We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number $a$ of ancill\ae\ and $n$ input qubits, we give a tradeoff between $a$ and the required depth, that results in a non-constant lower bound for fanout when $a = n^{1-o(1)}$. We also show that, regardless of the number of ancill\ae\, arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.
We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number a of ancill\ae\ and n input qubits, we give a tradeoff between a and the required depth, that results in a non-constant lower bound for fanout when a=n1−o(1). We also show that, regardless of the number of ancill\ae, arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.
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.
We show that the quantum parity gate on n>3 qubits cannot be cleanly simulated by a quantum circuit with two layers of arbitrary C-SIGN gates of any arity and arbitrary 1-qubit unitary gates, regardless of the number of allowed ancilla qubits. This is the best known and first nontrivial separation between the parity gate and circuits of this form. The same bounds also apply to the quantum fanout gate. Our results are incomparable with those of Fang et al. [3], which apply to any constant depth but require a sublinear number of ancilla qubits on the simulating circuit.
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.
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.
The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice QNC0/qpoly , and AC0[p] , which is the class of classical constant-depth circuits with unbounded-fan in and MODp gates. Additionally, we show a separation between QAC0 , the circuit class containing Toffoli gates with unbounded control, and AC0[p] , when QAC0 is augmented with additional mid-circuit measurements and classical fanout. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fanout gates, while relying solely on finite quantum gate sets. Equivalently, this yields a separation between AC0[p] and [QNC0,AC0]2 , i.e., shallow quantum circuits interleaved with simple classical computation. Secondly, we consider QNC0 circuits with infinite-size gate sets. We show that these circuits, along with quantum prime modular gates or classical prime modular gates in combination with classical fanout, can implement threshold gates, showing that QNC0[p]=QTC0 . Finally, we also show that in the infinite-size gate set case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.