The survivor count of a binary linear code: exact structure, counting complexity, and the bridge to Clifford+T circuits

To predict what a quantum computer will output you have to work out an astronomical sum, and almost every term in it cancels to nothing. This work counts how many actually survive, and shows that number is out of reach of the standard tools: not the Tutte polynomial, not the usual weight counts, not the usual recursions. A binary linear code is a set of bit strings closed under bitwise addition — the object behind error-correcting codes. This paper studies a counting quantity attached to such a code, its survivor count. Where it comes from. Take a quantum circuit built from Clifford gates and T gates and ask for the expected value of its output. That expectation is a signed sum of exponentially many terms — quadratic Gauss sums over the two-element field — indexed by the codewords of a binary linear code. What that sum comes to is a well-studied quantity. This paper asks a different question about the same sum: not what it adds up to, but how many of its terms are really there. The number that survive is the survivor count of the code. What was already known, and what is taken up here. Whether one individual term survives is classical: a quadratic Gauss sum vanishes unless its form is trivial on the radical of its polar form, and is otherwise a power of two carrying the Arf sign. The weight attached to a single support is published as well, as the local isotropic-nullity weight of Bouchet's isotropic systems and of the restricted Tutte–Martin polynomial; the paper treats it as background rather than as a result of its own. What the paper takes up is the whole family at once. The census runs over all 2k codewords of a code, and the questions asked are what determines that total, what exact evaluation costs, which parameters and moduli buy an algorithm, and how far into the planar case each side reaches. What the paper proves. An exact criterion, and a closed form. The class sum at a support vanishes unless a certain half-weight form vanishes on a radical; where it does not vanish, the survivors are counted exactly by a power of two read off the dimensions of the code. One parity obstruction can remove them all. Consequently the survivor count depends only on the code's binary matroid. Separations. It is determined by none of the following: the Tutte polynomial; the cycle and cut weight enumerators taken together on planar graphs; the local marginals. It is not invariant under duality, and it obeys no constant-coefficient deletion–contraction recursion. The standard machinery that classifies graph and matroid invariants therefore does not reach it, and every line drawn below between easy and hard is drawn from the definition instead. A complexity map. Exact evaluation is #P-complete under deterministic polynomial-time Turing reductions, both on explicit generators and on graph cycle codes, and stays hard on four restricted graph families, on linear complementary dual codes, and at each fixed hull dimension. It is polynomial at bounded pathwidth, at bounded treewidth by a proved but unimplemented join step, on several code classes, and modulo 2, 3 and 6. Planar graphs. An exact algorithm running in time 2O(√n log n), conditional on three imported constructors, together with a #P-hardness statement conditional on one named planar Tutte evaluation being #P-hard. A bridge to quantum circuits, in both directions. Every Clifford+T circuit compiles to a single code and a single integer h whose output expectation is exactly (A + B√2)/2h+1, an identity holding exactly in the ring Z[√2], where (A,B) is that code's mod-8 weight census. Separately, a reverse construction makes the sign of such a bias hard for PromiseBQP at an exponentially small threshold, while at inverse-polynomial thresholds the same sign is decided classically with bounded error. Scope, and what is not claimed. Every statement carries exactly the status its proof supports and never a stronger one. Results resting on an unproved hypothesis carry that hypothesis at the head of their own statement, and no hypothesis is discharged anywhere in the paper. Nothing here claims, implies or denies any equality or separation of complexity classes. Where a comparison with the literature was not made, the paper says so at the statement it touches; nothing here asserts priority. Supplementary material. The consolidated record, and the proof files named in the paper's appendix, are deposited separately at doi:10.5281/zenodo.22804078. Abstract, as printed in the paper A Clifford+T circuit's output expectation is a signed sum of quadratic Gauss sums over the codewords of a binary linear code; counted here is how many survive. For C in F_2^m, let K_F be the dual subcode supported in a codeword F and R_F its radical: the survivor count S(C) counts, over all F, the cosets of K_F in the even-weight subspace on F with nonzero Gauss sum. An exact criterion is proved: the class sum at F vanishes unless the half-weight form twisted by the class representative s vanishes on R_F, and survivors at nonzero F number 2^(|F|-1-dim K_F-dim R_F+[F in the dual]), or none under a parity obstruction. So S is a binary-matroid invariant determined by none of the Tutte polynomial, the cycle and cut weight enumerators together on planar graphs, or the local marginals, not duality-invariant, and obeying no constant-coefficient deletion-contraction recursion. Exact evaluation is #P-complete under polynomial-time Turing reductions, and stays hard on four restricted graph families, on linear complementary dual codes and at each fixed hull dimension; it is polynomial at bounded pathwidth, at bounded treewidth by a proved but unimplemented join step, on several code classes and modulo 2, 3, 6. For planar graphs an exact 2^(O(sqrt(n) log n)) algorithm is given, conditional on three imported constructors, with hardness conditional on one named planar Tutte evaluation being #P-hard. Every such circuit compiles to one code and an integer h with output expectation exactly (A+B sqrt 2)/2^(h+1) in Z[sqrt 2], for (A,B) that code's mod-8 weight census; a reverse construction makes that sign PromiseBQP-hard at an exponentially small threshold, while inverse-polynomial ones are decided classically with bounded error. No complexity-class equality or separation is claimed or denied.