Tensor network non-zero testing

# Tensor network non-zero testing

## Abstract

Tensor networks are a central tool in condensed matter physics. In this paper, we initiate the study of tensor network non-zero testing (TNZ): Given a tensor network , does represent a non-zero vector? We show that TNZ is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic -SAT problem on -dimensional qudits is in NP for and . This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic , (2) constant , and (3) for arbitrary interaction graphs.

## 1 Introduction

One of the central aims of condensed matter physics is the study of ground spaces of local Hamiltonians. Here, a -local Hamiltonian is a sum of Hermitian operators , each of which act non-trivially on subsets of (out of a total of ) qudits. Such operators typically govern the evolution of quantum systems in nature, and in particular, their ground space (i.e. the eigenspace of corresponding to its smallest eigenvalue) characterizes the state of the corresponding quantum system at low temperature. Thus, the theoretical study of ground spaces of local Hamiltonians is crucial to understanding (e.g.) exotic phases of matter, such as superfluidity, which manifest themselves at low temperatures.

To this end, one of the key tools used by the condensed matter physics community is that of tensor networks (see e.g. Reference [CV09] for a survey). Specifically, tensor networks allow one to succinctly represent certain non-trivially entangled quantum states. As such, they play a crucial role in the study of ground spaces of local Hamiltonians. For example, in the early 1990’s, White developed the celebrated DMRG heuristic [Whi92, Whi93], which is nowadays recognized [ÖR95, RÖ97, VPC04, VMC08, WVS09] as a variational algorithm over 1D tensor networks known as Matrix Product States (MPS). The intuitive reason why DMRG works so well is that for 1D gapped Hamiltonians, the ground state turns out to be well-approximated by an MPS [Has07]. Due in part to the success of DMRG, over the last two decades, a number of generalizations of MPS to higher dimensions have also been developed, such as Projected Entangled Pair States (PEPS) [VC04, VWPGC06] and Multiscale Entanglement Renormalization Ansatz (MERA) [Vid07, Vid08]; such networks are able to represent larger classes of entangled states. Unfortunately, with this additional expressive power comes a price: Contracting an arbitrary tensor network is P-complete [SWVC07]. (Here, contracting a network roughly means determining its value on a given input.)

#### Motivation.

Given that tensor networks play a fundamental role in condensed matter physics, and that contracting general networks is P-complete, here we ask a simpler question: Given a tensor network , how difficult is it to decide whether represents a non-zero vector?

Our original motivation for studying this question came from the following well-known open problem: Given a -local Hamiltonian whose terms pairwise commute, what is the complexity of estimating its ground state energy? This is known as the commuting -local Hamiltonian problem (-CLH). Note that although asking for local terms to commute may intuitively make the problem seem “classical”, such Hamiltonians can nevertheless have highly entangled ground states with exotic properties such as topological order [Kit03].

For general and local dimension , the best known upper bound on -CLH is Quantum-Merlin-Arthur (QMA). However, the following special cases are known to be in NP: for local dimension  [BV05], with (as well as with a “nearly Euclidean” interaction graph) [AE11], with on a square lattice [Sch11], special cases of on a square lattice with polynomial in the number of qudits [Has12], and the case where the interaction graph is a good locally-expanding graph [AE13]. In particular, implicit in the approach of Schuch [Sch11] is a simple tensor network representation of the ground space of any commuting -local Hamiltonian ; thus, the ability to verify in NP whether is non-zero would place -CLH into NP for and .

#### Results.

The decision problem we study in this paper is formally stated as follows. Below, denotes the number of physical edges in the network, each of which is assumed to have dimension . (See Section 2 for definitions.)

###### Problem 1 (Generalized Tensor Network Non-Zero Testing (gTNZ)).

Given a classical description of a tensor network and threshold parameters such that ,

• if there exists an input such that , output YES, and

• if for all , , output NO.

For convenience, we use the shorthand TNZ to refer to gTNZ with parameters and . Note that the key parameter here is , and there is no loss of generality in setting . This is because in this paper, we assume the entries of the input tensor network are specified as complex numbers with rational real and imaginary parts. Since the value of on any input is given by a polynomial in the entries of the nodes with terms, it follows that the gap in any instance of TNZ can be trivially amplified to by multiplying by an appropriate scalar based on the size of the network and the precision used to encode .

Our main results are as follows.

1. (Theorem 2) gTNZ is -hard.

2. (Theorem 3) TNZ is not in unless the Polynomial Hierarchy (PH) collapses to . Here, denotes the th level of PH (see Section 2 for definitions).

3. (Theorem 4) TNZ with the additional restriction that ’s nodes contain only non-negative entries is NP-complete, even when is given by a -regular graph with edges of bond dimension .

4. (Theorem 5 and Theorem 7) If ’s nodes represent injective linear maps, then is non-zero. Conversely, there exists a non-zero tensor network which does not have a “geometrically equivalent” injective tensor network representation . This implies that injective networks cannot exactly represent a state with long-range correlations (Observation 2).

5. (Lemma 3 and Corollary 8) The commuting variant of the Stoquastic Quantum -SAT problem is in NP for any and local dimension .

Remarks: The non-commuting variant of the Stoquastic Quantum -SAT problem is known to be MA-complete [BT09]. Injective tensor networks have previously been studied, e.g., in the translationally invariant case in [PGSGG10].

#### Significance.

Although we do not fully resolve the complexity of the commuting local Hamiltonian problem (-CLH), the strength of our approach is that, to the best of our knowledge, our line of attack on -CLH is the first which does not rely on Bravyi and Vyalyi’s Structure Lemma [BV05]. In fact, it is purely this novel viewpoint which allows us to easily place the Stoquastic Quantum -SAT problem into NP for any (Corollary 8). Moreover, although Theorem 3 suggests that testing whether an arbitrary tensor network is non-zero is unlikely to be in NP, it is entirely plausible that the simple structure of the specific network which arises in the context of -CLH (see Lemma 3) can be exploited to allow non-zero verification in NP.

Finally, as tensor networks are ubiquitous in condensed matter physics, it is crucial to understand their strengths and limitations. Result (2) shows that even the simple task of determining whether a given network represents a non-zero vector is in general very difficult. This underscores the need for cleverly designed classes of tensor networks such as MERA, which both manage to represent physically meaningful states, as well as allow efficient computation of local expectation values. To this end, we hope that our findings help guide the search for new key properties which make certain classes of tensor networks “manageable”. For example, the fact that TNZ on non-negative or injective networks lies in NP suggests that perhaps there are other physically relevant types of computations which can be performed on such networks “easily” (i.e. in a complexity class below P).

#### Organization of this paper.

This paper is organized as follows. In Section 2, we formally define tensor networks and the Polynomial-Time Hierarchy. Section 3.1 shows complexity-theoretic hardness results for TNZ. In Section 3.2, we study easier special cases of TNZ which fall into NP, such as non-negative and injective tensor networks. Section 4 discusses applications of TNZ to the commuting local Hamiltonian problem. We conclude with open questions in Section 5.

#### Notation.

We define . Let and denote the sets of non-negative real numbers and natural numbers, respectively. For operator , let and denote the null space of and the orthogonal complement of , respectively. The notation denotes the set of unitary operators mapping to itself.

## 2 Definitions

In this section, we introduce definitions used throughout this article. We begin with a brief introduction to tensor networks, which are a standard tool in condensed matter physics.

#### Tensor Networks.

There are two views of tensor networks we discuss here: The vector and linear map views. To introduce the first, we begin by thinking of a tensor simply as a -dimensional array; given inputs through , outputs a complex number. We call such an object a -dimensional tensor, where . Given two tensors, it is possible to “compose” them by “matching up” certain inputs; this is called edge contraction, and is best depicted via a simple but powerful graph theoretic framework, shown in Figure 1 [GHL14]. In Figure 1(a), the vertex corresponds to the tensor , and each edge corresponds to one of the input parameters or indices of .

In Figure 1(b), the edge denotes the contraction of and on their second index, the result of which is a -dimensional tensor defined as

 P(i1,i3,j1,j3)=∑kM(i1,k,i3)N(j1,k,j3).

Since is -dimensional, i.e. has inputs, it is depicted as having four “legs” (i.e. edges with only one endpoint) in Figure 1(b).

By composing multiple tensors, we obtain a tensor network.

Figure 2 depicts such a network. Here, open edges or legs are called physical edges, whereas contracted edges are called virtual edges. These names are physically motivated as follows. Recall that thus far, we have defined tensors as multi-dimensional arrays. The network in Figure 2 is such an array taking in inputs ; for each set of inputs, outputs a complex number . The name vector view now follows: can be thought of as representing a vector such that given computational basis state , outputs amplitude , i.e. . Why the names physical and virtual edges then? Typically in condensed matter physics, one thinks of the vertices in as corresponding to -dimensional quantum systems. Then, each node of would have a physical edge of dimension . The contracted edges, on the other hand, represent entanglement between systems; as such, they are called virtual edges. Their dimension is an important parameter known as the bond dimension of the network.

Some further terminology: A network without physical edges is called a closed network, and represents a complex number which can be computed by contracting the network. Given a closed network, a labeling of its (virtual) edges means setting each index of every tensor to some fixed value, such that indices sharing an edge are set to the same value.

Finally, we present the linear map view of tensor networks, which is perhaps best illustrated via the network in Figure 1(b). In this view, rather than thinking of all physical edges as being inputs, we can instead partition them into a set of inputs (say, edges and ) and a set of outputs (say, edges and ). Fix some values to inputs and . Then, the result is a new network with two remaining physical edges, and . But can now be thought of as a vector with inputs and , just as in our first viewpoint! In other words, any input to and is mapped to a -dimensional vector on inputs and . By extending this action linearly over all basis states , we have that acts as a linear map from inputs and to outputs and , as claimed.

#### The Polynomial-Time Hierarchy.

The Polynomial-Time Hierarchy (PH) [MS72] is defined as the union , where is defined as follows.

###### Definition 1 (Σpi).

A decision problem is in if there exists a polynomial time Turing machine such that given instance of ,

 x is a YES instance ⟺∃y1∀y2∃y3⋯Qiyi s.t. M accepts (x,y1,…,yi),

where if is odd, and if is even, and the are polynomial-length strings or proofs.

## 3 Complexity of TNZ

In this section, we show complexity-theoretic hardness of TNZ (Section 3.1), as well as study special cases of TNZ which fall into NP (Section 3.2).

### 3.1 Hardness of TNZ

Tensor networks are powerful objects; recall that simply contracting an arbitrary network is P-complete [SWVC07]. Thus, here we ask the natural question: Is TNZ easier? For the general problem gTNZ, it is easy to answer this question in the negative using standard techniques by showing a polynomial time Turing reduction from the -complete problem , as we do now in Theorem 2 below. Here, recall that in one is given a 2-CNF formula and asked to output the number of satisfying assignments to . We remark that a similar construction was used in [AL10] to sketch -hardness of contracting tensor networks.

###### Theorem 2.

There exists a polynomial-time Turing reduction from to gTNZ.

###### Proof.

Our approach is to first encode an arbitrary instance of into a (closed) tensor network , such that contracting outputs the number of satisfying assignments . By scaling by appropriate multiplicative factors, we can then apply the standard idea of binary search to compute using a polynomial number of calls to a gTNZ oracle.

To construct from , let and denote the sets of variables and clauses in , respectively. For each variable and clause , we create nodes and in our tensor network, respectively. If variable occurs in clause , we connect and by an edge. Thus, the degree of is the number of clauses appears in (either as a positive or negative literal), and the degree of is precisely since each clause contains two literals. All edges have bond dimension . Next, we specify the action of ’s nodes. Let denote the set of edges incident on a node of . Let be a node such that . Then if all edges in are labeled with the same bit (i.e. either all or all ), then outputs ; else, outputs . This enforces to correspond to a consistent assignment to variable . Now let for , where suppose for example . Let and be its incident edges corresponding to variables and , respectively. Then, outputs one if either is labeled or is labeled . This forces to correspond to a satisfied clause. Thus, the contraction of yields , since each edge labeling of the network corresponding to a consistent and satisfying assignment contributes to the sum.

Given , to now use an oracle for gTNZ to compute , we claim that for any positive integer , solving gTNZ allows us to determine if or . Assuming this claim, we have that since the number of assignments is at most for the number of variables in , by invoking gTNZ at most times in conjunction with binary search, we can determine efficiently.

To thus see that gTNZ indeed allows us to distinguish versus , simply multiply each tensor in by the scalar to obtain network . It follows that if , then the contraction of yields value at least , whereas if , then yields value at most . Setting and , we thus have our claim by using the fact that to obtain that

 α−β=2t−2t(k−1)k=2tk≥1.

Theorem 2 tells us that general instances of gTNZ are highly unlikely to be tractable. However, the proof relies critically on the ability to set the thresholds and as needed. What if we fix and , i.e. the case of TNZ? Clearly, the proof of Theorem 2 implies that this problem is at least NP-hard. Is it also in NP? The following theorem suggests not.

###### Theorem 3.

If TNZ is in , then , i.e. the Polynomial Hierarchy collapses to the -nd level.

To show Theorem 3, we require two lemmas.

###### Lemma 1.

Let be a closed tensor network on nodes and edges, where edge has bond dimension for , and such that the contraction of outputs value . Then, for any , one can construct in (deterministic) polynomial time a closed tensor network satisfying the following properties:

• Contracting outputs , and

• has nodes and edges, where edge has bond dimension .

###### Proof.

We construct as follows. For any pair of nodes and in connected by edge , we increase the bond dimension of by ; this extra dimension will play the role of a “switch”. In particular, whenever is labeled with this “switch” value, we will say edge is set to SWITCH. To now describe how the vertices act on this extra dimension, fix some arbitrary node , and relabel each node as . Then, in our new network , the action of each is as follows:

• If all edges incident on are not set to SWITCH, then acts identically to .

• Else, if there exists a pair of edges incident on , such that precisely one edge is set to SWITCH, then outputs .

• Else, if , then outputs . If , then outputs .

Thus, in there are only two ways to label all edges to obtain a non-zero value. The first is when all edges are not set to SWITCH; contracting over all such labellings contributes value to the sum. The second is when all edges are set to SWITCH; in this case, is added to the sum. Thus, outputs , as desired. ∎

###### Lemma 2.

Given a 2SAT formula on variables and non-negative integer , let denote the problem of deciding whether has at least satisfying assignments. Then, .

###### Proof.

Let denote an oracle deciding TNZ. We construct a non-deterministic Turing machine with access to which decides in polynomial time. Suppose has satisfying assignments. Then, the action of on input is as follows:

1. Non-deterministically guess a value satisfying .

2. As done in the proof of Theorem 2, construct a tensor network encoding , i.e. whose contraction yields value .

3. Using Lemma 1, map to a network whose contraction yields value .

4. Call on input . If outputs YES, output NO. Else, output YES.

We now prove correctness. First, if we have a YES instance of , then in step 1, guesses . The network then yields value upon contraction, signifying that we have guessed correctly. Thus, oracle outputs NO, in which case we flip the answer to YES in step 4. Conversely, if we have a NO instance of , then any guess made by in step 1 will yield a network whose value yields . Thus, oracle outputs YES in step 4, and we flip the answer to NO. To complete the reduction, note that each step above runs in non-deterministic polynomial time. ∎

With Lemmas 1 and 2 in hand, we now prove Theorem 3.

###### Proof of Theorem 3.

Let be an oracle deciding language in the statement of Lemma 2. Then, note that

 P#2SAT⊆PL. (1)

Indeed, this holds since any call to a oracle can be simulated in polynomial time by applying binary search in conjunction with the oracle . Now, if , we have by Lemma 2 that

 PL⊆PNPTNZ⊆PNPΣpi⊆NPNPΣpi=Σpi+2. (2)

On the other hand, since is -complete, we have that

 P#2SAT=P#P⊇PH, (3)

where the last containment is given by Toda’s theorem [Tod91], which states that . Combining Equations (1), (2), and (3), the claim follows. ∎

### 3.2 Easier instances of TNZ

In general, Theorem 3 implies that it is highly unlikely for TNZ to lie in PH. In contrast, in this section, we study special cases of TNZ whose complexity is provably in NP.

#### Non-negative tensor networks.

The first case we consider is very simple, and yet finds a nice application in Section 4: The case in which the input tensor network’s nodes contain only non-negative real numbers. Call such networks non-negative. Then, defining TNZ+ as the problem TNZ with a non-negative tensor network as input, we have the following.

###### Theorem 4.

TNZ+ is in NP, and is NP-hard even when the input network is given by a -regular graph with all edges of bond dimension .

###### Proof.

It is easy to see that TNZ+ is in NP; indeed, suppose we have a YES instance , i.e. there exists an input such that . Since all tensors comprising consist of non-negative entries, it follows that if and only if there exists a labeling of the tensors’ virtual edges yielding a positive number. Such a labeling can be verified in polynomial-time, yielding the claim.

Note now that the proof of Theorem 2 immediately yields that TNZ+ is NP-hard. However, the degree of the graph in that construction can be large. To obtain the statement of our claim here, we instead observe a many-one reduction from the NP-complete problem Edge-Coloring (ECOL) to TNZ+. Specifically, recall that in ECOL, one is given a simple graph and a choice of colors, and asked whether there exists a coloring of the edges so that no two edges of the same color are incident on the same vertex. For this problem, our starting point is the fact that determining whether a simple -regular graph is edge-colorable with colors is NP-hard [Hol81]. Thus, suppose is a simple -regular graph. We construct a tensor network from as follows. For each vertex , create a tensor node . For each edge , connect the tensor nodes and with an edge. Finally, define each tensor such that if , , and , and otherwise. Note that this is a closed network which is -regular, has bond dimension on all edges, and all tensor entries are non-negative.

To finally see correctness, observe simply that each tensor acts as a “local check”, such that outputs if and only if all its adjacent edges are given distinct values or colors. Hence, the network evaluates to a non-zero value if and only if there exists a valid -edge-coloring of , i.e. we have reduced the problem to an instance of TNZ+. As the reduction clearly runs in polynomial time, this completes the proof. ∎

Theorem 4 shows that -regular non-negative networks suffice to achieve NP-hardness for TNZ. In contrast, it is well known that tensor networks on -regular graphs can be efficiently contracted (even in the presence of arbitrary complex entries). This is because such graphs are a union of cycles and paths, and the latter two can be contracted similar to how Matrix Product States are contracted. Finally, note that the proof of Theorem 4 also yields the following simple result.

###### Observation 1.

Contracting a non-negative, -regular, planar tensor network with bond dimension is -hard.

###### Proof.

This follows simply because the contraction of the network constructed in the proof of Theorem 4 yields the number of valid edge-colorings of . The latter problem is -hard for -regular planar graphs and colors [CGW14]. ∎

#### Injective tensor networks.

We now consider so-called injective tensor networks, which were studied for example in the translationally invariant case in [PGSGG10]. To define such networks, we first require some terminology: Given a tensor network on vertex set , let . Then, the subnetwork of induced by is the network consisting of all vertices in , as well as all edges (physical and virtual) incident on vertices in . An example is given by Figure 3.

###### Definition 2 (Injective tensor network).

Let be a tensor network. We call -injective for if can be partitioned into sets of nodes , such that for all , the subnetwork of induced by has the following properties:

1. is connected.

2. At least one node in has a physical edge.

3. Let denote the linear map from the virtual edges crossing the cut versus in (where is the vertex set of ) to the physical edges of . Then, is an injective map.

By exploiting the injective property of such networks, we can show the following.

###### Theorem 5.

If a tensor network is -injective for some , then is non-zero.

###### Proof.

Let be a -injective tensor network, and let be a partition of the nodes of as in Definition 2 with corresponding linear maps . Now, since any is injective, it follows that the adjoint map from the physical to virtual edges is surjective. Thus, for each , there exists an input to its physical edges such that the output along the virtual edges is the -qudit product state . In other words, there exists a physical input to the network such that the contraction of the network along each edge involves only inner products of the form . Thus, is non-zero, as claimed. ∎

An immediate corollary to Theorem 5 is the following.

###### Corollary 6.

TNZ for a -injective network in which each of the sets of nodes in the injective partition are of size is in NP. Here, is the number of nodes in the network, and we assume .

###### Proof.

The prover here specifies the sets in Definition 2. The claim then follows by Theorem 5 and the fact that a network of size takes time to contract, thus allowing us to check whether each map specified by the prover is injective in polynomial time. ∎

Corollary 6 gives us an efficiently verifiable condition which can certify that a non-zero vector represented by tensor network is indeed non-zero. It is thus natural to ask whether a suitably defined converse of this statement might hold. For example, given a non-zero vector , does there always exist some -injective representation of in which the size of the sets are logarithmic? This question is interesting for two reasons. First, injective tensor networks are generic (see, e.g., [PGSGG10]). Second, using the techniques in Section 4, a positive answer to this question might be a step towards resolving the long-standing open question of whether the commuting -local Hamiltonian for arbitrary is in NP in the affirmative.

To make progress on this question, we define the notion of geometrically equivalent tensor networks. Specifically, we say that networks and are geometrically equivalent if the parameters of their underlying graphs (e.g. number of nodes, placement of physical and virtual edges, physical dimension, bond dimension, etc…) are identical. In other words and differ only in the specifications of the tensors (i.e. nodes) themselves. Note that the notion of geometric equivalence is arguably well-motivated, as often in Hamiltonian complexity, given a local Hamiltonian with interaction graph , one fixes the geometry of the tensor network ansatz intended to represent the ground state of to match .

With this definition in hand, we now show the following.

###### Theorem 7.

For all , there exists a non-zero network which does not have a geometrically equivalent -injective representation.

###### Proof.

We proceed by constructing a non-zero matrix product state (i.e. 1D tensor network) which satisfies the claim. To begin, consider the -qubit state

 |ψ⟩=|0⟩|0⟩⊗n−2|0⟩+|1⟩|0⟩⊗n−2|1⟩,

which can be represented by an MPS of bond dimension as follows. There are nodes in the network, which we label as , where node corresponds to qubit of . Each node has a physical edge. Vertex is connected via a virtual edge to vertex if and to if . The nodes and output if all their edges (i.e. both physical and virtual) are labeled by , or if all edges are labeled by ; otherwise, they output . As for through , these output if their physical edge is set to and either both virtual edges are or both are ; otherwise, they output . Thus, the only edge labelings which produce a non-zero value are those with all edges are labeled , or when the physical edges are labeled and the virtual edges are all labeled . In both these cases, the network outputs . Thus, the MPS represents , as claimed.

Assume now, for sake of contradiction, that admits a geometrically equivalent -injective representation for block size . Since , there exists a block such that . See Figure 4 for an illustration. By definition of injective, we know that is a contiguous set of nodes . Let and . We denote the virtual edges connecting and to as and , respectively. Now, by definition of , if we input and on physical edges and , respectively, outputs . Then, suppose the nodes in all receive physical input , and the nodes in all receive physical input , with the exception of which receives . Let and denote the vectors output by and on the edges and . Since the map corresponding to is injective, there exists a physical input to the nodes in such that outputs on and on . But this implies is non-zero on this input, which is a contradiction. This yields the claim. ∎

Note that the method for obtaining the contradiction in the proof of Theorem 7 implies the following about the types of quantum states that an injective network can represent.

###### Observation 2.

An injective tensor network cannot (exactly) represent a quantum state with long-range correlations (e.g. such as a Bell pair between the first and last qubits of a chain of tensors).

We remark that the condition of geometric equivalence plays an important role in this statement, as otherwise the notion of “long-range” is ill-defined. (In other words, to define “long-range”, we assume the underlying physical systems are arranged according to some fixed geometry which is respected by the tensor network describing them.)

## 4 Connections to Hamiltonian complexity

We now discuss connections between TNZ and the commuting -local Hamiltonian problem (-CLH). Recall that in -CLH, one is given a set of -local Hermitian operators , which act on -dimensional qudits and which pairwise commute, as well as real parameters and such that . We are asked to decide whether the smallest eigenvalue of is at most or at least .

We first observe a connection between TNZ and -CLH. Specifically, we note that ground states of YES instances of -CLH have a succinct tensor network description. Using this description, we then deduce that the ability to solve certain cases of TNZ in NP would place -CLH in NP for arbitrary and . More generally, we have the following.

###### Lemma 3.

For any and , there exists a non-deterministic polynomial time mapping reduction from -CLH on -dimensional qudits to .

###### Proof.

We use a setup similar to that of Schuch [Sch11]. Specifically, Let be an instance of -CLH with ground state , and let be an oracle deciding TNZ. Since all pairwise commute, if we take a spectral decomposition of each , it follows that for all , there exists an eigenspace projector such that . Since all also pairwise commute (as they all diagonalize in the same basis as ), it follows that the ground space of is given by . With this description of the ground space in hand, the reduction proceeds as follows.

1. Non-deterministically guess string and projectors .

2. Checks that for each , indeed encodes some eigenspace of with corresponding eigenvalue such that . If not, reject.

3. Write down a tensor network representing the state .

4. Feed into the oracle for TNZ and returns ’s answer.

Note that Step (3) can be computed in polynomial time since the projectors are at most -local, and thus each can be represented by a tensor node with entries. (In other words, start with a trivial network encoding the state , and then simply connect the nodes representing each to the appropriate legs of the network in an arbitrary order.) Thus, this procedure runs in (non-deterministic) polynomial time.

Let us verify correctness. Suppose first that is a YES instance. Then, there must exist a state and eigenspace projections with corresponding eigenvalues for each such that (1) and (2) . There hence exists a computational basis state such that such that . In the YES case, we correctly guess in Step (1). Then, extending to an orthonormal basis for the space projected onto by , i.e. , we have

 DnΠH|x⟩=Dn(∑i|ψi⟩⟨ψi|)|x⟩=cDn|ψ⟩+|ψ⊥⟩

for , , and for and in space . Thus, represents a ground state whose norm is at least , implying we have a succinct tensor network for the ground state. Moreover,

 ⟨x|(DnΠH|x⟩)=1Dn(⟨x|ΠHDn)(DnΠH|x⟩)≥1.

It follows that , and in Step (4) the TNZ oracle returns YES on input , as desired.

Suppose now that is a NO instance. Then, either the projectors guessed in Step (1) do not correspond to a valid eigenvalue of of value at most , or . In the former case, step will reject. For the latter case, regardless of which string we guess in Step (1), we have . Thus, represents the zero-vector, and so in Step (4), the TNZ oracle returns NO on input , causing us to reject, as desired. ∎

Lemma 3 shows that if , then -CLH is in NP for . Unfortunately, we know from Theorem 3 that it is unlikely for arbitrary instances of TNZ to be solvable in NP. On the other hand, by exploiting the specific structure of the tensor network constructed in Lemma 3, it may be possible to check whether is non-zero in NP. Here is a simple example for which this can be done — the MA-complete Stoquastic -SAT problem [BBT06]. In this problem, the input is a set of -local orthogonal projection operators whose entries in the standard basis are all non-negative, and the question is whether there exists a state such that for all , .

###### Corollary 8.

The variant of stoquastic quantum -SAT in which all local projectors pairwise commute is in NP for any and .

###### Proof.

By definition of Stoquastic -SAT, the network constructed in Lemma 3 for such a Hamiltonian has all real non-negative entries; thus, the claim follows from Theorem 4. ∎

## 5 Conclusions

In this paper, we have initiated the study of tensor network non-zero testing (TNZ). We have shown that TNZ for arbitrary tensor networks is highly unlike to be in the Polynomial-Time Hierarchy. We next obtained (among other results) that special cases of TNZ, such as non-negative and injective networks, lie in NP. Via a simple application of the non-negative case, we obtained that the commuting stoquastic quantum k-SAT problem is in NP for and -dimensional systems for .

Two questions we leave open are as follows. First, can the specific structure of the tensor network obtained in Lemma 3 be exploited to place the commuting -local Hamiltonian problem into NP for ? Second, can the commuting stoquastic -local Hamiltonian problem also be placed into NP using our techniques? Note that unlike for the stoquastic quantum k-SAT problem, here the local interaction terms are not necessarily projectors.

## 6 Acknowledgements

SG acknowledges support from a Government of Canada NSERC Banting Postdoctoral Fellowship and the Simons Institute for the Theory of Computing at UC Berkeley. ZL is supported by ARO Grant W911NF-12-1-0541, NSF Grant CCF-0905626 and Templeton Foundation Grant 21674. SWS is supported by NSF Grant CCF-0905626 and ARO Grant W911NF-09-1-0440. GW is supported by NSF Grant CCR-0905626 and ARO Grant W911NF-09-1-0440.

### Footnotes

1. Simons Institute for Theoretical Computing, University of California, Berkeley, CA 94720, U.S.A.
2. Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, U.S.A.

### References

1. D. Aharonov and L. Eldar. On the complexity of commuting local Hamiltonians, and tight conditions for Topological Order in such systems. In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS 2011), pages 334–343, 2011.
2. D. Aharonov and L. Eldar. Commuting local Hamiltonians on expanders, locally testable quantum codes, and the qPCP conjecture. Available at arXiv.org e-Print quant-ph/1301.3407, 2013.
3. I. Arad and Z. Landau. Quantum computation and the evaluation of tensor networks. SIAM Journal on Computing, 39(7):3089–3121, 2010.
4. S. Bravyi, A. Bessen, and B. Terhal. Merlin-Arthur games and stoquastic complexity. Available at arXiv.org e-Print quant-ph/0611021v2, 2006.
5. S. Bravyi and B. Terhal. Complexity of stoquastic frustration-free Hamiltonians. SIAM Journal on Computing, 39(4):1462, 2009.
6. S. Bravyi and M. Vyalyi. Commutative version of the local Hamiltonian problem and common eigenspace problem. Quantum Information & Computation, 5(3):187–215, 2005.
7. J.-Y. Cai, H. Guo, and T. Williams. The complexity of counting edge colorings and a dichotomy for some higher domain holant problems. Available at arXiv.org e-Print cs.CC/1404.4020v1, 2014.
8. I. Cirac and F. Verstraete. Renormalization and tensor product states in spin chains and lattices. Journal of Physics A: Mathematical and Theoretical, 42(50):504004, 2009.
9. S. Gharibian, Y. Huang, and Z. Landau. Quantum Hamiltonian complexity. Available at arXiv.org e-Print quant-ph/1401.3916v1, 2014.
10. M. Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics, P08024(08), 2007.
11. M. B. Hastings. Matrix product operators and central elements: Classical description of a quantum state. Geometry & Topology Monographs, 18:115–160, 2012.
12. I. Holyer. The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4):718–720, 1981.
13. A. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2–30, 2003.
14. A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential time. In Proceedings of the 13th Symposium on Foundations of Computer Science, pages 125–129, 1972.
15. S. Östlund and S. Rommer. Thermodynamic limit of density matrix renormalization. Physical Review Letters, 75:3537–3540, 1995.
16. D. Pérez-García, M. Sanz, C. E. González-Guillén, M. M. Wolf, and J. I. Cirac. Characterizing symmetries in a projected entangled pair state. New Journal of Physics, 12:025010, 2010.
17. S. Rommer and S. Östlund. Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group. Physical Review B, 55:2164–2181, 1997.
18. N. Schuch. Complexity of commuting Hamiltonians on a square lattice of qubits. Quantum Information & Computation, 11:901–912, 2011.
19. N. Schuch, M. Wolf, F. Verstraete, and J. I. Cirac. The computational complexity of PEPS. Physical Review Letters, 98:140506, 2007.
20. S. Toda. PP is as hard as the Polynomial-Time Hierarchy. SIAM Journal on Computing, 20:865–877, 1991.
21. F. Verstraete and J. I. Cirac. Renormalization algorithms for quantum-many body systems in two and higher dimensions. Available at arXiv.org e-Print cond-mat/0407066, 2004.
22. G. Vidal. Entanglement renormalization. Physical Review Letters, 99:220405, 2007.
23. G. Vidal. A class of quantum many-body states that can be efficiently simulated. Physical Review Letters, 101:110501, 2008.
24. F. Verstraete, V. Murg, and I. Cirac. Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Advances in Physics, 57(2):143–224, 2008.
25. F. Verstraete, D. Porras, and J. .I. Cirac. Density matrix renormalization group and periodic boundary conditions: A quantum information perspective. Physical Review Letters, 93:227205, 2004.
26. F. Verstraete, M. Wolf, D. Pérez-García, and J. I. Cirac. Projected entangled states: Properties and applications. International Journal of Modern Physics B, 20:5142, 2006.
27. S. R. White. Density matrix formulation for quantum renormalization groups. Physical Review Letters, 69:2863–2866, 1992.
28. S. R. White. Density-matrix algorithms for quantum renormalization groups. Physical Review B, 48:10345–10356, 1993.
29. A. Weichselbaum, F Verstraete, U Schollwöck, J. I. Cirac, and J. von Delft. Variational matrix product state approach to quantum impurity models. Physical Review B, 80:165117, 2009.
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