Matrix product state approach to lossy boson sampling and noisy IQP sampling

TL;DR

MPS-based classical algorithms for lossy boson sampling and noisy instantaneous quantum polynomial-time (IQP) sampling are developed, showing that classical simulability emerges at transmission rates scaling as $O(1/\sqrt{N})$, reaching the known boundary of quantum advantage with a tunable and scalable method.

Abstract

Sampling problems have emerged as a central avenue for demonstrating quantum advantage on noisy intermediate-scale quantum devices. However, physical noise can fundamentally alter their computational complexity, often making them classically tractable. Motivated by the recent success of matrix product state (MPS)-based classical simulation of Gaussian boson sampling (Oh et al., 2024), we extend this framework to investigate the classical simulability of other noisy quantum sampling models. We develop MPS-based classical algorithms for lossy boson sampling and noisy instantaneous quantum polynomial-time (IQP) sampling, both of which retain the tunable accuracy characteristic of the MPS approach through the bond dimension. Our approach constructs pure-state decompositions of noisy or lossy input states whose components remain weakly entangled after circuit evolution, thereby providing a means to systematically explore the boundary between quantum-hard and classically-simulable regimes. For boson sampling, we analyze single-photon, Fock, and cat-state inputs, showing that classical simulability emerges at transmission rates scaling as $O(1/\sqrt{N})$, reaching the known boundary of quantum advantage with a tunable and scalable method. Beyond reproducing previous thresholds, our algorithm offers significantly improved control over the accuracy-efficiency trade-off. It further extends the applicability of MPS-based simulation to broader classes of noisy quantum sampling models, including IQP circuits.

Figures (10)

• Figure 1: Photon loss in boson sampling circuit. (a) Lossy boson sampling circuit. Blue rectangles represent ideal beam splitters and phase shifters, while orange squares represent photon loss channels. (b) Assuming a uniform transmission rate, we separate the loss channel from the ideal linear optics and locate it in front of the ideal optics. $\eta$ is defined as the transmission rate of the total loss channel.
• Figure 2: Setup of a noisy IQP circuit. (a) The circuit consists of $Z$-diagonal gates (blue rectangles) with dephasing noise (orange squares) applied at each gate. Measurements are performed in the $X$ basis, equivalently implemented by a Hadamard layer followed by $Z$-basis readout. (b) Because dephasing commutes with $Z$-diagonal gates, all dephasing channels can be propagated to the input and consolidated as $\mathcal{N}_{0,0,p_d}$ acting before the circuit.
• Figure 3: Setup of a depolarizing-noisy IQP circuit. (a) The IQP circuit comprises $Z$-diagonal gates (blue rectangles), with a single-qubit depolarizing channel (orange squares) applied at each gate. Measurements are performed in the $X$ basis. (b) Each depolarizing channel is represented as a probabilistic mixture of Pauli errors $X$, $Y$, and $Z$, each occurring with rate $q$.
• Figure 4: Simulation of lossy boson sampling using pure-state decomposition. (a) The Fock input $|1\rangle^{\otimes N}|0\rangle^{\otimes (M-N)}$ goes through the loss channel, yielding the product mixed state $\hat{\sigma}^{\otimes N}\otimes|0\rangle\langle 0|^{\otimes (M-N)}$, where $\hat{\sigma}$ is the lossy single-photon state. (b) For sampling, each $\hat{\sigma}$ is decomposed into $|\psi_+\rangle$ and $|\psi_-\rangle$, drawn with equal probability.
• Figure 5: Entanglement (von Neumann) entropy of the output state of lossy boson sampling in the worst case, $\theta=\pi/4$. A complexity transition, from efficient to inefficient, occurs between coefficient values 4 and 5.