Abstract
The disjointness of a stabilizer code is a quantity used to constrain the level of the logical Clifford hierarchy attainable by transversal gates and constant-depth quantum circuits. We show that for any positive integer constant $c$, the problem of calculating the $c$-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete. We provide bounds on the disjointness for various code families, including the Calderbank-Shor-Steane codes, concatenated codes, and hypergraph product codes. We also describe numerical methods of finding the disjointness, which can be readily used to rule out the existence of any transversal gate implementing some non-Clifford logical operation in small stabilizer codes. Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.
Highlights
Designing fault-tolerant schemes is an essential step toward scalable universal quantum computation [1,2,3,4]
Systematic approaches to finding transversal logical gates for generic quantum error-correcting codes are not known, for stabilizer codes [32] we can rule out the possibility of implementing certain logical operations
We remark that Proposition 3 asserts that the level of the logical Clifford hierarchy attainable by transversal logical gates for the concatenated code S1 S2 cannot exceed the bounds in Eq (1) for the stabilizer codes S1 and S2
Summary
Stabilizer codes are an important class of quantum errorcorrecting codes. A stabilizer code is defined by its stabilizer group S, i.e., an Abelian subgroup of the Pauli group that does not contain −I. In what follows we identify the stabilizer code with its stabilizer group. Logical Pauli operators, which are the elements of the the normalizer of the stabilizer group in the. We write L to denote the set of all nontrivial logical Pauli operators. A stabilizer code is a CSS code iff there exists a choice of stabilizer generators such that every generator is either a Pauli X - or Z-type operator. S1 and S2 with parameters [[n1, k1, d1]] and [[n2, 1, d2]], respectively, we can concatenate them to obtain a new stabilizer code S1 S2. Given two full-rank binary matrices H1 and H2 of size m1 × n1 and m2 × n2, respectively, the corresponding hypergraph product code is specified by the following binary matrix: H1 ⊗ Im2 Im1 ⊗ H2 0n1 n2 ,n1 m2 +n2 m1.
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