Complexity of the XY antiferromagnet at fixed magnetization

Complexity of the XY antiferromagnet at fixed magnetization

Andrew M. Childs amchilds@umd.edu David Gosset dngosset@gmail.com  and  Zak Webb zakwwebb@gmail.com Department of Combinatorics & Optimization, University of Waterloo Institute for Quantum Computing, University of Waterloo Department of Computer Science, Institute for Advanced Computer Studies, and Joint Center for Quantum Information and Computer Science, University of Maryland Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology Department of Physics & Astronomy, University of Waterloo
Abstract.

We prove that approximating the ground energy of the antiferromagnetic XY model on a simple graph at fixed magnetization (given as part of the instance specification) is QMA-complete. To show this, we strengthen a previous result by establishing QMA-completeness for approximating the ground energy of the Bose-Hubbard model on simple graphs. Using a connection between the XY and Bose-Hubbard models that we exploited in previous work, this establishes QMA-completeness of the XY model.

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1. Introduction

Kitaev pioneered the study of quantum constraint satisfaction problems, where the goal is to approximate the ground energy of a Hamiltonian [5]. Many examples of such ground energy problems are known to be complete for the complexity class QMA, a quantum analogue of NP (see, e.g., [2]).

In this paper, we focus on computational problems defined by graphs. For example, MAX-CUT is a classical constraint satisfaction problem defined by a graph. The goal is to find a subset of vertices that maximizes the number of edges between that subset and its complement. This can be rephrased as a ground energy problem: it is equivalent to minimizing the energy of the Ising antiferromagnet on the graph, where there is a bit for every vertex and a constraint penalizing adjacent bits that agree. Equivalently, we may consider a qubit at every vertex and a interaction for every edge.

To obtain a genuinely quantum constraint satisfaction problem defined by a graph, we consider interaction terms with nonzero off-diagonal matrix elements. Natural choices include the antiferromagnetic Heisenberg model, with an interaction for each edge, and the antiferromagnetic XY model, with an interaction for each edge. The complexities of the ground energy problems for these models are unknown, although some variants have been studied [8, 4, 3]. Recent work has established QMA-completeness for the antiferromagnetic XY model with coefficients that vary throughout the graph and depend on system size [7]. This result holds even when the graph is a triangular lattice in two dimensions. Our result is incomparable because while we consider general (simple) graphs, we restrict the coefficient for every edge to be . In addition, the restriction to fixed magnetization is not present in reference [7].

Our main result concerns the antiferromagnetic XY model on a graph. For a given simple graph with vertex set and edge set , the Hamiltonian has the form

(1)

where denote Pauli matrices and a subscript indicates which qubit is acted on. Note that this Hamiltonian commutes with the total magnetization operator , so it decomposes into sectors for each eigenvalue of . We prove that approximating the ground energy of this Hamiltonian in a sector with fixed magnetization is a QMA-complete problem.

Our result is a natural extension of our previous work [3]. The difference is that we previously considered Hamiltonians defined by graphs that may have self-loops. In that context we established QMA-completeness of the ground energy problem for the Bose-Hubbard model (at fixed particle number), a system of bosons hopping on a graph with a repulsive on-site interaction. Then, using the relationship between hard-core bosons and spins, we also established QMA-completeness of a ground energy problem related to the XY model, where the Hamiltonian has an term associated with each edge (as in (1)) as well as a local magnetic field associated with each self-loop. In this paper we present a stronger result since we consider only simple graphs and Hamiltonians of the form (1).

Our proof relies heavily on machinery and results from reference [3]. We prove QMA-hardness of the ground energy problem for the Bose-Hubbard model on a simple graph at fixed particle number. Our starting point is the previous QMA-completeness result [3] which pertains to graphs with self-loops. Counterintuitively, we begin by increasing the number of self-loops: we modify a graph from the previous construction to obtain a new graph in which every vertex has a self-loop. The new graph is formed by taking two copies of , adding self-loops to each of them, and then adding some edges between the two copies. The modification is performed in such a way that the ground spaces of the Bose-Hubbard model on and (in the sector with a given number of particles) are simply related. Once we have a self-loop at every vertex, we then remove all self-loops to obtain another graph with no self-loops, which was our goal. This removal is equivalent to subtracting a term in the Hamiltonian that is proportional to the identity. We thus obtain a graph with no self-loops such that approximating the ground energy of the Bose-Hubbard model on is as hard as approximating the ground energy of , which is QMA-hard by our previous result.

Finally, we use a reduction presented in [3] showing that an instance of this ground energy problem for the Bose-Hubbard model on a graph is equivalent to an instance of the ground energy problem for the XY model at fixed magnetization on the same graph. Since this reduction preserves the graph, we establish QMA-completeness for the XY model as discussed above.

The remainder of this paper is organized as follows. In Section 2, we provide basic definitions and tools used in this paper. In Section 3, we review results from reference [3]. In Section 4, we describe a procedure that modifies graphs from the previous QMA-completeness result so that every vertex has a self-loop. We show a relationship between the ground energies before and after the modification. Finally, we remove all the self-loops from the modified graph, giving only a constant overall energy shift in the associated Hamiltonian. In Section 5, we use this strategy to establish that the ground energy problem for the antiferromagnetic XY model (on simple graphs, at fixed magnetization) is QMA-complete.

2. Preliminaries

In this paper, denotes a graph with vertex set , edge set , and adjacency matrix . Later we will be interested in the case where is a simple graph, but for now we allow the possibility that it has at most one self-loop per vertex. In other words, can be any symmetric 0-1 matrix.

2.1. The antiferromagnetic XY model on a graph

We define the antiferromagnetic XY model with local magnetic fields on to be the -qubit, two-local Hamiltonian

(2)
(3)

Note that the second term is only present if the graph has self loops; this term vanishes for simple graphs, giving the usual antiferromagnetic XY model. It is easy to see that this Hamiltonian (either with or without the second, local magnetic field term) conserves Hamming weight. Let

be the subspace with Hamming weight . We write for the smallest eigenvalue of within the sector with Hamming weight (i.e., the smallest eigenvalue of the restriction ).

2.2. The Bose-Hubbard model on a graph

We now review the Hamiltonian of the Bose-Hubbard model on a graph, as defined in [3]. We present only the “first quantized” formulation of this model that we use in this paper; see [3] for a broader discussion.

The Hilbert space of distinguishable particles that live on the vertices of is

Here each register represents the location of a particle. The Hilbert space of indistinguishable bosons on is the subspace of symmetric states

where

and is the symmetric group on elements.

For any operator , write

for the operator that acts on as on the th register and as the identity on all other registers. Define

(4)

where

is an operator that counts the number of particles at vertex . The Hamiltonian (4) acts on the distinguishable-particle Hilbert space. Since it is symmetric under permutation of the registers, the bosonic space is an invariant subspace for . The -particle Bose-Hubbard model is the restriction of this Hamiltonian to the bosonic subspace

Looking at equation (4), we see that the first term (the movement, or hopping, term) has smallest eigenvalue equal to times the smallest eigenvalue of , which we denote , while the second term (the interaction term) is positive semidefinite. Hence the smallest eigenvalue of is at least that of , which is at least .

It is convenient to subtract this constant to make the Hamiltonian positive semidefinite. Define

and write for its smallest eigenvalue. When the -particle ground space minimizes the energy of both terms (movement and interaction) separately, and we say the Hamiltonian is frustration free.

There is a connection between the Bose-Hubbard model on a graph and the XY model on the same graph. Consider the subspace of bosonic -particle states that have zero energy for the second (interaction) term in (4). (For example, any frustration-free state lives in this subspace.) States in this subspace have no support on basis states where more than one particle occupies any vertex of the graph. This is the subspace of hard-core bosons. Since every vertex can be occupied by at most one particle, this subspace can be identified with the Hamming weight subspace of qubits (each qubit represents a vertex, with basis states representing unoccupied and occupied states, respectively). Thus the action of the Bose-Hubbard model in the subspace of hard-core bosons is equivalent to a spin model. In fact, as discussed in [3], the restriction of (4) to the subspace of hard-core bosons is exactly equal to the restriction of (given by (3)) to the Hamming weight space .

2.3. Gate graphs

The QMA-completeness construction from reference [3] uses a class of graphs called gate graphs that we now review.

2.3.1. The graph

A -vertex simple graph denoted plays a central role in the construction. The graph is defined explicitly in reference [3] by specifying its adjacency matrix ; here we only describe the properties that we use in this paper. The vertices of are labeled by tuples

where . The adjacency matrix acts on the Hilbert space

The smallest eigenvalue of is . The corresponding eigenspace has an orthonormal basis given by the four states

(5)
(6)

where ,

and

Note that, depending on the value of in the second register, the first register of the state contains the output of a single-qubit computation where either the identity, Hadamard, or gate is applied to the state . (The state of the third register is in a product state with the first two, and is somewhat uninteresting; this register exists for technical reasons.)

2.3.2. Gate graphs and gate diagrams

A gate diagram is a schematic representation of a gate graph. To define gate graphs, we first define gate diagrams and then describe how a graph is associated with each of them.

The simplest gate diagrams are shown in Figure 1. These three basic gate diagrams are also called diagram elements. The diagram elements each represent the same gate graph which is just the -vertex graph described above. Each diagram element has a unitary label which is either , , or , as well as a set of eight circles that we call nodes. A node labeled is associated with the eight vertices of labeled with . Note that only half of the possible nodes appear in a given diagram element. Moreover, the half that does appear depends on the unitary label , , or .

A gate diagram is constructed by taking a set of diagram elements and adding edges between some pairs of nodes and self-loops to other nodes. Each node may have a self loop or an incident edge but never both (and never more than one edge or self-loop). If the gate diagram has diagram elements, then each node can be labeled with , and . We write for the set of nodes that have self-loops attached and we write for the set of pairs of nodes that are connected by an edge.

((a))

((b))

((c))
Figure 1. Diagram elements.

A gate diagram with diagram elements, self-loop set , and edge set is associated with a gate graph as follows. The vertex set of is just copies of the vertex set of , with vertices labeled with , , , and . The adjacency matrix acts on the Hilbert space

and is defined by

(7)
(8)
(9)

where we write and for the identity operators on the first and fourth registers, respectively. Note that and are always positive semidefinite, so the smallest eigenvalue of is lower bounded by , the smallest eigenvalue of . When these quantities are equal, we say is an -gate graph.

Definition 1.

A gate graph is an -gate graph if its smallest eigenvalue is .

Suppose is a ground state of the adjacency matrix of an -gate graph . For each , define

(where the states are defined in (5)(6)) and note (from equation (7)) that , where

(10)

and .

2.4. Spectral bounds for positive semidefinite matrices

Throughout our proof, we bound eigenvalue gaps of positive semidefinite matrices. For a positive semidefinite matrix , let denote the smallest nonzero eigenvalue.

Fact 1.

Let and be positive semidefinite matrices. Let have nonempty nullspace . Then .

Proof.

Let be an eigenvector of with eigenvalue .

Suppose first that the nullspace of is nonempty. In this case the nullspace of is equal to the nullspace of (since both and are positive semidefinite), and hence is orthogonal to this space. Thus

If instead the nullspace of is empty, then is the smallest eigenvalue of and again is variationally upper bounded by . ∎

A version of the following lemma was used but not explicitly stated in reference [6]. We gave a proof of this “Nullspace Projection Lemma” in [3] and used it extensively in that work. We recently became aware of another work [1] that proves a slightly stronger bound than the one from [3]. We quote the better bound here.

Lemma 1 (Nullspace Projection Lemma [1]).

Let . Suppose the nullspace of is nonempty and

Then

(11)

3. Previous results

In this Section we summarize results from reference [3] that are used in this paper.

In [3] we considered the problem of approximating the ground energy of the Bose-Hubbard model on a graph at fixed particle number. We also considered a special case of this problem where the goal is to determine whether the Hamiltonian is close to being frustration free (recall that in our setting, frustration freeness means ).

{mdframed}
Problem 1 (-Frustration-Free Bose-Hubbard Hamiltonian).

We are given a -vertex graph , a number of particles , and a precision parameter . The integer is provided in unary; the graph is specified by its adjacency matrix, which can be any symmetric - matrix. We are promised that either (yes instance) or (no instance) and we are asked to decide which is the case.

Here is given in unary so that the input size scales linearly with . An algorithm for this problem is considered efficient if it uses resources polynomial in .

The positive integer in the above definition parameterizes how close to frustration free the Hamiltonian is in the yes case. This definition slightly generalizes one presented in [3]; the definition of the “Frustration-Free Bose-Hubbard Hamiltonian” problem given in that paper corresponds to the choice . Here it is convenient to explicitly define computational problems for each positive integer . The proof of QMA-completeness presented in reference [3] applies to each (as noted on page 11 of [3]).

Theorem 1 ([3]).

For any positive integer , the problem -Frustration-Free Bose-Hubbard Hamiltonian is QMA-complete.

In fact, the results of reference [3] can be used to show that each of these problems is QMA-hard when restricted to a certain subset of instances. In particular,

Corollary 1.

For any positive integer , the problem -Frustration-free Bose-Hubbard Hamiltonian remains QMA-hard if we additionally promise that the graph is an -gate graph described by a gate diagram with diagram elements, satisfying

(12)

where is an absolute constant.

In Appendix A we prove this Corollary using the results of [3].

Using a reduction based on the connection between spins and hard-core bosons, we established QMA-completeness of the following problem.

{mdframed}
Problem 2 (XY Hamiltonian with local magnetic fields).

We are given a -vertex graph , an integer , a real number , and a precision parameter . The positive integer is provided in unary; the graph is specified by its adjacency matrix, which can be any symmetric - matrix. We are promised that either (yes instance) or else (no instance) and we are asked to decide which is the case.

Theorem 2 ([3]).

XY Hamiltonian with local magnetic fields is QMA-complete. Moreover, there is a direct reduction that maps an instance of -Frustration-Free Bose-Hubbard Hamiltonian specified by , , and , to an instance of XY Hamiltonian specified by , , , and , with the same solution.

This reduction is presented in Appendix B of [3]. For our purposes it is crucial that the reduction does not change the underlying interaction graph. In this paper we show that the -Frustration-Free Bose-Hubbard Hamiltonian on simple graphs is QMA-complete, and then we use the above reduction to show that the XY model (on simple graphs, i.e., without local magnetic fields) is QMA-complete.

4. Adding Self-Loops

In general, a gate graph is not a simple graph since it may have self-loops. From equations (8) and (9), we see that self-loops in the gate graph arise from both self-loops and edges in its gate diagram. An edge or a self-loop in the gate diagram is associated with self-loops or self-loops in , respectively. In this Section we describe a mapping from any -gate graph to a modified graph . The graph is not a gate graph. It is designed so that it has a self-loop on each of its vertices. We prove that certain properties of are related to those of . Ultimately our goal is to establish a relationship between the ground energies of the Bose-Hubbard models on these graphs.

4.1. Definition of

Consider an -gate graph described by a gate diagram with diagram elements and edge and self-loop sets and , respectively.

Define to be the set of vertices without self-loops in , i.e.,

(13)

Note that contains if and only if it contains for all , and further that for each , there exists some such that for all and .

The vertex set of is two copies of the vertex set of . We label the vertices of as

We define by its adjacency matrix

(14)

where

The first term of (14) is just two copies of ; the second term adds edges between the two copies as well as self-loops to both copies. Note that every vertex of has a self-loop.

4.2. Relationship between the adjacency matrices of and

Since commutes with , there is an eigenbasis for where each vector is of the form or . For states of the latter form, the second term in (14) vanishes. From this we see that, if is in the ground space of , then is in the ground space of and its ground energy is . We now prove that these states are a basis for the ground space.

Lemma 2.

Let be an -gate graph and let be the ground space of . Let

(15)

Then the ground space of is , and furthermore,

(16)
Proof.

To prove (15) it suffices to show that no state of the form is in the ground space of (since commutes with ).

Suppose (to reach a contradiction) that is in the ground space. Since the ground energy of is equal to , we must have

and hence

(17)

Now, since is in the -energy ground space of , it is contained in the space defined in equation (10). We now show that

(18)

which contradicts (17); this will show that no such state exists. It also establishes (16).

To establish (18), first observe that for we have , so the operator is block diagonal with a block for each (each block is a principal submatrix corresponding to the subspace spanned by the states with ). It is therefore sufficient to establish (18) for each block individually.

Focus on the block labeled by some . Now we use the fact about that is noted in the text following equation (13), namely, that there exists some such that for all and . Using this fact we can write

for some positive semidefinite operator . To finish the proof of equation (18), we show that the first term on the right-hand side is strictly positive within the block labeled by , which follows from

where in the last equality we used equations (5) and (6). ∎

Lemma 3.

Let be an -gate graph. Then

where is an absolute constant.

Proof.

We use the Nullspace Projection Lemma. Write as a sum of two positive semidefinite operators

Note that since is a projector. We will also need a bound on where is the nullspace of . Note that where is defined in equation (15). By Lemma 2, the nullspace of is . Using this fact, we see that is equal to the smallest eigenvalue of within the space , which we bound using equation (16):

Now applying the Nullspace Projection Lemma, we get

(19)

To complete the proof, we show that is upper bounded by an absolute constant. Looking at the general expression (7) for the adjacency matrix of a gate graph, we see that

This completes the proof: the right-hand side of this expression is an absolute constant since is a fixed 128-vertex graph. ∎

4.3. Relationship between the Bose-Hubbard models on and

We begin by defining a linear map from the Hilbert space of distinguishable particles on to the corresponding space for . This map is defined by its action on basis states as follows:

where is a superposition of the two vertices and of that are associated with vertex .

For a state we write for its image under this mapping, i.e.,

Clearly, if is normalized then so is . Furthermore, if (i.e., if it is symmetric under permutations of the registers) then .

Lemma 4.

Let . Then

Proof.

Writing and for the two vertices of corresponding to a vertex , we have

We can then rewrite each term in the sum as

Using this expression (and a similar expression for ), we see that

Summing both sides over and gives the claimed result. ∎

Lemma 5.

Let be an -gate graph. Then

(20)

If in addition is described by a gate diagram with diagram elements and satisfies for some absolute constant , then

(21)

where is an absolute constant.

Proof.

Let be a normalized state with minimal energy for , i.e., . The normalized state satisfies

for each , so

where in going from the second to the third line we used the fact that is positive semidefinite. Using this inequality and Lemma 4, we have

which establishes equation (20).

Now suppose and and consider the second part of the Lemma. Note that in this case has no nullspace, so . We write with positive semidefinite operators

and we use the Nullspace Projection Lemma.

To get a bound on , first note that every eigenvalue of is also an eigenvalue of the operator

(22)

(without the restriction to ), since this operator is permutation symmetric and preserves the symmetric subspace. Hence the smallest nonzero eigenvalue of is at least that of (22) and

(23)

where is an absolute constant (in the second step we used the fact that for any Hermitian matrix with smallest eigenvalue zero, and in the third step we used Lemma 3).

We also need a bound on , where is the nullspace of . Letting be the nullspace of

and using Lemma 2, we see that is equal to the image of under the mapping . Using this fact and Lemma 4, we get

Since has no nullspace, neither does the operator on the right-hand side of this equation. Hence and . Furthermore

(24)

where in the first inequality we used Fact 1.

Now using the bounds (23) and (24) along with the fact that and applying the Nullspace Projection Lemma, we get

Thus the result follows with . ∎

5. Removing self-loops

Our goal is to consider simple graphs, but so far we have described a method for mapping an -gate graph to a graph with self-loops on every vertex. We now remove all the self loops from to obtain a simple graph . The adjacency matrix of this graph is

and it has smallest eigenvalue .

Now consider the -particle Bose-Hubbard model on . We have

so

In particular, the smallest eigenvalues of these two Hamiltonians are equal:

(25)

We now use this relationship to show that the following problem is QMA-complete. {mdframed}

Problem 3 (-Frustration-Free Bose-Hubbard Hamiltonian on simple graphs).

We are given a -vertex simple graph , a number of particles , and a precision parameter . The integer is provided in unary. We are promised that either (yes instance) or (no instance) and we are asked to decide which is the case.

Theorem 3.

For any positive integer , the problem -Frustration-Free Bose-Hubbard Hamiltonian on simple graphs is QMA-complete.

Proof.

Let be a fixed positive integer. The problem is clearly contained in QMA since it is a special case of the QMA-complete problem -Frustration-Free Bose-Hubbard Hamiltonian. To show that it is QMA-hard, we reduce from another QMA-hard problem. Let and define “Problem A” to be the special case of -Frustration-Free Bose-Hubbard Hamiltonian where the graph is promised to be an -gate graph satisfying (12). Corollary 1 implies that Problem A is QMA-hard. We provide a reduction from Problem A to -Frustration-Free Bose-Hubbard Hamiltonian on simple graphs.

Let an instance of Problem A be given, specified by , , and . We assume that is smaller than some absolute constant . We show that any such instance of Problem A has the same solution as the instance of -Frustration-Free Bose-Hubbard Hamiltonian on the simple graph , with number of particles and precision parameter . This is sufficient to prove QMA-hardness of -Frustration-Free Bose-Hubbard Hamiltonian on simple graphs since there are only finitely many instances of Problem A (and of -Frustration-Free Bose-Hubbard Hamiltonian) with lower bounded by a constant.

First we check that , , and satisfy the conditions in the definition of the problem, i.e., that they specify a valid instance of -Frustration-Free Bose-Hubbard Hamiltonian. Let and . Then , and since , we have