Symbolic Reduction of Quantum Circuits Motivation In classical

Symbolic Reduction of Quantum Circuits

Motivation In classical computation, it is desirable to find a “minimal” circuit to compute a given function n In quantum computation this problem becomes essential, as longer circuits will necessarily be harder to insulate from decoherence. n

Background n n n Any quantum operator can be “simulated” using the controlled-not gate and (arbitrary) one qubit rotations. An arbitrary one qubit rotation can be written as S(w)T(x)R(y)T(z), where R, S, T are elementary gate families parameterized by angle. Furthermore, in a paper by Cybenko, an explicit decomposition of an arbitrary quantum operator into R, S, T and CNOT gates is given.

Symbolic Reduction The basic idea: take a circuit and a set of reduction rules – that is, transformations that preserve the operation performed by the circuit while decreasing the total number of gates – search the circuit for places to apply these. n Note that this is necessarily an iterative process; one reduction may allow another which was previously impossible. n

Some Reduction Rules It happens that R(0), S(0), and T(0) are all the identity. These gates, if seen, may be removed. n If two NOT (or CNOT) gates occur “next to” each other, remove them both. n For X in {R, S, T}, if an X(a) is “next to” X(b), combine X(a) and X(b) into X(a + b). n

Obstacles to Reduction n For specific circuits, it is evident that the biggest obstacles to reduction are long chains of CNOT gates.