Unlocking early fault-tolerant quantum computing with mitigated magic dilution

TL;DR

The paper introduces mitigated magic dilution (MMD), a framework that combines quantum error mitigation with a quasiprobability decomposition to synthesize small-angle single-qubit Z-rotations from noisy encoded magic states, addressing the resource bottlenecks of magic-state distillation in the early fault-tolerant era. By formulating a linear combination of channels (LCC) and climbing the Clifford hierarchy to include $T^{1/n}$ gates, the authors derive optimal decompositions with favorable sampling overheads, quantified by the $l_1$-norm $\Lambda_G$ and the resource-saving exponent $\gamma$. They show that dephasing noise on non-Clifford resources can be incorporated into the framework, allowing for better-than-quadratic or even higher-order savings depending on $n$ and noise level, particularly for small rotation angles. Applying MMD to the 2D Fermi-Hubbard model via a second-order Trotterisation demonstrates substantial reductions in magic-state consumption and sampling overhead compared with classical simulators and direct synthesis, with concrete gains on lattice sizes up to $8\times8$ and runtime penalties kept within feasible MegaQuOp scaling. Overall, MMD provides a practical pathway to leverage error-mitigated, high-level Clifford operations to enable meaningful quantum simulations on near-term, resource-constrained quantum devices.

Abstract

As quantum computing progresses towards the early fault-tolerant regime, quantum error correction will play a crucial role in protecting qubits and enabling logical Clifford operations. However, the number of logical qubits will initially remain limited, posing challenges for resource-intensive tasks like magic state distillation. It is therefore essential to develop efficient methods for implementing non-Clifford operations, such as small-angle rotations, to maximise the computational capabilities of devices within these constraints. In this work, we introduce mitigated magic dilution (MMD) as an approach to synthesise small-angle rotations by employing quantum error mitigation techniques to sample logical Clifford circuits given noisy encoded magic states. We explore the utility of our approach for the simulation of the 2D Fermi-Hubbard model. We identify evolution time regimes where MMD outperforms state-of-the-art synthesis techniques in the number of noisy encoded magic states required for square lattices up to size $8 \times 8$. Moreover, we demonstrate that our method can provide a practical advantage that is quantified by a better-than-quadratic improvement in the resource requirements for small-angle rotations over classical simulators. This work paves the way for early fault-tolerant demonstrations on devices supporting millions of quantum operations, the so-called MegaQuOp regime.

Figures (8)

• Figure 1: Action of $\mathcal{T}^{k}$ rotation channels (where $1 \leq k \leq 8$) on a density matrix $\rho$. The eight operations (in red) form the initial basis set $\mathcal{G}$. The target rotation channel $\mathcal{R}_z^{\theta}(\rho)$ is shown in blue.
• Figure 2: Generalised teleportation circuit to implement a $\sqrt[n]{T}$ gate using $i$ distinct magic states, where $n = 2^{i-1}$, and Clifford operations. Boxes and gates with a dashed line are classically controlled; they are only implemented if the measurement below obtains an outcome with eigenvalue $-1$.
• Figure 3: $\Lambda_{G}(\mathcal{R}_z^{\theta})$ as a function of target rotation angle $\theta$ for different optimal decompositions of the form $\{\mathcal{I}, \varepsilon(\mathcal{T}^\frac{1}{n}{}), \mathcal{Z}\}$ including the optimal Clifford decomposition of $\{\mathcal{I}, \mathcal{S}, \mathcal{Z}\}$. A dephasing error of $0.1\%$ is assumed for non-Clifford state preparation.
• Figure 4: Degree of magic resource saving $\gamma$ as a function of target rotation angle $\theta$ for different optimal decompositions of the form $\{\mathcal{I}, \varepsilon(\mathcal{T}^\frac{1}{n}), \mathcal{Z}\}$ relative to the optimal Clifford decomposition of $\{\mathcal{I}, \mathcal{S}, \mathcal{Z}\}$. A dephasing error of $0.1\%$ is assumed for non-Clifford state preparation.
• Figure 5: Expected number of magic states $E$ per sample as a function of target rotation angle $\theta$ for different optimal decompositions of the form $\{\mathcal{I}, \varepsilon(\mathcal{T}^\frac{1}{n}), \mathcal{Z}\}$ relative to the optimal Clifford decomposition of $\{\mathcal{I}, \mathcal{S}, \mathcal{Z}\}$. A dephasing error of $0.1\%$ is assumed for non-Clifford state preparation.