Classical Simulations of Low Magic Quantum Dynamics

TL;DR

This work develops a near-stabilizer classical simulation framework (LRSD) to efficiently simulate adaptive quantum circuits that remain close to the stabilizer manifold, focusing on states with low magic generated under frequent Pauli measurements. By representing the density matrix as a stabilized decomposition with a small set of logical Pauli operators and updating it through Clifford gates, measurements, and T gates, the authors derive explicit update rules and quantify magic via stabilizer nullity $M = G' - L$. They benchmark the approach on all-to-all monitored circuits with sub-extensive $T$-gate density and uncover measurement-induced phase transitions in purification and magic, including a four-phase structure for magic in Z-basis protocols and a purification transition with critical exponent $\nu_p \approx 0.45$ and dynamical exponent $z_p \approx 0.22$. The LRSD method, complemented by Bell sampling, enables scalable exploration of large systems, offering a complementary route to MPS-based methods for studying low-magic, high-entanglement dynamics and their transitions in nonlocal circuit geometries.

Abstract

We develop classical simulation algorithms for adaptive quantum circuits that produce states with low levels of ``magic'' (i.e., non-stabilizerness). These algorithms are particularly well-suited to circuits with high rates of Pauli measurements, such as those encountered in quantum error correction and monitored quantum circuits. The measurements serve to limit the buildup of magic induced by non-Clifford operations arising from generic noise processes or unitary gates, respectively. Our algorithms also allow a systematic truncation procedure to achieve approximate simulation. To benchmark our approach, we study the dynamics of all-to-all monitored quantum circuits with a sub-extensive rate of T-gates per unit of circuit depth, where we can simulate previously inaccessible system sizes and depths. We characterize measurement-induced phase transitions in the output wavefunction, including in the entanglement, purification, and magic. We outline the utility of our algorithms to simulate dynamics with low magic and high entanglement, complementary to the leading matrix-product state approaches.

Figures (16)

• Figure 1: (a) Circuit diagram for the evolution of the initial state that corresponds to the initial density matrix $\rho_o=\sum_{\ell} \lambda_{\ell} \sigma_{\ell} \rho_{S}$ with measurements, $T$-gates and Clifford unitaries (b) Schematic of the forking of the logicals $\sigma_{\ell}$ for the initial state under the circuit evolution in (a). The LRSD is represented diagrammatically, where each node represents $\sigma_{\ell}$ or $\lambda_{\ell}.$ Vertical edges connect terms from the initial state $\rho_o$ to the final state after the T-gate $T \rho_o T^{\dagger}$, or measurement of Pauli operator $P$, $\frac{I+P}{2} \rho_o \frac{I+P}{2}$. Each $\sigma_{\ell}$ is mapped to a new set of $\sigma_{\ell}$ attached to it by vertical lines. An X, or box with T, on a branch denotes measurement, or $T$-gate, respectively, in the entire circuit. (c) Equation for the LRSD and the effect of $T$-gates and measurements. (d) Each injection of a $T$-gate can fork each logical onto itself (Case I), all logicals into three new logicals (Cases II, IV), or certain logicals to two new logicals (Case III). The Pauli operator $\overline{P}$ is defined in Eq. \ref{['eq:psi_decomposition']}. The Pauli $\overline{P}_{(X/Y)}$ is defined by $\overline{P}_{X/Y}=\overline{P}_{1} \otimes \cdots \otimes (X/Y) \otimes \cdots \otimes \overline{P}_{L},$ where $X/Y$ and the $T$-gate is on the $i^{\mathrm{th}}$ qubit and likewise $\sigma_{\ell}^{(X/Y)}=\sigma_{\ell_{1}} \otimes \cdots \otimes (X/Y) \otimes \cdots \otimes \sigma_{\ell_{L}}.$ (e) Each measurement can fork each logical onto itself (Case I), eliminate all anticommuting logicals (Cases II), or update anticommuting logicals with the Pauli operator $\overline{P}$. Algorithm \ref{['alg:msm_near_clifford']} in Appendix \ref{['sec:algorithms']} provides details on measurement updates.
• Figure 2: (a) Circuit diagram for the single-pair all-to-all model. Each time step consists of a CZ gate applied between two random sites, followed by a measurement on a randomly chosen site in either the $X$ or $Z$ basis. Each dashed line represents the end of a single time step in the model. (b) Forking of the initial stabilizer wavefunction $|\psi\rangle=|+\rangle^{\otimes L}$ under circuit evolution with $T$-gates, measurements, and Clifford unitaries. Each injection of $T$-gate takes a stabilizer state $|\psi\rangle$ onto a superposition of at most two stabilizer states $|\psi'\rangle=c_{+}|\psi_{+}\rangle+c_{-}|\psi_{-}\rangle.$ Each branch represents a different stabilizer state. Measurements can collapse different branches onto a single branch. Each quantum trajectory is represented by a different color.
• Figure 3: (a) $\overline{\left|\{\lambda_l\}\right|}$ as a function of truncation error $\epsilon$ for $\epsilon$ in $[10^{-4},0.5]$. The $\overline{\cdots}$ denotes averaging over circuit trajectories. $|\{\lambda_{l}\}|$ represents the total number of logicals $\lambda_{l}$, where $\lambda_{l}$ is defined in Eq. \ref{['eq:LSRD_decomposition']}. Data is shown for the single-pair all-to-all model with Z-basis measurements, where Bell Sampling is performed on $|\psi\rangle \otimes |\psi\rangle$ for system size $L=50$, probability of $T$-gates $p_{T}=1/L$, and measurement probabilities $p_{m,Z} \in \{0.6, 0.7, 0.8, 0.9\}$, with a darker color denoting a larger measurement probability. (b) Stabilizer nullity $\overline{\mathcal{M}}$, as defined in Eq. \ref{['eq:magic_equation_main']}, as a function of the cutoff $\epsilon$ for several measurement probabilities $p_{m,Z}$. The dashed line denotes data obtained at zero truncation error. Data was obtained by averaging over $\mathcal{O}(10^{3})$ circuit realizations. For each set of $\epsilon$ at a fixed $p_{m,Z}$, the locations of unitary gates and measurements are fixed.
• Figure 4: (a) Scaling of the circuit-averaged number of entries $\overline{\mathcal{N}}$ with $L$, see Eq. \ref{['eq:NumberEntries']}, which is the total number of entries required to specify the state in our algorithm. $|\{\lambda_{l}\}|$ represents the total number of logicals $\lambda_{l}$, where $\lambda_{l}$ is defined in Eq. \ref{['eq:LSRD_decomposition']}. Data is shown for the single-pair all-to-all model with X-basis measurements. For the statevector simulation, $\mathcal{\overline{N}} = 2^{L}$, as shown by the black dashed line. Data is shown for measurement probabilities $p_{m,X} \in \{0.6,0.9\}$ (b) Data is shown for cutoff values $\epsilon \in \{1 \times 10^{-15}, 1 \times 10^{-10},1 \times 10^{-6}, 1 \times 10^{-5}, 5 \times 10^{-4} \}$, with a darker color denoting a larger cutoff. (c) Data is shown for $T$-gate probabilities $p_{T} \in \{1/L,0.75/L,0.5/L,0.25/L \}$. Data was obtained by averaging over $\mathcal{O}(10^{3})$ circuit realizations.
• Figure 5: (a) Phase diagram of the purification transition in the single-pair all-to-all model with $X$-basis only measurements as a function of measurement probability $p_{m,X}$ and probability of $T$-gates, $p_{T}$. The part of the phase diagram labeled mixed is the mixed phase of the purification transition, where $\overline{S_{Q}} \sim e^{t/\tau(L)}$ and $\overline{\tau(L)}$ grows exponentially in $L$. The part of the phase diagram labeled pure is the pure phase of the purification transition, where $\overline{\tau(L)}$ is the area law in $L$. $\overline{\cdots}$ denotes averaging over circuit trajectories. The critical measurement probability separating the mixed phase and pure phase is given by the critical purification (cp) rate $p_{m,X}^{cp} = 0.26(1)$, which is weakly dependent on $p_{T}$. The points represent the numerically calculated locations of the purification transition, where $\tau(L)$ grows logarithmically in $L$. The black dashed line is the phase boundary deduced by interpolation. (b) Purification time of $\overline{S_{Q}}$ as a function of system size $L$ at $p_T=0$ for different measurement probabilities $p_{m,X}$ from $0.18$ to $0.36$ in the steps of 0.02. The critical measurement probability is $p_{m,X}^{cp} = 0.26(1)$ with the dynamical exponent $z_{p}=0.22(2)$ (see Eq. \ref{['eq:tau']}). At $p_{m,X}^{cp}$, $\tau \sim L^{z_{p}}$, as indicated by the dashed line. (c) $\tau L^{-z_p}$ as a function of measurement probability $p_{m,X}$ for $L\in \{32, 64, 128\}$. The inset shows the data collapse with the estimated critical exponent $\nu_p=0.45(5)$. The ensemble size is 25000 circuits for each value of $p_{m,X}$ and $L$ in all three panels.