Symmetry-accelerated classical simulation of Clifford-dominated circuits

TL;DR

This work exploits symmetries in the computation of the stabilizer extent, proving that for real, diagonal, and real-diagonal unitaries, the optimization can be restricted to the corresponding subgroups of the Clifford group without loss of optimality.

Abstract

Classical simulation of quantum circuits plays a crucial role in validating quantum hardware and delineating the boundaries of quantum advantage. Among the most effective simulation techniques are those based on the stabilizer extent, which quantifies the overhead of representing non-Clifford operations as linear combinations of Clifford unitaries. However, finding optimal decompositions rapidly becomes intractable as it constitutes a superexponentially large optimization problem. In this work, we exploit symmetries in the computation of the stabilizer extent, proving that for real, diagonal, and real-diagonal unitaries, the optimization can be restricted to the corresponding subgroups of the Clifford group without loss of optimality. This ``strong symmetry reduction'' drastically reduces computational cost, enabling optimal decompositions of unitaries on up to seven qubits using a standard laptop--far beyond previous two-qubit limits. Additionally, we employ a ``weak symmetry reduction'' method that leverages additional invariances to shrink the search space further. Applying these results, we demonstrate exponential runtime improvements in classical simulations of quantum Fourier transform circuits and measurement-based quantum computations on the Union Jack lattice, as well as new insights into the non-stabilizer properties of multi-controlled-phase gates and unitaries generating hypergraph states. Our findings establish symmetry exploitation as a powerful route to scale classical simulation techniques and deepen the resource-theoretic understanding of quantum advantage.

Figures (11)

• Figure 1: Order of the Clifford group $\mathcal{C}_{n}$ and subgroups of interest. The figure presents the size of the Clifford group $\mathcal{C}_{n}$ in solid blue, the real subgroup $\mathcal{R}_{n}$ in dashed orange, the diagonal subgroup $\mathcal{D}_{n}$ in dash-dotted green, and the real-diagonal subgroup in dotted red.
• Figure 2: Stabilizer extent of two-qubit $\text{fSim}(\theta,\phi)$ gates as a function of the parameters $\theta$ and $\phi$. Color map depicting (a) the (true) stabilizer extent, $\xi \left(\text{fSim}(\theta,\phi) \right)$, (b) the ratio between the minimum $\ell_1$-norm squared obtained only with the transposition-invariant Cliffords and the (true) stabilizer extent, $\tilde{\xi}_{\mathcal{T}_{2}}\left(\text{fSim}(\theta,\phi) \right) /\xi\left(\text{fSim}(\theta,\phi) \right)$, and (c) the ratio between the minimum $\ell_1$-norm squared obtained using the permutation-invariant Cliffords and the (true) stabilizer extent, $\tilde{\xi}_{\mathcal{S}_{2}}\left(\text{fSim}(\theta,\phi) \right) /\xi\left(\text{fSim}(\theta,\phi) \right)\,.$ If the result in Theorem \ref{['theorem: Main result -- diagonals and reals']} were to hold for this particular example with the transposition and permutation invariances, the ratios depicted in (b) and (c) would have to be 1 for every pair $(\theta,\phi)$. That is eminently not the case, even though it is possible to expand $\text{fsim}(\theta,\phi)$ in terms of $\mathcal{C}_{2}^{\mathcal{T}_2}$ and $\mathcal{C}_{2}^{\mathcal{S}_{2}}$.
• Figure 3: Stabilizer extent of the multi-controlled-$P(\theta)$ gates as a function of $\theta$. The figure shows the stabilizer extent of the $C^{n-1}P (\theta)$ gates as defined in Eq. \ref{['eq: def multi-controlled Z']} for $n\in \{1,\dots,6\}$. The background of the plot is divided into regions colored in different shades of blue, highlighting values of the stabilizer extent between $\xi (T^{s-1})$ and $\xi (T^s)$ for $s\in \{1, 7\}$ and allow us to immediately identify the minimum number of $T$ gates needed to synthesize the different gates, as per Proposition \ref{['prop: gate synthesis']}.
• Figure 4: Submultiplicativity of the controlled-phase gates in the quantum Fourier transform subroutine. The blue circles depict the values obtained by multiplying the stabilizer extent of the individual $\{CP_{1,j}(2\pi i / 2^j)\}_{j=2}^{k+1}$ gates. Each circle corresponds to the value obtained for the largest unitary block, $U_k$, in a QFT subroutine with $(k+1)$ qubits. In yellow, we present the results obtained when splitting each $U_k$ into unitary blocks of the maximum size. Up to $k=4$, the results correspond to the true value of the stabilizer extent. For $5\leq k \leq 8$, each $U_k$ is split into two blocks whose stabilizer extent is multiplied. Finally, for $k\geq 9$, each $U_k$ is split into three blocks.
• Figure 5: The role of entanglement in the computation of $\xi(C^{n-1}Z)$. We show the minimum value found for the value of the $\ell_1$-norm squared of the decomposition of $C^{n-1} Z$ gates in terms of real-diagonal Cliffords. This value is plotted against the number of $CZ$ categories, $\mathfrak{C}_i$, used during the optimization. Note that given a fixed number of categories, several different combinations are possible; we tested all of these combinations and present only the minimum value in each case. Note that, for $n\leq 6$, the value of the stabilizer extent is achieved using only four $\mathfrak{C}_i$ categories. On the other hand, for $n=7$, five such sets are needed. However, this is still a large improvement from the total of 21 different sets $\mathfrak{C}_i$ that exist for that number of qubits.

Theorems & Definitions (19)

• Definition 1: Stabilizer extent for pure states Bravyi2019
• Definition 2: Stabilizer extent for matrices Bravyi2019
• Proposition 1
• Proposition 2: Exact gate synthesis
• Definition 3: $\mathcal{G}$-symmetric stabilizer extent
• Theorem 1: Strong $\mathcal{G}$-symmetry reduction
• Lemma 1
• proof : Proof of Theorem \ref{['theorem: Main result -- diagonals and reals']}
• Lemma 2: Weak $\mathcal{G}$-symmetry reduction
• Definition 4: Hypergraph quantum states Rossi2013HGs