In this work, we establish the strongest known lower bounds against QAC$^0$, while allowing its full power of polynomially many ancillae and gates. Our two main results show that:
(1) Depth 3 QAC$^0$ circuits cannot compute PARITY regardless of size, and require at least $Ω(\exp(\sqrt{n}))$ many gates to compute MAJORITY.
(2) Depth 2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage in depth $2$, regardless of size.
We present new techniques for simulating certain QAC$^0$ circuits classically in AC$^0$ to obtain our depth $3$ lower bounds. In these results, we relax the output requirement of the quantum circuit to a single bit (i.e., no restrictions on input preservation/reversible computation), making our depth $2$ approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC$^0$ circuits, which can compute PARITY exactly in depth $2$ using exponential size. Our proof techniques further suggest that, for inherently classical decision problems, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth $2$ QAC$^0$ circuits, regardless of size, cannot exactly synthesize an $n$-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth $2$ exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound).
Journal reference
STOC 2026: Proceedings of the 58th Annual ACM Symposium on Theory of Computing, 2199-2209
In this work, we establish the strongest known lower bounds against QAC0, while allowing its full power of polynomially many ancillae and gates. Our two main results show that:
(1) Depth 3 QAC0 circuits cannot compute PARITY regardless of size, and require at least Ω(exp(n)) many gates to compute MAJORITY.
(2) Depth 2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage in depth 2, regardless of size.
We present new techniques for simulating certain QAC0 circuits classically in AC0 to obtain our depth 3 lower bounds. In these results, we relax the output requirement of the quantum circuit to a single bit (i.e., no restrictions on input preservation/reversible computation), making our depth 2 approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC0 circuits, which can compute PARITY exactly in depth 2 using exponential size. Our proof techniques further suggest that, for inherently classical decision problems, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth 2 QAC0 circuits, regardless of size, cannot exactly synthesize an n-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth 2 exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound).
QAC0 is the family of constant-depth polynomial-size quantum circuits consisting of arbitrary single qubit unitaries and multi-qubit Toffoli gates. It was introduced by Moore [arXiv: 9903046] as a quantum counterpart of AC0, along with the conjecture that QAC0 circuits can not compute PARITY. In this work we make progress on this longstanding conjecture: we show that any depth-dQAC0 circuit requires n1+3−d ancillae to compute a function with approximate degree Θ(n), which includes PARITY, MAJORITY and MODk. We further establish superlinear lower bounds on quantum state synthesis and quantum channel synthesis. This is the first superlinear lower bound on the super-linear sized QAC0. Regarding PARITY, we show that any further improvement on the size of ancillae to n1+exp(−o(d)) would imply that PARITY ∈ QAC0. These lower bounds are derived by giving low-degree approximations to QAC0 circuits. We show that a depth-dQAC0 circuit with a ancillae, when applied to low-degree operators, has a degree (n+a)1−3−d polynomial approximation in the spectral norm. This implies that the class QLC0, corresponding to linear size QAC0 circuits, has approximate degree o(n). This is a quantum generalization of the result that LC0 circuits have approximate degree o(n) by Bun, Robin, and Thaler [SODA 2019]. Our result also implies that QLC0=NC1.
Bounds on the QAC0 Complexity of Approximating Parity
QAC circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. QAC0 circuits are QAC circuits of constant depth, and are quantum analogues of AC0 circuits. We prove the following: ∙ For all d≥7 and ε>0 there is a depth-d QAC circuit of size exp(poly(n1/d)log(n/ε)) that approximates the n-qubit parity function to within error ε on worst-case quantum inputs. Previously it was unknown whether QAC circuits of sublogarithmic depth could approximate parity regardless of size. ∙ We introduce a class of "mostly classical" QAC circuits, including a major component of our circuit from the above upper bound, and prove a tight lower bound on the size of low-depth, mostly classical QAC circuits that approximate this component. ∙ Arbitrary depth-d QAC circuits require at least Ω(n/d) multi-qubit gates to achieve a 1/2+exp(−o(n/d)) approximation of parity. When d=Θ(logn) this nearly matches an easy O(n) size upper bound for computing parity exactly. ∙ QAC circuits with at most two layers of multi-qubit gates cannot achieve a 1/2+exp(−o(n)) approximation of parity, even non-cleanly. Previously it was known only that such circuits could not cleanly compute parity exactly for sufficiently large n. The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which we name "nekomata" after an analogous cat yōkai.
QAC0 Contains TC0 (with Many Copies of the Input)
QAC0 is the class of constant-depth polynomial-size quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates. It is arguably the smallest natural class of constant-depth quantum computation which has not been shown useful for computing any non-trivial Boolean function. Despite this, many attempts to port classical AC0 lower bounds to QAC0 have failed.
We give one possible explanation of this: QAC0 circuits are significantly more powerful than their classical counterparts. We show the unconditional separation QAC0⊂AC0[p] for decision problems, which also resolves for the first time whether AC0 could be more powerful than QAC0. Moreover, we prove that QAC0 circuits can compute a wide range of Boolean functions if given multiple copies of the input: TC0⊆QAC0∘NC0. Along the way, we introduce an amplitude amplification technique that makes several approximate constant-depth constructions exact.
Bounds on the QAC0 Complexity of Approximating Parity
QAC circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. QAC0 circuits are QAC circuits of constant depth, and are quantum analogues of AC0 circuits. We prove the following:
∙ For all d≥7 and ε>0 there is a depth-dQAC circuit of size exp(poly(n1/d)log(n/ε)) that approximates the n-qubit parity function to within error ε on worst-case quantum inputs. Previously it was unknown whether QAC circuits of sublogarithmic depth could approximate parity regardless of size.
∙ We introduce a class of QAC circuits, including a major component of our circuit from the above upper bound, and prove a tight lower bound on the size of low-depth, mostly classical QAC circuits that approximate this component.
∙ Arbitrary depth-dQAC circuits require at least Ω(n/d) multi-qubit gates to achieve a 1/2+exp(−o(n/d)) approximation of parity. When d=Θ(logn) this nearly matches an easy O(n) size upper bound for computing parity exactly.
∙QAC circuits with at most two layers of multi-qubit gates cannot achieve a 1/2+exp(−o(n)) approximation of parity, even non-cleanly. Previously it was known only that such circuits could not cleanly compute parity exactly for sufficiently large n.
The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which we name nekomata after an analogous cat y\=okai.
In this work we study the robustness of QAC0 with respect to error tolerance and modifications to its gate-set. First, we investigate whether the non-zero error typically allowed for QAC0 circuits computing Boolean functions is truly necessary. We show that the error inherent in the parallel W-test of \cite{grier_morris_wu} can be eliminated entirely via a novel application of exact amplitude amplification in the many-copies context. Consequently, we find that QAC0 can \textit{exactly} simulate TC0 with polynomially many copies of the classical input and that for every fixed prime p exact QAC0, EQAC0, can compute total Boolean functions outside of AC0[p].
Second, we ask to what extent the computational power of QAC0 follows from the fact that arbitrary single-qubit gates may be used at any point in the circuit. We find that QAC0 is in fact robust to restrictions on which single-qubit gates are permitted: every QAC0 circuit can be approximately implemented by a QAC0 circuit consisting of just generalized Toffoli, S, and Hadamard gates. Moreover, this approximating circuit can be constructed efficiently from a classical description of the original circuit.
Low-degree approximation of QAC0 circuits
QAC0 is the class of constant-depth quantum circuits with polynomially many ancillary qubits, where Toffoli gates on arbitrarily many qubits are allowed. In this work, we show that the parity function cannot be computed in QAC0, resolving a long-standing open problem in quantum circuit complexity more than twenty years old. As a result, this proves QAC0⫋QACwf0. We also show that any QAC circuit of depth d that approximately computes parity on n bits requires 2Ω(n1/d) ancillary qubits, which is close to tight. This implies a similar lower bound on approximately preparing cat states using QAC circuits. Finally, we prove a quantum analog of the Linial-Mansour-Nisan theorem for QAC0. This implies that, for any QAC0 circuit U with a=poly(n) ancillary qubits, and for any x∈{0,1}n, the correlation between Q(x) and the parity function is bounded by 1/2+2−Ω(n1/d), where Q(x) denotes the output of measuring the output qubit of U∣x,0a⟩. All the above consequences rely on the following technical result. If U is a QAC0 circuit with a=poly(n) ancillary qubits, then there is a distribution D of bounded polynomials of degree polylog(n) such that with high probability, a random polynomial from D approximates the function ⟨x,0a∣U†Zn+1U∣x,0a⟩ for a large fraction of x∈{0,1}n. This result is analogous to the Razborov-Smolensky result on the approximation of AC0 circuits by random low-degree polynomials.