Quantum advantage from measurement-induced entanglement in random shallow circuits

TL;DR

The paper investigates how measurement induced entanglement (MIE) in random 2D quantum circuits can drive a quantum advantage in classically simulating their output distributions. It establishes a link between subsystem anticoncentration and long-range MIE, and develops both a shallow classical simulator for circuits with short-range MIE and unconditional quantum advantage results for random shallow Clifford circuits. A coarse-grained two-layer architecture with gates acting on $\tau\times\tau$ blocks (with $\tau=O(\sqrt{\log n})$) is proven to exhibit long-range MIE and unconditional quantum advantage, and these results extend to approximate two-qubit gate compilations that maintain the same entanglement properties. The work thus demonstrates a phase-transition like behavior in the MIE and provides rigorous evidence of quantum advantage in canonical random circuit sampling tasks using Clifford and coarse-grained architectures. This advances understanding of when constant-depth quantum circuits can outperform shallow classical circuits in realistic, locally interacting geometries.

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.

Figures (6)

• Figure 1: A 2D grid of qubits, partitioned into three regions $A,B$ and $C$. The purple region indicates the single qubit $A$. The red region indicates the square shielding region $B$ with side length $L$ which in this example is $L=7$. All other qubits are in $C$.
• Figure 2: Classical simulation of MIE in 2D random Clifford circuits in the brickwork architecture (see \ref{['fig:brickwork']}). Here $A$ is the qubit at the centre of the grid, $C$ contains all the qubits on the boundary of the grid, and $B$ contains all the other qubits. Each data point is estimated from $1040$ circuit instance samples. The simulation is performed using Stim gidney2021stim.
• Figure 3: 2D brickwork architecture on a $6\times6$ grid of qubits. Edges with the same color indicate two-qubit gates that can be applied simultaneously in a single layer. A circuit of depth $d$ applies a sequence of $d$ layers that rotates through the four colors.
• Figure 4: Coarse-grained circuit with two layers of gates for $m=4$ and $\tau=4$. The shaded region denotes the support of the first layer gates $V_1,\dots, V_{9}$. The gates with green boundaries represent the second-layer gates $W_1,\dots, W_{16}$.
• Figure 5: A cluster $\mathcal{G}(P)$ indicated in gray, with size $\lvert\mathcal{G}(P)\rvert = 11$ and perimeter$\mathrm{Per}(\mathcal{G}(P)) = 19$. Note edges of the cluster running along the exterior of the grid do not contribute to the perimeter.

Theorems & Definitions (45)

• Theorem 1: Informal
• Theorem 2: Informal
• Theorem 3: Informal
• Theorem 4
• Theorem 5
• Lemma 1
• proof
• Theorem 6: Formal version of \ref{['res:anticoncentration']}
• proof : Proof of \ref{['thm:anticonc']}
• Lemma 2