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arXiv:2610.01024

Exact TTT-counts of Toffoli layers from an isotropy bound

Arul Rhik Mazumder

quant-phcs.CCSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
v1 · Oct 1, 2026v2 · Oct 4, 2026v3 · Oct 29, 2026
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Abstract

The $T$-count is the dominant cost of fault-tolerant Clifford+$T$ computation. We prove that a layer of $m$ disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, needs exactly $6m+1$ $T$ gates in every Hadamard-free Clifford+$T$ circuit with clean ancillas. Campbell and Howard gave the matching construction. To our knowledge this is the first proof that it is optimal for general $m$ (for $m=1$ the value $7$ is classical, and Campbell and Howard state the value $13$ at $m=2$). On every controlled unitary the same floor comes within one of their exact count, and it recovers their $4m+3$ for a fan-out of $m$ Toffolis from one control. The proof rests on an isotropy constraint: for a pure-cubic phase, the vectors recording which $T$ gates touch each qubit span a totally isotropic subspace. In general the constraint gives the isotropy floor $δ\ge2(n-d^{\ast})-r$ for every diagonal level-three gate, computed from the phase polynomial in polynomial time. The floor is never below stabilizer nullity $ν$, which equals $n-d^{\ast}$ on this class, and a separate parity argument raises it to $2ν+1$ on non-Clifford pure-cubic gates. On the output of the TODD optimizer for the $24$ benchmark circuits it completes, the floor certifies $193$ of its $311$ merged phase-polynomial blocks optimal for their Hadamard layering ($186$ to $193$ across five optimizer seeds), against $113$ for nullity. The floor also holds, under stated conditions, for circuits whose only internal Hadamards form unitarily uncomputed temporary AND blocks, while under adaptive feedforward only $t\geν$ is proved.

Comments
69 pages, 3 figures, 9 tables

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The TTT-count is the dominant cost of fault-tolerant Clifford+TTT computation. We prove that a layer of mmm disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, needs exactly 6m+16m+16m+1 TTT gates in every Hadamard-free Clifford+TTT circuit with clean ancillas. Campbell and Howard gave the matching construction. To our knowledge this is the first proof that it is optimal for general mmm (for m=1m=1m=1 the value 777 is classical, and Campbell and Howard state the value 131313 at m=2m=2m=2). On every controlled unitary the same floor comes within one of their exact count, and it recovers their 4m+34m+34m+3 for a fan-out of mmm Toffolis from one control. The proof rests on an isotropy constraint: for a pure-cubic phase, the vectors recording which TTT gates touch each qubit span a totally isotropic subspace. In general the constraint gives the isotropy floor δ≥2(n−d∗)−rδ\ge2(n-d^{\ast})-rδ≥2(n−d∗)−r for every diagonal level-three gate, computed from the phase polynomial in polynomial time. The floor is never below stabilizer nullity ννν, which equals n−d∗n-d^{\ast}n−d∗ on this class, and a separate parity argument raises it to 2ν+12ν+12ν+1 on non-Clifford pure-cubic gates. On the output of the TODD optimizer for the 242424 benchmark circuits it completes, the floor certifies 193193193 of its 311311311 merged phase-polynomial blocks optimal for their Hadamard layering (186186186 to 193193193 across five optimizer seeds), against 113113113 for nullity. The floor also holds, under stated conditions, for circuits whose only internal Hadamards form unitarily uncomputed temporary AND blocks, while under adaptive feedforward only t≥νt\geνt≥ν is proved.
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In fault-tolerant Clifford+T circuits, a Boolean oracle costs T gates and extra qubits, and on four or more qubits every exact Toffoli gate needs an extra qubit. Gosset, Kothari and Zhang showed that sampling a fresh random circuit for each call makes the T-count of the Toffoli gate independent of its size, but a direct implementation of their sampler uses about one extra qubit per bit of precision. We show that computing the sampled parities in place, on the gate's own controls, removes every extra qubit once the gate is large enough: each random circuit then uses about 8 T gates per bit of precision and borrows only the gate's own idle inputs. This is one T gate more per circuit than the best measurement-free variant that keeps one clean ancilla. The same holds for exclusive-ors of indicators of affine subspaces, such as equality tests and checks of linear equations. For randomized circuits that do not measure, this is within a factor of eight of optimal up to an additive constant, even with extra qubits, and when each random circuit computes a Boolean function exactly, closing the factor asymptotically is equivalent to determining the leading constant in the T-count of exact multi-controlled Toffoli gates. Finally, the oracle of the indicator of a random set of fixed density needs exponentially many T gates at constant error, while that of an affine subspace of the same density needs a constant number, and extra qubits reduce the exponential cost only quadratically.
Fault-tolerant quantum computation requires substantial resources for non-Clifford gates. Their count also enters exponentially into the runtime of several classical simulation methods. We present algebraic methods for minimizing the non-Clifford CCZ (Toffoli) count in {CNOT,CCZ} phase circuits using tensor decomposition over F2\mathbb{F}_2F2​. On standard cubic benchmarks, these methods match or improve prior automated Toffoli counts, with most instances optimized in under a minute on a single CPU. With additional computation, the decompositions provide effective initializations for existing T-count methods, improving T counts on several circuits. We also present algebraic methods for CNOT-count optimization at fixed Toffoli count. This gives competitive ancilla-free finite-field multiplication circuits and reduces the CNOT count of XOR AES S-box circuits.
We determine the non-Clifford TTT-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+T+T+T model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an nnn-qubit operator with aaa clean ancillas and at most sss TTT gates can be compressed to use at most min⁡{a,n+2s}\min\{a,n+2s\}min{a,n+2s} ancillas, without increasing the absolute error or TTT-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error εεε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n2log⁡(n4/ε))Ω(n^2\sqrt{\log(n^4/ε)})Ω(n2log(n4/ε)​), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on nnn spins, we obtain independent lower bounds Ω(n)Ω(n)Ω(n) from stabilizer nullity and Ω(log⁡(1/ε))Ω(\log(1/ε))Ω(log(1/ε)) from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling Θ(n+log⁡(1/ε))Θ(n+\log(1/ε))Θ(n+log(1/ε)). As an application, we evaluate the TTT-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation TTT-count scales as the query count times the optimal TTT-count per query.
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Prior work of Beverland et al. has shown that any exact Clifford+TTT implementation of the nnn-qubit Toffoli gate must use at least nnn TTT gates. Here we show how to get away with exponentially fewer TTT gates, at the cost of incurring a tiny 1/poly(n)1/\mathrm{poly}(n)1/poly(n) error that can be neglected in most practical situations. More precisely, the nnn-qubit Toffoli gate can be implemented to within error εεε in the diamond distance by a randomly chosen Clifford+TTT circuit with at most O(log⁡(1/ε))O(\log(1/ε))O(log(1/ε)) TTT gates. We also give a matching Ω(log⁡(1/ε))Ω(\log(1/ε))Ω(log(1/ε)) lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use Ω(n)Ω(n)Ω(n) TTT gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the TTT-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue.