Quantum advantage from measurement-induced entanglement in random shallow circuits (2407.21203v1)

Abstract: We study random constant-depth quantum circuits in a two-dimensional architecture. While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth is at least a constant critical value. For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a quantum advantage phase transition in the classical hardness of sampling from the output distribution. Here we provide evidence for a quantum advantage phase transition in the setting of random Clifford circuits. Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth-d 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth O(d). Finally, we introduce a two-dimensional, depth-2, "coarse-grained" circuit architecture, composed of random Clifford gates acting on O(log n) qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage.

• The paper identifies a critical circuit depth (d*=6) where long-range measurement-induced entanglement emerges, drastically increasing the classical hardness of simulation.
• The paper rigorously proves that random Clifford circuits exhibiting long-range entanglement confer an unconditional quantum advantage over efficiently simulatable circuits.
• The paper introduces a probabilistic simulation algorithm for shallow 2D circuits, effectively approximating quantum output distributions under short-range entanglement conditions.

Quantum Advantage from Measurement-Induced Entanglement in Random Shallow Circuits

Introduction

The exploration of quantum advantage—where quantum computers surpass classical ones in computational tasks—is a primary objective in quantum information science. This paper examines the phenomenon using random constant-depth quantum circuits in a two-dimensional (2D) architecture, focusing specifically on the role of measurement-induced entanglement (MIE). The authors conjecture that long-range MIE proliferates when circuit depth exceeds a constant critical value $d^*$. The paper posits that this transition correlates with a phase transition in the classical hardness of sampling from these circuits' output distributions. The paper's contributions extend to proof-based analyses for random Clifford circuits and offer new insights into the classical simulation capacities of such quantum systems.

Main Contributions

Measurement-Induced Entanglement and Quantum Advantage

The authors explore MIE in 2D random quantum circuits consisting of Haar-random two-qubit gates. They introduce critical depth $d^*$ beyond which long-range MIE manifests, making classical simulation inefficacious. They provide numerical and heuristic evidence linking this phase transition to the quantum advantage—i.e., the classical hardness of sampling from the output distribution increases significantly for depths $d \geq d^*$.

Unconditional Quantum Advantage in Clifford Circuits

The paper extends existing theoretical separations between the computational powers of constant-depth quantum and classical circuits. By focusing on random Clifford circuits, the authors provide rigorous evidence of quantum advantage through long-range MIE. Specifically, they demonstrate that any 2D quantum circuit of depth $d$, satisfied with a short-range MIE property, can be classically simulated efficiently. In contrast, circuits with long-range MIE afford an unconditional quantum advantage.

Simulation Algorithms

The paper introduces a classical simulation algorithm that efficiently handles shallow 2D circuits exhibiting short-range MIE. This algorithm features a probabilistic method capable of parallelizing the classical gates to depth $O(d)$, showcasing the effective classical simulation of low-depth quantum circuits with a structured distribution of entanglement.

Results Overview

Numerical Evidence of MIE

The authors present numerical simulations indicating a shift in MIE behavior at the conjectured critical circuit depth $d^* = 6$, transitioning from short-range to long-range entanglement. These results imply a computationally significant phase transition that limits classical simulation in higher-depth regimes.

Classical Simulation Algorithm

The core of their approach lies in a modified gate-by-gate simulation method. This algorithm probabilistically simulates depth-$d$ quantum circuits, maintaining computational efficiency even as it scales with the number of qubits ($n$). Under the short-range MIE condition, the authors prove the classical simulation algorithm achieves a sampling that closely approximates the quantum circuit’s output distribution.

Proof of Long-Range MIE

A notable contribution is establishing a theoretical framework for proving long-range MIE in Clifford circuits. The authors extend these findings to propose a generalized architecture with “coarse-grained” random circuits. For instance, in a two-layer quantum circuit with gate operations on $O(\log(n))$ qubits, the existence of long-range MIE is rigorously proven, thus supporting the assertions of unconditional quantum advantage.

Implications and Future Directions

Practical Implications

The results are significant for near-term quantum computing, particularly in benchmarking experiments and quantum device characterization. The insights into MIE and classical hardness thresholds pave the way for identifying architectures and tasks that are suitable candidates for demonstrating quantum supremacy.

Theoretical Speculations

The work sparks several theoretical open questions: Precisely proving the $d^*$ phase transition, extending analysis to universal gate sets beyond Clifford circuits, and exploring the broader implications of these transitions in quantum computational complexity. Future research could focus on further refining the criteria that distinguish efficiently simulatable quantum circuits from those that confer a classical intractability due to emergent long-range entanglement properties.

Conclusion

This paper provides a multi-faceted exploration of the interplay between MIE and quantum advantage in 2D random shallow quantum circuits. Through numerical evidence, rigorous proofs, and algorithmic innovations, it establishes significant groundwork in understanding the classical simulation limits and heralds new territories for demonstrating quantum computational supremacy.

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