Stability of Topological Order at Zero Temperature

From today’s quant-ph arXiv listing we find the following paper:

Stability of Frustration-Free Hamiltonians, by S. Michalakis & J. Pytel

This is a substantial generalization of one of my favorite results from last year’s QIP, the two papers by Bravyi, Hastings & Michalakis and Bravyi & Hastings.
In this new paper, Michalakis and Pytel show that any local gapped frustration-free Hamiltonian which is topologically ordered is stable under quasi-local perturbations. Whoa, that’s a mouthful… let’s try to break it down a bit.
Recall that a local Hamiltonian for a system of n spins is one which is a sum of polynomially many terms, each of which acts nontrivially on at most k spins for some constant k. Although this definition only enforces algebraic locality, let’s go ahead and require geometric locality as well by assuming that the spins all live on a lattice in d dimensions and all the interactions are localized to a ball of radius 1 on that lattice.
Why should we restrict to the case of geometric locality? There are at least two reasons. First, spins on a lattice is an incredibly important special case. Second, we have very few tools for analyzing quantum Hamiltonians which are k-local on a general hypergraph. Actually, few means something closer to none. (If you know any, please mention them in the comments!) On cubic lattices, we have many powerful techniques such Lieb-Robinson bounds, which the above results make heavy use of [1].
We say a Hamiltonian is frustration-free if the ground space is composed of states which are also ground states of each term separately. Thus, these Hamiltonians are “quantum satisfiable”, as a computer scientist would say. This too is an important requirement, since it is one of the most general classes of Hamiltonians about which we have any decent understanding. There are several key features of frustration-free Hamiltonians, but perhaps chief among them is the consistency of the ground space. The ground states on a local patch of spins are always globally consistent with the ground space of the full Hamiltonian, a fact which isn’t true for frustrated models.
We further insist that the Hamiltonian is gapped, which in this context means that there is some constant γ>0 independent of the system size which lower bounds the energy of any eigenstate orthogonal to the ground space. The gap assumption is extremely important since it is again closely related to the notion of locality. The spectral gap sets an energy scale and hence also a length scale, the correlation length.  For two disjoint regions of spins separated by a length L in the lattice, the connected correlation function for any pair or local operators decays exponentially in L.
The last property, topological order, can be tricky to define. One of the key insights of this paper is a new definition of a sufficient condition for topological stability that the authors call local topological order. Roughly speaking, this new condition says that ground states of the local Hamiltonian are not distinguishable by any (sufficiently) local operator, except up to small effects that vanish rapidly in a neighborhood of the support of the local operator. Thus, the ground space can be used to encode quantum information which is insensitive to local operators! Since nature presumably acts locally and hence can’t corrupt the (nonlocally encoded) quantum information, systems with topological order would seem to be great candidates for quantum memories. Indeed, this was exactly the motivation when Kitaev originally defined the toric code.
Phew, that was a lot of background. So what exactly did Michalakis and Pytel prove, and why is it important? They proved that if a Hamiltonian satisfying the above criteria is subject to a sufficiently weak but arbitrary quasi-local perturbation then two things are stable: the spectral gap and the ground state degeneracy. (Quasi-local just means that strength of the perturbation decays sufficiently fast with respect to the size of the supporting region.) A bit more precisely, the spectral gap remains bounded from below by a constant independent of the system size, and the ground state degeneracy splits by an amount which is at most exponentially small in the size of the system.
There are several reasons why these stability results are important. First of all, the new result is very general: generic frustration-free Hamiltonians are a substantial extension of frustration-free commuting Hamiltonians (where the BHM and BH papers already show similar results). It means that the results potentially apply to models of topological quantum memory based on subsystem codes, such as that proposed by Bombin, where the syndrome measurements are only two-body. Second, the splitting of the ground state degeneracy determines the dephasing (T2) time for any qubits encoded in that ground space. Hence, for a long-lived quantum memory, the smaller the splitting the better. These stability results promise that even imperfectly engineered Hamiltonians should have an acceptably small splitting of the ground state degeneracy. Finally, a constant spectral gap means that when the temperature of the system is such that kT<<γ, thermal excitations are suppressed exponentially by a Boltzmann factor. The stability results show that the cooling requirements for the quantum memory do not increase with the system size.
Ah, but now we have opened a can of worms by mentioning temperature… The stability (or lack there of) of topological quantum phases at finite temperature is a fascinating topic which is the focus of much ongoing research, and perhaps it will be the subject of a future post. But for now, congratulations to Michalakis and Pytel on their interesting new paper.

[1] Of course, Lieb-Robinson bounds continue to hold on arbitrary graphs, it’s just that the bounds don’t seem to be very useful.

9 Replies to “Stability of Topological Order at Zero Temperature”

  1. Martin, thanks for bringing this result to my attention. I think that beyond the work of Hastings, which includes the Lieb-Robinson bounds and quasi-adiabatic evolution, another great tool which is currently underutilized is the detectability lemma. My hope is that one day we will be able to understand clearly how all these tools are connected. I think we will need to in order to answer some of the questions posed at the end of the stability paper. And Steve, thank you for your kind words. I think this result was a simple extension of the work with Bravyi and Hastings, but I am excited about some of the future directions this may lead towards.

  2. In an upcoming paper with Fernando Brandao and Michal Horodecki (arXiv:1009.xxxx, inshallah), we make use of the detectability lemma for an application that a priori appears unrelated.

  3. Can someone answer this question for me:
    What presently-known Hamiltonians are known *rigorously* to exhibit Local Topological Quantum Order (L-TQO)? In particular, is it known *rigorously* that all Levin-Wen string nets [1] exhibit L-TQO?
    [NOTE: Everything that follows is just explanation of the above question. So, if you already know why the answer the above question is yes, no, or nobody-yet-knows, you needn’t read any further.]
    I ask the above question due to the following remark that Bravyi, Hastings, and Michalkakis made in the first paper [2] on L-TQO—a remark they explicitly emphasize is not rigorous:
    [First, before quoting the remark, here’s some context on its notation:
    Let $H_0$ be the Hamiltonian for the entire system, which we assume to be some lattice of spins, possibly with nontrivial topology.
    Let $P$ be the projector onto the ground states of $H_0$.
    Let $A$ be any topologically simple (i.e., genus-0) subset of the system.
    Let $A^c$ be the complement of $A$ (i.e., the rest of the system not in $A$.)
    Let $B$ be the union of $A$ with the nearest-neighbors to the boundary of $A$.
    Let $P_B$ be the projector onto the ground states of the Hamiltonian containing all the terms from $H_0$ that act only on degrees of freedom wholly contained in $B$.
    Let $\rho_A = Trace_{A^c}{P}$ be the reduced density matrix describing the ground state(s) $P$ of the entire Hamiltonian when just the degrees of freedom in $A$ are observed.
    Let $\rho_A^{(B)} = Trace_{A^c}{P_B}$ be the reduced density matrix for the ground state(s) $P_B$ of the Hamiltonian restricted to $B$ when just the degrees of freedom in $A$ are observed.
    Finally, let “Condition TQO-2” be an extra condition on plain ol’ topological order that means the ground states of the whole Hamiltonian $H_0$ and the ground states of just that subset of $H_0$ terms restricted to $B$ are essentially indistinguishable by measurements just on $A$. In the case of frustration-free Hamiltonians whose terms are all mutually commuting that’s treated in this first L-TQO paper [2], the reduced density matrices for these 2 cases, $\rho_A$ and $\rho_A^{(B)}$, need to have the exact same kernels.

    Remark 2. Condition TQO-2 can be easily ‘proved’ if the excitations of $H_0$ are anyons (since the latter assumption lacks a rigorous formulation, the argument given below is not completely rigorous either). Indeed, in this case we can choose a complete basis of the excited subspace $Q$ such that the basis vectors correspond to various configurations of anyons. For non-abelian theories one may have several basis vectors for a fixed configuration of anyons that describe different fusion channels, see [3]. Note that any state $\psi \in P_B$ is a superposition of configurations with no anyons inside $B$. Since $A$ is a topological trivial region, any such configuration can be prepared from the vacuum P by some unitary operator $U_{A^c}$ acting on complementary region $A^c = \Lambda \backslash A$. Thus $\psi = U_{A^c} \psi_0$ for some ground state $\psi_0 \in P$. Since all ground states $\psi_0$ have the same reduced matrix on $A$, it means that $\psi$ and $P$ have the same reduced matrix on $A$. This implies TQO-2. The above arguments suggest that TQO-2 holds for all 2D models of TQO that can be described by commuting frustration-free [Hamiltonians] such as quantum double models [4] and Levin-Wen string-net models [1].

    I must confess I don’t understand why this argument isn’t rigorous for any family of Hamiltonians like the Levin-Wen string-net models where one explicitly knows what degrees of freedom upon which one must act in order to generate a quasiparticle excitation associated with any given location.
    [1] M.A. Levin and X-G. Wen, “String-net condensation: A physical mechanism for topological phases” Phys. Rev. B 71, 045110 (2005); also available as
    [2] S. Bravyi, M. Hastings, and S. Michalkakis, “Topological quantum order: stability under local perturbations” J. Math. Phys. 51, 093512 (2010); also available as
    [3] A. Kitaev, “Anyons in an exactly solved model and beyond,” Ann. Phys. 321, 2-111 (2006); also available as
    [4] A. Kitaev, “Fault-tolerant quantum computation by anyons”, Ann. Phys. 303, 2-30 (2003); also available as

  4. Bill, you are right about the fact that the intuition is correct as far as Levin-Wen models and the like are concerned. Since we did not write explicitly what the assumption was for a system to have anyonic excitations, we did not provide a rigorous proof, but only a sketch of a proof for each case we know about. In the end, all one has to do to check L-TQO is compute the reduced density matrices of pairs of local groundstates and show that they are all the same up to boundary errors (Corollary 3 in this paper). One of the open problems is to create a rigorous connection between correlation length and the decay in Local-TQO.

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