Quantum computation using PT-symmetric gates

The uniqueness of the inner product associated to a quantum system has come under scrutiny following the advent of PT-symmetric quantum mechanics. In this work we aim to ascertain the advantage offered by PT-symmetric gates for computation. To this end, we first show how a changing Hilbert-space inner product can be used to implement unbroken PT-symmetric evolutions. Next, we construct a model of computation in which the allowed gateset includes inner-product changing quantum operations in addition to unitary gates. This construction is based on the operational framework for changing the Hilbert-space inner product of a quantum system developed in S Karuvade, A. Alase and B. C. Sanders, Phys. Rev. Research 4, 013016 (2022). We also show how the new model allows for the implementation of non-unitary gates generated by unbroken PT-symmetric Hamiltonians. Furthermore, we show that the class of computational problems that are efficiently solved by the new model coincides with the complexity class bounded-error quantum polynomial time (BQP). To prove this result, we devised a technique to simulate inner-product changing quantum operations using unitary gates and measurements without changing the Hilbert-space representation of the underlying quantum system. Our work shows that exponential speedup for quantum computation cannot be achieved by changing the Hilbert-space inner product or by implementing PT-symmetric evolutions.