Contextuality and memory cost of simulation of Majorana fermions

Contextuality has been reported to be a resource for quantum computation [1], analogous to non-locality which is a known resource for quantum communication and cryptography [2]. In recent work, Karanjai et al. [3] show that the presence of contextuality places a lower bound on the spatial complexity of classically simulating quantum processes. In particular, they bound the memory cost of simulating Clifford operators. The method to derive lower bound works for a general family of classical simulations that can be framed as stochastic evolution in ontological models (hidden variable models): For instance, it generalizes the GottesmanKnill [7] and Wigner function simulation methods. However, the techniques in [3] are limited to quantum processes that of closed sub-theories, i.e., where products of measurable quantities are measurable, which are a very limited class of quantum circuits. Consequently, it excludes processes where measurements are loyal. In this work, we generalize this connection to non-closed circuit families with a broader concept of contextuality, namely, event-based contextuality [4]. We show that the presence of “event-based” contextuality places new lower bounds on the memory cost for simulating restricted classes of quantum computation. We apply this result to the simulation of the restricted model of quantum computation based on the braiding of Ising anyons known as topological quantum computation (TQC) model [5]. This model is the first known scheme of magic states distillation [6], a leading paradigm in fault-tolerant quantum computing. It is also of fundamental interest in the study of quantum resources that power quantum computation, as it lies at the intersection of two classically simulable sub-theories: FLO and Clifford circuits. For the TQC model, we prove that the lower bound in the memory required in a simulation is Ω(n log2 n), where n is the number of fermionic modes. This bound is extended to fermionic linear optics (FLO), a fermionic analogous of bosonic linear optics. Lower bound in the memory cost Since our goal is to simulate quantum statistics, the ontological model used for the classical simulation will reproduce the Born rule probabilities of a quantum subtheory. In the classical simulation the density matrix is represented by a probability distribution μρ(λ) over the state space Λ and the measurements become sub-stochastics maps, ΓO(λ, k|λ). After a measurement the probability distribution μρ(λ) is updated to μρ′(λ) with probability Pr(O, k|ρ, λ) = ∑ λ,λ′ ΓO(λ, k|λ)μρ(λ). The internal state λ ∈ Λ contains all the information necessary to characterize the statistics of all measurements allowed in the sub-theory. The lower bound in the space complexity is obtained by finding a lower bound in the size of the state space Λ required to simulate the sub-theory.