The computational power of $\mathsf{QAC}$, 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 $\mathsf{QAC}$ 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 $\mathsf{2D\text{-}QAC}$ circuits are universal: any $\mathsf{QAC}$ circuit can be exactly simulated by a two-dimensional geometrically local $\mathsf{QAC}$ circuit. Therefore, we focus on the computational power of $\mathsf{2D\text{-}QAC}$ circuits with grid width $w$. We prove that $\mathsf{PARITY}$ computation has a tight width--depth tradeoff of $w=n^{Θ(1/d)}$ in the constant-depth regime if the circuit allows arbitrary ancilla. Moreover, for polynomial-size circuits, a width lower bound of $n^{ω(1/d)}$ for $\mathsf{PARITY}$ would imply $\mathsf{PARITY}\notin\mathsf{QAC}^0$. We also establish Fourier tail and sensitivity bounds, together with tight width lower bounds and tight depth lower bounds in $\mathsf{1D\text{-}QAC}$ circuits for $\mathsf{MAJORITY}$ and $\mathsf{MOD}$ 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 $\mathsf{2D\text{-}QAC}^0$ circuit increases this variance by at most a factor depending only on its width, yielding a variance bound for every $\mathsf{2D\text{-}QAC}^0$ state. The high local measurement variance of cat and Dicke states then gives the width lower bounds.
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.
We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant δ>0, it ε-approximates an arbitrary single-qubit gate using O(log1+δ(1/ε)) clean ancillae, Hadamard and T single-qubit gates, O(log(1/ε))-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, T, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth O(loglog(1/ε)), and we prove a matching Ω(loglog(1/ε))-depth lower bound. Overall, we establish that Θ(loglog(1/ε))-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis.
These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-d QAC circuit can be simulated in depth O(d) using only Hadamard, X, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity∈/QAC0 to proving a Parity lower bound against circuits consisting only of Hadamard, X, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.
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.
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.
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
Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth.
We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential.
The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy.
Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension d and M≥d2−1 ports, we construct a unitary circuit of depth O(d), independent of M and including resource preparation and port selection, with entanglement fidelity at least (1−(d2−1)/(2M))2. Thus, at fixed d, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on d can also be removed remains open.