General conditions for universality of Quantum Hamiltonians

Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the result concerns the theory of analogue Hamiltonian simulation - a promising application of near-term quantum technology - the proof relies on abstract complexity theoretic concepts and the theory of quantum computation. As well as providing simplified proofs of previous Hamiltonian universality results, and offering a route to new universal constructions, the results in this paper give insight into the origins of universality. For example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.


I. INTRODUCTION
Recent work has precisely defined what it means for one quantum system to simulate the full physics of another 1 , and demonstrated that-within very demanding definitions of what it means for one system to simulate another-there exist families of local Hamiltonians that are universal, in the sense that they can simulate all other Hamiltonians to any accuracy desired. This rigorous mathematical framework of Hamiltonian simulation not only gives a theoretical foundation for describing analogue Hamiltonian simulation -one of the most promising applications of quantum computing in the NISQ (noisy, intermediate-scale quantum) era. It also unifies many previous Hamiltonian complexity results, and implies new ones 1 . And it has even found applications in constructing the first rigorous holographic dualities between local Hamiltonians, providing toy models of AdS/CFT duality in quantum gravity 2 .
However, previous constructions of universal Hamiltonians have relied heavily on using perturbation gadgets, and constructing complicated 'chains' of simulations to prove that simple models are indeed universal. For example, the original paper on universal quantum Hamiltonians used a chain of more than 10 simulations and perturbation gadget constructions to prove universality of the 2D general Heisenberg and XY models. This work was recently extended, with the first construction of a translationally invariant universal family of quantum Hamiltonians 3 , again using a complex, carefully engineered construction.
In 4 , the present authors developed a new method of proving universality and used it to construct the first universal models acting in one dimension. This new method pointed to a connection between universality and complexity. This connection is not entirely surprising. Indeed, a rigorous complexity theoretic characterisation of universal Hamiltonians has already been demonstrated in the classical case 5 . Essentially, 5 showed that if a family of classical Hamiltonians has a ground state energy problem that is NP-hard, then it is necessarily also capable of simulating the complete physics of any other classical Hamiltonian. The converse implication is immediate.
In the quantum setting, there were hints that a similar result might hold. The classes of two-qubit interactions that are universal for simulating all, stoquastic, and classical Hamiltonians, respectively, were fully characterised in 1 , and turned out to coincide precisely with the classes of interactions that have QMA-, StoqMA and NP-complete ground state energy problems. However, the proofs of these two classifications were independent, and it was certainly possible this coincidence only applied in the case of qubits, as the proof techniques relied critically on having only two-qubit interactions.
Furthermore, the more complicated non-commutative structure of quantum Hamiltonians made it impossible to replicate the classical approach of 1 to proving a relationship between complexity and simulation.
In this paper, by extending the simulation technique developed in 4 , we resolve this. We derive necessary and sufficient complexity-theoretic conditions for a family of Hamiltonians to be an efficient universal model, relating this directly to complexity-theoretic properties of the ground state.

II. MAIN RESULTS
Our main result is a complexity theoretic classification of which families of Hamiltonians are efficient universal models: The second condition, closure, relates to combining different Hamiltonians from the same model. We say a model, M, is closed if, given (1) , (2) ∈ M, acting on (possibly overlapping) sets of qudits , respectively, there exists a Hamiltonian (3) ∈ M which can simulate (1) + (2) .
Furthermore, in this paper we provide a recipe for modifying history state Hamiltonians so that the canonical reduction from a QMA-problem to the history state Hamiltonian is faithful.
Therefore, all that remains to show that a family of history state Hamiltonians is universal is to demonstrate closure.
We also derive two corollaries, giving complexity-theoretic conditions for families of Hamiltonians to be universal models which aren't efficient in the sense of 1 , but are nonetheless interesting.
These corollaries, along with the main theorem, give a complete classification of all known universal models.

III. DISCUSSION
The most obvious implication of our results is that they provide a new route for proving universality of families of Hamiltonians. All previous universality results 1,3,4,6 relied on explicit constructions, tailored to particular universal models. These constructions showed that given some arbitrary target Hamiltonian target , a Hamiltonian from the universal model could be constructed which simulated target . These constructions typically drew on techniques from Hamiltonian complexity theory, such as perturbation gadgets and history states, but the proofs of universality required substantial additional work.
That additional work can now in many cases be side-stepped, by using the extensive existing work classifying the complexity of the local Hamiltonian problem 7-10 , along with our main result (as well as our recipe for modifying history state Hamiltonians). The remaining step is to demonstrate closure -for some families of Hamiltonians this step is trivial (it follows from the definition of the model), for others it will require some work -however demonstrating closure will always be simpler than demonstrating universality, since closure requires the ability to simulate a very limited class of Hamiltonians.
It should be noted that this method of proving universality is not constructive, in the sense that it doesn't tell you how to simulate a given target Hamiltonian with a Hamiltonian from the a universal model. However, we do not view this as a drawback when compared to previous methods of proving universality. The previous methods for proving universality were theoretically constructive -in the sense that for any target Hamiltonian they provided a mathematical description of a Hamiltonian from the universal model which could simulate it. However, in reality the cost of leaving the target Hamiltonian completely general in the previous proofs was that the simulations constructed had to be very complex -putting them out of reach of current experimental limitations.
The benefit of our work is that it gives a simple route to proving universality, so now the work on constructing explicit simulations can focus on simulations which are experimentally feasible. The other direction our results could be used in is investigating the complexity of models which are currently used as analogue simulators, to determine whether there is hope of using them to construct universal models. One platform that is currently used for analogue Hamiltonian simulation is Rydberg atoms [12][13][14] . It has been suggested 12 that this might have promise as a universal simulator as in certain regimes the platform naturally encodes the -Hamiltonian, which is known to be universal 1 . However, the proof of universality can not be used to construct a universal simulator using Rydberg atoms as it involves complicated chains of perturbative simulations, requiring precise control beyond the reach of current experiments. An alternative approach to investigating the use of Rydberg atoms as universal simulators could be to investigate the complexity of the ground state energy problem of Rydberg interactions where the control over interaction strengths is limited to what is experimentally feasible. As outlined above, demonstrating complexity in this regime is likely to be more straightforward than directly proving universality. If, within the limitations on interaction strength, it is possible to demonstrate hardness of the ground state energy problem, that motivates attempts to look for simple universal constructions. If it is not possible to demonstrate hardness of the ground state energy problem within current experimental limitations, it may be possible to determine how much experimental techniques have to advance in order to overcome the barrier and achieve universality.
Finally, the relationship between complexity and universality is interesting from a fundamental physics standpoint. It was already clear that universality implied complexity -since universal models must be able to simulate all quantum many body physics. However the reverse direction was not obvious. Our results show that if the problem of deciding whether a Hamiltonian in M has a low energy ground state is hard for a quantum computer, then M must be rich enough to capture all quantum many body physics.
The remainder of the paper contains the technical proofs of our main results. In Section IV we cover some necessary technical background on universal Hamiltonians and complexity theory.
The notion of a faithful reduction is outlined in detail in Section V; in this section we also present a family of Hamiltonians which we demonstrate is QMA-complete under faithful reductions. Finally our main theorem is proved in Section VI.

A. Universal Hamiltonians
In 1 a rigorous definition of what it means for one quantum system to simulate another was developed: ii.
where an encoding is a map of the form E ( ) = ⊗ + ⊗ † for an isometry, and a local encoding maps local observables in the target system to local observables in the simulator system, defined as: Definition IV.2 (Local subspace encoding (Definition 13 from 1 )). Let be a subspace encoding. We say that the encoding is local if for any operator ∈ Herm(H ) there exists ′ ∈ Herm(H ′ ) such that Note that the role ofẼ in Definition IV.1 is to provide an exact simulation as per 1 (Def. 20).
However, it might not always be possible to construct this encoding in a local fashion. The local encoding E in turn approximatesẼ, such that the subspaces mapped to by the two encodings deviate by at most . controls how much the eigenvalues are allowed to differ.
In 1 it is shown that approximate Hamiltonian simulation preserves important physical properties.
We recollect the most important ones in the following. i. Denoting with ( ) (resp. ( ′ )) the th -smallest eigenvalue of (resp. ′ ), then for all ii. The relative error in the partition function evaluated at satisfies iii. For any density matrix ′ in the encoded subspace for which E (1) ′ = ′ , we have If we are interested in whether an infinite family of Hamiltonians can be simulated by another, the notion of overhead becomes interesting: if the system size grows, how large is the overhead necessary for the simulation, in terms of the number of qudits, operator norm or computational resources? We capture this notion in the following definition.
Definition IV.4 (Simulation, 1 (Def. 23)). We say that a family F ′ of Hamiltonians can simulate a family F of Hamiltonians if, for any ∈ F and any , > 0 and where and are separated by an exponentially small gap is the class PreciseQMA. PreciseQMA is known to be equal to the class PSPACE 15 .

We say that the simulation is efficient if, in addition, for acting on qudits and
and only if there exists a and a quantum exponential time verifier such that for all and all ∈ {0, 1} : QMA EXP ( , ) where and are separated by an inverse exponential is the class QMA EXP .
The canonical problem in Hamiltonian complexity is -H .

Promise:
The ground state energy min ( ) either ≥ , or ≤ .
We can also consider the special case where the set of interaction terms and / or the geometry of the interaction graph is restricted (which can implicitly constrain the family's locality ).

A. Faithful Hamiltonian reductions
The acceptance operator, ( ), of a QMA-verification circuit, , is defined as 17 (Th. 3.6): where requires ancillas, initialised in the |0 state. In other words, Definition V.1 means that any state with acceptance probability below the completeness threshold already lies significantly below it, namely 1/poly bounded away.
Note there is a subtle difference between the promise gap and the question of whether or not the acceptance operator is gapped. For any NO instance of a problem ∈ QMA the definition of QMA trivially implies that the acceptance operator is gapped (since the acceptance probability is below the soundness threshold ≤ , and − > 1/poly by definition). However, for YES instances it is possible to have an acceptance operator which is not gapped. We will see that for YES instances the question of whether or not the acceptance operator is gapped is related to the spectral gap of a Hamiltonian, rather than the promise gap.
The idea of requiring a gap in the spectrum of proof systems has arisen before in the Hamiltonian complexity literature, first in 18 in the definition of the class PGQMA (Polynomially Gapped QMA). 19 The notion of a gap in the spectrum of the proof system is again seen to be related to the • Π S denotes the projector onto the subspace S, • E is some local encoding (independent of the problem being encoded), (ii). The spectral gap above the subspace S 0 is Ω(1/poly( )).

B. The modified Kitaev Hamiltonian
The Hamiltonian we use to prove necessity of the faithfulness condition is a modification of the 5-local Hamiltonian shown to be QMA-complete in 10 . Note that this choice is convenient, but the procedure we set out here to demonstrate faithfulness could be applied to any history-state Hamiltonian in the literature.
The original 5-local Hamiltonian is a "circuit-to-Hamiltonian" mapping, given by where the Hamiltonian is acting on the Hilbert space and where the correspond to the gates applied at time in the circuit being encoded.
The Hamiltonian without the output penalty, has a degenerate ground space spanned by states of the form for arbitrary where where | 0 ( ) is the state of the quantum circuit at time if the input state of the ancillas and flag qubit correspond to the binary string 0 = 0 1+| | , and the input state of the witness is given by | .

C. The K-H problem is QMA-complete under faithful reductions
Let K be the family of Hamiltonians of form Eq. (14). We begin by showing that K-H is QMA-complete, then show that we can always choose the reductions to be faithful.
Proof. The proof that K-H is QMA-complete is essentially unchanged from the proof of QMA-completeness in 10 . We sketch the argument here very briefly. Assume the circuit being encoded is a QMA-verification circuit with completeness parameter and soundness parameter .
First consider the YES instances. By definition, there exists a witness such that the verification circuit accepts with probability at least . It follows that the ground state of MK has energy less than (1− ) +1 . For the NO cases we use the following geometrical lemma.
Lemma V.5 (Geometrical lemma, Lemma 14.4 10 ). Let 1 , 2 be two Hamiltonians with ground energies 1 , 2 respectively. Suppose that for both Hamiltonians the difference between the energy of the (possibly degenerate) ground space and the next highest eigenvalue is larger than Λ, and that the angle between the two ground spaces is . Then the ground energy of 1 + 2 is at least We apply Lemma V.5 to MK with 1 = in + out and 2 = prop + clock . We have 1 = 2 = 0. The smallest non-zero eigenvalue of 1 is (since in and out are commuting projectors). The smallest non-zero eigenvalue of 2 scales as Ω(1/ 2 ) (see 10 for proof). The angle between the ground spaces satisfies Again, the proof of this is unchanged from 10 as the ground space of 1 is equal to the ground space of in + out .
Therefore in NO instances the ground energy of MK is lower bounded by To show that we can always choose the reduction to be faithful, we first prove a lemma about the spectrum and low energy subspace of MK .

Lemma V.6. Consider a modified Kitaev-Hamiltonian, MK , encoding the verification circuit
of some QMA problem. Let ( ) be the acceptance operator for a verifier circuit for some where is the completeness parameter of the problem, and let ≔ − where is the largest eigenvalue of ( ) which is less than , as in Definition V.1.
If > 2 3 ( + 1) , then there exists a unitary transformation such that the subspace S 0 defined by Π S 0 ≔ † Π C 0 is the low energy subspace of MK : and the spectral gap above S 0 is given by Ω( +1 − 3 2 ).
Proof. It is a standard result that the zero-energy ground state subspace G of 0 is spanned by history states (0, ) for all . The spectral gap of 0 is Ω(1/ 3 ) 21 .
Since out = 1 3 < 1 2 3 , the Hamiltonian MK | G can be approximated by the Schrieffer-Wolff perturbative expansion (see Appendix A). Let Π G be the projector onto G.
Lemma V.7. The K-H problem is QMA-complete under faithful reductions.
Proof. For any verification circuit, , of any problem in QMA, we can require that the computation 'idles' in its initial state for time steps before carrying out its verification computation ("idling to enhance coherence" 21 ).
The history state of the computation for the first time steps will be given by The rest of the history state is captured in So the total history state is given by The encoding E ( ) = † defined via the isometry where the | are computational basis states, is local. (This can be verified by direct calculation, Moreover, we have that can be made arbitrarily small by increasing .
The result follows immediately from Lemma V.6 and the triangle inequality.

VI. GENERAL CONDITIONS FOR UNIVERSALITY
In order to state our main theorem we require one more definition. is closed. We will explicitly construct a universal model, solely based on these conditions. Consider the following problem:

Input:
A -local Hamiltonian target acting on spins with local dimension .

Question: Output YES
This problem is (clearly) trivial. But we can choose to construct a non-trivial QMA verification circuit for it. We will choose a verification circuit which picks out a particular subspace that allows us to prove universality. By Definition V.3 there must be a faithful reduction with respect to this verification circuit from Y -H to M-H .
The verification circuit we choose, and the subspace it picks out, are captured in the following.

Lemma VI.2. Y -H can be verified by a circuit with gapped acceptance operator
(31) Proof. The verifier circuit, , acts on the witness and two ancilla registers, , ′ . It will be helpful to divide the witness into two separate registers: An register, which is -dimensional qudits. And an ′ register, which consists of qutrits with orthonormal basis states |# , |0 and |1 , where = log 2 ( ). The register is the same size as the ′ register. The ′ register consists of a single qubit.
The verifier operates as follows: The entire procedure takes time = O(poly( , , target )/ ).
be the result of applying the phase estimation algorithm on with respect to = target . Then is an eigenvector of ( ) with eigenvalue 1, and all eigenvectors of ( ) with eigenvalue 1 are in span{ }. Moreover, and The next largest eigenvalue of ( ) is 1 2 .
It follows immediately from Eq. (36) that for any encoding E.
It follows from the triangle inequality and Definition V.2(i) that for any instance of Y -H there exists LS ∈ M with low energy subspace S 0 ≔ span{| : where E = ⊗ + ⊗ † is some local encoding and can be chosen to be arbitrarily small. The spectral gap above S 0 is Ω(1/poly( )).
Another trivial problem (that is therefore also evidently in QMA) is:

Input:
Classical description of a one-qudit state | Question: Output YES.
For this problem we will use a faithful reduction with respect to the non-trivial verification circuit which simply measures a single qudit in the basis. So, for any single qudit state | , there exists ∈ M such that: for some local encoding E state , where ( ) can be efficiently computed. (Since the problem size is (1) for a state | that can be described in (1) bits.) Wlog we will take ( ) = .
Consider a Hamiltonian acting on spins: and E ′ (1) = Π S 0 . Moreover, where is a unitary satisfying and We need the following technical lemma.

Lemma VI.3 (First-order simulation 26 [Lemma 14
). ] Let 0 and 1 be Hamiltonians acting on the same space and Π be the projector onto the ground space of 0 . Suppose that 0 is zero on Π and the next smallest eigenvalue is at least 1. Let be an isometry such that † = Π and Let sim = Δ 0 + 1 . Then there exists an isometry˜ onto the the space spanned by the eigenvectors of sim with eigenvalue less than Δ/2 such that We will apply Lemma VI.3 with 1 = = ′ +1 2 −( ′ +1) (1) and 0 = LS where = O(poly( )). We have that in Π S 0 , LS has energy zero and by Definition V.2(ii) the spectral gap above S 0 scales as Ω(1/poly( )) so 0 = LS has next smallest eigenvalue at least 1. Moreover, 1 = target . Note that ′ is an isometry which maps onto the ground state of 0 , S 0 . By construction we have that the spectrum of target is approximated to within by 1 restricted to Using that the operator norm is unitarily invariant, and that ′ = gives We also have where we have used 1 (Lemma 18) in the penultimate step. So Lemma VI.3 therefore implies that there exists an isometry˜ that maps exactly onto the low energy space of sim such that ˜ − ′ ≤ O target /(Δ/ ) = O target /Δ . By the triangle inequality and Eq. (40), we have The second part of the lemma implies that Therefore, the conditions of Definition IV.1 are satisfied for a (Δ ′ , ′ , ′ )-simulation of ′ , with By definition we can choose to be arbitrarily small. We can also make O −1 arbitrarily small. By increasing , we can also make arbitrarily small. Therefore, by choosing Δ such that we can construct sim which is a (Δ ′ , ′ , ′ )-simulation of ′ with arbitrarily small ′ , ′ . Since sim is a sum of Hamiltonians which are all in M, by the closure property there exists univ ∈ M which can efficiently simulate sim . Therefore, since simulations compose 1 (Lemma 17) univ can simulate ′ .
Finally, we show that ′ = is itself a simulation of target . Consider the local encoding where = |0 , and the non local encoding We have that So by increasing we can make the norm arbitrarily small. We also have that Ẽ′ = ′ , so condition (i) from Definition IV.1 is met. The spectrum of ′ is exactly the spectrum ofẼ ′ ( target ), so condition (ii) of Definition IV.1 is also met. Therefore ′ is a simulation of target .
Using the composition of simulations again, we have that univ can simulate target . We have left target arbitrary, so M is a universal model.
Finally we consider efficiency. The simulation of target by ′ is clearly efficient. To see that the simulation of ′ by sim is efficient, note that the number of qudits in the simulation, , must be polynomial in and target as LS is in QMA. Furthermore, sim = Ω(Δ) = poly( ′ , target , 1/ ′ , 1/ ′ ) = poly( , target , 1/ ′ , 1/ ′ ). Thus sim is an efficient simulation of ′ .
There are two corollaries about universal Hamiltonians which aren't efficient in the sense of 1 , but which are nonetheless interesting universal models which are better suited to some applications. To prove universality, note that in 4 (Theorem 3.6) a construction is given of a universal Hamiltonian, succ , (with exponential overhead in terms of number of spins and norm of simulating system) which can be described succinctly.
Since there exists a Hamiltonian in M which can simulate succ (for any values of the parameters in succ ), and since simulations compose, it follows that M is a universal model. When simulating general (non-succinct) Hamiltonians, the universal model, M, inherits an exponential overhead in terms of the numbers of qudits and the norm of the simulating system from succ .
QMA EXP is a more powerful complexity class than QMA so it may seem odd that it appears to be less efficient as a simulator. However, there are some situations where using a family of Hamiltonians meeting the conditions of Corollary VI.5 will give a more efficient simulator than Consider simulating the family of Hamiltonians K using the model M to exponential accuracy in . This demonstrates that P -M-H is PSPACE-complete (including under exponentially faithful reductions), but it does not contradict the statement that M-H is not QMA-complete. This is because the -H problem requires that the terms in the Hamiltonian are of order 1, which requires dividing each term in the simulator system by the simulating system norm, which by assumption is exponential in the size of the system. This leads to a Hamiltonian with an exponentially small spectral gap, which attenuates the promise gap too fast to maintain QMA-completeness, but gives PreciseQMA-completeness (and therefore PSPACE-completeness).
An example of a universal Hamiltonian meeting the conditions of Corollary VI.7 is given in 4 .
It is a translationally invariant universal model, but includes a phase parameter which encodes information about the target system, so the interactions are not fixed.

Appendix A: The Schrieffer-Wolff expansion
Consider a finite dimensional Hilbert space decomposed into a direct sum: Let Π − be the projector onto H − and Π + be the projector onto H + . Let 0 and 1 be Hermitian operators acting on H such that 0 is block diagonal with respect to the direct sum. Assume all the eigenvalues of 0 on H − are in the range [0, 0 ] for 0 < 1.
Consider the perturbed Hamiltonian˜ = Δ 0 + 1 where Δ ≫ 1 and < Δ 2 . The Schrieffer-Wolf transformation is a unitary rotation which is used to perturbatively diagonalise˜ . It satisfies the following properties: The effective low-energy Hamiltonian eff acting on H − is given by Define R ≔ | : |˜ | ∈ 0 − Δ 2 , 0 + Δ 2 , and let Π R be the projector onto R. Then The operators and eff can be expressed as Taylor series: A systematic method for calculating the Taylor coefficients is given in 27 (Section 3.2). We will only need the first two coefficients: The size of the operators in the Taylor expansion can be bounded (see 27 (Lemma 3.4)): This implies 26