On computational complexity and average-case hardness of shallow-depth boson sampling

TL;DR

This work explores the viability of achieving quantum computational advantage through boson sampling with shallow-depth linear optical circuits and obtains the average-case hardness for logarithmic-depth Fock-state boson sampling subject to lossy environments and for the logarithmic-depth Gaussian boson sampling.

Abstract

Boson sampling, a computational task believed to be classically hard to simulate, is expected to hold promise for demonstrating quantum computational advantage using near-term quantum devices. However, noise in experimental implementations poses a significant challenge, potentially rendering boson sampling classically simulable and compromising its classical intractability. Numerous studies have proposed classical algorithms under various noise models that can efficiently simulate boson sampling as noise rates increase with circuit depth. To address this issue particularly related to circuit depth, we explore the viability of achieving quantum computational advantage through boson sampling with shallow-depth linear optical circuits. Specifically, as the average-case hardness of estimating output probabilities of boson sampling is a crucial ingredient in demonstrating its classical intractability, we make progress on establishing the average-case hardness confined to logarithmic-depth regimes. We also obtain the average-case hardness for logarithmic-depth Fock-state boson sampling subject to lossy environments and for the logarithmic-depth Gaussian boson sampling. By providing complexity-theoretical backgrounds for the classical simulation hardness of logarithmic-depth boson sampling, we expect that our findings will mark a crucial step towards a more noise-tolerant demonstration of quantum advantage with shallow-depth boson sampling.

Figures (6)

• Figure 1: Outlines of our result
• Figure 2: Schematics of the butterfly circuit architectures in Definition \ref{['butterfly']} and their unitary matrix form, for mode number $M = 2^4 = 16$.
• Figure 3: Schematics of an mode number $M = 16$ circuit $C$ in $\mathcal{B}\mathcal{B}^*$ which contains a given $M_0 = 8$ mode circuit $C_0$ in $\mathcal{B}\mathcal{B}^*$
• Figure 4: (a) Schematics of the unit brickwork qubit circuit for measurement-based quantum computation raussendorf2001onebroadbent2009universalchilds2005unified. Each vertex (i.e., black dot) represents a $\ket{+}$ qubit, and each edge represents the CZ gate applied on two qubits. After CZ gate operations, each vertex is measured with a rotated $\ket{+}$ basis, where the rotation angle is explicitly given for measurement-based quantum computation. (b) Schematics of the 4-depth linear optical circuit proposed by brod2015complexity corresponding to the unit brickwork qubit circuit in (a) implemented by the KLM scheme (for more details, see Ref. brod2015complexity). Each vertex (i.e., white dot) represents zero or one photon Fock-state, and each edge represents the beam splitter with arbitrarily chosen coefficients. The initial beam splitter layer (red) corresponds to the input qubit (and ancillas), the following two beam splitter layers (yellow and green) correspond to the CZ gate implementation knill2002quantum, and the last beam splitter layer (blue) corresponds to the qubit rotation required for the measurement-based quantum computation. After the beam splitter operations, each vertex is measured with the Fock-state basis. (c) Schematics of the unit brickwork linear optical circuit in (b) but mapped in 3-dimensional grid linear optical circuit architecture. Here, the beam splitters are applied in parallel along each dimension, where the application sequence is along the direction $x \rightarrow y \rightarrow z \rightarrow x$.
• Figure 5: Schematics of $\mathcal{B}$ constructing (a) 1-dimensional grid linear optical circuit and (b) 2-dimensional grid linear optical circuit under a proper mode index permutation, for mode number $M = 16$. The grey-colored edges indicate trivial (i.e., identity) gates, and the other colored edges indicate non-trivial gates that compose the grid linear optical circuit we aim to construct.

Theorems & Definitions (44)

• Theorem 1: Informal
• Theorem 2: Informal
• Definition 1: Linear optical circuit architecture
• Definition 2: Butterfly circuit architecture
• Definition 3: Kaleidoscope circuit architecture
• Lemma 1: Dao et al dao2020kaleidoscope
• Definition 4: Local random circuit ensemble
• Definition 5: Random permutation circuit ensemble
• Lemma 2: Bosonic birthday paradox for the product circuit ensemble
• Remark 1: Conditions for the bosonic birthday paradox