Simplifying the simulation of local Hamiltonian dynamics

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I. INTRODUCTION
A quantum simulator (QS) aims at replicating the physics of another quantum system, the properties of which are extremely difficult to obtain [1].Currently, most of the existing QS are analog, i.e., naturally realise the physics of the target system and mimic its properties.Moreover, they are highly specialized experimental platforms acting as single-purpose simulators, and all of them, independently of the physical architecture on which they are implemented, face some problems regarding applicability, scalability, complexity, state preparation, control, and measurement (see e.g.[2] and references therein).It is desirable and expected that in the forthcoming years, some analog QSs will evolve into highly flexible devices, and also digital QSs, i.e., circuitbased quantum computers, will be further developed.Both classes of QS will be capable of simulating different quantum systems, preparing specific quantum states on demand, or analyzing their dynamics.This also includes the current NISQ (Noise Intermediate Scalable Quantum) platforms such as superconducting qubits, cold atoms and ions, Rydberg atoms, etc.Such versatility demands the development of theoretical frameworks capable of assessing the possibilities and applications of QS in areas as diverse as condensed-matter, high-energy physics, quantum chemistry, quantum gravity, out-ofequilibrium quantum physics or others [2][3][4][5][6].
As it is reflected in the literature [4,[7][8][9][10], there are different approaches to define what are the general condi-tions that a QS should fulfill.Those are obviously linked to the particular requirements needed to achieve a given goal, which in turn can also depend on the specific platform on which the QS will be implemented.
On the one hand, there is the strongest notion of a "universal" QS: a quantum many-body system simulates another one if it is able to reproduce its entire physics, i.e., it replicates its eigenstates, full energy spectrum, time evolution, any local noise process, correlation functions, observables, and thermal properties.This is the path developed in a series of very remarkable works [9,[11][12][13] based on a perturbative approach (perturbative gadgets) proposed initially in [14].Within this approach the authors proved that the physics of any quantum many-body system can be replicated by certain 2local spin-lattice models.The latter should be embedded in a Hilbert space significantly larger than that of the system to be replicated.Even a simple translationally invariant spin chain in 1D turns out to be a universal simulator in the above sense [11].Nevertheless, such an approach is not constructive and comes with the additional challenge of requiring a scale-up of the simulation platforms, either by increasing the local dimension of the many-body system acting as a simulator and/or demanding an exquisite control over the interactions, which cannot be implemented in current and/or near future experiments.On the other hand, there are less demanding definitions of quantum simulation where the spell is put on simulating either the ground state or the low energy physics of a complex quantum many-body system or some restricted dynamics.Simulating the dynamics of many-body quantum systems is, in general, a very difficult task for a classical or even for a NISQ device.Some advances have been recently proposed in this direction.For example, the purpose of [8,15] is to find a QS, whose time evolution in a restricted subspace is identical to that of the target system.In [8], the goal is to increase the locality of the QS, and this is achieved by requiring an additional qubit on each k-body interaction.On the contrary, in [15], the dimension of the QS does not increase, but the locality of both Hamiltonians remains the same.
Our work is aligned with the latter ones in that our interest is placed on simulating dynamics of a target klocal Hamiltonian using a k ′ -local Hamiltonian for k ′ < k, i.e., the simulator is more local than the target.We do not attempt to embed the QS in a Hilbert space larger than the one associated to the target Hamiltonian, given that adding qudits to an experimental platform remains a challenging task.Also, for simplicity, we restrict our Hamiltonians to be 1D.Our purpose is to investigate the simulation of quantum many-body dynamics with the following two aims: simplifying complexity while maintaining the local dimension of the parties and the total dimension of the Hamiltonian.
Our work is organized as follows.After presenting some preliminary concepts, in Section III we focus on exact dynamics and show, by means of an example, how some dynamics of a target Hamiltonian, H T , can be replicated exactly by a simulator, H QS , with simplified interactions in the same Hilbert space.In this example both Hamiltonians commute, which is a quite exceptional condition.Nevertheless, when considering non-commuting Hamiltonians, i.e., [H T , H QS ] ̸ = 0, it is still possible to reproduce some dynamical behaviour on a certain energy subspace, providing that both Hamiltonians have some eigenstates in common.It is hard to search for such shared subspaces by diagonalizing large Hamiltonians, and thus we present a generic upper bound to the dimension of such subspaces.In Section IV, we consider the most realistic case of non-exact simulation.There, we provide an upper bound to the error up to which any Hamiltonian is able to simulate another one, irrespectively of their structure.In an attempt to understand how such upper-bound depends on locality, we present an interesting case suggesting that the precision with which a certain 2-local QS simulates a broad family of target Hamiltonians increase while the locality of the target Hamiltonian decreases.This suggests that, at least in certain cases and in reduced dimensionalities, a k-local H T could be simulated better by a 2-local H QS as k increases, or in other words, the less local H T is.Finally, after showing that the worst-case error of simulation decreases when considering short times as expected, we develop a simple algorithm to find the k ′ -local Hamiltonian that best simulates a given k-local Hamiltonian in the short-time regime.

II. PRELIMINARY CONCEPTS
Most of quantum many-body systems are described by local Hamiltonians, and a vast majority of them only involve 2-body interactions.However, there are well known examples of relevant many-body Hamiltonians (lattice gauge theories, Kitaev topological models, etc.) where interactions are not 2-body and whose properties are very difficult to analyze.To fix concepts, we denote by H N an N -dimensional Hilbert space and by B(H N ) the set of its bounded operators.An n-qudit k-local Hamiltonian H ∈ B(H d n ) is of the form H = M i=1 h i , where h i is a Hamiltonian acting non-trivially on at most k parties, and M is some positive integer.An n-qudit k-local Hamiltonian with k = n is dubbed nonlocal.Furthermore, a k-local Hamiltonian is said to be more local than a k ′ -local Hamiltonian if k < k ′ .Note that k-locality does not necessarily imply interaction among k next neighbouring objects, and that there is no restriction on spatial locality.
We denote by H T ∈ B(H N T ) the Hamiltonian describing the target quantum system, the time evolution of which is to be replicated.The system achieving such quantum simulation will, in turn, be described by a Hamiltonian H QS ∈ B(H N QS ).As discussed previously, here we impose that N := N T = N QS .We hereby introduce our precise definition of simulation (ℏ = 1 is set throughout the whole manuscript): with ϵ ∈ (0, 1].
Notice that ϵ = 0 implies that the fidelity is 1, meaning that H QS simulates H T exactly in a subspace S ⊂ H N , where |ψ⟩ belongs.Maximizing ϵ over initial states and times informs us about the minimal performance of a given H QS at simulating a certain H T .The following definition can be put forward: Definition 2 The worst-case fidelity of simulation for given Hamiltonians H QS and H T is defined as

III. EXACT SIMULATION
In this section we address the scenario of error-free simulation, i.e., when ϵ = 0. We illustrate this scenario by showing examples of both commuting and noncommuting Hamiltonians which generate the same dynamics despite their different locality.

A. Commuting Hamiltonians
Using the Baker-Campbell-Hausdorff (BCH) formula, one can write condition (1) as with , where the remaining terms involve higher order commutators of H QS and H T .In order for the BCH series to converge absolutely in this case, note that H QS and H T must fulfill ∥H QS ∥+∥H T ∥ < π/t, for a given submultiplicative norm ∥•∥ [16,17].Given a target Hamiltonian H T , one can now ask for which H QS , initial states |ψ⟩ and times t, condition (3) is fulfilled.An answer was provided in [15], which we briefly sketch in what follows: consider H QS such that [H QS , H T ] = 0, and define the connector h := H QS −H T .Now write the initial state in the basis {|ϕ j ⟩} N j which diagonalizes h: |ψ⟩ = N j=1 a j |ϕ j ⟩, where a j = ⟨ϕ j |ψ⟩.

Then condition (3) becomes
the previous condition holds if h is fully degenerate, implying that H QS is simply H T plus some multiple of the identity.In order to avoid this trivial scenario, we naturally require that |ψ⟩ is only spanned by the degenerate eigenvectors of h, i.e., a j ̸ = 0 for all j such that |ϕ j ⟩ is degenerate and a j = 0 otherwise.All in all, any Hamiltonian H QS that commutes with a target Hamiltonian H T is able to simulate H T exactly at any time t and at any initial state spanned by the degenerate eigenvectors of the corresponding connector.Note that e ith converges absolutely, as does the exponential of any square matrix.
As an example of this approach, in [15] it was shown that an infinite-range-interaction (2-body all with all interactions) Hamiltonian can be simulated using a nearest-neighbor-interaction model with a staggered field.Notice that, in this case, both H T and H QS are 2-local Hamiltonians.Here, we demonstrate, by means of a toy model, that the same approach can be employed to find instances of Hamiltonians with different localities that also lead to the same dynamics.
As an ideal terrain to understand such dynamics, let us consider a simple case in which only 4-qubits are involved.Let the 3-local Hamiltonian describing a one-dimensional system with periodic boundary condition, be our target Hamiltonian.By engineering the (2-local) Heisenberg XYZ model we are able to obtain a QS Hamiltonian which commutes with H T and such that the corresponding connector has some degeneracy.Notice that is the Hilbert-Schmidt (HS) norm of an m-dimensional matrix X.Therefore, choosing J := J x = J y = J z (i.e., the Heisenberg XXX model), commutativity between H T and H QS is granted.By tuning J (see that J 3 and h x are fixed) and comparing the eigenvalues of the resulting H QS with those of H T , it is possible to create degeneracy in the connector.The evolution under H T of any initial state belonging to the subspace where the connector is degenerate can thus be exactly reproduced by the engineered H QS .A particular example is displayed in Appendix A 1. Clearly, the same approach can also be taken for higher spatial dimensional systems, where spins have a larger number of next neighbours and thus exhibit a greater richness of interactions.However, since this approach requires diagonalization, implementing it in large systems is not viable.As a more systematic way, we introduce a method to search for the best simulator Hamiltonian in any dimension and with fixed locality in Proposition 6 by allowing for an error ϵ in the fidelity of simulation, and by restricting to the short-time regime.

B. Non-commuting Hamiltonians
Even if [H QS , H T ] ̸ = 0, simulation is feasible in some cases.If both Hamiltonians share part of their eigenstates, it naturally follows from the previous approach that simulation can still be performed on the energy subspace spanned by those eigenstates.Let the set of shared eigenstates be Θ = {|φ j ⟩} NΘ j=1 (note that N Θ depends on the basis on which each Hamiltonian is expressed) and let the projected connector be h Then the simulatable states |ψ⟩ will be those living in the subspace where h (Θ) is degenerate.We illustrate with an example how two non-commuting Hamiltonians with different localities may generate the same dynamics in Appendix A 2. There again, we consider the H T of the previous section as our 3-local target Hamiltonian.Our QS is now a (2-local) Heisenberg XXX model with 1-body terms, where these 1-body terms prevent HQS from commuting with H T .Yet a far more complex question is whether two given non-commuting high-dimensional Hamiltonians give rise to the same dynamics.The answer is generally negative, since the subset of Hamiltonians with only one common eigenstate is already of measure zero [18].Moreover, computing the shared eigenstates entails diagonalizing such large Hamiltonians, which costs computational resources.Below, we derive an upper bound to the maximum number of such shared eigenstates, the calculation of which does not require diagonalization or computationally demanding techniques: Lemma 3 The maximum number r of shared eigenstates of two N -dimensional non-commuting Hamiltonians H T and H QS is bounded as where ∥X∥ 2 = max ∥v∥=1 (∥Xv∥) is the spectral norm of an n-dimensional matrix X, with v an n-dimensional vector and ∥v∥ Proof.Consider two arbitrary N -dimensional square matrices A X and B X , where X is the basis in which they are expressed.Their commutator can be written as [A X , B X ] = 0 ⊕r X ⊕ C X , with r X ∈ N, 0 ⊕p := 0 ⊕ ... ⊕ 0 p , and C X some traceless (N − r X )-dimensional matrix (without loss of generality, we have assumed that A X and B X share the first r X eigenvectors).It then holds that ∥[A X , B X ]∥ * = ∥C X ∥ * , for any chosen matrix norm * .Let us consider the Hilbert-Schmidt norm, ∥•∥ HS , and the spectral norm, ∥•∥ 2 .Now, recall that ∥X∥ HS ≤ √ m∥X∥ 2 , with m = dim X [19].If matrix X is a finite dimensional commutator (thus traceless), then the inequality can get saturated for even m if X is proportional to a diagonal matrix with half of their entries being 1, and half of them being −1.For n qubits (thus m = 2 n ), the inequality is saturated when choosing . Now, the maximum number of shared eigenvectors between A X and B X is given by since the considered norms are unitarily invariant.This completes the proof.■ Note that the calculation of the spectral norm is efficient as it is expressible by a semidefinite program [20].This upper bound, which is tight for even dimension N , informs about the maximal size that the shared subspace could have, helping to decide whether it is still worth trying to search for the shared eigenstates even by using diagonalization.Moreover, computing the upper bound to r can aid in the processes of figuring out the parameters of the corresponding H QS .Suppose one can only prepare initial states |ψ⟩ ∈ S ⊆ H N , where dim S ≤ N .Recall that exact simulation is granted in a subspace of at least dimension dim S (Θ) , where S (Θ) is the subspace in which h (Θ) is degenerate.Hence, dim In order for the dynamics to be simulatable in this case, it is required that dim S ≤ dim S (Θ) .If the current choice of the parameters of H QS < dim S, it holds that dim S > dim S (Θ) and therefore one must search for a different set of parameters.
In addition, we prove a necessary condition for a state to be an eigenstate of two Hamiltonians with different localities and leave its derivation in Appendix D.

IV. APPROXIMATE SIMULATION
We now focus on the realistic scenario of approximate quantum simulation, i.e., the case when H QS simulates H T up to some tolerable error ϵ ̸ = 0. Contrary to the setting with ϵ = 0, looking at condition (1) does not reveal in a straightforward way how both Hamiltonians should be related in order for one to ϵ-simulate the other.Nonetheless, we are able to provide an upper bound to the error up to which such a simulation can be performed: Theorem 4 [21] Every Hamiltonian H QS ϵ * -simulates any target Hamiltonian H T at any state |ψ⟩ and time t, with Here, ∆ h is the spectral diameter of the connector, h := Therefore, and the proof is finished.■ From here we see that the worst-case fidelity of simulation is never smaller than 1 − ϵ * .We emphasize that the calculation of the error ϵ * does not require diagonalization.The spectral diameter ∆ h is the difference between the largest and the smallest eigenvalue of h and can be obtained by means of a semidefinite program, as discussed later.As shown in Appendix E, other valid bounds can be derived for the worst-case fidelity of simulation, but they are smaller than the proved 1 − ϵ * .
Notice that 1 − ϵ * is greater than zero if and only if t∆ h < 2. In this regime, for fixed time t, we see that the smaller ∆ h , the higher 1 − ϵ * .Now note that since H QS is a k ′ -local Hamiltonian and H T is a k-local Hamiltonian (with k ′ < k), h is a proper k-local Hamiltonian.One could ask at this point how the spectral diameter of a k-local Hamiltonian depends on k.To answer this open question, research in the direction of [22] would be required.They study the extremal eigenvalues of k-local Hamiltonians (with k = O(1)) acting on n qubits, such that each qubit participates in at most l terms.Instead of showing, as they do, how such extremal eigenvalues change with the interaction l, one would need to examine how these vary with the locality k.A numerical route can be taken to explore such behaviours for particular cases.
Here we ask what kind of H QS yields the smallest spectral diameter ∆ h , when H T is a Hamiltonian with k-body nearest neighbour interactions in the z-direction describing a one-dimensional array of qubits.For instance, for k = 3 we force the target to be H T = C 3 j σ z j σ z j+1 σ z j+2 , where C 3 is just a normalization constant.As shown below, this problem can be cast as a semidefinite program: Hamiltonian with nearest neighbour interactions in the zdirection describing a one-dimensional array of n qubits: where C k is a normalization constant, Λ are the generators of all possible j-body interactions, and β ̸ = 0 is the strength of the k ′ -body interactions of H QS .
Note that the second constraint prevents H QS from having local terms with localities larger than k ′ .Also, the third constraint forces H QS to present non-vanishing k ′body interactions.Fig. 1 shows the minimal spectral diameter ∆ h when H T is a Hamiltonian with k-body nearest neighbour interactions (k = 3, 4, 5) describing an array of 5 qubits, and H QS has only 2-local interactions of strength larger than or equal to β = 0.01.Periodic boundary conditions are enforced on H T .The minimal value of ∆ h decreases when H T is less local, suggesting that such kind of target Hamiltonians are simulated to a better precision the less local they are.
Fixing now ∆ h in t∆ h < 2, simulators are expectedly more precise when restricting to short times.Here, we where Λ (j) i are the generators of all possible j-body interactions, with j > k ′ .
Notice that the states (1−itH X ) |ψ 0 ⟩ are not normalized.It is possible to find a solution for this conic program using state-of-the-art solvers like [23].In Fig. 2 we show the minimum HS norm between states |ψ(t)⟩ = e −itH T |ψ 0 ⟩ and |ϕ(t)⟩ = e −itH QS |ψ 0 ⟩, for several paradigmatic initial states |ψ 0 ⟩.Here, H T is a random 3-qubit 3-local Hamiltonian in 1D, and H QS is a 3-qubit 2-local Hamiltonian in 1D.The evolution of the W state can be reproduced exactly in this case.For the rest of the initial states, the plotted distance scales linearly with time in this short time regime.

V. DISCUSSION
We have explored the dynamic simulation of k-local Hamiltonians using more local Hamiltonians acting on the same Hilbert space, an approach that aligns effectively with the present experimental limitations: as argued in the introduction, scaling up simulation platforms to accommodate remarkably larger simulator Hamiltonians is still a challenging task.In the exact simulation scenario, we have confirmed that some Hamiltonians with different localities can produce the same dynamics at certain subsets of states even if they do not commute.Further, we have analyzed the more realistic scenario where the dynamics is reproduced up to some error and provided a lower bound to the worst-case fidelity with which any Hamiltonian can simulate another one.Based on this, we have numerically shown evidence that the spectral diameter decreases as the locality increases, and this suggests that the simulation can be performed more precisely.Moreover, we have presented a program to find the Hamiltonian that best simulates a given klocal Hamiltonian in the short-time regime and solved it for a particular physical scenario.
The relation between the worst-case fidelity of simulation and the spectral diameter of the corresponding connector has been unveiled.This has allowed us to study dynamic simulatability without requiring diagonalization, which becomes computationally expensive when considering large systems.In turn, the relation between locality and spectral diameter needs to be further investigated in the line of [22], which would help us understand how to better engineer quantum simulation settings.Also, further research could be conducted in the spirit of [18] to calculate the relative volume of the simulatable sets of states, which would shed light on the potential of simulatability of each particular scenario.First, we search for the shared subspace.We refer the details to Appendix C, but note that the size of the shared subspace corresponds to the nullity of the commutator, i.e., the number of vanishing eigenvalues of the commutator.We choose the parameter set {b x , b y , b z } such that the nullity is large, while J is not involved in the eigenvalues of the commutator due to the fact that [H QS , H T ] = 0. Thus, we take {b x /J 3 , b y /J 3 , b z /J 3 } = {−4, 0, 1} and in this case the number of shared eigenstates is 12.The set of the shared eigenstates is obtained by constructing a proper linear combination of the eigenstates of the commutator (see Appendices B C).
The only thing left is to create degeneracy in h = H QS − HT in the corresponding subspace as we do in Appendix A 1. By tuning J, we have created some degeneracy in h.Fig. 4 displays the parameter J/J 3 and the degenerate eigenstates of h of such cases.The following theorem can be derived: Here, ∆ h is the spectral diameter of the connector, h := H QS − H T , i.e., ∆ h = max ij |λ i − λ j |, where {λ i } N i=1 are the eigenvalues of h.

FIG. 1 .
FIG. 1. Spectral diameter ∆ h minimised by the SDP (13) as a function of the locality of HT .

FIG. 3 .
FIG.3.Coefficients of two degenerate eigenstates of h and value of the parameter J/J3.The x label denotes the spin basis, where "0" means spin down and "1" means spin up.

FIG. 4 . 1 r=1(− 1 )
FIG.4.Coefficients of two eigenstates degenerate in h and the value of parameter J/J3.The x label denotes the spin basis, where "0" means down spin and "1" means up spin.