Invested and Potential Magic Resources in Measurement-Based Quantum Computation

Abstract

Magic states and magic gates are crucial for achieving universal quantum computation, but important questions about how magic resources should be implemented to attain maximal quantum advantage have remained unexplored, especially in the context of measurement-based quantum computation (MQC). This work bridges the gap between MQC and the resource theory of magic by introducing the key concepts of "invested" and "potential" magic resources. The former quantifies the magic cost associated with MQC, serving as both a resource witness and a feasible upper bound for the practical realization, and is gate-order independent; The latter represents the maximal achievable magic resource in a given graph structure defining MQC. We utilize both concepts to analyze the quantum Fourier transform (QFT) and provide a fresh perspective on the universality of MQC, highlighting the crucial role of non-Pauli measurements in injecting magic. In particular, we theoretically prove that high-dimensional graphs can generate an exponential advantage of MQC compared to classical computing. We demonstrate experimentally our theoretical findings in a high-fidelity four-photon setup, surpassing conventional magic state injection (MSI) methods in both qubit efficiency and resource utilization. Our findings pave the way for future research exploring magic resource optimization and novel distillation schemes within the MQC framework, advancing fault-tolerant universal quantum computation.

Figures (3)

• Figure 1: Cartoon showing the relationship between invested magic resources ($\mathcal{M}$), potential magic resources ($\mathcal{P}$), and reserved magic resources ($\mathcal{R}$) using a water-pouring analogy.
• Figure 2: Demonstration of invested magic resources in QFT. (a) and (b) show the distribution of invested magic resources across different frequencies $k$, with a concentration of resources in the lower-frequency range. (c) illustrates the scaling of invested magic resources with the number of qubits, with the 'T injection' curve representing MSI with T-counts equal to $n$. This provides a direct comparison between the two methods for generating magic states.
• Figure 3: (a) Experimental generation of the $|T\rangle$ state using a 1D graph in three steps. The angles of $M^\theta$ are ${0,\theta_m,\pi/4}$ for steps 1, 2, and 3, respectively. The plot on the right displays the total invested magic resources $\mathcal{M}_2$ and the reserved magic resource $\mathcal{R}$ at each step. (b) Experimental realization of QFT using a 2D graph. The left side of the figure depicts the 2D graph structure employed for implementing the QFT circuit of $n=2$ (left bottom) and indicates the order of measurements performed on the qubits (steps 1 to 4). The generation of $|\text{CS}\rangle$ from MQC requires at least 6 qubits, as the figure shows. Step 3 on the 2'th qubit and step 4 on the 3'th qubit are equivalent to the local rotation for the physical qubits 2nd and 3rd qubit. According to Eq.\ref{['eq: Box']}, implementing $M_1^0$, $M_4^{(\pi/8,0)}$, $\hat{T}_2$, and $\hat{T}_3$ corresponds to the steps 1 to 4, in which $M_4^{(\pi/8,0)}$ is for non-standard measurement $M^{(\phi,\theta)}$, projection onto $\{\cos\phi|0\rangle\pm e^{-i\theta}\sin\phi|1\rangle\}$, and $\hat{T}$ is T-gate $\hat{T}=\text{diag}[1,e^{i\pi/4}]$. Move experimental results see Sec. IX and Sec. X in supp.