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Stabilizer Slicing: Coherent Error Cancellations in Low-Density Parity-Check Stabilizer Codes
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The stabilizer rank of a quantum state ψ is the minimal r such that | ψ ⟩ = ∑ j = 1 r c j | φ j ⟩ for c j ∈ C and stabilizer states φ j . The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n -th tensor power of single-qubit magic states.We prove a lower bound of Ω ( n ) on the stabilizer rank of such states, improving a previous lower bound of Ω ( n ) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ , the stabilizer rank of any state which is δ -close to those states is Ω ( n / log n ) . This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F 2 n , and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.more » « less
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How complex is the structure of quantum geometry? In several approaches, the spacetime atoms are obtained by the SU(2) intertwiner called quantum tetrahedron. The complexity of this construction has a concrete consequence in recent efforts to simulate such models and toward experimental demonstrations of quantum gravity effects. There are, therefore, both a computational and an experimental complexity inherent to this class of models. In this paper, we study this complexity under the lens of stabilizer entropy (SE). We calculate the SE of the gauge-invariant basis states and its average in the SU(2) gauge invariant subspace. We find that the states of definite volume are singled out by the (near) maximal SE and give precise bounds to the verification protocols for experimental demonstrations on available quantum computers.more » « less
