A Approximate Quantum Error Correcting Codes

# Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains

## Abstract

Quantum error correction was invented to allow for fault-tolerant quantum computation. Systems with topological order turned out to give a natural physical realization of quantum error correcting codes (QECC) in their groundspaces. More recently, in the context of the AdS/CFT correspondence, it has been argued that eigenstates of CFTs with a holographic dual should also form QECCs. These two examples raise the question of how generally eigenstates of many-body models form quantum codes. In this work we establish new connections between quantum chaos and translation-invariance in many-body spin systems, on one hand, and approximate quantum error correcting codes (AQECC), on the other hand. We first observe that quantum chaotic systems exhibiting the Eigenstate Thermalization Hypothesis (ETH) have eigenstates forming approximate quantum error-correcting codes. Then we show that AQECC can be obtained probabilistically from translation-invariant energy eigenstates of every translation-invariant spin chain, including integrable models. Applying this result to 1D classical systems, we describe a method for using local symmetries to construct parent Hamiltonians that embed these codes into the low-energy subspace of gapless 1D quantum spin chains. As explicit examples we obtain local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and the Motzkin spin chain model with periodic boundary conditions, thereby yielding non-stabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models.

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## I Introduction

Quantum error correcting codes (QECC) were originally designed for fault-tolerant quantum computation [1]. The idea is to cleverly encode the quantum information into entangled states in a way that the information is inaccessible locally. At first sight, it may seem the conditions for quantum error correction are very different from everything we have normally in nature, and that it would take very special engineered quantum systems to realize it. This intuition turned out to be wrong; QECCs appear naturally in the groundspace of topological ordered systems [2]. This connection has lead to many insights both in the study of quantum error correction [3]; [4] and of topological order [5]; [6] in the past 20 years.

In a different direction, in recent years there have been ongoing efforts of connecting the holographic correspondence to quantum error correction. In the Anti-de Sitter (AdS)/Conformal Field Theory (CFT) correspondence [7]; [8], it has been understood to a certain degree that, bulk local operators in AdS are dual to nonlocal operators on the boundary CFT [9]. Quantum error correction has recently been used [10] for explaining seemingly puzzling facts about this correspondence. It was argued that bulk local operators, reconstructed on the boundary, should commute with boundary local operators only within a certain subspace of the full boundary CFT Hilbert space. Interpreting this subspace as the code subspace of an error correcting code not only clears the apparent puzzles but also gives a new information-theoretic perspective to the AdS/CFT correspondence. Since then quantum error correction has served as a guiding feature for the application of tools from quantum information to the challenge of constructing explicit realizations of AdS/CFT duality [11]; [12]; [13]. Understanding holographic codes from the perspective of the CFT continues to be a major open challenge [14]; [15].

In this Letter, we explore one-dimensional physical systems through the lens of AQECC. Guided by the codes found in the ground space of topologically ordered gapped Hamiltonians and the expectation of good codes in eigenspaces of certain CFTs (motivated by AdS/CFT correspondence), we ask what other physical conditions lead to good quantum codes. First we observe a connection between quantum chaos and quantum error correction, pointing out that the Eigenstate Thermalization Hypothesis (ETH) [16] can be interpreted as saying that eigenstates with close-by energies form an AQECC. This observation directly supports the QECC view of the AdS/CFT correspondence, as the CFTs considered there are expected to be chaotic. Then we show that merely translation-invariance of the Hamiltonian already implies that most (translation-invariant) eigenstates in a subextensive energy window of finite energy density form AQECCs. This general result also applies to integrable models and even to non-interacting Hamiltonians. In some of these cases we show that it is possible to use local symmetries of the states to generate an interacting Hamiltonian that embeds the finite energy eigenstates (i.e., the codespace) of the noninteracting Hamiltonian into the groundspace or low-lying energy subspace of gapless 1D quantum systems. As examples we show how this procedure can give rise to the Heisenberg and Motzkin models. For these systems we confirm the AQECC performance of the low energy eigenspace by direct calculations, thereby showing that non-stabilizer codes can appear at low energy in physically plausible 1D models. The precise statements about the distance, the dimension of the codespace and the scaling of the error of the AQECC, are given for each case.

## Ii Approximate QECC

We start with a brief description of the features of approximate quantum error correction. For exact quantum error correction, Knill and Laflamme gave a convenient set of necessary and sufficient conditions for a code being able to correct a noisy channel [17]. Similar conditions for the approximate case were found by Beny and Oreshkov [18], which we now review. We consider qubits arranged in a line and assume that errors are local. We say that a subspace of a -dimensional vector space is a approximate quantum error correction code (AQECC) if and for every channel acting on at most consecutive qubits, we have

 min|ψ⟩∈C⊗2maxD⟨ψ|(D∘Λ⊗I)(|ψ⟩⟨ψ|)|ψ⟩≥1−ε, (1)

where the maximum is over decoding channels , and the minimum is over pure entangled states acting on and a reference system (we denote the tensor product space of and the reference by above). In words, this condition states that one can correct, up to error , the effect of local noise on at most qubits. If Eq. (1) only works for a particular , we say the code is -correctable under .

In this work we find it convenient to consider a set of codewords that span the code space, , and show that these codewords satisfy an approximate version of the Knill-Laflamme conditions,

 ⟨ψi|E|ψj⟩=CEδij+εij. (2)

Corollary 5 of the Appendix shows that if this condition is satisfied, then the error of the code as defined in (1) can be bounded as . For many-body systems with sites, it is natural to seek so that the probability of recovering the logical state converges to 1 quickly with increasing system size.

## Iii AQECC from ETH

The Eigenstate Thermalization Hypothesis (ETH) states that thermalization in a quantum system takes place already on the level of eigenstates. Given the Hamiltonian , with being energy eigenstates with eigenvalue (ordered as ), Srednicki proposed the following version of ETH [16]: There are constants such that for every in the bulk of the spectrum and for any local observable ,

 |⟨El|O|El⟩−⟨El+1|O|El+1⟩|≤exp(−c1N), (3)

and

 |⟨Ek|O|El⟩|≤exp(−c2N). (4)

Indeed Eq. (3) tells us that the energy eigenstates around are locally indistinguishable from each other, and therefore also from the thermal state of the same energy. They ensure that the long-time average of any local observable is thermal. Eq. (4), in turn, guarantees that the fluctuations around the long-time average is small. Comparing the ETH condition Eq. (3) to the AQECC condition Eq. (2), we observe that:

###### Remark.

ETH implies that any region of the spectrum with finite energy density have eigenstates forming approximate error correcting codes.

Note that the distance of the code is given by the range of locality for which ETH holds in the system. This is expected to vary depending on the model, and can be as large as a constant fraction of the size of the system [19]. From Eq. (2) and Corollary 5 of the Appendix, we find that the codes have constant rate, i.e. , and exponentially small error. Note that, these are very good codes for highly chaotic systems in which ETH holds for -local observables with .

However, a major drawback is that the codewords are exponentially close to each other in energy, hence it is not clear at all if the Hamiltonian can help with encoding and decoding. One way forward is to split the codewords in energy by sacrificing either the dimension of the codespace or the error of the code. We leave to future work to investigate whether the locality of the Hamiltonian leads to good ways of encoding and decoding in this case.

Notice that ETH codes introduced above are somewhat analogous to random subspace codes (in terms of the parameters achieved) [20]. This is no coincidence. One of the ways of understanding quantum chaos is that apart from a few conserved quantities (e.g., energy), the physics of the model mimics the one of a fully random system. Here we give a coding perspective of this view.

An important application of the observation is in connection to the recent proposal of interpreting some aspects of the AdS/CFT correspondence as an error correcting encoding of the AdS bulk into the boundary CFT [10]. It is expected that holographic CFTs are chaotic and thus satisfy ETH [21]. Therefore our observation provides strong evidence in favor of the proposal in Ref. [10]. However, ETH is a claim about eigenstates with finite energy density, whereas the error correcting properties of eigenstates of holographic CFTs are expected to hold even at zero energy. We will partially address this point later in the paper, constructing specific examples of gapless spin chains with AQECC in their low-lying spectrum. The connection of ETH and AQECC that we point out also suggests that such holographic CFTs might be chaotic in an extreme sense of satisfying ETH at all energies.

## Iv AQECC from Translation-Invariance

Although ETH is expected to hold for a large class of systems, its range of validity is still not completely understood. Our next result shows that even just from translation invariance we can already get codes from eigenstates of local models (albeit with worse parameters). Consider a 1D translation invariant Hamiltonian with sites. Let be the set of energy eigenvalues close to : , and define the microcanonical state of energy as

 τMC(E):=1|SE|∑k:Ek∈SE|Ek⟩⟨Ek|. (5)

Note that in one-dimension the correlation length is a function of mean energy only, and it is a constant independent of system size when is too. The choice for the energy window is arbitrary; all we need is that the associated microcanonical ensemble has finite correlation length, which is true as long as it is subextensive and larger than [22].

We prove that:

###### Theorem 1.

Let be a 1D translation invariant local Hamiltonian and be such that the microcanonical state at energy has finite correlation length (independent of system size). Pick uniformly independently at random from , where is a basis of translation-invariant eigenstates of , and . Then with high probability they form an AQECC with and

 d=min(Ω(log(N)),minp≠q∈[L]|Eip−Eiq|−O(log(N))). (6)

Note that by choosing for sufficiently small , the minimum energy gap will be of order , and thus the distance of the code is with high probability.

The proof in Section B builds upon two results. First, the result of [23] establishes a weak version of the eigenstate thermalization hypothesis (ETH) for 1D translation invariant systems (see Lemma 7): The fraction of the nonthermal energy eigenstates around the microcanonical energy is exponentially small with the system size . This means that with high probability, randomly chosen codewords do look like the thermal state, and hence are locally indistinguishable. Second, the result from [24] states that eigenstates of general (not necessarily translation-invariant) local Hamiltonians with different energies cannot be “connected” by local operators, in the sense that the off-diagonal matrix elements of the local operator in energy eigenbasis drop off exonentially with the energy gap (see Lemma 9 in Section B). This tells us to choose the codewords sufficiently far apart in energy so that we have the desired distance for the code.

Translation invariance is crucial in the proof of the results. Technically, it allows us to replace the local observable by an extensibe observable, given by a sum of trnaslations of the original one. Then we can use techniques of large deviation bounds on the measurement of extensive observables in non-critical spin systems to obtain the result. Intuitively, translation invariance guarantees that the information of the codewords is spread to the whole system “uniformly”, and hence cannot be corrupted locally by noise.

Note that in addition to translation invariance, the only feature of 1D systems we use in the proof is that the microcanonical states at finite energy density always have a finite correlation length. Therefore the theorem generalizes to higher dimensions for eigenstates with finite energy densities (albeit with a worse scale of the error of the code).

## V AQECC from the Low-Energy Eigenspace of Gapless Models

So far we have considered eigenstates at finite energy density. Here we show they are also relevant to the low-lying spectrum of gapless models. We first apply Theorem 1 to noninteracting models, and map the codewords at finite energy eigenstates to low-energy eigenspace of interacting models. We then further analyze the performance of these specific codes by explicitly revealing the working code subspace.

Classical Models: Consider a 1-local Hamiltonian on a system of qubits,

 H=N∑i=112(I−σzi), (7)

which has eigenvalues . Theorem 1 implies that with high probability a subset of randomly chosen translation invariant eigenstates of Eq. (7) with energies in will be an AQECC with distance. As eigenstates, we can take uniform superpositions of -basis states , where , with a particular magnetization ,

 |hNm⟩=1√(NN/2+m/2)∑s:M(s)=m|s⟩. (8)

Mapping to Low-Lying Eigenstates: Although Theorem 1 only applies to states with finite energy density (when the correlation length of the microcanonical state is finite), it turns out that the excited state AQECC in the example above can be embedded into low energy states of a different local model. This connection is based on the fact that the permutation symmetric energy eigenstates (8) of the spin-1/2 model (7) also span the ground space of the ferromagnetic Heisenberg model,

 H=−12N∑j=1(σxjσxj+1+σyjσyj+1+σzjσzj+1). (9)

For ease of notation we consider the version of this model with periodic boundary conditions (PBCs). In the Appendix we choose codewords with magnetization in the range and show the following proposition by explicit calculation.

###### Proposition 2.

For any with the ground space of the spin 1/2 ferromagnetic Heisenberg model with sites and PBCs contains an AQECC with , , and .

Specifically, we prove Proposition 2 in terms of Eq.(2). A -local error can change the magnetization by at most , so for different codewords, i.e. the case , we have zero error in Eq.(2). Furthermore, the -body reduced density matrix of different codewords are indistinguishable in the thermodynamic limit, i.e. this gives the error for the cases in Eq.(2). Note that the AQECC parameters achieved in Proposition 2 are asymptotically equivalent to those in Theorem 1, though one difference is that in Proposition 2 the codewords are chosen deterministically. Finally, we note that the existence of error correcting codes in the ground space of Heisenberg models has been observed before [25]; [26], although the choices of code words as well as the QEC parameters differ in that work from the ones presented here.

Just as finite energy density codes of (7) can be embedded in the ground space of the Heisenberg model, one can also consider the spin 1 version of (7),

 H=N∑i=112(I−Szi). (10)

The permutation invariant eigenstates of (10) are uniform superpositions of basis states , where , with a particular magnetization ,

 |gNm⟩=1|gNm|∑w:M(w)=m|w⟩. (11)

By Theorem 1 a randomly chosen subset of states of the form (11) with magnetization will with high probability form an AQECC with distance . Just as a finite energy density AQECC of (7) was turned into a ground space AQECC of (9), we seek a parent Hamiltonian which contains the states (11) in its ground space.

Such a parent Hamiltonian can be constructed by using the connection between classical random walks (and more generally reversible Markov chains) and stoquastic frustration free local Hamiltonians [27]; [28]; [29]. The following rules applied to any pair of consecutive basis labels (with periodic boundary conditions) suffice to connect all of the basis states at each energy,

 |1,−1⟩↔|0,0⟩,|0,1⟩↔|1,0⟩,|0,−1⟩↔|−1,0⟩.

These local moves can be adjusted into a local Hamiltonian such that the states constructed as the uniform superposition of basis states of the same energy become the ground states:

 H=N∑j=1(|F⟩⟨F|j,j+1+|U⟩⟨U|j,j+1+|D⟩⟨D|j,j+1), (12)

with , , where the labels is replaced by . This model is called the spin-1 Motzkin chain with periodic boundary conditions (PBCs) [30]; [31]. Using the well-studied analytical properties ofthe ground states of these models, we prove the following proposition in the Appendix.

###### Proposition 3.

For any with the ground space of the spin 1 Motzkin model on sites with PBCs contains an AQECC with , , and .

The intuitive explanation and the calculations are similar to those for the Heisenberg model. These results also hold for the degenerate Heisenberg and Motzkin chains with open boundary conditions with the restriction that errors are only applied far from the endpoints of the chain. Finally, we note that it is possible to perturb the model with a local translation invariant field in such a way that that is the unique ground state, with an inverse polynomial gap to the first excited state [31]. With this perturbation the states gain an energy that increases with the magnitude of , but which vanishes in the thermodynamic limit. This variant of the Motzkin chain is of interest in the present context because it shows that it is possible for models with a unique ground state to be part of a code space that includes gapless excitations.

## Vi Conclusions

In this Letter we have given new examples of approximate quantum error correction against local noise in the energy eigenstates of physical systems, which goes beyond the well-studied ground states of gapped topologically ordered systems. To be more specific, we have explicitly showed that energy eigenstates packed around some finite energy density eigenstate of systems exhibiting ETH, and almost all translation invariant finite energy eigenstates of 1D translation invariant local Hamiltonians, construct approximate error correcting codes. We applied the latter result to noninteracting local Hamiltonians to map the finite-energy-density codes to the low-energy subspace of interacting Hamiltonians, eg. Heisenberg model and spin-1 Motzkin chain. We studied the ground states of these models with periodic boundary conditions and further detailed the parameters of the approximate error correcting code that can be found in their low energy.

One can interpret our results from many perspectives. One perspective may be that it is not unusual to find error correcting codes in physical systems; it is indeed a generic phenomena as shown by our results of AQECC from systems with ETH and translation invariance. Another point of view which builds upon the first one is that even though error correcting codes can be found easily in Hamiltonian systems, their varying performance under different types of errors may be a way to characterize different properties of these physical systems. For example, the Motzkin spin-1 model that we analyzed is gapless, however the gap closes as on the contrary to observed in 1D lattice models whose critical points are effectively described by CFTs. To pursue its potential relevance to AdS/CFT(-like correspondence), one shall follow [10]; [32] where certain properties of AdS/CFT such as radial commutativity, subregion duality and Ryu-Takayanagi formula have been matched to operator algebra quantum error correcting codes.

There are numerous other questions one can ask building upon our work. Hence, our results shall best be taken as a first step to elucidate the role of error correcting codes in physical systems, from topological order to ETH, AdS/CFT, and gapless quantum systems. The performance of these codes under specific noise channels must be intimately connected to the physical properties manifested by these systems.

## Vii Acknowledgments

We thank Xi Dong, Tarun Grover, Nick Hunter-Jones, Robert Koenig, John Preskill for discussions. E.C. is grateful for support provided by the Institute for Quantum Information and Matter, with support of the Gordon and Betty Moore Foundation (GBMF-12500028). M.B.S. acknowledges the support from Simons Qubit fellowship provided by Simons Foundation through It from Qubit collaboration. F.B, E.C. and M.B.S. were supported by an NSF Physics Frontiers Center (NSF Grant PHY-1125565). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915.

## Appendix A Approximate Quantum Error Correcting Codes

Here we give a brief description of the features of approximate quantum error correction that we use. We follow closely [[18]]. We say that a subspace of a -dimensional vector space of qubits arranged in a line is a approximate quantum error correction code (AQECC) against -local errors if and for every channel acting on at most consecutive qubits, we have

 min|ψ⟩∈C⊗2maxD⟨ψ|(D∘N⊗I)(|ψ⟩⟨ψ|)|ψ⟩≥1−ε, (13)

where the maximum is over decoding channels , and the minimum over pure entangled states acting on and a reference system (which altogether we denoted by above). In words, the condition above says that one can correct, up to error , the effect of any noise on at most neighboring qubits. If 13 only works for a particular noise channel , we say that the code is -correctable under .

For exact quantum error correction, Knill and Laflamme gave a convenient set of necessary and sufficient conditions for a code being able to correct a noisy channel . Similar conditions for the approximate case were found by Beny and Oreshkov [[18]].

For two channels and , Let be the Bures metric, where the fidelity of the two channels is defined as follows:

 F(N,M):=max|ψ⟩F(I⊗N(|ψ⟩⟨ψ|),I⊗M(|ψ⟩⟨ψ|)), (14)

with the maximiztion over all bipartite states of the input of the channel and a vector space isomorphic to it.

Then we have:

###### Proposition 4.

[Corollary 2 of [[18]]] A code defined by the projector is -correctable under a noise channel if and only if

 PEyiEjP=λijP+PBijP, (15)

where are the components of a density operator, and , where and .

In the proposition the projector is the projector onto the support of the subspace defining the code. An easy consequence of Proposition 4 is the following:

###### Corollary 5.

Let be an orthogonal set of states in such that for all and any -local operator ,

 ⟨ψi|E|ψj⟩=CEδij+εij, (16)

with a constant (only depending on ). Then forms a AQECC.

###### Proof.

Let . We find Eq. (15) to be true with and

 Bij=∑k≠l⟨ψl|EiEj|ψk⟩|ψl⟩⟨ψk|+∑k(⟨ψk|EiEj|ψk⟩−⟨ψ1|EiEj|ψ1⟩)|ψk⟩⟨ψk|. (17)

We now note two facts: (1) the Bures metric is upper bounded by the trace norm; and (2) for every two channels (see Lemma 23 of [[33]]). Then

 d(N+B,N)≤2k∥B∥1/21≤2k⎛⎝2d∑i,j∥Bij∥1⎞⎠1/2≤2d+kmaxi,j∥Bij∥1/21. (18)

From Eq. (16) we can bound the square of the latter as follows

 ∥Bij∥1≤2k∑k≠lεkl+2k∑k=1εkk≤22kε. (19)

## Appendix B Codes from Translation-Invariance

Here we give a proof of Theorem 1.

### Diagonal Elements are Close

We say a state on a finite dimensional lattice has correlation length if

 maxX,Ztr(ρX⊗Z)−tr(ρX)tr(ρZ)∥X∥∥Z∥≤exp(−dist(X,Z)/ξ), (20)

with the maximimization over all Hermitian matrices .

The next lemma, due to Anshu [[34]], gives a large deviation principle for the measurement of the energy, according to a local Hamiltonian , on a state with a finite correlation length.

###### Lemma 6.

[Theorem 1.1 of [[34]]] Let be a quantum state with correlation length and be the average energy of . Let be the projection onto the eigenspace of with eigenvalues .

For it holds that

 tr(ρΠ≥⟨H⟩ρ+Na)≤e2Dkξe−c2(Na2ξ)1D+1Dξ. (21)

for an universal constants .

Given a Hamiltonian with spectral decomposition , let be the the set of eigenvalues close to :

 SE:={Ek:Ek∈[E−√N,E+√N]}. (22)

Define the microcanonical state of energy as

 τE:=1|SE|∑k:Ek∈SE|Ek⟩⟨Ek|. (23)

We note that in one dimension the correlation length is a function of mean energy only, and it is a constant independent of system size when is a contant as well. The choice for the energy window is arbitrary. All we need is that the associated microcanonical ensemble has finite correlation length, which is true as long as it is subextensive and larger than [[22]].

The following is a quantitative version of the main result of [[23]], which established a weak version of ETH (only concerned with diagonal elements and only applying to most eigenstates). Using the large deviation principle of Lemma 6 we can give a finite version of it, with error bounds (in contrast, the result of [[23]] concerns asymptotics). It is here that the assumption of having translation-invariant eigenstates is used.

###### Proposition 7.

There is a constant such that the following holds. Let be a 1D local Hamiltonian on qubits and be an observable acting non-trivially only on a connected region of length . Then for any ,

 Missing or unrecognized delimiter for \right (24)

for a universal constant , with the correlation length of the microcanonical state of energy .

###### Proof.

For a local observable , define , with the average over the microcanonical state, i.e. . Define with denoting a translation by sites. Following [[23]], for any we have:

 Pr|Ek⟩∈SE(⟨Ek|O|Ek⟩≥⟨O⟩τE+δ) = Pr|Ek⟩∈SE(eλ⟨Ek|¯¯¯¯¯O′|Ek⟩≥eλδN) ≤ e−λδNE|Ek⟩∈SE(eλ⟨Ek|¯¯¯¯¯O′|Ek⟩) ≤ e−λδNE|Ek⟩∈SE(⟨Ek|eλ¯¯¯¯¯O′|Ek⟩) = Missing or unrecognized delimiter for \right

where the first inequality follows from Markov’s inequality and the second from the inequality: , valid for any Hermitian matrix and state .

The key step of the proof is the first line of the equation above where we used that . This relation holds because the eigenstates are translation invariant. Therefore, we can replace the expectation of a local observable by the expectation value of an extensive observable, which allows us to bring the well-developed machinery of large deviation bounds for spins systems, which we now employ.

Let be the spectral decomposition of . We have

 Missing or unrecognized delimiter for \right

We can upper bound the first term as follows:

 e−λδN∑i:oi≤δN/2eλoitr(τEPi)≤e−λδN/2. (25)

For the second term, we have

 ∑i:oi>δn/2eλoitr(τEPi) = 4N∑j=0⎛⎝∑i:δN/2+(j+1)≥oi>δN/2+jeλoitr(τEPi)⎞⎠ (26) ≤ 4N∑j=0eλ(δN/2+j+1)tr(τEP≥δN/2+j):=X.

with .

Using Lemma 6,

 X≤4N∑j=0eλ(δN/2+j+1)e2Nαξe−c(N(δ2+kN)2ξ)1/2. (27)

Choosing

 λ:=O(ξ3/2N−1/2) (28)

we find

 X≤e2Nαξexp(−c′δN1/2/ξ3/2). (29)

Eqs. (29) and (25) gives the statement. ∎

A direct consequence of the proposition above is the following:

###### Corollary 8.

Let be a 1D local Hamiltonian on qubits and be a connected region with less than sites. Then

 Pr|Ek⟩,|El⟩∈SE(∥tr∖Z(|Ek⟩⟨Ek|)−tr∖Z(|El⟩⟨El|)∥1≥δ)≤exp(−cδN1/2/ξ3/2) (30)

for a constant , with the correlation length of the microcanonical state of energy . We denote by the partial trace over the complement of .

###### Proof.

Proposition 7 gives that for a fixed Hermitian matrix with over sites:

 Pr|Ek⟩,|El⟩∈SE(tr(O(tr∖Z(|Ek⟩⟨Ek|)−tr∖Z(|El⟩⟨El|)))≥δ/2)≤exp(−cδN1/2/ξ3/2). (31)

Consider a -net over the set of all Hermitian matrices of unit operator norm. We have . Using the union bound

 Pr|Ek⟩,|El⟩∈SE(∥tr∖Z(|Ek⟩⟨Ek|)−tr∖Z(|El⟩⟨El|)∥1≥δ/2) (32) ≤ Pr|Ek⟩,|El⟩∈SE(maxO∈Nδtr(O(tr∖Z(|Ek⟩⟨Ek|)−tr∖Z(|El⟩⟨El|)))≥δ) ≤ Missing or unrecognized delimiter for \left ≤ |Nδ|exp(−cδN1/2/ξ3/2)≤exp(−c′δN1/2/ξ3/2),

for a constant sufficiently small.

### Off-Diagonal Elements are Small

The next lemma, from Arad, Kuwahara and Landau [[24]] (and attributed to Hastings), shows that for a local Hamiltonian, eigenstates well separated in energy are not connected by local operators. Its proof uses similar ideas to the proof of the Lieb-Robinson bound:

###### Lemma 9.

[Theorem 2.1 of [[24]]] Let and be projectors onto the subspaces of energies of that are and , respectively. For an operator , let be a subset of interactions terms such that , and let . Then

 ∥Π[ε′,∞]OΠ[0,ε]∥≤∥O∥.e−λ(ε′−ε−2R), (33)

with with the locality of and an upper bound on the number of local terms involving each particle.

The result has a straightforward corollary, which we state for future use:

###### Corollary 10.

Let and be two eigenstates of a local Hamiltonian and a connected region. Then

 ∥tr∖Z(|Ek2⟩⟨Ek1|)∥1≤e−λ(|Ek1−Ek2|−2|Z|) (34)

with the size of and a universal constant.

### Proof of Theorem 1

Theorem 1. Let be a 1D translation invariant local Hamiltonian and be such that the microcanonical state at energy has finite correlation length (independent of system size). Pick uniformly independently at random from , where is a basis of translation-invariant eigenstates of , and . Then with high probability they form an AQECC with and

 d=min(Ω(log(N)),minp≠q∈[L]|Eip−Eiq|−O(log(N))). (35)
###### Proof.

The theorem is a consequence of Corollary 8, Corollary 10 and Corollary 5. Indeed, the union bound and Corollary 8 show that with high probability, for every and with

 ∥tr∖Z(|Eip⟩⟨Eip|)−tr∖Z(|Eiq⟩⟨Eiq|)∥1≤N−25. (36)

Corollary 10, in turn, gives that for every and with

 ∥tr∖Z(|Eip⟩⟨Eiq|)∥1≤e−λ(|Eip−Eiq|−Ω(log(N))). (37)

These two conditions and Corollary 5 allows us to bound the error of the code as

 ε≤22k+d(e−λ(|Eip−Eiq|−Ω(log(N)))+N−25)1/2≤O(1/N)1/4, (38)

choosing the several constants appropriately. ∎

Observation 1: One drawback of the theorem is that if the minimum energy gap is less than , then the distance is zero. With high probability this will not be the case (since the energy window is , we pick elements uniformly at random, and the energy distribution of eigenvalues of a random model is normal [[22]]). However, if we want to make sure that this bad case will not happen, we can consider a variant of the theorem in which we consider energies and pick each state uniformly from for .

Observation 2: The only feature we used of being in one dimension is that the microcanonical states at finite energy density always have a finite correlation length. Therefore the theorem generalizes to higher dimensions for eigenstates with finite energy densities (albeit with a worse scale of the error of the code).

## Appendix C Parent Hamiltonians from local symmetries

Let be a 1D translation invariant classical Hamiltonian with periodic boundary conditions acting on qudits with local dimension . The statement that is classical means that there is some tensor product basis , with for each , such that we can express as

 HC=∑s∈BHC(s)|s⟩⟨s|

Let be a locally generated group of symmetries of . Since is translation invariant these generators are described by a set of -local invertible linear maps , with for each , and their translations. The action of on sites is expressed by . Since describes a symmetry of it follows that

 HC|s1...sj...sj+k...sN⟩=HC|s1...ri(sj...sj+k)...sN⟩

for all and for all . Furthermore, for each it follows that all of the states in the orbit also have the same energy with respect to . These orbits partition into subsets .

For each local symmetry generator and each site we can define a local projector

 Πrj=12∑sj,...,sj+k(|sj...sj+k⟩⟨sj...sj+k|+|r(sj...sj+k)⟩⟨r(sj...sj+k)| −|r(sj...sj+k)⟩⟨sj...sj+k|−|sj...sj+k⟩⟨r(sj...sj+k)|), (39)

and if is the Hamiltonian defined by the sum of all these projectors

 H=N∑j=1p∑r=1Πrij.

then the ground space of is -fold degenerate, and it is spanned by states which are uniform superpositions of the states in each orbit,

 |ψm⟩=1|Bm|∑|s⟩∈Bm|s⟩. (40)

## Appendix D Heisenberg spin chain AQECC

The Hamiltonian of the spin 1/2 quantum Heisenberg chain on sites with periodic boundary conditions is

 H=−12N∑j=1(σxjσxj+1+σyjσyj+1+σzjσzj+1). (41)

To study this system we use the -basis consisting of states , where and each labels the eigenvalue of . Let be the magnetization operator and define . Since the model (41) has an -fold degenerate ground space, with ground states that are labeld by magnetization values . We denote these ground states by , where is the magnetization and the system size is made explicit because we will also consider ground states of the Heisenberg chain on connected subsets of the sites. As is well known from the exact solution of (41) the state can be expressed in terms of the -basis states as

 |hNm⟩=1√(NN/2+m/2)∑s:M(s)=m|s⟩. (42)

As part of the verification of the error correcting conditions we will use expressions for the -body connected reduced density matrices of these ground states. By taking to always be asymptotically smaller than we can express the Schmidt decomposition of (42) along the cut between the spins where the error acts nontrivially and its compliment,

 |hNm⟩=d∑r=−d|hdr⟩|hN−dm−r)⟩⎡⎢⎣(dd/2+r/2)(N−dN/2−d/2+m/2−r/2)(NN/2+m/2)⎤⎥⎦1/2. (43)

It follows from (43) that the reduced density matrix on sites that (note that refers to the trace over the -neighboring sites) is given by

 ρd(m,N)=d∑r−d|hdr⟩⟨hdr|⎡⎢⎣(dd/2+r/2)(N−dN/2−d/2+m/2−r/2)(NN/2+m/2)⎤⎥⎦. (44)

### Proof of Proposition 2

###### Proof.

We separate the approximate error correction condition into two parts. First is the nonexistence of a -body operator that maps different codewords to each other. More precisely, this corresponds to the case in Eq. (2). Note that by construction whenever and are distinct codewords, since and is the error operator supported on a connected region of -sites, so it can change the magnetization by at most . Noting that a -local operator can change the magnetization of by at most , define and define the code space to be

 C=Span({|hNm⟩:m=−mmax,−mmax+I,…,mmax−I,mmax}). (45)

where is a multiple of to be chosen later. The second approximate error correction condition is the local indistinguishability of the -body reduced density matrices of the codewords. More precisely, this corresponds to the case in Eq. (2). Below we show that an arbitrary observable has approximately the same expectation value for all of the codewords by computing the trace distance between the local reduced density matrices