Does qubit connectivity impact quantum circuit complexity?

TL;DR

This paper demonstrates that quantum circuits for arbitrary unitaries can be efficiently implemented under various qubit connectivity constraints, including 1D chains, with near-optimal depth and size, and explores the impact of connectivity on specific unitary families.

Contribution

It shows that connectivity constraints like 1D chains do not fundamentally limit the efficiency of quantum circuit implementation for general unitaries, extending results to various graph structures and special unitary families.

Findings

Quantum circuits for any unitary can be implemented with near-optimal depth and size under 1D chain connectivity.

Connectivity constraints can be mitigated with ancilla qubits, enabling polynomial depth circuits.

Special unitary families like diagonal and block diagonal unitaries admit nearly optimal implementations despite connectivity limitations.

Abstract

Some physical implementation schemes of quantum computing can apply two-qubit gates only on certain pairs of qubits. These connectivity constraints are commonly viewed as a significant disadvantage. For example, compiling an unrestricted $n$-qubit quantum circuit to one with poor qubit connectivity, such as a 1D chain, usually results in a blowup of depth by $O(n^2)$ and size by $O(n)$. It is appealing to conjecture that this overhead is unavoidable -- a random circuit on $n$ qubits has $\Theta(n)$ two-qubit gates in each layer and a constant fraction of them act on qubits separated by distance $\Theta(n)$. While it is known that almost all $n$-qubit unitary operations need quantum circuits of $\Omega(4^n/n)$ depth and $\Omega(4^n)$ size to realize with all-to-all qubit connectivity, in this paper, we show that all $n$-qubit unitary operations can be implemented by quantum circuits of $O(4^n/n)$ depth and $O(4^n)$ size even under {1D chain} qubit connectivity constraint. We extend this result and investigate qubit connectivity in three directions. First, we consider more general connectivity graphs and show that the circuit size can always be made $O(4^n)$ as long as the graph is connected. For circuit depth, we study $d$-dimensional grids, complete $d$-ary trees and expander graphs, and show results similar to the 1D chain. Second, we consider the case when ancillary qubits are available. We show that, with ancilla, the circuit depth can be made polynomial, and the space-depth trade-off is not impaired by connectivity constraints unless we have exponentially many ancillary qubits. Third, we obtain nearly optimal results on special families of unitaries, including diagonal unitaries, 2-by-2 block diagonal unitaries, and Quantum State Preparation (QSP) unitaries, the last being a fundamental task used in many quantum algorithms for machine learning and linear algebra.