Hierarchies of resources for measurement-based quantum computation

Abstract

For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where non-locality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.

The power of qutrits for non-adaptive measurement-based quantum computing

“…If someone were to show that the qu-p-it GHZ state is the optimal resource for the induced sets of Bell inequalities by p-NMQC, our findings and any conjectures discussed below would be more powerful. However, the striking mathematical similarities between 2-NMQC and 3-NMQC, exemplified by our findings and the additional ones in [46], make it seem likely that the qu-p-it state does indeed play such a significant role for generalised (n, p, p) Bell inequalities. Using unitary p-valued observables m with the property mp = 1 as an ansatz could be promising too, as the dichotomic observables utilised by Werner and Wolf [31] in the case of qubit Bell inequalities fulfil m2 = 1, which is crucial for their proofs.…”

“…A natural extension of our work is general p-NMQC, where the goal is to compute a function f : F n p → F p . Interestingly, in [46], the authors show-independently to our work-that the qu-p-it GHZ state can indeed be used to compute all functions f : F n p → F p with p-NMQC, p prime, elevating any classical pre-processor to universality.…”

“…The associated generalised Bell inequalities are maximally violated by qutrit GHZ states. We showed that the minimum size of the qutrit GHZ state that is required to compute a function that differs from a constant function at only one input value scales exponentially with n. Independently of this work, theorem 1 and its generalisation to all p prime was also recently shown in [46], using a similar, but different approach.…”

“…While finishing the manuscript we became aware of independent work [46], where the authors show that the qu-p-it GHZ state can indeed be used to compute all functions f : F n p → F p with p-NMQC, p prime. They use a more general ansatz that takes into account phase relations and could be helpful to address the open problems mentioned in this outlook.…”