Practical framework for simulating permutation-equivariant quantum circuits

Abstract

Understanding which subclasses of quantum circuits are efficiently classically simulable is fundamental to delineating the boundary between classical and quantum computation. In this context, it is well known that certain tasks based on permutation-equivariant unitaries-i.e., $n$-qubit circuits whose action commutes with the qubit-permuting representation of the symmetric group $S_n$-can be simulated in polynomial time. However, existing approaches scale as $O(n^7)$, and can rapidly become prohibitively expensive. In this work, we introduce a practical algorithm for simulating $S_n$-equivariant circuits under the assumption that the gate generators are at most $k$-local, with $k\in O(1)$. The resulting method runs in $O(n^{ω+1})$ time for constant depth, where $ω$ is the matrix multiplication exponent, significantly lowering the polynomial degree compared to existing techniques. Finally, we numerically validate this scaling by simulating the dynamical evolution of the Lipkin-Meshkov-Glick model, and show that for $n=512$ spins, a standard laptop can compute the concurrence of the evolved state in under two minutes.

Figures (6)

• Figure 1: Schematic representation of matrices in the basis that block-diagonalizes the group's action. Each of the dashed blocks corresponds to the irreps labeled by an index $\lambda = (n-m, m)$ where $m = 0,\dots, \lfloor \frac{n}{2}\rfloor$. The irreps $\lambda$ of $U$ are of the dimension $d_\lambda$ and multiplicity $m_\lambda$, while the irrep of $R(\sigma )$ have a dimension $m_\lambda$ and multiplicity $d_\lambda$. Note that as illustrated by the gray region in (c), the initial states $\rho$ need not be block-diagonal as it is not necessarily $S_n$-equivariant.
• Figure 2: Order parameter $m$ as a function of the longitudinal field, $h_z$. The dashed line corresponds to the analytical expression in the thermodynamic limit, given by Eq. \ref{['eq:order_params_th']}. The result is shown for $J = 1$ and $\gamma = 0.5$, however, the order parameter is independent of $J$ and $\gamma$, and thus identical for all anisotropy values.
• Figure 3: Rescaled concurrence as a function of the longitudinal field, $h_z$. The analytical concurrence in the thermodynamic limit is computed using Eq. \ref{['eq:CR_th']} with $J = 1$. The numerically simulated results converge toward the analytical prediction as the number of qubits $n$ increases for both (a) $\gamma = 0.5$ and (b) $\gamma = 0.8$, with the largest deviation from the thermodynamic limit value around $h_z = 1$, where the phase transition takes place.
• Figure 4: Runtime of digitized AQC as a function of the number of qubits. In this setting, the circuit depth scales as the number of Trotter steps, therefore, linearly with the number of qubits, i.e., $L \in \order{n}$, which results in the theoretical bound scaling as $\order{n^4}$ when the dynamics are confined in the fully symmetric sector. The measured runtime exhibits better scaling behavior than the theoretical upper bound.
• Figure 5: Simulation time required to compute the matrix elements of $\mathcal{T}_{S_n}(P_{\boldsymbol{k}})$ in the Schur basis for different choices of $\boldsymbol{k}$ with $k \in \order{1}$ using Algorithm \ref{['alg:schur_pauli_matrix_elements']}. All the simulations were performed using the Julia package and factorial evaluations inside the algorithm were optimized by precomputing log-factorials at the start. The observed simulation time scales at most as $\order{n^2}$, in agreement with the claim of Corollary \ref{['cor:cost_matrix_calculation']}, with a slightly higher runtime observed in (b). This discrepancy is due to additional practical overhead, primarily arising from the evaluation of bound conditions.

Theorems & Definitions (12)

• Definition 1: $S_n$-equivariant operator
• Theorem 1: Complexity of Heisenberg evolution
• Corollary 1
• Theorem 2
• Lemma 1: Quantum circuit and sample complexity for generic states
• Lemma 2: Quantum circuit and sample complexity for $S_n$-equivariant states
• Theorem 3: Action of symmetrized Pauli strings in the Schur basis
• proof
• Corollary 2: Computational complexity for the matrix element computation of symmetrized Pauli strings in the Schur basis
• proof