The Complexity of Tensor Rank


We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by Håstad (J. Algorithm. 11(4), 644–654 1990). The hardness proof also implies an algebraic universality result.

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Fig. 1


  1. 1.

    The complexity class \(\exists \mathbb {R}\) was introduced explicitly in [24, 26] and some other papers, but other researchers probably thought of \(\text {ETh}(\mathbb {R})\) as a complexity class before, e.g., Shor [28], and Buss, Frandsen and Shallit [8].

  2. 2.

    We are not aware of any stronger lower bounds on \(\exists \mathbb {F}\) for any field \(\mathbb {F}\). If we allow rings, then \(\exists \mathbb {Z}\), for example, is undecidable, its complexity equivalent to the halting problem \(\emptyset ^{\prime }\). This was shown in a famous series of results by Davis, Robinson, and Matiyasevic [10, 19].

  3. 3.

    Koiran’s result assumes the generalized Riemann hypothesis (GRH); as far as we know there is no unconditional upper bound on \({\exists \mathbb {C}}\) other than PSPACE.

  4. 4.

    If \({\mathbb {Z}}\) had an existential definition in \({\mathbb {Q}}\), then it would follow that \(\exists \mathbb {Q} \equiv \exists \mathbb {Z} \equiv \emptyset ^{\prime }\). Koenigmsann [15] gives some evidence that there is no such definition (implying that his universal definition of \({\mathbb {Z}}\) in \({\mathbb {Q}}\) is optimal), however, there may be other routes towards the undecidability of \(\exists \mathbb {Q}\), and it may be undecidable without being as hard as \(\emptyset ^{\prime }\).

  5. 5.

    In other models, e.g., the Blum-Shub-Smale model [7] this was well-known earlier.

  6. 6.

    There are other definitions of eigenvalues for tensors as well.

  7. 7.

    The proof in [25] yields a quartic systems, but that can be reduced to quadratic, by removing the final (unnecessary) squaring operation.

  8. 8.

    There is also a notion of minrank for matrices with entries in {+, −}. Given such a matrix is there a real matrix of rank at most 3 with that sign pattern? This problem turns out to be \(\exists \mathbb {R}\)-hard as well [2, 3], but does not seem to be related to our minrank problem.


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Schaefer, M., Štefankovič, D. The Complexity of Tensor Rank. Theory Comput Syst 62, 1161–1174 (2018).

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  • Computational complexity
  • Existential theory of fields
  • Tensor rank