A construction of combinatorial NLTS

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 networks²⁶ 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) conjecture¹ 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 $H=∑i=1mhi$ [each $0⪯hi⪯I$ 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 $TrHψ≤εm+λmin(H)$⁠.

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 product³⁴ 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.

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.

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 “Z₂ symmetry.” It was shown in Ref. 12 that locally testable quantum Calderbank-Shor-Steane (CSS) codes² 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.

The follow-up work⁵ 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.

Our starting point is the NLETS theorem shown in Ref. 22. It is based on the observation that the CAT state $12|00…0〉+12|11…1〉$ 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 $M=00⟩⟨00+11⟩⟨11$⁠. 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 $M+δI$ (for $δ≈1n$⁠). 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–Spielman³² (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:

1. 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.

2. 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.

See Sec. IV for full details. We note that tensor networks have previously been combined with local (quantum) codes to obtain global properties.²⁹ 

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.

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.

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 $H∈F2m×n$ 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 $λ=max|λ2|,|λn|$⁠, where λ₂, 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 k₀ and distance Δ₀. 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 ith bit of x_v is the value on the edge with the ith smallest number. Formally,

We will abbreviate T(C, G) as T for convenience. Since there are d − k₀ independent checks in C, the number of independent checks in T is at most n(d − k₀). Thus, the rank k of T is $k≥nd2−n(d−k0)=nk0−d2$⁠.

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 $ve,ve′$ come in pairs (Fig. 1). We will often denote the joint Hilbert space $Hve⊗Hve′$ as $He$ and abbreviate $|0〉ve|0〉ve′$ as |0⟩_e and $|1〉ve|1〉ve′$ as |1⟩_e. Thus, $|0〉ve|0〉ve′+|1〉ve|1〉ve′$ 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 ith bit of c and e_i is the ith 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

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:

Suppose ɛ|E| local terms from H are removed (see Fig. 2). Since each local term corresponds to an edge, let E₁ be the remaining edges and let $H1=∑e∈E1he$ be the Hamiltonian that remains. We will show that any state |ψ⟩ that is a ground state of H₁ has a large circuit complexity if ɛ is a sufficiently small constant.

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, …, 2ɛ|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:

Consider the following basis within $G$⁠:

where each |Ψ_b⟩ is normalized as $Ψb|Ψb=1$⁠. 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 $Pv⊗r∈R|br〉r=0$ 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 $δI$ 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 ${|Φb〉}b∈{0,1}|R|$ are mutually orthogonal. The following claim is analogous to Claim 5:

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.

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′.

Recall that |Φ_b⟩ is a superposition over the edge assignments to E₁ ∪ R. We now show that these edge assignments form distant clusters.

Let

be the projector onto the edge assignments in B_i. The following claim holds: