Universality Classes of Stabilizer Code Hamiltonians

TL;DR

This paper analyzes the finite temperature behavior of stabilizer code Hamiltonians by solving their partition functions using duality, revealing their universality classes and effective dimensions, with specific results on the 4D Toric Code and Haah's Code.

Contribution

It introduces a generic duality-based method to determine the universality class and effective dimension of stabilizer code Hamiltonians at finite temperature.

Findings

4D Toric Code belongs to 4D Ising universality class

Haah's Code exhibits dimensional reduction to 1D Ising class

Method provides insights into robustness and dynamics of quantum memories

Abstract

Stabilizer code quantum Hamiltonians have been introduced with the intention of physically realizing a quantum memory because of their resilience to decoherence. In order to analyze their finite temperature thermodynamics, we show how to generically solve their partition function using duality techniques. By unveiling each model's universality class and effective dimension, insights may be gained on their finite temperature dynamics and robustness. Our technique is demonstrated in particular on the 4D Toric Code and Haah's Code -- we find that the former falls into the 4D Ising universality class, whereas Haah's Code exhibits dimensional reduction and falls into the 1D Ising universality class.

Equations

$$\begin{array}{r}|{\psi }_0⟩=\prod _{s=1}^S\frac{1}{2}(\text{mathds}1+B_s)|+x⟩,\\ |{\varphi }_0⟩=\prod _{r=1}^R\frac{1}{2}(\text{mathds}1+A_r)|+z⟩,\end{array}$$

$${\mathcal{P}}_{ij}^{\mu \nu }\equiv \prod _{p\in P_{ij}^{\mu \nu }}{\sigma }_p^z$$

$${\mathcal{Q}}_{ij}^{\mu \nu }\equiv \prod _{p\in Q_{ij}^{\mu \nu }}{\sigma }_p^x$$

$${\mathcal{P}}_i^{\mu }\equiv \prod _{v\in P_i^{\mu }}{\sigma }_v^z$$

$${\mathcal{Q}}_i^{\mu }\equiv \prod _{v\in P_i^{\mu }}{\tau }_v^z$$

$$\prod _c{B}_c^{n_c}⟺\sum _c{n}_{c}c,n_c=0,1.$$

$${\partial }_3\text{quantity}[\sum _c{n}_{c}c]=\sum _c{n}_c{\partial }_{3}c=\sum _c{n}_c\sum _{p\in \partial c}p,$$

$$\prod _{c\in P}{B}_c=\text{mathds}1⟺{\partial }_3\sum _{c\in P}c=0$$

$${\mathcal{T}}_b=\sum _{\gamma \in \Gamma }{\mathsf{T}}_b^{|\gamma |}=\sum _{n=0}^{4L^4}{b}_n{\mathsf{T}}_b^n,$$

$$\begin{array}{rl}{\mathcal{Z}}_{\text{4DI}}& =2e^{4L^4\beta J}\sum _{{\gamma }^{\ast }\in {\Gamma }^{\ast }}\text{quantity}(e^{-2\beta J}{)}^{|{\gamma }^{\ast }|}\\ & =2e^{4L^4\beta J}\sum _{n=0}^{4L^4}{b}_n^{\ast }\text{quantity}(e^{-2\beta J}{)}^n,\end{array}$$

$$\Sigma =\sum _c{n}_{c}c+\sum _{m=1}^4{C}_m$$

$${\mathcal{T}}_b=\sum _{{\gamma }^{\ast }\in {\Gamma }^{\ast }}\text{quantity}{\mathsf{T}}_b^{|{\gamma }^{\ast }|}\text{quantity}[1+\sum _{m=1}^4{\mathsf{T}}_b^{L^3-|{\gamma }^{\ast }\cap C_m|}+\dots ],$$

$$\begin{array}{rl}{\mathcal{T}}_b& \le \sum _{{\gamma }^{\ast }\in {\Gamma }^{\ast }}\text{quantity}{\mathsf{T}}_b^{|\gamma |}\text{quantity}[1+15\sum _{n=0}^{4L^4}{\mathsf{T}}_b^n]\\ & =\text{quantity}[1+15\sum _{n=0}^{4L^4}{\mathsf{T}}_b^n]{\mathcal{T}}_b^{\ast },\end{array}$$

$$\frac{1}{L^4}\log {\mathcal{T}}_b\le \frac{1}{L^4}\log {\mathcal{T}}_b^{\ast }+\frac{1}{L^4}\log \text{quantity}[\frac{16-{\mathsf{T}}_b}{1-{\mathsf{T}}_b}]$$

$$\begin{array}{r}{A}_s=\prod _{\ell \in s}{\sigma }_{\ell }^x,\\ B_p=\prod _{\ell \in p}{\sigma }_{\ell }^z,\end{array}$$

$$H_0=-a\sum _s{A}_s-b\sum _p{B}_p.$$

$$\begin{array}{r}{A}_s=A_{i,j},1\le i,j\le L-1,\\ B_p=B_{i,j},0\le i,j\le L-1,\end{array}$$

$${\sigma }_{i+\frac{1}{2},j}^{\mu },0\le i\le L-1,0\le j\le L,$$

$${\sigma }_{i,j+\frac{1}{2}}^{\mu },0\le i\le L,0\le j\le L-1.$$

$$V_x=-\lambda \sum _{i=0}^{L-1}\sum _{j=0}^L{\sigma }_{i+\frac{1}{2},j}^x,$$

$$\begin{array}{r}\text{commutator}{A}_s{A}_{s^{\prime }}=\text{commutator}{B}_p{B}_{p^{\prime }}=\text{commutator}{A}_s{B}_p=\text{commutator}{A}_s{\sigma }_{i+\frac{1}{2},j}^x=0,\\ \text{anticommutator}{B}_{i,j}{\sigma }_{i+\frac{1}{2},j}^x=\text{anticommutator}{B}_{i,j}{\sigma }_{i+\frac{1}{2},j+1}^x=0.\end{array}$$

$$\begin{array}{rl}{B}_{i,j}& \to {\tau }_{i,j}^z{\tau }_{i,j+1}^z,\\ {\sigma }_{i+\frac{1}{2},j}^x& \to {\tau }_{i,j}^x.\end{array}$$

$$\begin{array}{rl}& H_0+V_x\to H_x^{\text{Dual}}\\ & =\sum _{i=0}^{L-1}\left[\sum _{j=0}^{L-1}{\tau }_{i,j}^z{\tau }_{i,j+1}^z+\sum _{j=0}^L{\tau }_{i,j}^x\right]+\sum _{j=1}^{(L-1{)}^2}{\tau }_{L,j}^z.\end{array}$$

$$V_y=-\lambda \sum _{i=0}^{L-1}\sum _{j=0}^L{\sigma }_{i+\frac{1}{2},j}^y.$$

$$\begin{array}{rl}\text{commutator}{A}_s{A}_{s^{\prime }}=\text{commutator}{B}_p{B}_{p^{\prime }}=\text{commutator}{A}_s{B}_p& =0,\\ \text{anticommutator}{B}_{i,j}{\sigma }_{i+\frac{1}{2},j}^y=\text{anticommutator}{B}_{i,j}{\sigma }_{i+\frac{1}{2},j+1}^y& =0,\\ \text{anticommutator}{A}_{i,j}{\sigma }_{i+\frac{1}{2},j}^y=\text{anticommutator}{A}_{i,j}{\sigma }_{i-\frac{1}{2},j}^y& =0.\end{array}$$

$$\begin{array}{r}{B}_{i,j}\to {\tau }_{i,j}^z{\tau }_{i,j+1}^z,\\ A_{i,j}\to {\rho }_{i,j}^z{\rho }_{i+1,j}^z,\\ {\sigma }_{i+\frac{1}{2},j}^y\to {\tau }_{i,j}^x{\rho }_{i,j}^x.\end{array}$$

$$\begin{array}{rl}& H_0+V_y\to H_y^{\text{Dual}}\\ & =\sum _{i=0}^{L-1}\left[\sum _{j=0}^{L-1}{\tau }_{i,j}^z{\tau }_{i,j+1}^z\right]+\sum _{j=1}^{L-1}\left[\sum _{i=1}^{L-1}{\rho }_{i,j}^z{\rho }_{i+1,j}^z\right]\\ & +\sum _{i=0}^{L-1}\sum _{j=0}^L{\tau }_{i,j}^x{\rho }_{i,j}^x.\end{array}$$

Full text

Universality Classes of Stabilizer Code Hamiltonians: Supplemental Material

Zack Weinstein

Department of Physics, Washington University, St. Louis, MO 63130, USA

  

Gerardo Ortiz

Department of Physics, Indiana University, Bloomington, IN 47405, USA

  

Zohar Nussinov

Department of Physics, Washington University, St. Louis, MO 63130, USA

I Gauge-like Symmetries

We briefly review the $d$-dimensional gauge-like symmetries of the 4DTC and Haah’s Code. These symmetries may act as logical operators on the ground state manifold, manipulating the encoded logical qubits. Two easily constructed ground states can be written in the language of the main paper:

[TABLE]

where $\ket{+x}$ and $\ket{+z}$ respectively denote the simultaneous $+1$ eigenstate of each $\sigma^{x}_{n}$ and $\sigma^{z}_{n}$.

I.1 4D Toric Code

The 4DTC displays two $d=2$ dimensional gauge-like symmetries Duivenvoorden et al. (2019). (i) Let $P^{\mu\nu}_{ij}$ with $1\leq i,j\leq L$ be a plane of plaquettes extending in the directions $\mu$ and $\nu$. The operators

[TABLE]

commute with all $A_{\ell}$ and all $B_{c}$ and act nontrivially on the ground state $\ket{\psi_{0}}_{\text{4DTC}}$. (ii) Similarly, let $Q^{\mu\nu}_{ij}$ for $1\leq i,j\leq L$ be the set of all $\mu-\nu$ plaquettes with fixed $\mu$ coordinate $i$ and $\nu$ coordinate $j$. The operators

[TABLE]

commute with all $A_{\ell}$ and $B_{c}$ and act nontrivially on the ground state $\ket{\phi_{0}}$. Note that $\mathcal{P}^{\mu\nu}_{ij}$ and $\mathcal{Q}^{\rho\sigma}_{k\ell}$ anticommute for $(\mu,\nu)=(\rho,\sigma)$ and commute otherwise. This allows us to generate the entire code space of dimension 64 from either ground state (1), in a manner completely analogous to that of the 2DTC’s $d=1$ gauge-like symmetries Kitaev (2003); Nussinov and Ortiz (2008).

I.2 Haah’s Code

Notably, Haah’s Code does not have any string-like logical operators, and therefore has no $d=1$ dimensional gauge-like symmetries. Instead, it has the following two $d=2$ dimensional gauge-like symmetries Haah (2011): (i) let $P^{\mu}_{i}$ for $1\leq i\leq L$ be a plane of vertices perpendicular to the $\mu$ direction. Then,

[TABLE]

commute with each $A_{v}$ and $B_{v}$ and are symmetries. These operators act nontrivially on $\ket{\psi_{0}}$. Similarly, the symmetries

[TABLE]

commute with $A_{v}$ and $B_{v}$ and alter the ground state $\ket{\phi_{0}}$. Unlike the 4DTC, these logical operators commute, and do not constitute all logical operators of the code Haah (2011).

II 4D Toric Code Duality

In this section, we more carefully prove the duality given by equation (12) of the main paper. We start by defining the model as in the main paper: let $\Lambda=\mathbb{Z}_{L}^{4}$ be a $d=4$ lattice of length $L$ in each direction, with sites $v=(v_{1},v_{2},v_{3},v_{4})\in\Lambda$. This lattice has $L^{4}$ vertices $v$, $4L^{4}$ 1D links $\ell$, $6L^{4}$ 2D plaquettes $p$, $4L^{4}$ 3D cubes $c$, and $L^{4}$ 4D hypercubes $h$. We associate with each plaquette a two-dimensional Hilbert space $\mathcal{H}_{p}=\mathbb{C}^{2}$, and with the full lattice a total Hilbert space $\mathcal{H}=\otimes_{p}\mathcal{H}_{p}$. We use the operators $\sigma^{\mu}_{p}$ with $\mu=x,y,z$ to denote the $\mu$-Pauli operator acting on $\mathcal{H}_{p}$. Finally, we define $A_{\ell}$, $B_{c}$, $H_{\text{4DTC}}$, $\mathcal{Z}_{\text{4DTC}}$, $\mathcal{T}_{a}$, and $\mathcal{T}_{b}$ as in equations (2) through (8) of the main paper.

We start by noting that the $A_{\ell}$ and $B_{c}$ algebras are identical. This is because the center of every plaquette in the lattice also lies at the center of every plaquette of the dual lattice, as can easily be verified. In the dual lattice, $A_{\ell}$ looks like the product of plaquettes in a cube, while $B_{c}$ looks like the product of plaquettes surrounding a link (c.f. the 2D toric code Kitaev (2003), for which the operators $A_{s}$ and $B_{p}$ are interchanged in roles on the dual lattice). As a result, $\mathcal{T}_{a}$ and $\mathcal{T}_{b}$ will be identical, with $a\leftrightarrow b$. We will focus on the $B_{c}$ algebra, which may be easier to visualize (despite its four-dimensionality).

The form of $B_{c}$ constraints are most easily described in the language of homology. For a brief pedagogical introduction to homology and their relation to toric codes, see Appendix A of Ref. Anwar et al. (2014). In short, each $B_{c}$ is associated with an oriented 3-cell (cube) $c$ on the 4-torus, and $B_{c}$ products are associated with 3-chains, formal sums of 3-cells over $\mathbb{Z}_{2}$:

[TABLE]

To obtain such a product in terms of $\sigma^{z}_{p}$ operators, we apply the boundary operator $\partial_{3}$, which yields a formal sum of 2-cells (plaquettes) $p$:

[TABLE]

A set of 3-cells $P$ is therefore a constraint exactly when its corresponding 3-chain is a 3-cycle – that is, a 3-chain with no boundary:

[TABLE]

Let $\Gamma$ be the set of all cellular 3-cycles on $\Lambda$. Then, $\mathcal{T}_{b}$ is given by:

[TABLE]

where $|\gamma|$ is the hypersurface area of the 3-cycle $\gamma$, and each $b_{n}$ is the number of cellular 3-cycles of hypersurface area $n$ (including the empty 3-cycle, so that $b_{0}=1$). The simplest such 3-cycle is $\partial h$, the boundary of a simple hypercube $h$. One can easily verify that the product of $B_{c}$ over the eight cubes in the hypercube yields the identity, as given by equation (13) of the main paper.

On the other hand, consider the 4D Ising model given by equations (9) and (10) of the main paper. For any spin configuration $\mathcal{C}$, we may draw a domain wall in the dual lattice around each connected set of flipped spins. These domain walls are also cellular 3-cycles on the lattice, with an extra condition: domain walls are necessarily the boundary of a 4-chain, the hypervolume of spins flipped. As a result, domain walls are required to be topologically trivial, or contractible, 3-cycles. Letting $\Gamma^{*}$ be the set of contractible 3-cycles, $\mathcal{Z}_{\text{4DI}}$ can be written:

[TABLE]

where $b_{n}^{*}$ is the number of contractible 3-cycles of hypersurface area $n$. We note that all contractible 3-cycles can be formed from the boundaries of simple hypercubes – that is, any domain wall can be formed by combining those from individual spin flips.

By contrast, $\mathcal{T}_{b}$ also contains topologically nontrivial 3-cycles, stretching across the periodic boundaries. However, these 3-cycles necessarily contain no less than $L^{3}$ 3-cells, in order to stretch across the torus and eliminate the 2-cell boundaries. We therefore find that $b_{n}=b_{n}^{*}$ for $n<L^{3}$, proving equation (12) of the main paper.

It is now necessary to show that the $\mathcal{O}(\mathsf{T}_{b}^{L^{3}})$ terms are indeed negligible in the thermodynamic limit. First, we remark that these terms vanish under different boundary conditions: had our lattice been endowed with three or fewer periodic boundaries, then all 3-cycles would be contractible, and the duality would be exact for all system sizes. Since the thermodynamic free energy should not depend on boundary conditions, we expect on physical grounds that the $\mathcal{O}(\mathsf{T}_{b}^{L^{3}})$ terms should not contribute in the thermodynamic limit. Indeed, since $\mathsf{T}_{b}^{L^{3}}\rightarrow 0$ very quickly for all $\beta$, any coefficients $b_{n}$ must scale quite pathologically with $L$ to have any hope of contributing in the thermodynamic limit.

To prove that the topologically nontrivial terms of $\mathcal{T}_{b}$ are trivial in the thermodynamic limit, we use the following standard observation from homology: there exist only four homologically distinct nontrivial 3-cycles on the 4-torus. Two nontrivial 3-cycles are considered homologically equivalent if they differ by the boundary of a 4-chain – that is, a contractible 3-cycle. As a result, every possible 3-cycle $\Sigma$ can be written as:

[TABLE]

where each $C_{m}$ is one of four representative nontrivial 3-cycles from each homology class. A more careful calculation of $\mathcal{T}_{b}$ then gives:

[TABLE]

where the ellipsis contains the additional 11 terms corresponding to two, three, or all four $C_{m}$. While it is difficult in general to determine the order of terms with nontrivial cycles, we can bound $\mathcal{T}_{b}$ above by replacing a given order of $\mathsf{T}_{b}$ with all orders of $\mathsf{T}_{b}$, giving a summable expression:

[TABLE]

where $\mathcal{T}_{b}^{*}=\sum b_{n}^{*}\mathsf{T}_{b}^{n}$. In the free energy, we therefore have:

[TABLE]

Since $\mathcal{T}_{b}$ is also bounded below by $\mathcal{T}_{b}^{*}$, we have proven that the two give the same thermodynamic free energy. This gives the desired 4DTC-Ising duality.

As a final remark, although the $\mathcal{O}(\mathsf{T}_{b}^{L^{3}})$ terms above appear due to the topology of the lattice, they do not contain any of the information regarding the topological degeneracy associated with the 4DTC. For example, the 2DTC on a simple torus of genus $g=1$ is dual to two Ising chains Nussinov and Ortiz (2008). Since each Ising chain is two-fold degenerate, the conventional 2DTC on a torus is four-fold degenerate. Further analysis employing the algebraic relations between the “bonds” Nussinov and Ortiz (2009a); Nussinov et al. (2012a); Nussinov and Ortiz (2009b); Cobanera et al. (2011); Weinstein et al. (2020); Nussinov et al. (2012b) and the Euler-L’Huillier relation reveals that for the 2DTC, on general $g>0$ manifolds, the degeneracy of all levels (not only of those in the ground state sector) is $4^{g-1}$ times that of two decoupled Ising chains Nussinov and Ortiz (2009a). This constant then correspondingly appears as a global prefactor in the dual partition function. We similarly expect that for the 4DTC, the topological degeneracy commonly associated with the ground state manifold indeed extends to all energy levels of the system.

III A duality mapping preserving the locality of heat bath couplings

A common criticism of using dualities to investigate dynamics is that local perturbations become generally nonlocal in the dual model. As a result, a coupling to a local heat bath modeled by a local magnetic field might in general become a highly nonlocal coupling in the dual model, and it may not be possible to infer dynamics of the original model from the dual model as a result. Here, we present a proof-of-concept dualization of a local heat bath to a local heat bath, in order to show that it is possible to construct dualities which should certainly preserve local dynamics. We do not show that all local heat baths can be mapped to a dual local heat bath – indeed, local heat baths will generically become highly nonlocal. Instead, by showing that these mappings are possible, we argue that the original and dual models in most dualities should have sufficiently similar thermalization on physical grounds.

We consider the ordinary 2D Toric Code Kitaev (2003) on a square lattice of side length $L$, with stabilizer generators given by:

[TABLE]

where each $s$ is a “star” of four links surrounding a given vertex, and each $p$ is a “plaquette” of four links forming a square. The unperturbed stabilizer Hamiltonian is given by:

[TABLE]

In the following, we will employ open (i.e., non-periodic) boundary conditions, in which the stars on the boundary will not be included in the sum (16). We note that the Toric Code is traditionally defined on a torus, but that the thermodynamics in the bulk of the system should not be affected by the boundary conditions in the thermodynamic limit.

For notational convenience, we will from now on relabel operators by their $(i,j)$ coordinates:

[TABLE]

where the $(i,j)$ coordinates of $s$ denote the lattice site of the star, and the $(i,j)$ coordinates of $p$ denote the bottom-left corner of $p$. Finally, we will denote the coordinates of spins on links as if they were placed in the center of their respective link. That is, Pauli operators on $x$-links will be denoted by:

[TABLE]

and Pauli operators on $y$-links will be denoted by:

[TABLE]

As a first simple example, we consider a local bath modeled by a perturbing magnetic field acting on only $x$-links, given by:

[TABLE]

where $\lambda>0$ gives the magnetic field strength. Each $\sigma^{x}$ in (20) anticommutes with two adjacent plaquettes, and commutes with all stars. The commutation relations of the full Hamiltonian $H_{0}+V_{x}$ is then given by:

[TABLE]

We note that a magnetic field acting only on $y$-links, or a field coupling to $\sigma^{z}$ instead of $\sigma^{x}$, would behave in very much the same manner.

To employ a bond-algebraic duality, we map each operator in (16) and (20) to another set of spin operators $\tau^{\mu}$ in such a way as to preserve the bond algebra (21). Towards this end, we map each vertical column of plaquettes to an open Ising chain, and map (20) to a transverse field on these chains:

[TABLE]

The stars can then each be mapped to auxiliary spins $\tau^{z}_{L,j}$. The resulting dual Hamiltonian is given by:

[TABLE]

We might also consider a local bath modeled by a perturbing field that anticommutes with both stars and plaquettes, given by:

[TABLE]

Each $\sigma^{y}_{i+\frac{1}{2},j}$ once again anticommutes with vertically adjacent plaquettes, but now also anticommutes with horizontally adjacent stars. The commutation relations are given by:

[TABLE]

Here, each vertical column of plaquettes is once again mapped to an open Ising chain of $\tau^{z}$ spins, while horizontal rows of stars are mapped to open Ising chains of $\rho^{z}$ spins. To anticommute with both $\tau$ chains and $\rho$ chains, each $\sigma^{y}$ is mapped to a transverse coupling between chains $\tau^{x}\rho^{x}$:

[TABLE]

The resulting dual Hamiltonian is then:

[TABLE]

At first glance, the above duality might seem highly nonlocal due to the coupling terms $\tau^{x}\rho^{x}$. However, if the $\tau$ Ising chains are visualized as vertical Ising chains, and the $\rho$ Ising chains are visualized as horizontal Ising chains lying across the $\tau$ Ising chains in a crossing pattern, then the couplings are localized to the points where the Ising chains cross.