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Efficient mutual magic and magic capacity with matrix product states

Poetri Sonya Tarabunga, Tobias Haug

TL;DR

This work develops scalable measures of nonstabilizerness (magic) in quantum many-body systems by introducing mutual von-Neumann SREs and magic capacity, both computable efficiently for matrix-product states and via Pauli sampling for statevectors. It defines mixed-state SREs through two Pauli-string distributions, establishes a framework for mutual magic that connects quantum and classical information through Pauli sampling, and ties magic capacity to the anti-flatness of the Pauli spectrum. The authors introduce two Metropolis-Hastings Monte-Carlo schemes and a Pauli-based statevector algorithm to compute mutual 2-SRE and M1/C_M up to sizable system sizes, and validate these tools on Clifford+T circuits and ground-state transitions in TFIM and Heisenberg models. The results show mutual SREs robustly detect criticality independent of local basis and reveal distinct scaling regimes for magic capacity across phase transitions, offering practical means to study the complexity of quantum many-body dynamics and circuits.

Abstract

Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or magic) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time $O(Nχ^3)$ for matrix product states (MPSs) of bond dimension $χ$. We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual $2$-SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with $O(8^{N/2})$ time and $O(2^N)$ memory, which we demonstrate for $24$ qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.

Efficient mutual magic and magic capacity with matrix product states

TL;DR

This work develops scalable measures of nonstabilizerness (magic) in quantum many-body systems by introducing mutual von-Neumann SREs and magic capacity, both computable efficiently for matrix-product states and via Pauli sampling for statevectors. It defines mixed-state SREs through two Pauli-string distributions, establishes a framework for mutual magic that connects quantum and classical information through Pauli sampling, and ties magic capacity to the anti-flatness of the Pauli spectrum. The authors introduce two Metropolis-Hastings Monte-Carlo schemes and a Pauli-based statevector algorithm to compute mutual 2-SRE and M1/C_M up to sizable system sizes, and validate these tools on Clifford+T circuits and ground-state transitions in TFIM and Heisenberg models. The results show mutual SREs robustly detect criticality independent of local basis and reveal distinct scaling regimes for magic capacity across phase transitions, offering practical means to study the complexity of quantum many-body dynamics and circuits.

Abstract

Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or magic) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time for matrix product states (MPSs) of bond dimension . We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual -SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with time and memory, which we demonstrate for qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.

Paper Structure

This paper contains 21 sections, 1 theorem, 56 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. SRE
  3. Mutual magic and magic capacity
  4. Mutual SRE
  5. Magic capacity and anti-flatness of the Pauli spectrum
  6. Efficient numerical methods by Pauli sampling
  7. Perfect sampling of mutual von-Neumann SRE for MPS
  8. Monte Carlo algorithms for mutual 2-SRE
  9. Statevector simulation for Pauli sampling, magic capacity and SRE
  10. Applications to many-body phenomena
  11. Clifford+T circuits
  12. Hamiltonian ground state transitions
  13. Discussion
  14. Proof of marginals for probability distribution $q_\rho(P)$
  15. Comparison of mutual SREs
  16. ...and 6 more sections

Key Result

Theorem E.1

Given an $N$-qubit state $\ket{\psi}=\sum_{i=1}^{2^N}a_i\ket{i}$, there is an algorithm to sample $\sigma$ from the probability distribution $p(\sigma)=2^{-N}\vert\bra{\psi}\sigma\ket{\psi}\vert^2$ in $O(8^{N/2})$ time and $O(2^{N})$ memory within the statevector representation of $\ket{\psi}$.

Figures (15)

  • Figure 1: Mutual 2-SRE $\mathcal{I}_2$ of the groundstate of the (a) the Heisenberg model with bond dimension $\chi=20$, and (b) TFIM with $N=16$. For the TFIM, the Monte Carlo results are obtained with $\chi=8$ and $\chi'=2$. The mutual SRE is calculated in respect to two subsystems $A,B$ of size $N/4$ located at the respective boundaries of the chain.
  • Figure 2: Von-Neumann SRE density $m_1$ for Clifford circuits doped with T-gate density $z=N_\text{T}/N$. a) Average $m_1$ against $z$ for different qubit numbers $N$. b)$\partial_z m_1$ against $z$. Curves of $\partial_z m_1$ intersect for all $N$ at saturation transition $z_{\text{c},1}^\text{fit}\approx 2.05$, which is marked by the dashed black line. c)$\partial_z m_1$ against rescaled $(z-z_{\text{c},1}^\text{fit})N^\gamma$, where we find scaling factor $\gamma\approx0.7$. After rescaling, the curves collapse for all $N$, showing the universality of $\partial_z m_1$.
  • Figure 3: Magic capacity $C_\text{M}$ for Clifford circuits doped with T-gate density $z=N_\text{T}/N$. a) Magic capacity $C_\text{M}$ against $z$. Curves for all $N$ intersect at a transition point $z_{\text{c},C_\text{M}}\approx 2.35$, which is marked by vertical dashed line. We used $K=10^5$ Pauli samples for $N=8$ and $N=12$, while $K=10^3$ for $N=16$ and $N=20$. Data is averaged over at least 100 random instances. b) Magic capacity density $C_\text{M}/N$, which is rescaled with $1/N$. c) Derivative of magic capacity density $\partial_z C_\text{M}/N$ against $z$. Curves for $N\geq12$ intersect at transition point $z_{\text{c},C_\text{M}}\approx 2.35$. b)$\partial_z C_\text{M}/N$ against rescaled $(z-z_{\text{c},C_\text{M}})N$. Around the transition point $z_{\text{c},C_\text{M}}$, curves for $N\geq12$ can be made to collapse to a single curve.
  • Figure 4: Von-Neumann SRE for groundstate of Heisenberg model with anisotropy $\Delta$. a)$m_1=M_1/N$, b) mutual von-Neumann SRE $\mathcal{I}_1^{[q]}$ for bipartitions of size $N/2$. c) Magic capacity density $C_M/N$ against $\Delta$. d) Magic capacity $C_M$ rescaled with $1/N^2$. We use $K=10^5$ samples to estimate $m_1$ and $C_M$.
  • Figure 5: Von-Neumann SRE for groundstate of TFIM against field $h$, where we choose the Hamiltonian in the standard computational basis. a)$m_1=M_1/N$ and b)$C_M$ rescaled by $1/N$. c) Mutual von-Neumann SRE $\mathcal{I}_1^{[q]}$ and d) Mutual $2$-SRE for bipartitions of size $N/2$. We use $K=10^5$ Pauli samples to estimate $m_1$, $C_M$ and $\mathcal{I}_1^{[q]}$.
  • ...and 10 more figures

Theorems & Definitions (1)

  • Theorem E.1: Simulation of Pauli sampling in statevector representation