A major open problem in understanding shallow quantum circuits (QAC$^0$) is whether they can compute Parity. We show that this question is solely about the Fourier spectrum of QAC$^0$: any QAC$^0$ circuit with non-negligible high-level Fourier mass suffices to exactly compute PARITY in QAC$^0$. Thus, proving a quantum analog of the seminal LMN theorem for AC$^0$ is necessary to bound the quantum circuit complexity of PARITY.
In the other direction, LMN does not fully capture the limitations of AC$^0$. For example, despite MAJORITY having $99\%$ of its weight on low-degree Fourier coefficients, no AC$^0$ circuit can non-trivially correlate with it. In contrast, we provide a QAC$^0$ circuit that achieves $(1-o(1))$ correlation with MAJORITY, establishing the first average-case decision separation between AC$^0$ and QAC$^0$. This suggests a uniquely quantum phenomenon: unlike in the classical setting, Fourier concentration may largely characterize the power of QAC$^0$.
PARITY is also known to be equivalent in QAC$^0$ to inherently quantum tasks such as preparing GHZ states to high fidelity. We extend this equivalence to a broad class of state-synthesis tasks. We demonstrate that existing metrics such as trace distance, fidelity, and mutual information are insufficient to capture these states and introduce a new measure, felinity. We prove that preparing any state with non-negligible felinity, or derived states such as poly(n)-weight Dicke states, implies PARITY $\in$ QAC$^0$.
A major open problem in understanding shallow quantum circuits (QAC0) is whether they can compute Parity. We show that this question is solely about the Fourier spectrum of QAC0: any QAC0 circuit with non-negligible high-level Fourier mass suffices to exactly compute PARITY in QAC0. Thus, proving a quantum analog of the seminal LMN theorem for AC0 is necessary to bound the quantum circuit complexity of PARITY.
In the other direction, LMN does not fully capture the limitations of AC0. For example, despite MAJORITY having 99% of its weight on low-degree Fourier coefficients, no AC0 circuit can non-trivially correlate with it. In contrast, we provide a QAC0 circuit that achieves (1−o(1)) correlation with MAJORITY, establishing the first average-case decision separation between AC0 and QAC0. This suggests a uniquely quantum phenomenon: unlike in the classical setting, Fourier concentration may largely characterize the power of QAC0.
PARITY is also known to be equivalent in QAC0 to inherently quantum tasks such as preparing GHZ states to high fidelity. We extend this equivalence to a broad class of state-synthesis tasks. We demonstrate that existing metrics such as trace distance, fidelity, and mutual information are insufficient to capture these states and introduce a new measure, felinity. We prove that preparing any state with non-negligible felinity, or derived states such as poly(n)-weight Dicke states, implies PARITY ∈ QAC0.
In this work, we prove the strongest known lower bounds for QAC0, allowing polynomially many gates and ancillae. Our main results show that:
(1) Depth-3 QAC0 circuits cannot compute PARITY, and require Ω(exp(n)) gates to compute MAJORITY.
(2) Depth-2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage, regardless of size.
We develop new classical simulation techniques for QAC0 to obtain our depth-3 bounds. In these results, we relax the output requirement of the quantum circuit to a single bit, 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 (exp size). Our techniques further suggest that, for boolean total functions, 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). Finally, we argue that approximating PARITY in QAC0, with significantly better than 1/poly(n) advantage on average, is just as hard as computing it exactly. Thus, extending our techniques to higher depths would also rule out approximate circuits for PARITY and related problems
The circuit class QAC0 was introduced by Moore (1999) as a model for constant depth quantum circuits where the gate set includes many-qubit Toffoli gates. Proving lower bounds against such circuits is a longstanding challenge in quantum circuit complexity; in particular, showing that polynomial-size QAC0 cannot compute the parity function has remained an open question for over 20 years. In this work, we identify a notion of the Pauli spectrum of QAC0 circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical AC0 circuits. We conjecture that the Pauli spectrum of QAC0 circuits satisfies low-degree concentration, in analogy to the famous Linial, Mansour, Nisan (LMN) theorem on the low-degree Fourier concentration of AC0 circuits. If true, this conjecture immediately implies that polynomial-size QAC0 circuits cannot compute parity. We prove this conjecture for the class of depth-d, polynomial-size QAC0 circuits with at most nO(1/d) auxiliary qubits. We obtain new circuit lower bounds and learning results as applications: this class of circuits cannot correctly compute the n-bit parity function on more than (1/2 + 2−Ω(n1/d))-fraction of inputs, and the n-bit majority function on more than (1/2 + O(n−1/4))-fraction of inputs. Additionally we show that this class of QAC0 circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for QAC0 circuits. More broadly, our results add evidence that “Pauli-analytic” techniques can be a powerful tool in studying quantum circuits.
Polylogarithmic-Weight Dicke States in QAC0 and Arbitrary Symmetric States in QACf0
An n-qubit Dicke state of weight k, is the uniform superposition over all n-bit strings of Hamming weight k. Dicke states are central to quantum algorithms exhibiting speedups, such as Decoded Quantum Interferometry (Jordan et al., \emph{Nature}, 2025). In the NISQ era, quantum hardware is constrained by both depth and locality, motivating the question of which global operations suffice to prepare such states. QAC0, the quantum analogue of AC0, minimally extends local O(1)-depth quantum circuits by allowing arbitrary-width Toffoli (reversible AND) gates.
We show that Dicke states of polylog(n) weight can be prepared in QAC0. This gives the first QAC0 construction of any super-constant-weight n-qubit Dicke state, since previous constructions relied on the much more powerful FANOUTn gate. In general, we show that any weight-k Dicke state can be constructed using FANOUTmin(k,n−k) gates. Combined with recent hardness results, this yields a tight characterization: for k≤n/2, a n-qubit weight-k Dicke state can be prepared in QAC0 if and only if FANOUTk∈ QAC0. We develop a limited-fanout state-synthesis toolkit for QAC0 that yields further constant-depth, poly(n)-ancilla constructions:
1. Every n-qubit symmetric state supported on Hamming weight ≤k can be prepared using FANOUTk gates.
2. Every O(logn)-qubit state can be prepared using quantum random-access memory (QRAMn), which refers to a coherent indexing gate. QRAMn is a potentially weaker resource than FANOUTn and can be implemented in QACf0.
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.
The computational power of QAC, which are constant-depth quantum circuit families consisting of arbitrary single-qubit unitaries and n-qubit generalized Toffoli gates, has gained tremendous focus recently. In this work, we initiate the study of the computational complexity of geometrically local QAC circuits, where the qubits are arranged in a high-dimensional grid, and all the generalized Toffoli gates act on contiguous qubits along one direction. We prove that 2D-QAC circuits are universal: any QAC circuit can be exactly simulated by a two-dimensional geometrically local QAC circuit. Therefore, we focus on the computational power of 2D-QAC circuits with grid width w. We prove that PARITY computation has a tight width--depth tradeoff of w=nΘ(1/d) in the constant-depth regime if the circuit allows arbitrary ancilla. Moreover, for polynomial-size circuits, a width lower bound of nω(1/d) for PARITY would imply PARITY∈/QAC0. We also establish Fourier tail and sensitivity bounds, together with tight width lower bounds and tight depth lower bounds in 1D-QAC circuits for MAJORITY and MOD computation.
Our main tool is \emph{local measurement variance}, the maximum variance of normalized sums of observables on separated regions. We use a hybrid argument to show that each layer of a 2D-QAC0 circuit increases this variance by at most a factor depending only on its width, yielding a variance bound for every 2D-QAC0 state. The high local measurement variance of cat and Dicke states then gives the width lower bounds.
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.