Surface codes, quantum circuits, and entanglement phases

Surface codes—leading candidates for quantum error correction (QEC)—and entanglement phases—a key notion for many-body quantum dynamics—have heretofore been unrelated. Here we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to $(1+1)\mathrm{D}$ free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state $|{\mathrm{\Psi }}_{\infty }\rangle$. Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for $|{\mathrm{\Psi }}_{\infty }\rangle$ via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of ($d+1$)D topological codes and $d$-dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest.

Figure 1

Phase diagrams of surface-code QEC, 1D entanglement, and 2D scattering networks. Bit-flip errors give real Ising couplings $J$ (a); coherent $exp\left[i\varphi X\right]$ errors yield complex couplings $J=-(1/2)ln(itan\varphi )$ (b). The RBIM bond signs, related to $X$ error configurations, are swapped with probability $p$. The solid lines sketch the phase boundaries, the black dots mark computed phase transition points. The dashed line shows (a) the Nishimori line representing incoherent-error QEC and (b) the $p={sin}^2\varphi$ “partial Pauli twirl” line for coherent errors. The error-correcting phase (QEC✓) is dual to a circuit yielding a topologically nontrivial 1D area law with entanglement-spectrum zero modes (${\mathrm{AL}}_{\mathrm{top}}$), and to an insulating network with topological invariant $\mathcal{I}=-1$ ($I_{-1}$). Above threshold (QEC✗) we find (a) a topologically trivial 1D area law (no zero modes, ${\mathrm{AL}}_{\mathrm{triv}}$) and a $\mathcal{I}=1$ insulator ($I_1$), or (b) a logarithmic entanglement phase (LL) and a metallic network.

Figure 2

(a) Bulk patch of the toric code, with physical qubits marked by black dots, and $X$ and $Z$ stabilizers by white and gray disks, respectively. (b) The code is mapped to a RBIM with real couplings for incoherent and complex couplings for coherent errors; $X$ stabilizers map to Ising spins ${\sigma }_v$. The nearest-neighbor couplings have sign ${\eta }_{v{v}^{\prime }}$.

Figure 3

Fermionic quantum circuits acting on Majorana fermion lines. The “time” direction, along the cylinder, is upwards. (a) Circuit for real Ising couplings $J$ (from incoherent bit flips). If ${\eta }_{n,i}^{\left(h\right)}<0$, the $H_{n,i}$ gates involve unitary double braids (blue) and imaginary time evolution (dark red). (b) For complex Ising couplings $J=-(1/2)ln(itan\varphi )$, from coherent errors, the gates $H_{n,i}$ are always unitary (blue), while the $V_{n,i}$ consist of unitary braids (blue) and imaginary time evolution (dark red).

Figure 4

Network model for the (a) real and (b) complex Ising couplings from incoherent and coherent errors, respectively. At each Ising bond in Fig. 2 we now have junction matrices $H_{n,i}$ and $V_{n,i}$. They scatter directed Majorana modes residing on the networks' links. Translating $H_{n,i}$ and $V_{n,i}$ into junction scattering matrices requires different link direction layouts for real and complex Ising couplings: counterpropagating pairs of links for real couplings [60] and copropagating pairs for complex Ising couplings [57].

Figure 5

(a) Entanglement spectrum and (b) entanglement entropy for real $J$ on the Nishimori line. We use $L=5M$ and cylinder circumferences from $M=20$ (black) to $M=160$ (light orange). We averaged over 16–128 configurations of $\eta$; error bars ($2\times \mathrm{standard}$ error) are imperceptible. The $p=0.1093$ vertical dashed line marks the Nishimori point [60], i.e., the QEC threshold [42]. The horizontal dashed line in panel (b) marks the $ln2$ bound in the $\mathcal{I}=-1$ phase.

Figure 6

Dimensionless conductivity $g$ for complex couplings at $p={sin}^2\varphi$ as a function of the rescaled system length $L/\ell \left(\varphi \right)$ for wide systems with $M=5L$, averaged over 100–${10}^4$ configurations of $\eta$; error bars ($2\times \mathrm{standard}$ error) are imperceptible. For angles above a critical ${\varphi }_c$, the conductivity increases with $L/\ell \left(\varphi \right)$ (metallic phase; dashed black line shows $g\propto (1/\pi )ln[L/\ell (\varphi \left)\right]$). Below the transition, it decreases exponentially to zero (localized phase, dashed gray line shows the exponential tail).

Figure 7

(a) Entanglement spectrum and (b) entanglement entropy for complex $J=-(1/2)ln(itan\varphi )$ and $p={sin}^2\varphi$ for $L=5M$ and cylinder circumferences from $M=20$ (black) to $M=160$ (light orange). We averaged over $2^5$–$2^8$ configurations of $\eta$; error bars ($2\times \mathrm{standard}$ error) are imperceptible. The $\varphi =0.095\pi$ dashed line marks the entanglement transition; the horizontal dashed line in panel (b) marks $S=ln2$.

Figure 8

Entanglement entropy $S_{M/2}$ in the metallic phase ($\varphi >{\varphi }_c$) for complex couplings at $p={sin}^2\varphi$ as a function of the rescaled circumference $M/m\left(\varphi \right)$. We averaged over $2^5$–$2^8$ configurations of $\eta$; error bars are $2\times \mathrm{standard}$ error, and the gray dashed line serves as a guide for the eye showing a logarithmic increase. The inset shows ${\lambda }_0$, the exponentially decaying zero mode; the decay length increases with $\varphi$.

Figure 9

Dimensionless conductivity $g$, averaged over 100–${10}^4$ configurations of $\eta$ (error bars are $2\times \mathrm{standard}$ error), as a function (a) of $\varphi$ for different sizes and (b)–(c) of $L$ for different angles $\varphi$. In (b), we show the conductivity for the insulating case, and in (c) the metallic case.