The role of cohomology in quantum computation with magic states

Robert Raussendorf1,2, Cihan Okay3, Michael Zurel1,2, and Polina Feldmann1,2

1Department of Physics & Astronomy, University of British Columbia, Vancouver, Canada
2Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, Canada
3Department of Mathematics, Bilkent University, Ankara, Turkey

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Abstract

A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function – an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces – possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n$\geq$2. We establish that the obstructions to the existence of such Wigner functions are cohomological.

Which essential property of quantum theory is required to harness its potential for computation? Many candidates have been suggested, such as entanglement, superposition, interference, and the largeness of Hilbert space, as well as quasiprobability. Each of these properties has its own domain of applicability and limitations, but collectively, the insights they provide extend our understanding of the theory of quantum computation.

In the scheme of quantum computation with magic states, it has been established that a quantum speedup can arise only if a quasiprobability function representing the initial state takes negative values. However, not all quasiprobability functions can serve as reliable indicators. When the dimension of the Hilbert space is odd, Gross' Wigner function can be used for this purpose. In contrast, when the dimension is even, Gross' Wigner function doesn't exist, and the process of constructing an appropriate indicator becomes more involved. In this work, we identify a mathematical reason for this dichotomy: cohomological obstructions occur in the even-dimensional case, but not in the odd-dimensional case.

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