Stabilizers for Compiling Logical Circuits under Hardware Constraints
For quantum computing researchers, this work provides a novel method to simplify compilation by exploiting redundancy in error-correcting codes, potentially reducing circuit overhead.
The paper addresses the problem of compiling quantum circuits under hardware constraints while maintaining fault-tolerance. It shows how error-correcting codes can be leveraged to choose logically equivalent but physically distinct operators, reducing the need for costly swap operations. The approach reduces the problem to a least squares problem with a closed-form solution, demonstrated using the [[4,2,2]] code.
To implement quantum algorithms on a quantum computer, we must overcome the twin problems of fault-tolerance -- how can we realize a relatively noiseless computation by cleverly combining noisy components? -- and compilation -- how can we realize an arbitrary quantum algorithm given the basic operations available on the quantum device at hand? We show how treating the former problem via error-correcting codes enables greater flexibility in resolving the latter. Specifically, we explicitly leverage the fact that error-correcting codes introduce redundancy which renders physically distinct operators logically indistinguishable. In terms of computation, it suffices to implement any operator logically equivalent to some target, yet from a compilation perspective, certain choices may be preferable to others. Our novel contribution is making this intuition precise in the general setting of the special unitary group. In particular, we describe how to reduce the problem of making a compilation-ideal choice to a least squares problem and provide a closed form solution thereof. Using our framework, it is possible to circumvent inserting costly swaps to adhere to hardware connectivity; instead, we could realize the logical target through a distinct physical Hamiltonian that is natively accessible. We elucidate our approach using the $[[4,2,2]]$ code. We discuss connections to compressed sensing that may pave the way to efficient compilation leveraging physical degrees of freedom.