Optimal and improved gate decompositions for accelerated classical simulation of near-Gaussian fermionic circuits

Abstract

Fermionic Gaussian circuits can be simulated efficiently on a classical computer, but become universal when supplemented with non-Gaussian operations. Similar to stabilizer circuits augmented with non-stabilizer resources, these non-Gaussian circuits can be simulated classically using rank- or extent-based methods. These methods decompose non-Gaussian states or operations into Gaussian ones, with runtimes that scale polynomially with measures of non-Gaussianity such as the rank and the extent -- quantities that typically grow exponentially with the number of non-Gaussian resources. Current fermionic rank- and extent-based simulators are limited to Gaussian circuits with magic-state injection. Extending them to mixed states and non-unitary channels has been hindered by the lack of known extent-optimized decompositions for physically relevant gates and noisy channels. In this work, we address this gap. First, we derive analytic decompositions for key non-Gaussian gates and channels, including decompositions for arbitrary two-qubit fermionic gates which are provably optimal for diagonal gates or those acting on Jordan-Wigner-adjacent qubit pairs. Second, we show that stochastic Pauli noise can reduce the effective extent of non-Gaussian rotation gates, but that fermionic magic is substantially more robust to such noise than stabilizer magic. Finally, we demonstrate how these decompositions can accelerate classical sampling from the output distribution of a quantum circuit. This involves a generalization of existing sparsification methods, previously limited to convex-unitary channels, to circuits involving intermediate measurements and feed-forward. Our decompositions also yield speedups for emulating noisy Pauli rotations with quasiprobability simulators in the large-angle/arbitrary-strength-noise and small-angle/low-noise parameter regimes.

Figures (6)

• Figure 1: The gates $\mathrm{SWAP}$ and $H$ have unitary extent $\xi(\mathrm{SWAP}) = \xi(H) = 2$. The gate $C(\theta)$ has unitary extent $C(\theta) = 1+|\sin(\theta/2)|$, the gates $R_{ZZ}(\theta)$ and $R_Y(\theta)$ have unitary extent $\xi(R_{ZZ}(\theta)) = \xi(R_Y(\theta)) = 1+|\sin\theta|$.
• Figure 2: The gadget in the picture uses the magic state $\ket{v} = (I \otimes V \otimes I) \ket{\psi^+}^{\otimes 2}$ to implement the associated magic gate $V$ (see Hebenstreit_2019). The gadget consists of two Bell measurements which can be implemented using the matchgate $G(H,H)$ and computational basis measurements Hebenstreit_2019. Postselecting the measurement outcome consisting of all zeros gives $V (\ket{\phi_1} \otimes \ket{\psi_2})$ at the output of the circuit. Instead of postselecting, one can apply a circuit of Gaussian operations which uses ancillas initialized in computational basis states to correct when obtaining a measurement outcome different from all-zeros, see Hebenstreit_2019.
• Figure 3: Bloch sphere illustrating the decomposition of the Choi-equivalent state $\psi = \mathcal{N}_\mathcal{Y}(| 0 \rangle \! \langle 0 |)$ of the noisy rotation channel $\mathcal{N}_\mathcal{Y} = (1-p) \mathcal{R_Y}(\theta) + p \mathcal{R_Y}(\theta + \pi)$ for $\theta = \pi/5$ and $p=0.08$. A feasible (non-optimal) decomposition is the convex combination $\psi = (1-p) \psi(\theta) + p \psi(\pi + \theta)$ of the states $\psi(\theta) = R_Y(\theta) \ket{0}$ and $\psi(\pi + \theta) = R_Y(\pi + \theta) \ket{0}$ shown in red.
• Figure 4: Upper bound for the channel extent $\Xi(\mathcal{N_P})$ of the noisy rotation channel $\mathcal{N_P} = \mathcal{E_P}(p) \circ \mathcal{R_P}(\theta)$ for any Pauli operator $P$, for different values of the error probability $p$. This upper bound is tight for the channels $\mathcal{N}_Y$ and $\mathcal{N}_{ZZ}$, i.e., the figure shows $\Xi(\mathcal{N_Y})$ and $\Xi(\mathcal{N_{ZZ}})$.
• Figure 5: The full lines correspond to $1 + (1-2/3p)|\sin (2 \theta)|$, the upper bound obtained in \ref{['eq:FNLNprime']} for the fermionic nonlinearity $\Phi(\mathcal{N}')$ of the channel $\mathcal{N}' = \mathcal{E}(p) \circ \mathcal{R_{ZZ}}(\theta)$ defined in \ref{['eq:Nprime00']}, for different values of the error probability $p$. The circles correspond to the upper bound for $\Phi(\mathcal{N}')$ obtained in Hakkaku_2022 (see Fig. 2 therein) -- the data was kindly provided by the authors.

Theorems & Definitions (48)

• Lemma 1: Ensemble sampling lemma for adaptive circuits
• Lemma 2: Sparsification variance bound
• Definition 1: Convex-Gaussian channels
• Lemma 3: Lemma III.1 in Ref. PhysRevA.84.022310
• Lemma 4: see cudby2024gaussiandecompositionmagicstatesreardonsmith2024fermioniclinearopticalextent
• Definition 2: Rank, analogous to the definition of stabilizer rank in PhysRevX.6.021043
• Definition 3: Pure state extent, analogous to the Definition 3 of stabilizer extent in Bravyi_2019
• Definition 4: Unitary extent and unitary rank, cf. Definition 6 in Bravyi_2019
• Definition 5: Convex-unitary channel
• Definition 6: Convex-unitary channel extent, analogous to Definition 4.19 of stabilizer channel extent in seddonThesis