The NLTS (No Low-Energy Trivial State) conjecture [M. H. Freedman and M. B. Hastings, Quantum Inf. Comput. **14**, 144 (2014)] posits that there exist families of Hamiltonians with all low energy states of high complexity (with complexity measured by the quantum circuit depth preparing the state). Here, we prove a weaker version called the combinatorial no low error trivial states (NLETS), where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms. This generalizes the prior NLETS results [L. Eldar and A. W. Harrow, in *2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)* (IEEE, 2017), pp. 427–438] and [Nirkhe *et al.*, in *45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), Leibniz International Proceedings in Informatics (LIPIcs)*, edited by Chatzigiannakis *et al.* (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018), Vol. **107**, pp. 1–11]. Our construction is obtained by combining tensor networks with expander codes [M. Sipser and D. Spielman, IEEE Trans. Inf. Theory **42**, 1710 (1996)]. The Hamiltonian is the parent Hamiltonian of a perturbed tensor network, inspired by the “uncle Hamiltonian” of Fernández-González *et al.* [Commun. Math. Phys. **333**, 299 (2015)]. Thus, we deviate from the quantum Calderbank-Shor-Steane (CSS) code Hamiltonians considered in most prior works.

## I. INTRODUCTION

The approximation of the ground energy of a local Hamiltonian continues to be a leading goal of quantum complexity theory and quantum many-body physics. While a generic, accurate, and efficient approximation method is unlikely, due to the seminal result of Kitaev,^{3} physically motivated *Ansätze*, such as tensor networks^{26} and low depth quantum circuits,^{6,18,30} continue to explore the low energy spectrum of many interesting Hamiltonians.

A fundamental question on the power of low-depth quantum circuits is the no low-energy trivial state (NLTS) conjecture,^{14} which posits the existence of local Hamiltonians with all low energy states having high quantum circuit complexity. This is a necessary consequence of the quantum probabilistically checkable proofs (PCP) conjecture^{1} under the reasonable assumption that QMA ≠ NP. We refer to the reader to existing works^{1,7,8,12,22} for a detailed discussion on the NLTS conjecture and its close connection with quantum error correction, robustness of entanglement, and the power of variational quantum circuits.

To formally define the NLTS conjecture, we introduce a *n*-qubit local Hamiltonian *H* as a sum of local terms $H=\u2211i=1mhi$ [each $0\u2aafhi\u2aafI$ is supported on *O*(1) qubits and each qubit participates in *O*(1) local terms] with *m* = Θ(*n*). The ground states of *H* are the eigenstates with eigenvalue *λ*_{min}(*H*). An *ɛ*-energy state *ψ* satisfies $TrH\psi \u2264\epsilon m+\lambda min(H)$.

*(NLTS*^{14} *)*. There exists a fixed constant *ɛ* > 0 and an explicit family of *O*(1)-local Hamiltonians ${H(n)}n=1\u221e$ such that for any family of *ɛ*-energy states {*ψ*_{n}}, the circuit complexity CC(*ψ*_{n}) grows faster than any constant.

Here, CC(*ψ*) is the *quantum circuit depth*, the depth of the smallest quantum circuit that prepares *ψ*. An interesting property of any (potential) NLTS Hamiltonian is that it must live on an expanding interaction graph, ruling out all the finite-dimensional lattice Hamiltonians that have been very well studied in quantum many-body physics. The same holds for (potential) Hamiltonians that may witness the quantum PCP conjecture.^{1}

A weaker version of this conjecture is known, called the NLETS theorem. A local Hamiltonian *H* (as defined above) is frustration-free if *λ*_{min}(*H*) = 0. A state *ψ* is called *ɛ**-error* if there exists a set *S* of qubits of size at least (1 − *ɛ*)*n* such that *ψ*_{S} = *ϕ*_{S}, where *ϕ* is some ground state of *H* and the subscript *S* means that we take a partial trace over the qubits in [*n*]\*S*.

Note that the *ɛ*-energy states include the set of *O*(*ɛ*)-error states, but the reverse direction is not true. The NLETS theorem was first proved by Ref. 12, by considering the hypergraph product^{34} of two Tanner codes on expander graphs.^{32} In the follow-up work,^{22} the authors constructed an NLETS Hamiltonian that, in fact, lived on a one-dimensional lattice. In the recent work,^{7} super-constant circuit lower bounds were shown for *o*(1)-energy states [such as $O(1logn)$-energy] of all quantum code Hamiltonians that have near-linear rank or near-linear distance. Interestingly, such lower bounds are again possible with the two-dimensional, punctured toric code, showing that expansion of the underlying interaction graph is not needed for circuit lower bounds on “almost constant” energy states.

The authors of Refs. 12 and 22 identified the intermediate question of *combinatorial NLTS* (cNLTS), which aims at finding frustration-free Hamiltonians with super-constant circuit lower bounds for states *ψ* that satisfy at least 1 − *ɛ* fraction of local terms. The main interest in this question stems from the fact that any (potential) cNLTS Hamiltonian must also live on an expanding interaction hypergraph, hence exhibiting the geometric features of an NLTS Hamiltonian. Here, we provide the first construction of a cNLTS Hamiltonian.

*There exists a fixed constant*

*ɛ*> 0

*and an explicit family of*

*O*(1)

*-local frustration-free Hamiltonians*${H(n)}n=1\u221e$

*, where*$H(n)=\u2211i=1mhi(n)$

*acts on*

*n*

*particles and consists of*

*m*= Θ(

*n*)

*local terms, such that for any family of states*{

*ψ*

_{n}}

*satisfying*

*the circuit complexity*CC(

*ψ*

_{n})

*is*Θ(log

*n*)

*.*

The set of states that satisfy 1 − *ɛ* fraction of local terms also include *O*(*ɛ*)-error states. Thus, the above family of Hamiltonians is also NLETS.

### A. Other related results

In Ref. 11, thermal states of certain quantum codes were shown to have circuit lower bounds. The authors of Ref. 8 showed circuit lower bounds for quantum states with a “*Z*_{2} symmetry.” It was shown in Ref. 12 that locally testable quantum Calderbank-Shor-Steane (CSS) codes^{2} of linear distance are NLTS. Such codes are not known to exist, with the best distance thus far being $n$.^{19} However, the dramatic recent progress in quantum codes^{9,15,27,28} opens up the exciting possibility that such codes may exist.

#### 1. New results

The follow-up work^{5} supersedes the main result here, as it shows the NLTS property for good quantum codes.^{19,28} It uses a different Hamiltonian family, but the underlying connection is that it proves a quantum analog of Theorem 9. We believe that the Hamiltonian family in this work are also NLTS when the parameter *δ* is set to a constant.

### B. Outline of the construction

Our starting point is the NLETS theorem shown in Ref. 22. It is based on the observation that the CAT state $12|00\u20260\u3009+12|11\u20261\u3009$ is close to the unique ground state of Kitaev’s clock Hamiltonian. This clock Hamiltonian is obtained from the circuit preparing the CAT state and then padding with identity gates. We observe that yet another Hamiltonian can be constructed by viewing the CAT state as a Matrix Product State (MPS). The MPS representation of the CAT state is obtained by starting with $n2$ Einstein−Podolsky−Rosen (EPR) pairs,

and then projecting qubits *i*, *i* + 1, for even *i*, with the projector $M=00\u27e9\u27e800+11\u27e9\u27e811$. Most MPSs are the unique ground states of a parent Hamiltonian (such MPSs are called injective). However, the CAT state MPS clearly does not have this privilege, since *M* is not an invertible map. However, inspired by Ref. 13, we can perturb *M* to consider a state obtained by mapping qubits *i*, *i* + 1 with $M+\delta I$ (for $\delta \u22481n$). This is an invertible map, which makes the resulting MPS injective. Using the corresponding parent Hamiltonian, we obtain another construction of the NLETS Hamiltonian.^{23}

Since any cNLTS Hamiltonian must be on an expanding interaction graph, an approach to construct the desired Hamiltonian is to write down a tensor network for the CAT state on an expanding graph, perturb the tensors, and then take the parent Hamiltonian. Unfortunately, this argument seems not to work, since the tensor network for the CAT state is extremely brittle. If we remove one EPR pair and allow arbitrary inputs to the tensors acting on this EPR pair, we can produce the states |00…0⟩ or |11…1⟩. This brittleness reflects in the nearby parent Hamiltonian, and there are product states that violate just one local term. Our second observation is that the CAT state tensor network can be viewed as a *repetition Tanner code* on an expanding graph. Thus, we can generalize the tensor network and look at Tanner codes defined on expander graphs, as proposed by Sipser–Spielman^{32} (Sec. III). The tensor network state is now a uniform superposition over all the codewords of this Tanner code. With (1) linear distance and (2) linear rank (and with a suitable choice of parameters), such a code protects us from two sources of brittleness:

Removing an

*ɛ*fraction of EPR pairs (analogously, local terms of the Hamiltonian) weakens the expansion properties of the underlying graph. Linear distance ensures that the codewords, while no longer far away from each other, are partitioned into distant groups for a small constant*ɛ*.Removing an

*ɛ*fraction of EPR pairs (analogously, local terms of the Hamiltonian) can drastically reduce the number of strings appearing in the superposition. Linear rank ensures that the number of strings is large enough if*ɛ*is a small constant.

### C. Local systems and non-isotropic Gauß’s laws

Tanner codes can be understood in terms of homology with local systems,^{33} where differentials take values in the space of local checks.^{21} A trivial example is the toric code, where the differential at each vertex detects violations of $Z2$-flux conservation or, in other words, violations of a local parity-check code (this is, of course, nothing but the usual simplicial $Z2$-homology). The family of Hamiltonians that we construct can be understood in terms of differentials defined from more complicated local codes. The Hamiltonians ensure that these differentials are zero for ground states, which means that they enforce a non-isotropic Gauß’s law that takes the directionality of the incoming fluxes into account. Together with the expansion of the underlying graph, this leads to the cNLTS property.

### D. Tensor networks and quantum complexity

Kitaev’s clock construction is a powerful method to map quantum computations to the ground states of local Hamiltonians. It turns out that the tensor networks provide a similar mapping. As shown in Ref. 31, any measurement-based quantum computation can be mapped onto a tensor network. One could thus imagine a form of circuit-to-Hamiltonian mapping different from Kitaev’s: perturb the above tensor network and consider its parent Hamiltonian. A standard objection to this approach is that the mapping also works for post-selected quantum circuits, which is far more powerful than QMA. However, this objection is not expected to apply to our case, as injective tensors cannot post-select on events of very small probability. We leave an understanding of the promise gap of this mapping for future work.

## II. TANNER CODE

In this section, we review a construction of linear codes from regular graphs called Tanner codes. A (classical) linear code *C* of length *n* and rank *k* is a subspace of dimension *k* of the vector space $F2n={0,1}n$. A linear code *C* can be defined by specifying a parity check matrix $H\u2208F2m\xd7n$ such that ker *H* = *C*. We call the *m* rows of *H* checks of the code *C*.

Consider a regular graph *G* = (*V*, *E*) with degree *d* and *n* = |*V*| vertices. For *S*, *S*′ ⊂ *V*, we denote the number of edges between *S* and *S*′ as *E*(*S*, *S*′) (we count an edge {*u*, *v*} twice if *u*, *v* ∈ *S* ∩ *S*′). Let $\lambda =max|\lambda 2|,|\lambda n|$, where *λ*_{2}, *λ*_{n} are the second largest and the smallest eigenvalues of the adjacency matrix, respectively.

A Tanner code *T*(*C*, *G*) ⊂ {0, 1}^{|E|} is defined using the graph *G* and a classical linear code *C* ⊂ {0, 1}^{d} of rank *k*_{0} and distance Δ_{0}. We imagine bits on edges and checks on the vertices. Let the edges be numbered using the integers {1, 2, …|*E*|} in some arbitrary manner. Given a string *x* ∈ {0, 1}^{|E|} and a vertex *v*, let *x*_{v} ∈ {0, 1}^{d} be the restriction of *x* to the edges incident to *v*, where the *i*th bit of *x*_{v} is the value on the edge with the *i*th smallest number. Formally,

We will abbreviate *T*(*C*, *G*) as *T* for convenience. Since there are *d* − *k*_{0} independent checks in *C*, the number of independent checks in *T* is at most *n*(*d* − *k*_{0}). Thus, the rank *k* of *T* is $k\u2265nd2\u2212n(d\u2212k0)=nk0\u2212d2$.

*Suppose* Δ_{0} ≥ 2*λ**. The distance of* *T* *is lower bounded by* $n\Delta 024d=|E|\Delta 022d2$*.*

*x*∈

*T*be the non-zero code-word of the smallest Hamming weight, and let

*E*

_{x}be the edges where

*x*takes value 1. Let

*S*be the set of all vertices on which at least one edge in

*E*

_{x}is incident. Since the distance of

*C*is Δ

_{0}, at least Δ

_{0}edges from any vertex in

*S*stay within

*S*(and those edges belong to

*E*

_{x}). Thus, |

*E*(

*S*,

*S*)| ≥ |

*S*|Δ

_{0}. However, the expander mixing lemma

^{35}(Lemma 4.15) ensures that

*S*is associated with at least Δ

_{0}edges in

*E*

_{x}and every edge in

*E*

_{x}is associated with two vertices in

*S*), the lemma concludes.□

## III. INJECTIVE TENSOR NETWORK FROM THE CODE *T*(*C*, *G*)

Let |EPR⟩ = |00⟩ + |11⟩ be an unnormalized EPR pair. Given *G*, we consider a Hilbert space consisting of *nd* qubits, with *d* qubits for each vertex *v* ∈ *V*. For a *v* ∈ *V*, we identify each qubit with a unique edge incident on *v* and label the qubit as *v*_{e}. As a result, given an edge *e* = (*v*, *v*′), qubits $ve,ve\u2032$ come in pairs (Fig. 1). We will often denote the joint Hilbert space $Hve\u2297Hve\u2032$ as $He$ and abbreviate $|0\u3009ve|0\u3009ve\u2032$ as |0⟩_{e} and $|1\u3009ve|1\u3009ve\u2032$ as |1⟩_{e}. Thus, $|0\u3009ve|0\u3009ve\u2032+|1\u3009ve|1\u3009ve\u2032$ will be referred to as |EPR⟩_{e}. Define the unnormalized state

For each vertex *v*, define the projector that only accepts the codewords of the local code at *v*,

where *c*_{i} is the *i*th bit of *c* and *e*_{i} is the *i*th edge incident on *v* (in the ascending numbering specified on the edges). The tensor network state is obtained by projecting the *d* qubits on each vertex using these projectors,

Note that the normalization follows since there are 2^{k} codewords in *T*. We can think of the string *x* as an *edge assignment* and |Φ⟩ as a uniform superposition over edge assignments from *T*. Now, we can make the tensor network “injective” by defining

_{0}| ⊗

_{v∈S}

*P*

_{v}|Θ

_{0}⟩. This is essentially the number of codewords when parity checks only act on the vertices in

*S*. If we were to include the parity checks in

*V*\

*S*as well, we would obtain the original code. Since there are at most (

*d*−

*k*

_{0})(

*n*− |

*S*|) independent checks, the following inequality holds:

The nice property of |Ψ⟩ is that it is the unique ground state of a local Hamiltonian. For *e* = (*v*, *v*′), define

where “span” means that *h*_{e} is the projector onto the image of *g*_{e}. Since *g*_{e}|Ψ⟩ = 0, we have *h*_{e}|Ψ⟩ = 0. Let

Then, |Ψ⟩ is a ground state of *H* with ground energy 0. In fact, we have the following claim, which is well known about injective tensor networks:

|Ψ⟩ is the unique ground state of *H*.

Suppose |Ψ′⟩ is a ground state of *H*. Then, it is also a ground state of *∑*_{e}*g*_{e}. Write |Ψ′⟩ = ⊗_{v∈V}*Q*_{v}|Θ′⟩ for a (possibly unnormalized) quantum state |Θ′⟩. This is possible since ⊗_{v∈V}*Q*_{v} is invertible. Observe that |Θ′⟩ is a ground state of $\u2211e(I\u2212EPR\u27e9\u27e8EPRe)$. This is possible only if |Θ′⟩ = |Θ_{0}⟩, which proves the claim.□

## IV. THE HAMILTONIAN *H* IS cNLTS

Suppose *ɛ*|*E*| local terms from *H* are removed (see Fig. 2). Since each local term corresponds to an edge, let *E*_{1} be the remaining edges and let $H1=\u2211e\u2208E1he$ be the Hamiltonian that remains. We will show that any state |*ψ*⟩ that is a ground state of *H*_{1} has a large circuit complexity if *ɛ* is a sufficiently small constant.

### A. Structure of the ground space of *H*_{1}

Let *W* ⊂ *V* be the set vertices on which the removed edges were incident. Among the *d*|*W*| qubits in these vertices, some qubits were associated with the removed edges. We will call these qubits “residual” and denote their set by *R* = {1, 2, …, 2*ɛ*|*E*|}. We are free to choose any “assignment” |0⟩, |1⟩ to the residual qubits. Since we have been thinking of assignments as occurring on the edges, we will sometimes refer to *R* as a set of edges (Fig. 2). Thus, we will continue using the terminology of “edge assignment.” Note the following cardinality bounds:

Let

We have the following claim, which is analogous to Claim 6:

*H*

_{1}is

*ω*⟩ = ⊗

_{v∈V}

*Q*

_{v}|

*τ*⟩ be a ground state of

*H*

_{1}for some |

*τ*⟩, which is possible since ⊗

_{v∈V}

*Q*

_{v}is invertible. Note that

*e*∈

*E*

_{1}, $I\u2212EPR\u27e9\u27e8EPRe|\tau \u3009=0$. This shows that |

*τ*⟩ belongs to the space spanned by the vectors ${|\Theta 1\u3009\u2297r\u2208R|br\u3009r,\u2009such\u2009thatb\u2254b1,\u2026b|R|\u2208{0,1}|R|}$, as there are no constraints on the residual qubits. This completes the proof.□

Consider the following basis within $G$:

where each |Ψ_{b}⟩ is normalized as $\Psi b|\Psi b=1$. Note that this is an orthonormal basis, as the residual qubits are fixed according to *b* (the operators *Q*_{v} do not change any computational basis state). Along the lines of Claim 5, we would expect that this state is close to the following state (ignoring normalization):

However, we have to be careful: if $Pv\u2297r\u2208R|br\u3009r=0$ for any *v* ∈ *W*,^{24} the above state is 0, whereas |Ψ_{b}⟩ is non-zero for all *b*. With this in mind, we let *W*_{b} ⊂ *W* denote all the vertices with which *b* is consistent [Fig. 3 (left)]. We observe that

since *Q*_{v} acts as $\delta I$ at any *v* ∈ *W*\*W*_{b} (above *Z*_{b} is a normalization constant) and define

where *k*_{b} will be determined shortly. Note that the states ${|\Phi b\u3009}b\u2208{0,1}|R|$ are mutually orthogonal. The following claim is analogous to Claim 5:

*k*

_{b}. Note that |Φ

_{b}⟩ is simply a superposition over edge assignments that satisfy the Tanner code with checks on (

*V*\

*W*) ∪

*W*

_{b}, where we condition the edges in

*R*to have fixed edge assignments according to

*b*. Conditioning the edge assignments in

*R*to be

*b*leads to a set of parity check over edges in

*E*

_{1}(some of these checks may also impose a parity of 1 on the edge assignments in

*E*

_{1}). Each vertex in

*V*\

*W*contributes to

*d*−

*k*

_{0}checks. Each vertex in

*W*

_{b}contributes to anywhere between 0 to

*d*−

*k*

_{0}independent checks. If we define

*c*

_{v}as the number of independent checks due to

*v*∈

*W*

_{b}that involve edges in

*E*

_{1}, we have

*c*

_{v}≤

*d*−

*k*

_{0}and

*W*

_{b}⊆

*W*, a lower bound on

*k*

_{b}is

*S*(with the condition that edges in

*R*are assigned

*b*). If we were to add parity checks on remaining vertices in (

*V*\

*W*)\

*S*and

*W*

_{b}\

*S*, we would obtain the $2kb$ codewords accounted for in Eq. (3). The number of such linearly independent parity checks that are added is at most $(d\u2212k0)(|V\W|\u2212|S\W|)+\u2211v\u2208Wb\Scv$. This gives us the upper bound

*S*⊂ (

*V*\

*W*) ∪

*W*

_{b}. Thus,

*V*\

*W*) ∪

*W*

_{b}| ≤

*n*.□

Thus, if *δ* is small enough, it suffices to understand the properties of the space,

From Claim 8, each |Φ_{b}⟩ is a superposition over $2kb$ edge assignments. Moreover, each such edge assignment satisfies all the checks in *V*′ ≔ *V*\*W*. Thus, let us understand the properties of edge assignments that satisfy such checks. Define a new Tanner code *T*′ ≔ *T*(*C*, *G*′), where *G*′ = (*V*′, *E*′ ∪ *F*), *E*′ is the set of edges in the subgraph induced by *V*′, and *F* is the set of edges that connected *V*′ with *W*. We will think of each edge in *F* as “free,” being incident to just one vertex in *V*′ (Fig. 4). Theorem 9 shows that the Hamming distance between the codewords of *T*′ is either small or large.

### B. Properties of the Tanner code *T*′

Let *I* ⊂ *V*′ be the set of vertices on which an edge in *F* is incident (Fig. 4). From Eq. (2), note the following bounds:

Since *G*′ may no longer be an expander, we do not have any guarantees on the distance of *T*′. However, we can show some structure in the codewords of *T*′.

*Suppose* $\epsilon \u226416d$ *and* Δ_{0} ≥ 4*λ**. The Hamming distance between the codewords of* *T*′ *is either* $\u22648d2\epsilon |E\u2032\u222aF|$ *or* $\u2265\Delta 024d2\u22c5|E\u2032\u222aF|$*.*

*T*′ is a linear code, we show that the Hamming weight of a non-zero codeword

*x*is either $\u22648d2\epsilon |E\u2032\u222aF|$ or $\u2265\Delta 028d2\u22c5|E\u2032\u222aF|$. Let

*J*

_{x}⊂

*E*′ ∪

*F*be the set of edges on which

*x*assigns 1. If |

*J*

_{x}| ≤ 8

*d*

^{2}

*ɛ*|

*E*′ ∪

*F*|, then we are done, or else

*J*

_{x}| must be significantly larger. Let

*S*⊂

*V*′ be the vertices on which the edges in

*J*

_{x}are incident. We can apply the expander mixing lemma to the

*original graph*

*G*and obtain

*E*(

*S*,

*S*)| as follows: Every vertex in

*S*\

*I*has degree at least Δ

_{0}and each edge incident to such a vertex belongs to

*E*(

*S*,

*S*) (since such an edge is not in

*F*, both its endpoint are in

*S*). Edges incident to vertices in

*S*∩

*I*may not belong to

*E*(

*S*,

*S*), and thus, we will not count them (see Fig. 4). Overall, we have

*J*

_{x}| ≥ 4

*d*

^{2}

*ɛ*|

*E*|, we can naively bound $|S|\u2265|Jx|d\u22654d\epsilon |E|$. Thus,

*S*is associated with at least Δ

_{0}edges in

*J*

_{x}and every edge in

*J*

_{x}is associated with at most 2 vertices in

*S*. This completes the proof.□

### C. Structure of the states |Φ_{b}⟩

Recall that |Φ_{b}⟩ is a superposition over the edge assignments to *E*_{1} ∪ *R*. We now show that these edge assignments form distant clusters.

*Suppose* $\epsilon \u2264\Delta 02300d4$ *and* Δ_{0} ≥ 4*λ**. There are disjoint sets* $B1,B2,\u2026\u2282{0,1}|E1\u222aR|$ *such that for any* *x*, *y* ∈ *B*_{i}*, the Hamming distance between* *x* *and* *y* *is* $\u226410d2\epsilon |E1\u222aR|$, *and for any* *x* ∈ *B*_{i} *and* *y* ∈ *B*_{j} *with* *i* ≠ *j**, the Hamming distance between* *x*, *y* *is* $\u2265\Delta 0210d2\u22c5|E1\u222aR|$*. Furthermore, the states* ${|\Phi b\u3009}b\u2208{0,1}|R|$ *are uniform superpositions over some edge assignments in* ∪_{i}*B*_{i}*.*

*x*,

*y*appearing in |Φ

_{b}⟩, when restricted to the edges in

*G*′ (which we denote

*x*

_{G′},

*y*

_{G′}), belong to

*T*′. The edge assignment

*x*is obtained from

*x*

_{G′}by appending assignments to the edges in (

*E*

_{1}∪

*R*)\(

*E*′ ∪

*F*). There are at most $d|W|\u2264Eq.(2)2d\epsilon |E|$ such edges. Thus, invoking Theorem 9, the Hamming distance between

*x*,

*y*is either at most

*x*,

*z*is $\u226420d2\epsilon |E1\u222aR|<\Delta 0210d2\u22c5|E1\u222aR|$, which, in turn, requires that the Hamming distance between

*x*,

*z*is $\u226410d2\epsilon |E1\u222aR|$. This forces $x,z\u2208R$. The sets

*B*

_{1},

*B*

_{2}, … are the equivalence classes formed by this relation, which completes the proof.□

Let

be the projector onto the edge assignments in *B*_{i}. The following claim holds:

Let $\delta 2\u22642\u2212d10000n$, $\epsilon \u2264120000d2$, and *k*_{0} ≥ 0.55*d*. For any *i* and any *b*, *b*′ ∈ {0, 1}^{|R|} (with *b* ≠ *b*′), $\u3008\Psi b|\Pi Bi|\Psi b\u3009\u2264150$ and $\u3008\Psi b|\Pi Bi|\Psi b\u2032\u3009=0$.

_{b}⟩ is a uniform superposition over $2kb$ edge assignments and the size of each

*B*

_{i}is at most $|E1\u222aR|10d2\epsilon |E1\u222aR|\u226422d10\epsilon |E1\u222aR|\u226428d\epsilon |E|$, we have

*E*| is larger than some constant. Now, Claim 8 ensures that

_{b}⟩ is a superposition over edge assignments with the fixed

*b*on the edges in

*R*. That is, $|\Psi b\u3009=|\cdots \u3009\u2297r\u2208R|br\u3009r$. Thus, $\Pi Bi$, being a projector onto computational basis states, satisfies $\Pi Bi|\Psi b\u3009=|\cdots \u3009\u2297r\u2208R|br\u3009r$. Similarly, $|\Psi b\u2032\u3009=|\cdots \u2032\u3009\u2297r\u2208R|br\u2032\u3009r$. Since

*b*≠

*b*′, the claim follows.□

### D. Circuit lower bound

We will assume that $\epsilon \u2264120000d2,\Delta 02300d4$; Δ_{0} ≥ 4*λ*; *k*_{0} ≥ 0.55*d*, and $\delta 2\u22642\u2212d10000n$. Note that these conditions can be met with constant *k*_{0}, Δ_{0}, *d*, which ensures that *ɛ* is a constant (see Sec. V). Our main theorem is below, which proves that *H* is cNLTS. The argument is directly inspired by the quantum circuit lower bound argument in Ref. 12, based on the partitioning of quantum codewords. However, we consider a simpler argument based on the tight polynomial approximations to the AND function,^{4,10,16} inspired by Ref. 17.

*Let*|

*ρ*⟩ =

*U*|0⟩

^{⊗m}

*on*

*m*≥

*nd*

*qubits, where*

*U*

*is a depth*

*t*

*quantum circuit, such that*$12\Vert \Gamma \u27e9\u27e8\Gamma \u2212\rho \u27e9\u27e8\rho \Vert 1\u22640.1$

*for some ground state*

^{25}|Γ⟩

*of*

*H*

_{1}

*. It holds that*

*m*≤ 2

^{t}

*nd*without loss of generality as

*H*

_{1}acts on

*nd*qubits [see Ref. 7 (Sec. 2.3) for a justification based on the light cone argument]. We can expand

*μ*

_{b}⟩ act on

*m*−

*nd*qubits outside

*V*. Using Claim 11, we find that for any

*i*,

*E*

_{1}∪

*R*appearing in |Γ⟩ belong to some

*B*

_{i}. In other words, $\u2211i\u3008\Gamma |\Pi Bi|\Gamma \u3009=1$. Thus, we can find two disjoint sets of indices

*M*,

*M*′ such that

*B*

_{M}= ∪

_{i∈M}

*B*

_{i},

*B*

_{M′}= ∪

_{i∈M′}

*B*

_{i}, $\Pi M=\u2211i\u2208M\Pi Bi$, and $\Pi M\u2032=\u2211i\u2208M\u2032\Pi Bi$. From Theorem 10, the Hamming distance between the sets

*B*

_{M}and

*B*

_{M′}is $\u2265\Delta 0210d2\u22c5|E1\u222aR|\u2265\Delta 0220d2\u22c5nd$. On the other hand, we just established that $\u3008\Gamma |\Pi M|\Gamma \u3009\u226513$ and $\u3008\Gamma |\Pi M\u2032|\Gamma \u3009\u226513$. From Refs. 4, 10, and 16, there exists a

*f*· 2

^{t}-local operator

*L*such that

_{M}

*L*Π

_{M′}= 0, we have

## V. EXPLICIT CONSTRUCTION OF THE TANNER CODE *T*(*C*, *G*)

In this section, we give an explicit construction of a suitable family of Tanner codes ${T(C,Gi)}i=1\u221e$ from which the family of cNLTS-Hamiltonians ${H(i)}i=1\u221e$ of Theorem 3 is obtained.

For the graphs *G*_{i} underlying the Tanner codes, we employ a construction of spectral Cayley-expanders due to Lubotzky, Phillips, and Sarnak.

(Ref. 20). *Assume that* *p* *and* *q* *are distinct, odd primes such that* $q>2p$ *and* *q* *is a square modulo* *p**. Then, there exists a symmetric generating set* Γ *of* $PSL2(Fq)$ *such that the Cayley graph* $Cay(PSL2(Fq),\Gamma )$ *is a non-bipartite,* *p* + 1*-regular expander graph with* $\lambda <2p$*.*

By fixing a suitable prime *p*, we obtain a family of regular graphs *G*_{i} of degree *d* = *p* + 1 and order *q*(*q*^{2} − 1)/2 with spectral bound $\lambda <2d\u22121$. We mention in passing that by a result due to Alon and Boppana, this is the best possible bound that any family of regular graphs can achieve and such families are called *Ramanujan graphs*.

Furthermore, we require a linear, binary code *C*. More specifically, for the construction of the cNLTS-Hamiltonians to go through, we require the existence of a classical linear binary code *C* of block size *d* encoding at least *k*_{0} ≥ 0.55*d* bits with distance Δ_{0} ≥ 4*λ*. As the degree *d* of the graphs is constant, it suffices to show that a suitable code *C* exist, as a brute-force search has time-complexity bounded by a constant *O*(1). The existence of a suitable code is guaranteed by the Gilbert–Varshamov bound [see, e.g., Ref. 36 (Chap. 5) for a proof].

(Gilbert–Varshamov). *Let* 0 ≤ *μ* ≤ 0.5 *and let* 0 ≤ *R*_{0} ≤ 1 − *H*_{2}(*μ*)*, then there exists a binary, linear code of block size* *d**, rank* *k*_{0} = *R*_{0}*d*, *and distance* Δ_{0} = *μd**.*

We can now give an explicit construction of a suitable code family ${T(C,Gi)}i=1\u221e$. By Theorem 14, there exists a code *C* encoding *k*_{0} ≥ 0.55*d* bits when *μ* ≤ 0.09. We further require that *C* has distance Δ_{0} = *μd* ≥ 4*λ*. This is the case via Theorem 13 by choosing, for example, *μ* = 0.09 and prime *p* = 7901.

## ACKNOWLEDGMENTS

We thank Chinmay Nirkhe for helpful discussions. We also thank Robbie King and the anonymous referee for carefully reading our manuscript. A.A. acknowledges support through the NSF award QCIS-FF: Quantum Computing and Information Science Faculty Fellow at Harvard University (Grant No. NSF 2013303). N.P.B. acknowledges support through the EPSRC Prosperity Partnership in Quantum Software for Simulation and Modeling (Grant No. EP/S005021/1).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Anurag Anshu**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Nikolas P. Breuckmann**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

One may note that the local terms of the resulting parent Hamiltonian do not have norm $\u22641$ and may scale as $1\delta $. However, we can fix this by the standard trick of replacing each local term by the projector onto its image. The frustration-free property of the parent Hamiltonian ensures that the ground state is unchanged.

This would happen iff the edge assignment *b* were simply inconsistent with the codewords allowed at *v*. For example, *v* may connect with *d* − 1 edges in *R* and *b* may assign Δ_{0} − 2 number of ‘1’s on those edges.

While $\Gamma $ may act on more qubits than included in *V*, we say that $\Gamma $ is a ground state of *H*_{1} if its reduced state on the qubits in *V* belongs to the ground space of *H*_{1}.