Error-correcting codes for adiabatic quantum computation

Abstract

Recently, there has been growing interest in using adiabatic quantum computation as an architecture for experimentally realizable quantum computers. One of the reasons for this is the idea that the energy gap should provide some inherent resistance to noise. It is now known that universal quantum computation can be achieved adiabatically using two-local Hamiltonians. The energy gap in these Hamiltonians scales as an inverse polynomial in the number of quantum gates being simulated. Here we present stabilizer codes which can be used to produce a constant energy gap against one-local and two-local noise. The corresponding fault-tolerant universal Hamiltonians are four-local and six-local, respectively, which are the optimal result achievable within this framework.

Performance of two different quantum annealing correction codes

“…Because H P is a part of physical problem Hamiltonian H I it inherits the latter's time-dependence, i.e., is turned on via the annealing schedule B(t). This aspect of QAC differs from standard error suppression [26].…”

“…This is unavoidable in the setting of the D-Wave device, which (also unlike Ref. [26]) prevents H X from being encoded, as this would require many-body X ⊗n terms, which are experimentally unavailable. Because of this tension there is an optimal penalty value γ that depends on α, the problem instance, and other variables.…”

“…QAC is designed to work within the present technological restrictions of available quantum annealers whereby only the problem Hamiltonian (H I ) and not the transverse field Hamiltonian (∑ i X i ) can be encoded. This differs from the standard scheme of error-suppression via energy penalties for AQC wherein the entire Hamiltonian is encoded [26,30]. This type of encoding in principle allows for the suppression of arbitrary errors, but requires k-local interactions with k ≥ 3 [31].…”

“…Therefore, any scalable quantum annealing architecture will require quantum error correction [25]. Unfortunately, theoretical progress in quantum error correction for adiabatic quantum computing and quantum annealing has not enjoyed the same success as that of other quantum computing paradigms, in spite of recent advances [26][27][28][29][30]. Physical constraints, such as locality of the interaction terms in the Hamiltonian [31,32], and a nogo theorem constraining what can be achieved with commuting two-local interactions [33], remain stubborn hurdles.…”

Adiabatic quantum simulators

“…[32][33][34] However, for small systems this might not be necessary indicating that our method is feasible for immediate experimental technologies. For this reason, we will examine how robust a small scale implementation of our protocol would be.…”