Quantum advantage with shallow circuits (1704.00690v1)

Abstract: We prove that constant-depth quantum circuits are more powerful than their classical counterparts. To this end we introduce a non-oracular version of the Bernstein-Vazirani problem which we call the 2D Hidden Linear Function problem. An instance of the problem is specified by a quadratic form q that maps n-bit strings to integers modulo four. The goal is to identify a linear boolean function which describes the action of q on a certain subset of n-bit strings. We prove that any classical probabilistic circuit composed of bounded fan-in gates that solves the 2D Hidden Linear Function problem with high probability must have depth logarithmic in n. In contrast, we show that this problem can be solved with certainty by a constant-depth quantum circuit composed of one- and two-qubit gates acting locally on a two-dimensional grid.

• The paper introduces the 2D Hidden Linear Function problem, proving that classical circuits need logarithmic depth to solve it with high probability.
• The paper shows that shallow quantum circuits using constant-depth, localized gates can deterministically solve the HLF problem via quantum entanglement.
• The paper rigorously establishes lower bounds for classical circuits, underscoring the practical advantage of utilizing quantum nonlocality in computation.

Analyzing Quantum Advantage with Shallow Circuits

The paper "Quantum Advantage with Shallow Circuits" by Bravyi, Gosset, and König presents a rigorous demonstration of quantum circuits' superiority over their classical counterparts, specifically in the context of constant-depth computational tasks. The researchers introduce the 2D Hidden Linear Function (HLF) problem as a novel benchmark to establish this quantum-classical separation.

The paper leverages the limitations inherent in classical circuits associated with the $\NC^0$ complexity class—constant-depth circuits with bounded fan-in. In contrast, they illustrate how Shallow Quantum Circuits (SQCs), which also operate within a constant depth using localized one- and two-qubit gates on a two-dimensional grid, can solve certain functions inaccessible to classical counterparts.

Core Contributions and Claims

1. 2D Hidden Linear Function Problem: The paper introduces a non-oracular version of the Bernstein-Vazirani problem called the 2D Hidden Linear Function problem. In this problem, the objective is to identify a linear Boolean function embedded in a quadratic form $q$ over binary fields, which further maps $n$-bit strings to integers modulo four. The vital claim here is that classical circuits require depth that is logarithmic in $n$ to satisfactorily solve this problem with high probability.
2. Quantum Superiority with SQCs: Contrastingly, the authors demonstrate that a quantum approach using constant-depth circuits suffices to deterministically solve the HLF problem. The solution utilizes the entanglement inherent in quantum mechanics, facilitating operations that defy classical emulation within similar constraints.
3. Technical Results and Proofs: A significant portion of the paper confronts classical circuits' limitations via lower bound proofs. It establishes that classical probabilistic circuits cannot resolve the 2D HLF problem using only constant depth. The quantum advantage does not rest on conjectures or assumptions about unresolved complexity class separations (e.g., $\BQP$ vs. $\BPP$) but is rigorously demonstrated using specific problem instances.

The paper's key results are anchored on demonstrable quantum correlations that defy classical simulation. Specifically, the authors explore measurements on cluster states exhibiting quantum nonlocality, concluding that classical circuits can't replicate the SQCs' input-output correlations. Moreover, the construction of a simple quantum circuit consisting of two layers of single-qubit gates and a unitary involving nearest-neighbor interactions substantiates the practical feasibility of their claims.

Implications and Future Directions

From a theoretical standpoint, the paper's findings propel the narrative of quantum computing's capability to achieve operational feats using minimalistic yet potent resources—shallow circuits. The implications of this research are profound for next-generation quantum algorithms and could accelerate the momentum toward fully realized quantum computing systems.

Practically, the insights into quantum versus classical power dynamics could inform hardware architecture, specifically in crafting efficient quantum processors that capitalize on shallow circuits' advantages. It also introduces a path for further studies into constant-time quantum algorithms and their unique characteristics bypassing classical circuit limitations.

Future research should explore scaling these principles, optimizing resource allocations further, and examining the full scope of shallow circuits for extensive quantum algorithms. Additionally, extending these results to sampling problems remains an open and promising avenue for understanding quantum computational superiority more comprehensively.

In summary, this work enriches the landscape of quantum computation by delineating specific instances where quantum approaches, particularly shallow circuits, transcend classical possibilities, opening doors for innovations in both theoretical and practical applications of quantum technologies.