Worst-case depth hierarchy for shallow quantum circuits

2026-06-15

AI Summary

This work resolves the long-standing open problem of establishing a depth hierarchy theorem within constant-depth quantum circuits (QNC⁰). By constructing a family of two-round interactive problems and leveraging tools from constraint systems, nonlocal games, and group theory, the authors reduce the uniqueness of operator-valued solutions to multi-controlled phase operations into a method for proving circuit depth lower bounds. They demonstrate that for every $d \geq 12$, there exists a computational task that cannot be approximately solved with near-perfect accuracy by any quantum circuit of depth $d-1$, yet is efficiently solvable by a slightly deeper bounded-fan-in quantum circuit; moreover, no classical circuit of sublogarithmic depth can perfectly solve this task. This constitutes the first unconditional, explicit worst-case depth hierarchy in QNC⁰ and exhibits a provable quantum advantage of QNC⁰ over classical NC⁰.

Abstract

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 $\mathsf{QNC}^0$. For each $d\ge 12$, 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 $\mathsf{QNC}^0$ over $\mathsf{NC}^0$. 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 $\mathsf{QNC}^0$, establishing a genuinely quantum hierarchy.

Problem

• quantum circuits
• circuit depth
• depth hierarchy
• QNC⁰
• computational power

Innovation

• quantum circuit depth hierarchy
• QNC⁰
• nonlocal games
• operator-valued constraint systems
• unconditional quantum advantage