Exponential T-count hardness and stabilizer rank from syndrome decoding
Year: 2026
Type: preprint
Abstract: Lower bounds of the form 2^Ω(t) on the strong classical simulation of Clifford+T circuits with t T gates, and exponential lower bounds on efficiently computable stabilizer decompositions of the magic state |T⟩^⊗t, have been derived from the Exponential Time Hypothesis (ETH). Two manuscripts of October 2026 claim deterministic subexponential algorithms for #k-SAT, which would refute ETH. Under one of these claims, Tulsyan showed that the witness circuits of the ETH-based bounds, which use O(t) Clifford gates, are simulated exactly in time 2^O(t/√log t), and recorded 2^Ω(t) hardness with unboundedly many Clifford gates as open. We give an unconditional reduction showing that such hardness follows from the exponential hardness of syndrome decoding of binary linear codes, a standard assumption of code-based cryptography that does not depend on ETH. Syndrome decoding at length N reduces, with N+1 adaptive queries, to deciding whether ⟨0|U|0⟩ ≠ 1 for Clifford+T circuits U with 8N+O(log N) T gates and O(N²) gates. The Clifford gates prepare a coset of the code in superposition, and the T gates pay only for a Hamming-weight test built from relative-phase Toffoli gates. Consequently, a strong simulator running in time 2^(αt)·poly(g) on circuits with g gates yields a decoder running in time 2^(8αN+o(N)). If decoding requires circuits of size 2^(δN) at every length N, then χ(|T⟩^⊗t) ≥ 2^((δ/16)t − O(log t)) for all large t. Together with Tulsyan's conditional bound, this indicates that the density of the Clifford part, not the T-count alone, governs the fine-grained complexity of simulating Clifford+T circuits.
Author: Pei, Yan Ru
Institution: University of California San Diego
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225408853