We focus on the depth optimization of CNOT circuits on hardware with limited connectivity. We adapt the algorithm from Kutin et al. that implements any n -qubit CNOT circuit in depth at most 5n on a Linear Nearest Neighbor architecture. Our proposal is a block version of Kutin et al.’s algorithm that is scalable with the number of interactions available in the hardware: the more interactions we have, the less the depth. We derive better theoretical upper bounds and provide a simple implementation of the algorithm. Overall, we achieve better depth complexity for CNOT circuits on some realistic quantum hardware like a grid or a ladder. For instance, the execution of an n -qubit CNOT circuit on a grid can be done in depth 4n+8 .
We present a 2D nearest-neighbor quantum architecture for Shor's algorithm to factor an
n-bit number in
O(log3n) depth. Our implementation uses parallel phase estimation, constant-depth fanout and teleportation, and constant-depth carry-save modular addition. We derive upper bounds on the circuit resources of our architecture under a new 2D model which allows a classical controller and parallel, communicating modules. We provide a comparison to all previous nearest-neighbor factoring implementations. Our circuit results in an exponential improvement in nearest-neighbor circuit depth at the cost of a polynomial increase in circuit size and width.
Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy
ε, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an
n-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time
T≥1, with circuit depth
O(Tlog(nT/ε)) and gate count
O(nTlog(nT/ε)). This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization.
Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time
T in the stated regime and sufficiently small fixed error
ε, any nearest-neighbor quantum circuit simulating these dynamics requires depth
Ω(logn/loglogn). This matches our
O(logn) upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.
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.
Circuit depth is a central resource in complexity theory. While bounded-depth classical circuits admit well-understood hierarchy theorems, the internal structure of constant-depth quantum computation remains comparatively unexplored.
We prove an explicit depth hierarchy theorem for
QNC0. For each
d≥12, we construct a family of two-round interactive problems on which no depth-
(d−1) quantum circuit can achieve near-perfect success, regardless of gate set, circuit size, or ancillary qubits. In contrast, we prove that our construction admits realizations by simple bounded fan-in quantum circuits of depth larger than
d by a small constant factor. Moreover, all bounded fan-in classical circuits of sublogarithmic depth (in the input size) fail to achieve perfect success on these tasks for every
d, yielding a hierarchy of problems that show unconditional quantum advantage of
QNC0 over
NC0.
A key obstacle is the scarcity of lower bound techniques for quantum circuits. To address this, we develop methods to analyze how depth affects a circuit's ability to realize nonlocal correlations amongst its output qubits in a fine-grained manner. Our approach exploits the correspondence between constraint systems and nonlocal games, translating group-theoretic constructions into rigid operator-valued constraint systems and then into non-local games. In particular, we construct constraint systems whose unique faithful operator-valued solutions require every perfect strategy, and every near-perfect strategy to a fixed precision, to implement multi-controlled phase operations. This reduces to a nonlocal unitary-synthesis problem, yielding depth lower bounds for both shallow quantum and classical circuits.
These results show that increasing depth strictly increases computational power within
QNC0, establishing a genuinely quantum hierarchy.