Distinguishing phases via non-Markovian dynamics of entanglement in topological quantum codes under parallel magnetic field

We investigate the static and the dynamical behavior of localizable entanglement and its lower bounds on nontrivial loops of topological quantum codes with parallel magnetic field. Exploiting the connection between the stabilizer states and graph states in the absence of the parallel field and external noise, we identify a specific measurement basis, referred to as the canonical measurement basis, that optimizes localizable entanglement when measurement is restricted to single-qubit Pauli measurements only, thereby providing a lower bound. In situations where computing even the lower bound is difficult, we propose an approximation of the lower bound that can be computed for larger systems according to the computational resource in hand. Additionally, we compute a lower bound of the localizable entanglement that can be computed by determining the expectation value of an appropriately designed witness operator. We investigate the behavior of these lower bounds in the vicinity of the topological to nontopological quantum phase transition of the system, and perform a finite-size scaling analysis. We also investigate the dynamical features of these lower bounds when the system is subjected to Markovian or non-Markovian single-qubit dephasing noise. We find that in the case of the non-Markovian dephasing noise, at large time, the canonical measurement-based lower bound oscillates with a larger amplitude when the initial state of the system undergoing dephasing dynamics is chosen from the nontopological phase, compared to the same for an initial state from the topological phase. On the other hand, repetitive collapses followed by revivals to high value with time are observed for the proposed witness-based lower bound in the nontopological phase, which is absent in the topological phase. These features can be utilized to distinguish the topological phase of the system from the nontopological phase in the presence of dephasing noise.

Figure 1

(a) A Kitaev code on a square lattice of $3\times 3$ architecture. Each of the $N_E$ edges in the square lattice of $N_P$ plaquettes contains a qubit, represented by the circles, constituting an $N$-qubit ($N=N_E=2N_P=18$ in the present case, with $N_p^h=N_P^v=3$) system with periodic boundary conditions in both the horizontal and vertical directions. Here $N_E \left(N_P\right)$ is the cardinality of ${\mathcal{N}}_E$ (${\mathcal{N}}_P$) (see Sec. 2). A plaquette, denoted by the index $p$ (shown by a shaded square), and a vertex, denoted by the index $v$ (shown by thick intersecting lines on a shaded square), consist of four qubits each. The nontrivial loops in the vertical and horizontal directions, representing respectively the logical operators $L_{h,v}^z$ and $L_{h,v}^x$, have been marked with the thick lines consisting of three qubits each. The $L_h^x$ ($L_h^z$) and the $L_v^z$ ($L_h^x$) operators intersect each other at a single qubit. A periodic boundary condition is assumed in both the horizontal and vertical directions. (b) Color code on a hexagonal lattice. The lattice is constituted of $N_P$ plaquettes (in the present case, $N_P=6$ in a $3\times 2$ architecture) and $N$ (in this example, $N=12$) qubits. Red ($=c_1=r$), green ($=c_2=g$), and blue ($=c_3=b$) signify three different colors of the plaquettes. The thick vertical and horizontal lines represent respectively the vertical and horizontal nontrivial loops representing the operators $L_{h,r}^{\alpha }$ and $L_{v,g}^{\beta }$ [$\alpha =x\left(z\right)$ if $\beta =z\left(x\right)$]. Periodic boundary condition is assumed in both the horizontal and vertical directions. (c) A torus of genus 1. The horizontal and the vertical nontrivial loops are shown by the white (horizontal) and gray (vertical) lines.

Figure 2

Dephasing rate. Variation of $\gamma \left(t\right)$ as a function of $t$ for different values of $s$, across the Markovianity to non-Markovianity transition point $s=2$. Both axes are dimensionless.

Figure 3

Bounds of localizable entanglement in Kitaev code. (a, b) Variations of $E_{\mathrm{\Omega }}^L, E_{\mathrm{\Omega }}^{RL}, E_{\mathrm{\Omega }}^{\prime }, E_{\mathrm{\Omega }}^{\prime \prime }$, and $E_{\mathrm{\Omega }}^w$ (vertical axes) as functions of the parallel magnetic field strength $g$ (horizontal axes) in the cases of a Kitaev code of $N_P=4$ plaquettes, where entanglement is localized over a nontrivial loop corresponding to (a) $L_h^x$ and (b) $L_h^z$. The labels “T” and “NT” represent the “topological” and the “nontopological” phases on the $g$ axis, and the dashed vertical line indicates the topological to nontopological QPT point $g_c=0.328474\left(3\right)$. (c) The dependence of $E_{\mathrm{\Omega }}^{\prime \prime }$ and $E_{\mathrm{\Omega }}^w$ (vertical axis) on $g$ (horizontal axis) is shown for a Kitaev code with $N_P=9$ plaquettes, corresponding to the nontrivial loops representing $L_h^x$ and $L_h^z$. (d) The variation of the difference between the two lower bounds, $|E_{\mathrm{\Omega }}^{\prime \prime }-E_{\mathrm{\Omega }}^w|$ (vertical axis), as a function of $g$ (horizontal axis) in the case of a Kitaev code of nine plaquettes, for nontrivial loops representing $L_h^x$ and $L_h^z$. Both the horizontal and the vertical axes in all figures are dimensionless.

Figure 4

Canonical measurement setup for Kitaev code. If the nontrivial loop represents a $L_h^x$ ($L_h^z$) operator, then the qubits that are not on $L_h^x$ ($L_h^z$), but are on the plaquette operators (vertex operators) through which $L_h^x$ ($L_h^z$) passes, are measured in ${\sigma }^z$ (${\sigma }^x$) basis. The rest of the qubits that are not on these plaquette (vertex) operators are measured in the ${\sigma }^x$ (${\sigma }^z$) basis. This is demonstrated for (a) $L_h^x$ (qubits 7, 8, 9, 13, 14, 15 are measured in ${\sigma }^z$ basis, and the rest in ${\sigma }^x$ basis) and (b) $L_h^z$ (qubits 10, 11, 12, 16, 17, 18 are measured in ${\sigma }^x$ basis, and the rest in ${\sigma }^z$ basis) in the figures. The gray (black) color of the qubits signifies ${\sigma }^z$ (${\sigma }^x$) measurement, while the unmeasured qubits are represented by white circles.

Figure 5

Estimated maximum error for $E_{\mathrm{\Omega }}^{\prime \prime }$. Variation of ${\epsilon }_m$ as a function of $N$ for Kitaev code, where localizable entanglement is computed over $\mathrm{\Omega }\equiv L_h^x$ and $L_h^z$. The maximum error occurs in the case of nine-plaquette toric code, although all values of ${\epsilon }_m\sim {10}^{-6}$ or lower. Both the horizontal and the vertical axes are dimensionless.

Figure 6

QPT using lower bound of localizable entanglement in Kitaev code. Variations of $|\frac{d{E}_{\mathrm{\Omega }}^{\prime \prime }}{dg}|$ as a function of $g$ across the QPT point of a Kitaev code under parallel magnetic field for different system sizes $N$, when $\mathrm{\Omega }$ is chosen to be a nontrivial loop representing (a) $L_h^x$ and (b) $L_h^z$. The Kitaev code is considered over a lattice of $N_P^h\times N_P^v$ architecture (see Sec. 2a1) with $N_P^{h,v}$ being the number of plaquettes in the horizontal and vertical directions, where the system size $N$ takes the values 8 $(2\times 2)$, 12 $(2\times 3)$, 16 $(2\times 4)$, 18 $(3\times 3)$, and 20 $(2\times 5)$. (Insets of a and b) The variation of $ln|g_m\left(N\right)-g_c\left(\infty \right)|$ as a function of $lnN$. The dotted line represents the fitted curve according to Eq. (24), with (a) $\alpha =0.38\left(6\right), \nu =0.58\left(8\right)$ for $\mathrm{\Omega }\equiv L_h^x$ and (b) $\alpha =0.10\left(2\right), \nu =0.16\left(0\right)$ for $\mathrm{\Omega }\equiv L_h^z$. Both the horizontal and the vertical axes in all figures are dimensionless.

Figure 7

Constructing the local witness operators in Kitaev code. The plaquette and the vertex stabilizer operators, a subset of which can potentially contribute in building a local witness operators corresponding to a nontrivial loop representing (a) $L_h^x$ and (b) $L_h^z$, are shown for a nine-plaquette Kitaev code. The qubits denoted by the white circles are the ones on which the witness operators may have support. The labels of the plaquettes are denoted by boxed numbers, while numbers alone denote the vertices. See Eqs. (1) for the descriptions of the plaquette and the vertex stabilizer operators.

Figure 8

QPT using witness-based lower bound in Kitaev code. Variations of $|\frac{d{E}_{\mathrm{\Omega }}^w}{dg}|$ as a function of $g$ across the QPT point of a toric code under parallel magnetic field for different system sizes $N$, when $\mathrm{\Omega }$ is chosen to be a nontrivial loop representing (a) $L_h^x$ and (b) $L_h^z$. The toric code is considered over a lattice of $N_P^h\times N_P^v$ architecture (see Sec. 2a1), and the system sizes are taken to be same as in Fig. 6. (Insets of a and b) The variation of $ln|g_m\left(N\right)-g_c\left(\infty \right)|$ as a function of $lnN$. The dotted line represents the fitted curve according to Eq. (28), with (a) $\beta =0.44\left(8\right), \delta =0.65\left(5\right)$ for $\mathrm{\Omega }\equiv L_h^x$, and (b) $\beta =0.29\left(7\right), \delta =0.43\left(5\right)$ for $\mathrm{\Omega }\equiv L_h^z$. Note that in (b), $d{E}_{\mathrm{\Omega }}^w/dg$ data are plotted only until the value of $g$ beyond which $E_{\mathrm{\Omega }}^w=0$ (see Fig. 3). Both the horizontal and the vertical axes in all figures are dimensionless.

Figure 9

Lower bounds of localizable entanglement in color codes. Variations of $E_{\mathrm{\Omega }}^L, E_{\mathrm{\Omega }}^{RL}, E_{\mathrm{\Omega }}^{\prime }$, and $E_{\mathrm{\Omega }}^{\prime \prime }$ (vertical axes) as functions of the parallel magnetic field strength $g$ (horizontal axes) in the case of a color code of six plaquettes, where entanglement is localized over a nontrivial loop corresponding to $L_{h,r}^x$ in the bipartition (a) $1:\text{rest}$ and (b) $2:\text{rest}$. In the case of localizable entanglement computed over $2:\text{rest}$ partition of the nontrivial loop, we have used normalized negativity as the seed measure for ease of comparison between (a) and (b). Both the horizontal and the vertical axes in all figures are dimensionless.

Figure 10

Markovian and non-Markovian dynamics of localizable entanglement in Kitaev code. Variations of $E_{\mathrm{\Omega }}^{\prime \prime }$ and $E_{\mathrm{\Omega }}^w$ as functions of $t$ in the case of Markovian ($s=1$) and non-Markovian ($s=3$) dephasing noise, when the initial states of the dynamics are taken from the topological (a and b, $g={10}^{-1}$) and nontopological (c and d, $g=8\times {10}^{-1}$) phases of the Kitaev code in the presence of parallel magnetic field. The Kitaev model is defined on a four- ($N_P=4,2\times 2$) and eight-plaquette ($N_P=8,4\times 2$) lattice, and $E_{\mathrm{\Omega }}^{\prime \prime }$ and $E_{\mathrm{\Omega }}^w$ are computed over a nontrivial loop representing $L_h^x$. Both the horizontal and the vertical axes in all figures are dimensionless.

Figure 11

Entanglement collapse time. The variations of ECTs, ${\tau }_{nm}^3$ and ${\tau }_m^{1,3}$, as functions of $g$ across the QPT point of the Kitaev model in a parallel magnetic field on $2\times 2$ and $3\times 3$ square lattices, with $\mathrm{\Omega }\equiv L_h^z$. The value of $E_c^3$ for computing the ECTs for the Markovian and the non-Markovian dephasing noise is $E_c^3=0.14$ for the $2\times 2$ square lattice, and $E_c^3=0.11$ for the $3\times 3$ square lattice. Both axes are dimensionless.

Figure 12

Distinguishing topological phase from the nontopological phase via dynamics. Variation of (a) $E_{\mathrm{\Omega }}^{\prime \prime }$ and $E_{\mathrm{\Omega }}^w$ as functions of $g$ and $t$. The behaviors of $E_{\mathrm{\Omega }}^{\prime \prime }$ and $E_{\mathrm{\Omega }}^w$ as functions of $g$ across the QPT point, for fixed values of $t$, are demonstrated in (c) and (d), respectively. All the axes in all figures are dimensionless.

Figure 13

Markovian and non-Markovian dynamics of localizable entanglement in color code. Variation of $E_{\mathrm{\Omega }}^{\prime \prime }$ as functions of $t$ in the case of a color code on a hexagonal lattice of six plaquettes in parallel magnetic field, where the initial states of the dynamics are chosen from the topological and the nontopological phases. Both axes are dimensionless.

Figure 14

Examples of canonical measurement setups for Kitaev code. In the case of the four-plaquette Kitaev code, the local unitary equivalent graph state $|{\psi }_G\rangle$ can be obtained by application of the Hadamard operator on a set of control qubits. The corresponding graph $G$ has a connected subgraph $G_{\mathrm{\Omega }}$ over $\mathrm{\Omega }$, made of qubits 4 and 8 (i.e., the subsystem $\mathrm{\Omega }\equiv L_v^z$) connected by a link. An optimal measurement setup that does not disturb $G_{\mathrm{\Omega }}$ during the measurement corresponds to ${\sigma }^x$ measurements on qubits 1,7, and ${\sigma }^z$ measurements on the rest. This is equivalent to ${\sigma }^z$ measurements on qubits 3,7, and ${\sigma }^x$ measurements on the rest in the ground state $|{\psi }_g\rangle$ of the Kitaev code, thereby constructing the canonical measurement setup described in Sec. 3a (see also Fig. 4). The same for a nine-plaquette Kitaev code has also been demonstrated, leading to a connected subgraph on qubits 6, 12, and 18 (constructing $L_v^z$) after measurement. Note that in $G_{\mathrm{\Omega }}$ corresponding to the nine-plaquette Kitaev code, qubit 18 serves as the hub. Once measurements are made on the graph state, the postmeasure states of the system is $|{\psi }_{G_{\mathrm{\Omega }}}\rangle {\otimes }_{i\in \overline{\mathrm{\Omega }}}|{+}_i\rangle$ up to local unitary operations, where $|{\psi }_{G_{\mathrm{\Omega }}}\rangle$ is the state corresponding to the star graph on $\mathrm{\Omega }$.