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arXiv:2104.05822 · Quantum Information and Computation

On the extremal points of the Lambda polytopes and classical simulation of quantum computation with magic states

Cihan Okay, Michael Zurel, Robert Raussendorf

Published Sep 1, 2021• 0 citations
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Abstract

We investigate the $\Lambda$-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, $\Lambda_n$, for every number $n$ of qubits. We establish two properties of the family $\{\Lambda_n, n\in \mathbb{N}\}$, namely (i) Any extremal point (vertex) $A_\alpha \in \Lambda_m$ can be used to construct vertices in $\Lambda_n$, for all $n>m$. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage $A_\alpha$. In addition, we describe a new class of vertices in $\Lambda_2$ which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of $\Lambda_n$, the above results extend efficient classical simulation of quantum computations beyond the presently known range.

DOI
10.26421/qic21.13-14-2

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Recently, a class of objects, known as ΛΛΛ-polytopes, were introduced for classically simulating universal quantum computation with magic states. In ΛΛΛ-simulation, the probabilistic update of ΛΛΛ vertices under Pauli measurement yields dynamics consistent with quantum mechanics. Thus, an important open problem in the study of ΛΛΛ-polytopes is characterizing its vertices and determining their update rules. In this paper, we obtain and describe the update of all degenerate vertices of Λ2Λ_{2}Λ2​, the 222-qubit ΛΛΛ polytope. Our approach exploits the fact that Λ2Λ_{2}Λ2​ projects to a well-understood polytope MP\text{MP}MP consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope MP‾\overline{\text{MP}}MP, which is MP\text{MP}MP intersected by the polytope defined by a set of Clauser-Horne-Shimony-Holt (CHSH) inequalities. Owing to a duality between CHSH inequalities and vertices of MP\text{MP}MP we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of MP‾\overline{\text{MP}}MP.
We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides probability estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.
Abstract Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits nnn, even approximately. We reduce the problem to solving a 333-\class{SAT} instance on n2n^2n2 variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.