Universal qudit Hamiltonians

# Universal qudit Hamiltonians

Stephen Piddock  and Ashley Montanaro
School of Mathematics, University of Bristol, UK
stephen.piddock@bristol.ac.uk.ashley.montanaro@bristol.ac.uk.
###### Abstract

A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (-level systems):

• We completely characterise the -qudit interactions which are universal, if augmented with arbitrary 1-local terms. We find that, for all and all local dimensions , almost all such interactions are universal aside from a simple stoquastic class.

• We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the and Heisenberg interactions are universal for all local dimensions (spin ), implying that a quantum variant of the Max--Cut problem is QMA-complete. We also show that for all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model.

• We prove universality of any interaction proportional to the projector onto a pure entangled state.

## 1 Introduction

What does it mean to say that a class of (quantum-)physical systems is complex? One perspective is to look at the physical phenomena displayed by that type of system. If these phenomena are rich and complex, then the system arguably can be said to be complex itself. Another perspective is to look at the computational power of the system: the ability to build a universal computer using the system would serve as strong evidence that the system is complex.

Interestingly, in some cases these notions of complexity are equivalent. Recent work by us, together with Cubitt, introduced and characterised the notion of universality in many-body quantum Hamiltonians [17]. A family of Hamiltonians is said to be universal if any other quantum Hamiltonian can be simulated arbitrarily well by some Hamiltonian in that family. By “simulate”, we mean the following (see Section 2 below for a formal definition): Hamiltonian simulates Hamiltonian if the low-energy part of is close to in operator norm, up to a local isometry (i.e. a map which associates each subsystem of the system with a discrete set of subsystems of the system).

This notion of simulation is very strong, as it implies that the low-energy part of reproduces all physical properties of (such as eigenvalues, ground states, partition functions, correlation functions, etc.) [17]. Universality is correspondingly a very strong notion. As a universal family of Hamiltonians can simulate any other quantum Hamiltonian, any physical phenomenon that can occur in a quantum system must occur within Hamiltonians picked from . This implies that the ability to implement Hamiltonians in allows universal “analogue” simulation of arbitrary quantum systems [20, 14]. In addition, if one also assumes that the simulation can be computed efficiently (as is usually the case), universal families of Hamiltonians are computationally universal, in a number of senses [17]. First, they can be used to perform arbitrary quantum computations, either by preparing a simple initial state, evolving according to for some time and measuring, or via adiabatic evolution. Second, the problem of approximately computing the ground-state energy of Hamiltonians from is QMA-complete, where QMA is the quantum analogue of the complexity class NP [7, 21], and hence expected to be computationally hard.

A natural way to classify physical systems is in terms of the types of interactions that they are built from. Let be a set of interactions on up to qudits (-level subsystems), i.e. each element of is a Hermitian operator on for some . Then we say that an -qudit Hamiltonian is an -Hamiltonian if

 H=∑iαiH(i), (1)

where for all , and the non-trivial part of is picked from . That is, for some . is a so-called -local Hamiltonian. We stress that the coefficients can (usually) be either positive or negative. We also say that is an -Hamiltonian with local terms if it can be written in the form (1) by adding arbitrary 1-local operators. The form (1) encompasses a vast array of the Hamiltonians studied in condensed-matter physics, such as the general Ising model () and the general Heisenberg model (). In the case where for some , we just call an -Hamiltonian.

Determining the complexity of -Hamiltonians is a natural quantum generalisation of the long-running programme in classical complexity theory of classifying constraint satisfaction problems (CSPs) according to their complexity. Beginning with Schaefer’s famous 1978 dichotomy theorem for boolean CSPs [40], which has been extended in many different directions since (see e.g. [15, 42] for references), this project aims to pinpoint, for each possible set of constraints , the complexity of a CSP that uses only constraints from (perhaps weighted, to give an optimisation problem). A quantum generalisation of this question is to determine the complexity of approximately computing the ground-state energy of -Hamiltonians up to precision [21]. This problem, which we call simply -Hamiltonian, is a special case of the Local Hamiltonian problem, which in general is QMA-complete [27, 29] when contains all -qubit interactions for any fixed . The classical special case of the -Hamiltonian problem corresponds to containing only diagonal interactions; such problems are known as “valued” or “generalised” CSPs, and a full complexity classification of these was only obtained in 2016, by Thapper and Živný [42].

A full classification was given in [16] of the computational complexity of the -Hamiltonian problem in the special case where all interactions in are on at most 2 qubits; this was sharpened by [10], which showed that one complexity class in the classification was equivalent to the previously studied class StoqMA [8]. It was later shown in [17] that each of the classes in [16] corresponds to a physical universality class. These results can be summarised as follows:

###### Theorem 1 ([30, 25, 16, 17, 10]).

Let be any fixed set of two-qubit and one-qubit interactions such that contains at least one interaction which is not 1-local. Then:

• If there exists such that locally diagonalises , then -Hamiltonians are universal classical Hamiltonian simulators [30] and the -Hamiltonian problem is NP-complete [25, 16];

• Otherwise, if there exists such that, for each 2-qubit matrix , , where and , are arbitrary single-qubit interactions, then -Hamiltonians are universal stoquastic Hamiltonian simulators [17] and the -Hamiltonian problem is StoqMA-complete [10, 16];

• Otherwise, -Hamiltonians are universal quantum Hamiltonian simulators [17] and the -Hamiltonian problem is QMA-complete [16].

A stoquastic Hamiltonian is one whose off-diagonal elements in the standard basis are all nonpositive. Here we sometimes generalise this terminology slightly by also calling stoquastic if there exists a local unitary such that is stoquastic.

### 1.1 Our results

Here we continue the programme of classifying universality of Hamiltonians – and hence the computational complexity of the -Hamiltonian problem – by generalising from qubit interactions to qudit interactions, i.e. local dimension , or equivalently spin . As well as being a natural next step from the perspective of computational complexity, this framework includes many important models studied in condensed-matter theory [1, 6, 24, 28, 31, 33, 39]. However, it is significantly more difficult than the qubit case. One reason for this is that in the case of qubits, there was a simple “canonical form” into which any 2-qubit interaction could be put by applying local unitaries [16], which dramatically reduced the number of types of interaction that needed to be considered. No comparably simple canonical form seems to exist for  [32].

We first consider -Hamiltonians with local terms. This is a more general setting than just -Hamiltonians, and hence easier to prove universality results. From a computer science point of view, allowing free local terms corresponds to allowing arbitrary constraints or penalties on individual variables in a CSP. For conciseness, we say that is LA-universal (“locally assisted universal”) if the family of -Hamiltonians with local terms is universal. Similarly, we say that is LA-stoquastic-universal if it can simulate any stoquastic Hamiltonian. Then our main result about universality with local terms is a complete classification theorem:

###### Theorem 2.

Let be a set of interactions, which are not all 1-local, between qudits of dimension . Then is:

• stoquastic and LA-stoquastic-universal, if there exists such that all interactions in are, up to the addition of 1-local terms, given by a linear combination of operators taken from the set ;

• LA-universal, otherwise.

We note some general consequences of this result for Hamiltonians assisted by local terms. First, we see that any nontrivial -qudit interaction can be used to simulate an arbitrary stoquastic Hamiltonian. Second, almost any -qudit interaction can actually be used to simulate arbitrary general Hamiltonians. Third, perhaps surprisingly, there exist Hamiltonians whose 2-local part is diagonal, but which are LA-universal.

We highlight some examples for . Consider

The single interaction in is equal to plus some 1-local terms, so is stoquastic and LA-stoquastic-universal. On the other hand, the interaction in cannot be decomposed in this way, so is LA-universal. So, for example, given access to interactions of the form of and arbitrary local terms, one can perform universal quantum computation.

Next we consider the more general -Hamiltonian problem, where the lack of “free” 1-local terms makes it much more challenging to prove universality results. Here we focus on qudit generalisations of the qubit Heisenberg (exchange) interaction (). Hamiltonians built from this interaction enjoy significant levels of symmetry, which made it one of the most difficult cases to prove universal in previous work [16, 17]. The most symmetric such generalisation in local dimension is the Heisenberg model (often known as “ Heisenberg model” in the literature [33, 6]), where the interaction is

 h=d2−1∑a=1Ta⊗Ta (2)

for some traceless Hermitian matrices such that . Up to adding an identity term and rescaling, is just the swap operator, or the projector onto the symmetric subspace of two qudits,

 Psym=14∑i,j(|ij⟩+|ji⟩)(⟨ij|+⟨ji|).

is invariant under conjugation by local unitaries, implying that the eigenspaces of any Hamiltonian built only from interactions inherit this property. Nevertheless, we have the following result:

###### Theorem 3.

For any , the Heisenberg interaction , where are traceless Hermitian matrices such that , is universal. This holds even if the weights in the decomposition (1) are restricted to be non-negative.

The special case of Theorem 3 was shown in [17]. As a corollary of Theorem 3, we obtain QMA-hardness of a quantum variant of the Max--Cut problem [18] (equivalently, a quantum generalisation of the (classical) antiferromagnetic Potts model [44]). In the Max--Cut problem, we are given a graph where each edge has a non-negative weight , and are asked to partition the vertices into sets, such that the sum of the weights of edges between vertices in different sets is maximised. That is, we find a map from each vertex to an integer such that is maximised. The natural “quantum” way of generalising this problem is to replace each vertex with a -dimensional qudit, and replace each weighted edge across two vertices with a weighted projector onto the symmetric subspace across the corresponding qudits (equivalently, an interaction ). Then the task is to approximate the ground-state energy of the corresponding Hamiltonian , up to precision . Call this problem Quantum Max--Cut.

To see why this is a suitable (and non-trivial) generalisation, note that gives an energy penalty to a pair of qudits that are both in the same computational basis state, similarly to the classical case, but that the behaviour of the quantum variant can sometimes be quite different. For example, consider the case , and four vertices arranged in an unweighted cycle. Classically, the vertices can clearly be partitioned into two sets such that there are no edges between vertices in the same set. However, there is no quantum state that is simultaneously in the ground space of all corresponding projectors . This is because the unique ground state of is maximally entangled, and each qubit cannot be maximally entangled with both of its neighbours simultaneously.

It is an immediate consequence of Theorem 3 that:

###### Corollary 4.

For any , Quantum Max--Cut is QMA-complete.

The special case of Corollary 4 was shown in [38].

Next, we consider the case where the interactions are of the form for an entangled two qudit state .

###### Theorem 5.

Let be the projector onto an entangled two-qudit state . Then -Hamiltonians are universal.

In fact, Theorem 5 holds even in the restrictive setting where all the interactions are required to sit on the edges of a bipartite interaction graph (see Section 6 for a precise statement). Entanglement is a very well studied property of quantum systems, and is well known to be fundamental to many interesting quantum phenomena. This result can be viewed as an intriguing and apparently tight link between entanglement and universality.

A perhaps more familiar, and also very well-studied, interaction we consider is another generalisation of the qubit Heisenberg interaction (e.g. [1, 37, 34]): the Heisenberg interaction in local dimension (often just called the “spin- Heisenberg interaction”, where ). Now the interaction is of the form

 h=Sx⊗Sx+Sy⊗Sy+Sz⊗Sz,

where , , generate a -dimensional irreducible representation of and correspond to the familiar Pauli matices , , (up to an overall scaling factor). Note that, although the Lie algebra involved is the same as for the qubit case, the interaction may have very different properties for higher ; for example, it has distinct eigenvalues (see equation (40) below). Nevetheless, this generalisation turns out to be universal too:

###### Theorem 6.

For any , the Heisenberg interaction , where , , are representations of the Pauli matrices , , , is universal.

Finally, we consider yet another well-studied generalisation of the Heisenberg model (see e.g. [2, 24, 28, 31]): the general bilinear-biquadratic Heisenberg model in local dimension (spin 1). Here the interaction used is

 h(θ):=(cosθ)h+(sinθ)h2,

where is an arbitrary parameter and is the spin-1 Heisenberg interaction, which can be written explicitly as

 h=X3⊗X3+Y3⊗Y3+Z3⊗Z3 (3)

where

 X3=1√2⎛⎜⎝010101010⎞⎟⎠,Y3=i√2⎛⎜⎝0−1010−1010⎞⎟⎠,Z3=⎛⎜⎝10000000−1⎞⎟⎠.

The special case corresponds to the famous Affleck-Kennedy-Lieb-Tasaki (AKLT) model [2]. Our result here is as follows:

###### Theorem 7.

Let , where is an arbitrary parameter and is the spin-1 Heisenberg interaction. For all , is universal.

We therefore see that, although different values of may correspond to very different physics [31], from a universality point of view they are all of equal power.

We remark that, in common with most previous work in this area [16, 17], we usually allow each interaction weight to be positive or negative. This can lead to physical systems built from the same interaction having very different physical properties (e.g. antiferromagnetism vs. ferromagnetism). It is sometimes possible to prove universality-type results for interactions whose weights all have the same sign [38]; we achieve this in Theorem 3, but in general leave this extension for future work. Another interesting direction is to prove universality for systems with simpler interaction patterns [36, 41, 38, 17], or with less heavily-weighted interactions [13].

### 1.2 Related work

There has been a substantial amount of work characterising the complexity of various types of qubit Hamiltonians from the perspective of QMA-completeness; see [17, 7, 21] for references. In the case of qudits, rather than general classification results, most work has considered carefully designed special cases where QMA-completeness can be achieved. Indeed, it is often the case that these results aim to reduce the local dimension of a QMA-complete construction that achieves some other desiderata. For example, Aharonov et al. [3] gave a QMA-complete family of local Hamiltonians on a 1D line with , later improved to by Hallgren, Nagaj and Narayanaswami [23]; Gottesman and Irani [22] gave a QMA-complete family of translationally invariant Hamiltonians on a 1D line with , later improved to by Bausch, Cubitt and Ozols [4]. The local dimension has been reduced even further to , for a translationally invariant Hamiltonian on a 3D lattice [5]. We refer to [7] for further examples, including the more general case where the local dimension can vary across the system being considered. In all these cases, one fixes the dimension and then carefully tunes the types of interactions used to achieve the desired result. Here, by contrast, we begin with a fixed set of interactions and attempt to determine the complexity of Hamiltonians based on these interactions.

### 1.3 Overview of proof of Theorem 2

We now give an informal discussion of our LA-universality classification result. The majority of the work to prove Theorem 2 is taken up by the special case of 2-local interactions, and sets containing only one interaction. To prove universality of an interaction , we use simulations: showing that an interaction known to be universal [16, 17] can be implemented using Hamiltonians consisting of terms and 1-local terms. Our simulations are all based on perturbative gadgets, as introduced in [27] and used for example in [10, 17, 36], to effectively implement one Hamiltonian within the ground space of another. For example, a type of gadget we often use is a so-called mediator gadget. In this type of gadget, one or more ancilla (“mediator”) qudits are added to the system. Strong interactions within the mediator qudits effectively project these qudits into a fixed state. Then weaker interactions between the mediator and original qudits implement effective interactions between the original qudits. The interactions produced are determined rigorously via perturbation theory.

First we consider the special case of diagonal interactions with 2-local rank , where the 2-local rank of an interaction is informally defined as follows: Writing , and for some basis of Hermitian matrices, the 2-local rank of is the rank of . (For example, has 2-local rank 1.) We can think of diagonal matrices symmetric under qudit interchange and with 2-local rank 2 as being of the form for some diagonal matrices and . To show that such interactions are universal (a similar argument works for non-symmetric interactions), we use our free 1-local terms to apply a heavy interaction to each qudit which effectively projects it into a 2-dimensional subspace. Note that even though and commute, this need not be the case for the corresponding projected qubit interactions. This allows us to generate a 2-qubit effective interaction within this subspace which is universal [17].

Remaining within the special case of diagonal interactions, the next step is to consider those with 2-local rank 1, which are of the form . To deal with this case, we split into two parts. When has at least 3 distinct eigenvalues, we design a gadget using an additional qudit to implement the effective interaction , which is universal from the previous case. When has 2 distinct eigenvalues, but is not of the form , we show that another gadget can be used to simulate an interaction where has 3 distinct eigenvalues. For the remaining diagonal case – interactions of the form for – we show that local unitary rotations can be used to transform any Hamiltonian built of such interactions into a stoquastic Hamiltonian, so we cannot expect this case to be universal.

We then move on to non-diagonal interactions. We first consider those of the form for some that does not commute with (otherwise we would be in the diagonal case). For all such interactions, we show there exists a gadget which projects the interaction onto a 2-qubit subspace on which the resulting interaction is universal. The non-commutativity makes this task simpler than in the diagonal case. The next step is interactions with 2-local rank , but not of the form . For these, we show that one can always produce an effective interaction of the form using two rounds of simulation.

All 2-qudit interactions can be handled using one of these lemmas. Considering the interaction formed by deleting the 1-local parts from , we know that is LA-universal if the 2-local rank of is . If not, then for some and . Either has 2-local rank , or is proportional to . Either way, we are in one of the previously considered cases.

The final step to complete the proof of Theorem 2 is to generalise to -local interactions for . To do so, we show that our free 1-local terms can be used to extract 2-local “sub-interactions” from the interactions we are given; this is a generalisation to of an analogous argument for qubits in [16]. Then either we can produce a universal sub-interaction, or all the sub-interactions of all interactions in are proportional to , up to 1-local terms. In the latter case, the overall interactions must all have been of the form , so the whole Hamiltonian is stoquastic.

### 1.4 Overview of proof of Theorems 3, 5, 6 and 7

The techniques required to prove universality of interactions without free local terms are very different, and in general this setting is much more challenging. Given the symmetry displayed by the interactions we consider, we need to consider some notion of encoding in order to implement arbitrary effective interactions. In the case of the Heisenberg interaction, we proceed by using a perturbative gadget to encode a qubit within the 2-dimensional ground space of a system of qudits; this generalises a similar (but significantly simpler) gadget used for the case in [17]. Interactions across pairs of qudits within the gadget implement effective and interactions, while interactions across two gadgets can be used to implement a non-trivial 2-qubit interaction, which is enough to prove universality using the results of [17, 38]. In order to analyse the gadget’s behaviour, we need to use the representation theory of the Lie algebra , and in particular analysis of quadratic Casimir operators [19], which are operators of the form for some representation of the generators of . The Hamiltonian corresponding to the Heisenberg interaction on the complete graph on qudits turns out to have a close connection to the Casimir operator corresponding to the representation , whose spectral properties are well-understood, and which has beautiful algebraic features that enable suitable gadget weights to be determined for any .

Theorem 5 is proven using a gadget that shows that, when is the projector onto an entangled state of two qudits, -Hamiltonians can simulate -Hamiltonians for some where either is an entangled state of two qubits, in which case universality follows from Theorem 1; or , in which case universality can be shown to follow from universality of the -Heisenberg interaction (Theorem 3).

The gadget for the Heisenberg interaction also relies on properties of the corresponding Casimir operator, but is more complicated than the case. Here the key technical step is to give a gadget that allows interactions to be simulated, given access to interactions; once this is achieved, it is not too hard to show that for any , this allows the Heisenberg interaction to be simulated in local dimension 3 (qutrits). Applying the gadget again, we can produce the interaction , which (in local dimension 3) is the same as the Heisenberg interaction, and hence universal. The analysis of this gadget depends on fourth-order perturbation theory, for which we need to prove a new general simulation lemma based on the Schreiffer-Wolff transformation [9]. Previous work gave general simulation lemmas for up to third-order perturbations [10], but extending this line of argument to fourth-order is more complex technically; in particular, there are non-trivial interference effects between different gadgets to take into account. We thus hope that this result will find other applications elsewhere.

We note that higher order perturbation theory has been considered before in the literature in slightly different settings, mostly in a framework where only the ground state energy is reproduced; for example [26] considers perturbation theory at arbitrary order. Although the contribution of the fourth order term in a Schreiffer-Wolff perturbative series has been considered before [12], we are not aware of any explicit demonstration of how the interactions must be chosen such that this fourth order term dominates as in Lemma 12. Cross gadget interference has previously been seen before for certain parameter regimes of low strength Hamiltonians [11], where it can be easily shown to disappear simply by increasing the strength of the interactions; whereas in Lemma 13, the cross gadget terms are independent of the strength of the Hamiltonian.

Finally, for the remaining bilinear-biquadratic Heisenberg interactions in dimension 3, we use different gadgets depending on the value of , which we can assume is within the range because we are free to choose the signs of interactions arbitrarily. When and , then there exists an entangled state which is either the unique ground state or the unique highest excited state of . Using a perturbative gadget to effectively project some qudits onto , we can obtain a new interaction for some . Taking a linear combination of these two interactions, we can simulate the Heisenberg interaction. When , has a 3-dimensional ground space. We encode a qutrit within this subspace of two physical qutrits, and use interactions across pairs of qutrits to simulate the Heisenberg interaction across logical qutrits. These ranges encompass all values of except . In this last special case, corresponds to the well-studied AKLT interaction [2]. Here the ground space of is 4-dimensional, but we are able to construct a mediator qutrit gadget which effectively projects 3 qutrits into the unique ground state of a 3 qutrit AKLT Hamiltonian. This again allows us to simulate the Heisenberg interaction.

## 2 Summary of techniques

Next, we give the required definitions to state our results formally, describe previous results that we use, and exemplify our results by giving a simple example of a simulation. We then proceed to a full technical presentation of the remainder of our results.

### 2.1 Definitions

We first formally define the notions of simulation and universality that we will use. For an arbitrary Hamiltonian , we let denote the orthogonal projector onto the subspace . We also let denote the restriction of some other arbitrary Hamiltonian to , and write and . We let denote the set of linear operators acting on a Hilbert space , and use the standard notation and for the commutator and anticommutator of and , respectively.

###### Definition 1 (Special case of definition in [17]; variant of definition in [10]).

We say that is a -simulation of if there exists a local isometry such that:

1. There exists an isometry such that and ;

2. .

We say that a family of Hamiltonians can simulate a family of Hamiltonians if, for any and any and (for some ), there exists such that is a -simulation of . We say that the simulation is efficient if, in addition, for acting on qudits, ; is efficiently computable given , , and ; and each isometry maps to qudits.

The first part of Definition 1 says that can be mapped exactly into the ground space of by some “encoding” isometry which is close to a local isometry . The second part says that the low-energy part of is close to an encoded version of . In [17] a more general notion of encoding was used, which allowed for complex Hamiltonians to be encoded as real Hamiltonians, for example; here we will not need this directly. (However, as we make use of the results of [17], we do use this notion of encoding indirectly.)

###### Definition 2 ([17]).

We say that a family of Hamiltonians is universal if any (finite-dimensional) Hamiltonian can be simulated by a Hamiltonian from the family. We say that the universal simulator is efficient if the simulation is efficient for all local Hamiltonians.

Here all simulations we develop will be efficient, so whenever we say “universal”, we mean “efficiently universal” in the above sense.

The main technique we will use to prove universality will be the remarkably powerful concept of perturbative gadgets [27]. Let be a Hilbert space decomposed as , and let denote the projector onto . For any linear operator on , write

 O−−=Π−OΠ−,O−+=Π−OΠ+,O+−=Π+OΠ−,O++=Π+OΠ+. (4)

Throughout, let be a Hamiltonian such that is block-diagonal with respect to the split , , and , where denotes the minimal eigenvalue of .

Slight variants of the following lemmas were shown in [10], building on previous work [36, 9]:

###### Lemma 8 (First-order simulation [10]).

Let and be Hamiltonians acting on the same space. Suppose there exists a local isometry such that and

 VHtargetV†=(H1)−−. (5)

Then -simulates , provided that the bound holds.

###### Lemma 9 (Second-order simulation [10]).

Let , , be Hamiltonians acting on the same space, such that: ; is block-diagonal with respect to the split ; and . Suppose there exists a local isometry such that and

 VHtargetV†=(H1)−−−(H2)−+H−10(H2)+−. (6)

Then -simulates , provided that .

###### Lemma 10 (Third-order simulation [10]).

Let , , , be Hamiltonians acting on the same space, such that: ; and are block-diagonal with respect to the split ; . Suppose there exists a local isometry such that and

 VHtargetV†=(H1)−−+(H2)−+H−10(H2)++H−10(H2)+− (7)

and also that

 (H′1)−−=(H2)−+H−10(H2)+−. (8)

Then -simulates , provided that .

We will often apply the simulation results in these lemmas to many individual interactions within a larger overall Hamiltonian, in parallel. For the gadgets we will use, it was shown in [17, Lemma 36] (following similar arguments in previous work, e.g. [36, 10]) that the overall simulation produced is what one would expect (i.e. a sum of the individual simulated interactions, without unexpected interference between the terms). In addition, the simulations that we use will either associate a fixed number of ancilla (“mediator”) qudits with each interaction, or encode each logical qudit within a fixed number of physical qudits. In each such case, the overall isometry is easily seen to be a tensor product of local isometries as required for Definition 1; for readability, we leave this isometry implicit.

Later on, we will need a new fourth-order simulation lemma. As this is more technical to state (and its proof has some additional complications involving interference), we defer it to Section 3.

### 2.3 Example: the AKLT interaction

To see how the above simulation results can be used to prove universality, we give a simple example of how the AKLT interaction [2] can simulate the Heisenberg interaction. The AKLT interaction is defined in local dimension (spin 1) by , where is the Heisenberg interaction defined in (3).

###### Lemma 11.

The AKLT interaction is universal.

###### Proof.

We will use a gadget construction to show that can simulate the invariant interaction , which is shown to be universal in Theorem 3. We will use Lemma 9 and construct a second-order mediator qutrit gadget involving 3 mediator qutrits labelled 3, 4, 5 that will result in an effective interaction between qutrits 1 and 2. Let , which has a unique ground state on qutrits 3, 4, 5

 |ψ⟩=1√6(|012⟩+|120⟩+|201⟩−|021⟩−|210⟩−|102⟩),

the completely antisymmetric state on 3 qutrits. Define to be the projector onto the ground space of , and let for some . Then one can check (either by hand or using a computer algebra package) that and

 −ΠH2H−10H2Π=−2λ2227(23h12+h212+1363I)Π.

Let for some so that . Then by Lemma 9, choosing and we can simulate

 ΠH1Π−ΠH2H−10H2Π=20(h12+h212)−2723I

which one can check is the Heisenberg interaction as desired, up to rescaling and deletion of an identity term. Note that this can only produce positively-weighted interactions, but Hamiltonians of this restricted form are indeed proven universal in Theorem 3. ∎

We will need the following lemma, which we prove for the first time here (and hence state a bit more generally than the above simulation lemmas, although we will only need on the right-hand side of (9)). The proof is technical, and hence (as with the subsequent lemma) deferred to Appendix A.

###### Lemma 12 (Fourth-order simulation).

Let , , , , be Hamiltonians acting on the same space, such that: ; and are block-diagonal with respect to the split ; . Suppose there exists a local isometry such that and

 ∥VHtargetV†−Π−(H1+H4H−10H2H−10H4−H4H−10H4H−10H4H−10H4)Π−∥⩽ϵ/2 (9)

and also that

 (H2)−−=Π−H4H−10H4Π− and (H3)−−=−Π−H4H−10H4H−10H4Π−. (10)

Then -simulates , provided that .

For fourth-order gadgets, unlike the gadgets analysed in previous work, it is unfortunately not the case that one can disregard interference between different gadgets applied in parallel; there are additional terms generated by interference between gadgets. We calculate this interference in the following lemma.

###### Lemma 13.

Consider a Hilbert space with multiple fourth-order mediator gadgets labelled by , each with heavy Hamiltonian which acts non-trivially only on , and interaction terms , , , which act non-trivially only on . Let denote the projector onto the ground space of , and . Suppose that for each , these terms satisfy the conditions of Lemma 12; in particular, , and are block diagonal with respect to the , split, and

 Π(i)−H(i)2Π(i)−=Π(i)−H(i)4(H(i)0)−1H(i)4Π(i)− and Π(i)−H(i)3Π(i)−=−Π(i)−H(i)4(H(i)0)−1H(i)4(H(i)0)−1H(i)4Π(i)−.

For each , let , and let .

Suppose there exists a local isometry such that is the ground space of and also , where

 M=∑i Π−(H(i)1+H(i)4(H(i)0)−1H(i)2(H(i)0)−1H(i)4−H(i)4(H(i)0)−1H(i)4(H(i)0)−1H(i)4(H(i)0)−1H(i)4)Π− +∑i≠jΠ−(H(i)4(H(i)0)−1H(j)4(H(j)0)−1H(j)4(H(i)0)−1H(i)4 −H(i)4(H(i)0)−1H(j)4(H(i)0+H(j)0)−1H(j)4(H(i)0)−1H(i)4 −H(i)4(H(i)0)−1H(j)4(H(i)0+H(j)0)−1H(i)4(H(j)0)−1H(j)4)Π−

and is the projector onto the ground space of .

Then simulates , provided that .

Note that the first line of the simulated Hamiltonian is what one would expect when summing the contributions of each of the gadgets separately. The other terms are in general not zero and may be thought of as the cross-gadget interference.

We will only need to use Lemma 13 via the following simplified corollary.

###### Corollary 14.

Suppose the conditions of Lemma 13 hold, and in addition for all (for example when is a projector). Then the expression for is given by

 M=∑iΠ−(H(i)1+H(i)4H(i)2H(i)4−H(i)4H(i)4(H(i)0)−1H(i)4H(i)4)Π−−12∑i
###### Proof.

For , by the additional assumption of the present corollary , so the expression for the cross-gadget interference from Lemma 13 simplifies to

 ∑i≠jΠ−(H(i)4H(j)4H(j)4H(i)4−12(H(i)4H(j)4H(j)4H(i)4+H(i)4H(j)4H(i)4H(j)4))Π− = 12∑i≠jΠ−(H(i)4H(j)4H(j)4H(i)4−H(i)4H(j)4H(i)4H(j)4)Π−=−12∑i

where we note that the sum over includes both cases and . ∎

## 4 LA-universal Hamiltonians

We first prove LA-universality (or otherwise) of various classes of interactions, before bringing these results together into a full classification theorem by showing that every interaction fits into one of these classes. Before embarking on the proof, we observe that for any interaction , we can delete its 1-local part by using our free 1-local terms. This corresponds to replacing with

 H′=H−Id⊗Tr1(H)−Tr2(H)⊗Id+Tr(H)I⊗Id2. (11)

We call the 2-local part of . For a fixed basis of Hermitian matrices, we can decompose for some real matrix . We define the 2-local rank of to be the rank of .

Note that this definition is independent of the choice of basis . Suppose we instead write for two other bases and of Hermitian matrices. Since these are bases there must exist invertible matrices and such that . Then

 H′ =∑a,b~MabSa⊗S′b =∑c,dMcdTc⊗Td=∑a,b(∑c,dRcaMcdRdb)Sa⊗S′b

and thus since and are both full rank.

We now move on to the first case of the proof, diagonal interactions.

### 4.1 Interactions diagonalisable by local unitaries

###### Lemma 15.

Let be a nonzero diagonal 2-qudit interaction. If the 2-local rank of is , then is LA-universal; otherwise, is LA-stoquastic-universal.

###### Proof.

First note that we can use 1-local terms to replace with its 2-local part, as in (11). This still results in a diagonal interaction and allows us to assume that . Let be given by for some matrix . Then the 2-local rank of is given by . Next observe that we can assume that the interaction is either symmetric or antisymmetric with respect to permuting the qudits on which it acts, because we can apply it in either direction, with positive or negative weights. So we obtain either or , corresponding to mapping either to or . This cannot affect the condition on the rank of , because

 rank(A)=rank((A+AT)+(A−AT))⩽rank(A+AT)+rank(A−AT);

if , then either , or ; but this latter possibility cannot occur because is skew-symmetric, so .

We will apply Lemma 8 by using heavily-weighted local terms to effectively project each subsystem on which acts into a 2-dimensional subspace, which will encode a qubit. Such a projection can be described by a matrix . We aim to produce an effective 2-qubit interaction which is universal. As we can apply arbitrary local terms, we can project each qudit onto an arbitrary 2-dimensional subspace by choosing a “heavy” Hamiltonian in Lemma 8 such that has as its ground space. The local isometry in the lemma is just given by .

The result of projecting is the 2-qubit interaction

for some real coefficients such that

 βik=12Tr[P|i⟩⟨i|P†σk].

Reordering the sums, we obtain

 H′=3∑k,ℓ=0(d∑i,j=1βikAijβjℓ)σk⊗σℓ=3∑k,ℓ=0⟨βk|A|βℓ⟩σk⊗σℓ,

where we define the unnormalised vector . We can write down explicit expressions for these vectors as

 βi1=Re(P∗1iP2i),βi2=Im(P∗1iP2i),βi3=12(|P1