Tensor Network Representations for Intrinsically Mixed-State Topological Orders

The gist The method presents a general protocol to construct fixed-point tensor network representations for intrinsically mixed-state topological phases, which exhibit nontrivial topological phenomena and do not have pure-state counterparts.

Tensor Network Representation for Pure States

Tensor networks are an efficient platform to represent interesting quantum states of matter as well as to compute physical observables and information-theoretic quantities. For many exactly solvable models of two and three-dimensional pure-state topological phases, there exist corresponding tensor network representations (see for example [37–41]) which are fixed points under renormalization. The ground state of the ZN toric code model is then expressed as the sum over all physical configurations weighted by the tensor network contracted over the virtual indices (which are suppressed in the equation below) where M = 2L squared is the number of edges.

Fixed-Point Tensor Network Construction

The method exploits the power of anyon condensation in Choi states and is applicable to cases where decoherence/disorders produce only Abelian excitations. The protocol involves several steps summarized in a flowchart in fig. 1 which outlines the general TN construction procedure for pure-state topological phase ψ⟩ decohered in Abelian channels, i.e. the Kraus operators Ka only create Abelian anyons from the topological phase. The post-channel Choi state is given by a formula involving a connecting tensor h which imposes strong decoherence.

Decoherence Channels in ZN Topological Order

The paper demonstrates general procedures for different types of decoherence channels.

1. Pure-flux decoherence considers a channel where each Kraus operator only acts on a single link, of the form: Nl(ρ) = 1/N squared X(-1)j=0 (Xa l)j ρ(Xa l)-j. The resulting tensor network has the same form as in figure 7a.

2. Dyonic decoherence involves channels of the form Xa l Zb l−a, which represent the most general form for the decoherence of an arbitrary individual anyon ma e b in the ZN topological phase. The resulting tensor network representation is given by equation (29) with this connecting tensor.

3. Fermion channel in the Z2 toric code is presented as a special example where the local channel acts as Nl[(·)] = 1/2 ((·) + XlZl−a(·)XlZl−a. The resulting tensor network representation of the density matrix is then given by equation (40).

Extensions to More Complex Systems

The construction extends beyond two-dimensional decohered topological phases to include more complex examples.

1. Decoherence in the S3 topological phase involves a non-Abelian S3 topological order, where the ground state is constructed by assigning a tensor gβαγδ = g-1αβδγ = O gαβγδ,O gαβγδ = L+,αβ(g)L+,γδ(g−1).

2. Generalization to arbitrary CSS codes is discussed by defining a TN for the channel ρA ∝ Yk 1 + Ak squared. The density matrix is expressed as ρ ∝ Xj,j′ "Yk T kT∗ Yl g jl g j'l∗ h l>.

3. A three-dimensional Z2 toric code renormalization group procedure is presented, which involves multiple distinct steps to yield the original TN as a fixed point of the RG procedure.

4. A fixed-point TN representation of chiral topological order is provided, which is unique in the mixed-state setting because it cannot arise from locally decomposable Hilbert spaces.

Summary and Outlook

The method can be readily generalizable beyond the examples presented, for example to string-net models. The fixed-point tensor networks can still be deformed to examine the stability of and the phase transitions into/out of mixed-state topological phases, similar to the pure-state cases in [45–49]. The tensor network formalism can be inspiring for the study of non-maximal decoherence and the critical error threshold. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The subtlety of generalizing the formalism to the channels which create non-Abelian excitations can be seen in the following example. The tensor network formalism can be inspiring for the study of non-maximal decoherence and the critical error threshold. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models. It can also be applied directly to the decoherence of states that are themselves the result of condensation, as given in the example of the chiral theory’s tensor network in section VI D. The scope of the method is readily generalizable beyond the examples presented, for example to string-net models.

1. Improve quantum error correction (QEC) by using fixed-point tensor network representations for intrinsically mixed-state topological phases, which are constructed from strongly decohered/disordered purestate topological phases. This allows AI systems to model and analyze the stability of topological phases under realistic noise models like pure-flux decoherence or dyonic decoherence, offering insights into their robustness.

2. Implement a Tensor Entanglement Renormalization Group (TERG) procedure for evaluating correlators, which is applicable because the TN states are fixed points under renormalization. This capability enables the AI to efficiently compute physical observables in large systems by exploiting the computational cost reduction mentioned in Section II.

3. Develop models for quantum computation using anyonic systems that do not satisfy braiding non-degeneracy, as these correspond to mixed-state topological phases which host nontrivial chiral central charge. This allows AI to explore novel computational paradigms beyond conventional pure states without relying on conventional symmetries.

4. Create a general protocol for representing the post-channel Choi state using the formula in Equation (28), which involves adding connecting tensor h between ancillas in bra and ket layers to impose strong decoherence. This enables the AI to simulate realistic noise channels, such as those modeled by generalized Pauli noise, and predict the resulting maximally decohered state (Equation 29).

5. Design a representation for fermionic systems using Projected Entangled Pair States (fPEPS) by modifying the projector Q in Equation (A6) to include an ancilla, allowing the TN to represent the decohered state of the Z2 toric code in a fermionic tensor network description. This provides a method for handling non-bosonic decoherence in quantum codes.

6. Extend fixed-point TN methods to arbitrary CSS codes by defining tensors T that enforce the Z2 Gauss law and showing how an RG procedure can be applied, which is relevant for analyzing decohered non-Abelian S3 topological order. This allows AI to study complex stabilizer codes beyond the simplest two-dimensional examples.

Tensor networks are an efficient platform for representing topological states of matter as well as computing both physical observables and information-theoretic quantities. We present a general protocol to construct tensor network representations for intrinsically mixed-state topological phases. Such phases naturally emerge in noisy long-range entangled quantum systems, and exhibit nontrivial topological phenomena without pure-state counterparts. The method exploits the isomorphism between anyon decoherence in the physical Hilbert space and anyon condensation in the corresponding Choi space. The protocol is applicable to a broad class of systems arising from decoherence of pure-state topological phases, where the incoherent noise creates excitations in the Choi space that have trivial monodromy with all excitations. Representative examples at renormalization group fixed points include m a e b decoherence in the N toric code, decohered non-Abelian quantum double of S 3 as well as pure Z / X decoherence of arbitrary CSS codes. Another notable example is that of the mixed-state chiral semion, whose tensor network representation has no pure-state analogue. We then generalize the formalism beyond RG fixed points and, as an example, examine the decoherence transition in a Z 3 toric code channel that suffers from sign problem when mapped to a statistical mechanical model. Together, these results establish tensor networks as a natural platform for analyzing noisy intrinsically mixed-state topological matter.