Many-Body Magic Via Pauli-Markov Chains—From Criticality to Gauge Theories

We introduce a method to measure many-body magic in quantum systems based on a statistical exploration of Pauli strings via Markov chains. We demonstrate that sampling such Pauli-Markov chains gives ample flexibility in terms of partitions where to sample from: in particular, it enables the efficient extraction of the magic contained in the correlations between widely separated subsystems, which characterizes the nonlocality of magic. Our method can be implemented in a variety of situations. We describe an efficient sampling procedure using tree tensor networks, that exploit their hierarchical structure leading to a modest $O(\log N)$ computational scaling with system size. To showcase the applicability and efficiency of our method, we demonstrate the importance of magic in many-body systems via the following discoveries: (a) for one-dimensional systems, we show that long-range magic displays strong signatures of conformal quantum criticality (Ising, Potts, and Gaussian), overcoming the limitations of full state magic; (b) in two-dimensional ${\mathbb{Z}}_2$ lattice gauge theories, we provide conclusive evidence that magic is able to identify the confinement-deconfinement transition, and displays critical scaling behavior even at relatively modest volumes. Finally, we discuss an experimental implementation of the method, which relies only on measurements of Pauli observables.

Popular Summary

In quantum computing, the property of nonstabilizerness, also known as magic, is an essential resource that would be required to outperform classical computers. Without magic, quantum computers can be easily simulated with classical computers. It is thus important to be able to measure the amount of magic that quantum states possess. This is however a notoriously difficult task, especially if we consider large quantum systems.

We introduce a general method to measure magic in quantum many-body systems based on Monte Carlo samplings of Pauli strings. We then apply this specifically with tensor network methods, which are the favorite tools to simulate many-body systems. Using this method, we study many-body magic in various models, both in one- and two-dimensional systems, and our numerical results suggest a deep connection between (long-range) magic and many-body properties. Furthermore, we have proposed an experimental protocol for measuring magic based on the same approach. This protocol is advantageous for its practicality since it only relies on measurements on a single copy of a state, making it applicable within the current experimental capabilities in various platforms.

Our work enables and motivates further study of magic in different contexts in many-body physics, with considerable potential to advance our understanding of quantum systems. Moreover, our proposed approach also holds significance in advancing numerical simulations of many-body systems.

Figure 1

Stabilizer entropies for qubit and qutrit. The SRE density $m_1$ and $m_2$ for single-qubit state (a) defined in Eq. (8), and for single-qutrit state defined in Eq. (9).

Figure 2

Schematics of partitions. (a) Full partition. (b) Two widely separated partitions for the calculation of long-range magic in Eq. (7). (c) Subleading term as in Eq. (24), as well as a sketch depicting the increment trick discussed in the main text.

Figure 3

Efficient Monte Carlo sampling using the tree tensor network. (a) Tree tensor network (TTN) representation of a many-body wave function $|\psi ⟩$, where tensors are depicted as circles arranged in a binary-tree structure. Each tensor is identified by a pair of zero-indexed integers $[l,n]$, representing its layer index $l$ and tensor index $n$ at that layer. The red circle at the top-most layer represents the root tensor having index $[0,0]$, where the isometry center of the TTN is taken. (b) To evaluate the expectation value of a tensor product of single-site operators $⟨O_1{O}_2\dots O_N⟩$, we first place each operator $O_i$ at the physical site it acts on in the TTN representation. Then, we compute the effective link operators, which live at the virtual links by the coarse-graining procedure as shown in the figure. The coarse-graining is performed iteratively from the physical sites to the top-most virtual links, which are directly connected to the root tensor. At each step, the link operators $O_{[l+1,2n]}$ and $O_{[l+1,2n+1]}$ are combined into $O_{[l,n]}$ by the $[l,n]$ tensor. The resulting link operator $O_{[l,n]}$ acts on the $[l-1,\lfloor n/2\rfloor ]$ tensor one layer above in the TTN structure. (c) The expectation value $⟨O_1{O}_2\dots O_N⟩$ is calculated from the contraction of the root $[0,0]$ tensor and the top-most link operators as shown in the figure. (d) Considering a modified operator, which differs only at a single site from the previous one, $O_1{O}_2\cdots O_i^{\prime }\cdots O_N$, we need only to recompute the link operators in the path from the modified physical site $i$ to the topmost link.

Figure 4

Efficient estimation of Rényi-2 SRE density in 1D quantum Ising chain. (a) The subleading term for the Rényi-2 SRE, $c_L=2M_2(L/2)-M_2(L)$, directly estimated using the efficient scheme specified in Sec. 3, for various system sizes in 1D quantum Ising chain. (b) The SRE density $m_2$ for the 1D quantum Ising chain near the critical point $h_c=1$ computed using the increment method using different subleading terms. (Inset) The sampling errors for $m_2$ at $h_c=1$ for various system sizes $L$ (in log scale). Clearly, the errors show even slower than logarithmic growth for the efficient sampling scheme. Here we consider TTN bond dimension $\chi =30$ and the number of samples is $N_S={10}^6$. Error bars represent 95% confidence interval.

Figure 5

Magic density in 1D quantum three-state Clock model. (a) The SRE density $m_1$ in the ground-state of the three-state Clock model as a function of $h$. (b) Finite-size scaling for $m_1$. Here. $m_{1,m}$ is the maximum $m_1$ at $h_c=1$. We extract the critical exponent $\nu \approx 0.844$ and $\gamma \approx 0.66$. The correlation-length exponent $\nu$ is close to the known ${\nu }_{\text{Potts}}=5/6$. We used bond dimension up to $\chi =36$ and the number of sample is $N_S={10}^6$. Error bars represent 95% confidence interval.

Figure 6

Magic density and long-range magic in spin-1 $XXZ$ chain. (a) The magic density $m_1$ and (b) long-range magic $L({\rho }_{AB})$ of the ground-state of the spin-1 $XXZ$ model with $\mathrm{\Delta }=1$ as a function of $D$. We consider bond dimension up to $\chi =60$ and the number of samples is $N_S={10}^6$. Error bars represent 95% confidence interval. The dashed vertical lines represent the best estimates available for the transition points.

Figure 7

Long-range magic in 1D quantum Ising chain. The long-range magic $L({\rho }_{AB})$ as in Eq. (7) in the ground state of 1D quantum Ising chain as a function of the transverse field $h$. It peaks at the critical point $h_c=1$. (Inset) $L({\rho }_{AB})$ at $h_c=1$ for various system sizes $L$ (in log scale). We consider TTN bond dimension up to $\chi =30$ and the number of samples is $N_S={10}^6$. Error bars represent 95% confidence interval.

Figure 8

Magic densities in 2D ${\mathbb{Z}}_2$ gauge theory. The SRE densities (a) $m_1$ and (b) $m_2$ of the ground state of ${\mathbb{Z}}_2$ gauge theory on $L\times L$ square lattice as a function of $h$. We use TTN bond dimension up to $\chi =60$ and the number of samples is $N_S={10}^6$. Error bars represent 95% confidence interval.

Figure 9

Finite-size critical scaling of SRE density in 2D ${\mathbb{Z}}_2$ gauge theory. (a) The SRE density $m_1$ near the critical point at the ${\mathbb{Z}}_2$ gauge theory. Even with small TTN bond dimension $\chi =30$, $m_1$ captures the transition very well: all the curves cross near the known critical point $h_c=3.04$. (b) Finite-size scaling of $m_1$. Here, $m_{1,\mathrm{cr}}$ is $m_1$ at $h=3.04$. We find the correlation length critical exponent $\nu =0.64\pm 0.05$. The extracted $\nu$ is remarkably close to the known ${\nu }_{\text{3D}}\simeq 0.63$ for 3D Ising universality class. Here, the number of samples is $N_S={10}^7$.

Figure 10

Simulated experiment to measure SREs. Simulation of experimental measurement of SREs in the ground state of 1D quantum Ising chain for $L=8$. Here, $N_M=500$ and $N_S={10}^4$.

Figure 11

The errors in SRE density for simulated experiments. The deviation $\delta m_n=|m_{n,\text{sim. expt.}}-m_{n,\text{exact}}|$ for $n=1,2$ in the ground state of 1D quantum Ising chain at the critical point $h=1$ for $L=16$. In (a), we fix $N_S=10000$ and vary $N_M$, while in (b), we fix $N_M=\{{10}^3,{10}^5\}$ and vary $N_S$.

Figure 12

Autocorrelation time and statistical errors in Monte Carlo sampling of SREs. (a) Integrated autocorrelation time ${\tau }_I$ at the ground state of 2D transverse-field Ising model with $h=3$ for various system sizes $N=L\times L$. It is linear for $m_1$ and saturates for $m_2$. (b) Standard deviation $\sigma$ for various system sizes. Inset shows $\sigma$ for $m_1$ in log-log scale. The solid line denotes a fit $\sigma =a{N}^{-b}$ for $L\ge 6$, with $b=0.503$. The standard deviation is obtained by error propagation.

Figure 13

Convergence of SRE with respect to bond dimension. SREs $m_1$ and $m_2$ at the ground state of 2D transverse-field Ising model with $h=3$ and $L=10$ for various bond dimension $\chi$.

Figure 14

The Binder cumulant across the critical point in the 2D quantum Ising model. Here we approximate the ground state of 2D quantum Ising model with TTN having bond dimension $\chi =30$, in parity with Fig. 9.

Figure 15

Monte Carlo sampling of Pauli strings.