gives the Jamiolkowski fidelity at one time. The complexity of this method will not increase rapidly as the number of noises increases in the circuit and when there is a lot of noises in the circuit, this method should be more efficient than other methods.