Translationally-Invariant Universal Quantum Hamiltonians in 1D

Recent work has characterised rigorously what it means for one quantum system to simulate another, and demonstrated the existence of universal Hamiltonians -- simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many body system. Previous universality results have required proofs involving complicated `chains' of perturbative `gadgets'. In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models, and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result, that the local Hamiltonian problem for translationally-invariant spin chains in one dimension with an exponentially-small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D--1D holographic duality between local Hamiltonians.


Introduction
Analog Hamiltonian simulation is one of the most promising applications of quantum computing in the NISQ (noisy, intermediate scale, quantum) era, because it does not require fully fault-tolerant quantum operations. Its potential applications have led to an interest in constructing a rigorous theoretical framework to describe Hamiltonian simulation.
Recent work has precisely defined what it means for one quantum system to simulate another [CMP18], and demonstrated that-within very demanding definitions of what it means for one system to simulate another-there exist families of Hamiltonians that are universal, in the sense that they can simulate all other quantum Hamiltonians. This work was recently extended, with the first construction of a translationally invariant universal family of Hamiltonians [PB20].
Previous universality results have relied heavily on using perturbation gadgets, and constructing complicated 'chains' of simulations to prove that simple models are indeed universal. In this paper we present a new, simplified method for proving universality. This method makes use of another technique from Hamiltonian complexity theory: history state Hamiltonians [KSV02]. Leveraging the fact that it is possible to encode computation into the ground state of local Hamiltonians, we show that it is possible to prove universality by constructing Hamiltonian models which can compute the energy levels of arbitrary target Hamiltonians.
In order to ensure that the universality constructions preserve the entire physics of the target system (and not just the energy levels), we make use of an idea originally from [Aha+07] and used recently in [AZ18a; NVY18;Aha+14]: 'idling to enhance coherence'. Before computing the energy levels of the target system, the computation encoded in the simulator system 'idles' in its initial state for time . By choosing to be sufficiently large, we can ensure that with high probability there is a fixed set of spins in the simulator system which map directly to the state of the target system.
As well as providing a route to simplifying previous proofs, this 'history-state simulation method' also offers more insight into the origins of universality, and the relationship between universality and complexity. The classification of two-qubit interactions by their simulation ability in [CMP18], which showed that the universal class was precisely the set of QMA-complete interactions, was suggestive of a connection between simulation and complexity. And a complexity theoretic classification of universal models already exists in the classical case [CC16]. But until now it was not clear whether a connection existed for general quantum interactions, or whether it was merely an accident in the two-qubit case. Previous methods for proving universality in the quantum case didn't offer a route to classifying universal models, and the more complicated non-commutative structure of quantum Hamiltonians meant that the techniques from the classical proof couldn't be applied. By demonstrating that it is possible to prove universality by leveraging the ability to encode computation into ground states, we have provided a route to showing that the connection between universality and complexity holds more generally. In a companion paper [Koh+21] we make this insight rigorous, by deriving a full complexity theoretic classification of universal quantum Hamiltonians.
We also use the 'history-state simulation method' to provide a simple construction of two new universal models. Both of these are translationally invariant systems in 1D, and we show that one of these constructions is efficient in terms of the number of spins in the universal construction (yet not in terms of the simulating system's norm): Theorem 1.1. There exists a two-body interaction ℎ (1) depending on a single parameter ℎ (1) = ℎ (1) ( ), and a fixed one-body interaction ℎ (2) such that the family of translationally-invariant Hamiltonians on a chain of length , is a universal model, where Δ, and are parameters of the Hamiltonian, and the first sum is over adjacent sites along the chain. The universal model is efficient in terms of the number of spins in the simulator system.
By tuning , and Δ, this model can replicate (in the precise sense of [CMP18]) all quantum many body physics. This is the first translationally invariant universal model which is efficient in terms of system size overhead. Its existence implies that, for problems which preserve hardness under simulation, complexity theoretic results for general Hamiltonians can also apply to 1D, translationally invariant Hamiltonians (though care must be taken when applying this, as the construction is not efficient in the norm of the simulating system). This is for instance the case for a reduction from a PreciseQMA-hard local Hamiltonian (LH) problem, for which the reduction to a translationally-invariant version preserves the correct promise gap scaling. This in turn implies that the local Hamiltonian problem remains PSPACE-hard for a promise gap that closes exponentially quickly, even when enforcing translational invariance for the couplings. This stands in contrast to a promise gap which closes as 1/poly in the system size, in which case the variant is either QMA (for non-translational invariance) or QMA EXP (for translational invariance) complete.
Furthermore, Theorem 1.1 allows us to construct the first toy model of holographic duality between local Hamiltonians from a 2D bulk to a 1D boundary, extending earlier work on toy models of holographic duality in [Pas+15] and [KC19a].
We also construct a universal model which is described by just two free parameters, but where the model is no longer efficient in the system size overhead: Theorem 1.2. There exists a fixed two-body interaction ℎ (3) and a fixed one-body interaction ℎ (2) such that the family of translationally-invariant Hamiltonians on a chain of length , is a universal model, where Δ and are parameters of the Hamiltonian, and the first sum is over adjacent sites along the chain.
By varying the size of the chain that this Hamiltonian is acting on, and tuning the Δ and parameters in the construction, this Hamiltonian can replicate (again in the precise sense of [CMP18]) all quantum many body physics. We are able to demonstrate that constructing a universal model with no free parameters is not possible, but the existence of a universal model with just one free parameter is left as an open question.
The remainder of the paper is set out as follows. In Section 2 we cover the necessary background regarding the theory of simulation, and encoding computation into ground states of QMA-hard Hamiltonians. In Section 3 we give an overview of the new method for proving universality, and our two new universal constructions. Reading these sections should be enough to gain an intuitive understanding of our approach and our results. The full proofs of our results are given in Section 4 -this section may be skipped on an initial reading if you are primarily interested in understanding the general approach, or the applications of the results. The complexity theory implications are discussed in Section 5, while in Section 6 the new toy model of holographic duality is constructed. Avenues for future research, are discussed in Section 7.

Hamiltonian Encodings
Any simulation of a Hamiltonian by another Hamiltonian ′ must involve "encoding" in ′ in some fashion. In [CMP18] it was shown that any encoding map E ( ) which satisfies three basic requirements i) E ( ) = E ( ) † for all ∈ Herm ii) spec(E ( )) = spec( ) for all ∈ Herm iii) E ( + (1 − ) ) = E ( ) + (1 − )E ( ) for all , ∈ Herm and all ∈ [0, 1] must be of the form where is an isometry, denotes complex conjugation, and and are orthogonal projectors. Moreover, it is shown that, under any encoding of the form given in Eq.
(3), E ( ) will also preserve the measurement outcomes, time evolution and partition function of . A local encoding is an encoding which maps local observables to local observables, defined as follows.
It is shown in [CMP18] that if an encoding E ( ) = ( ⊗ + ⊗ ) † is local, then the isometry can be decomposed into a tensor product of isometries = ⊗ , for isometries : H ⊗ → H ′ , for some ancilla system . 1 In this paper all of the encodings we work with are of the simpler form E ( ) = † .

Hamiltonian Simulation
Building on encodings, [CMP18] developed a rigorous formalism of Hamiltonian simulation, formalizing the notion of one many-body system reproducing identical physics as another system, including the case of approximate simulation and simulations within a subspace. We first describe the simpler special case of perfect simulation. If ′ perfectly simulates , then it exactly reproduces the physics of below some energy cutoff Δ, where Δ can be chosen arbitrarily large. For brevity, we abbreviate the low-energy subspace of an operator via ≤Δ( ) ≔ span{| : | = | ∧ ≤ Δ}. ii.
Note that the role ofẼ is to provide an exact simulation as per Definition 2.2. 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. 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.
We say that the simulation is efficient if, in addition, for acting on qudits and ′ acting on qudits, ′ = poly( , 1/ , 1/ , Δ) and = poly( , 1/ , 1/ , Δ); ′ is efficiently computable given , Δ, and ; each local isometry in the decomposition of is itself a tensor product of isometries which map to O(1) qudits; and there is an efficiently constructable state | such that | = | .
As already outlined, in [CMP18] it is shown that approximate Hamiltonian simulation preserves important physical properties. We recollect the most important ones in the following. iii. For any density matrix ′ in the encoded subspace for which E (1) ′ = ′ , we have

Circuit-to-Hamiltonian Mappings
The key idea behind our universal constructions is that it is possible to encode computation into the ground state of local Hamiltonians. This technique was first proposed by Feynman in 1985, and is the foundation for many prominent results in Hamiltonian complexity theory, such as QMA-hardness of the local Hamiltonian problem [Fey85;KSV02].
For the constructions we develop in this paper, we will make use of the ability to encode an arbitrary quantum computation into the ground state of a local Hamiltonian. These are often called "circuit-to-Hamiltonian mappings", though the mappings may involve other models of quantum computation than the circuit model. These Hamiltonians are typically constructed in such a way that their ground states are "computational history states". A very general definition of history states was given in [GGC18]; we will only require the simpler "standard" history states here. Note that need not necessarily be a gate in the quantum circuit model. It could also e.g. be one time-step of a quantum Turing machine, or even a time-step in some more exotic model of quantum computation [BCO17], or an isometry [UHB17]. In the particular constructions we make use of in this work, will be a time-step of a quantum Turing machine.

High-level outline of the construction
As mentioned in Section 2.2, the key technique we make use of in our universality constructions is the ability to encode computations into the ground states of local Hamiltonians. The model of computation we encode is the quantum Turing machine (QTM) model -standard techniques for encoding QTMs in local Hamiltonians give translationally invariant Hamiltonians [CPGW15b;GI09].
In both the constructions we develop in this work a description of the Hamiltonian to be simulated (the "target" Hamiltonian, target ) is encoded in the binary expansion of some natural number, ∈ N . Details of this encoding are given in Section 3.2. The natural number is then itself encoded in some parameter of the universal Hamiltonian (see Section 3.3 for two methods of encoding natural numbers in parameters of universal Hamiltonians).
The Hamiltonian we use to construct the universal model has as its ground state computational history states (cf Definition 2.7) which encode two QTMs ( 1 and PE ) which share a work tape. The two computations are 'dovetailed' together -the computation 1 occurs first, and the result of this computation is used as input for PE . The first QTM, 1 , extracts the binary expansion of from the parameter of the Hamiltonian. At the end of 1 's computation, the binary expansion of is written on the work tape which 1 shares with PE . An outline of the methods we use to extract and write it on the Turing machine tape are given in Section 3.3. The second QTM, PE reads in , which contains a description of target , from the work tape which it shares with 1 . It also reads in an input state | -this is unconstrained by the computation (it can be thought of as carrying out the same role as a witness in a QMA verification circuit). It then carries out phase estimation on | with respect to the unitary generated by target .
The Hamiltonian which encodes 1 and PE has a zero-energy degenerate ground space, spanned by history states with all possible input states | . In order to recreate the spectrum of target we need to break this degeneracy. We achieve this by adding one body projectors to the universal Hamiltonian which give the correct energy to the output of PE to reconstruct the spectrum of target .
With this construction the energy levels of the universal Hamiltonian recreate the energy levels of target . To ensure that the eigenstates are also correctly simulated, before 1 carries out its computation, it 'idles' in its initial state for some time . By choosing large enough, we show that this construction can approximately simulate any target Hamiltonian. A more detailed sketch of how we use idling and phase estimation to achieve simulation is given in Section 3.4, while rigorous proofs are given in Section 4.

A Digital Representation of a Local Hamiltonian
As discussed in Section 3.1, we need to encode a description of the target Hamiltonian target in some parameter of the universal Hamiltonian. In Section 3.3 we outline the two methods we use to encode a natural number in the parameter of a Hamiltonian. But how do we represent target = =1 ℎ in the binary expansion of a natural number ∈ N, irrespective of its origin?
We will assume that target is a -local Hamiltonian, acting on spins of local dimension . We emphasize that can be taken to be , i.e. the system size-and therefore we can simulate any Hamiltonian, not just local ones. However we keep track of the locality parameter as it is relevant when deriving the overhead of our simulations.
Every value needed to specify the -local simulated system target will be represented in Elias-′ coding, which is a simple self-delimiting binary code which can encode all natural numbers [Fen03;KC19b]. For the purpose of the encoding, we will label the spins in the system to be simulated by integers = 1, . . . , .
The encoding of target begins with the three meta-parameters (spin count), followed by (locality), and then (number of -local terms). Each of the -local terms in is then specified by giving the label of the spins involved in that interaction, followed by a description of each term of the × Hermitian matrix describing that interaction. Each such matrix entry is specified by giving two integers and . The matrix entry can be recovered by calculating √ 2 − , which is accurate up to a small error. 2 Specifying target to accuracy requires each such matrix entry to be specified to accuracy /( 2 ). Therefore the length of the description of target is 2 log Finally, the remaining digits of specify Ξ-the bit precision to with which the phase estimation algorithm should calculate the energies (i.e. we require QPE to extract Ξ binary digits), and -the length of time the system should "idle" in its initial state before beginning its computation.
So, the binary expansion ( ) of has the following form: Here ′ ( ) denotes in Elias-′ coding, and · denotes concatenation of bit strings. With regards to the identification of a real number = √ 2 − , we observe that it is clearly straightforward to recover from and (by performing basic arithmetic). The other direction works as follows.
Remark 3.1. Let ∈ N, and let Ξ ∈ N denote a precision parameter. Then we can find and the algorithm runs in O(poly(Ξ, log 2 )).
2 Note that by Weyl's equidistribution theory Proof. We solve 2 Ξ = ⌊2 Ξ √ 2⌋ − 2 Ξ as a linear Diophantine equation in the variables and , with largest coefficient O(2 Ξ ). This can be done in polynomial time in the bit precision of the largest coefficient, for instance by using the extended Euclidean algorithm [Fox00].
In Section 4, we describe a construction to (Δ ′ , , ′ )-simulate the Hamiltonian described by , but note that this will only give a (Δ ′ , , ′ + )-simulation of the actual target Hamiltonian target .

Encoding the target Hamiltonian in parameters of the simulator Hamiltonian
In Section 3.2 we described how we encode the information about the Hamiltonian we want to simulate, target in a natural number . Now we require a method to encode in some parameter of the universal Hamiltonian, and a method to write its binary expansion on the Turing machine tape shared by 1 and PE . We develop two constructions, building on the mappings in [CPGW15b] and [GI09]. The first construction is efficient in terms of the number of spins in the simulator system, while the second construction is not efficient, but requires less parameters to specify the universal model. In both cases the computation encoded in the ground state of the Hamiltonian is a QTM, and the mapping from a QTM to the Hamiltonian gives a translationally invariant Hamiltonian.

Encoding the target Hamiltonian in a phase of the simulator Hamiltonian
First we consider the construction building on the work in [CPGW15b]. Here, we encode the natural number ∈ N in a phase = /2 ⌈log 2 ⌉ of the Hamiltonian.
The Hamiltonian for this construction is given by = =1 ℎ ( , +1) where is the number of spins in the simulator system, and ℎ is a two-body interaction of the form [CPGW15a, Theorem 32]: where is a fixed Hermitian matrix and , are fixed non-Hermitian matrices. For a detailed construction of the terms in the Hamiltonian we refer the interested reader to [CPGW15a, Section 4].
The circuit-to-Hamiltonian map encodes two Turing machine computations "dovetailed" together, where the two Turing machines share a work tape. The first computation is a phase estimation algorithm. It extracts the phase from the Hamiltonian, and writes its binary expansion onto the work tape. The second computation will be outlined in Section 3.4.
In order to extract digits from a phase = 0. 1 2 · · · +1 · · · , we require a runtime of 2 . In our case, we have = | | = poly , , log( / ) , where | | denotes the number of digits in the binary expansion of . As our computation is encoded as a computational history state, this in turn means that the spectral gap of the history state Hamiltonian necessarily closes as O(2 − poly( , ,log ( / )) ) [BC18; CB17;GGC18]. This scaling of the spectral gap means that the universal model constructed via this method is not efficient in terms of the norm of the simulator system (see Theorem 4.5 for full discussion of the scaling). However, it is important to note that using the construction from [CPGW15b] it is possible to encode a computation with exponential runtime into a Hamiltonian on polynomially many spins. Details of the construction are given in [CPGW15a, Section 4.5] (in particular the relevant scaling is discussed on [CPGW15a, Page 81]). We will not give the details of the construction here, but note that it encodes a Turing machine which runs for O( exp( )) time steps in a Hamiltonian acting on spins [CPGW15a,Proposition 45]. Therefore, the universal model constructed via this method is efficient in terms of the number of spins in the simulator system.

Encoding the target Hamiltonian in the size of the simulator system
Our second construction builds on the mapping in [GI09]. Here, we encode the description of the target into the binary expansion of -the number of spins the universal Hamiltonian is acting on.
The circuit-to-Hamiltonian map encodes two Turing machine computations "dovetailed" together, where again the two Turing machines share a work tape. The first Turing machine is a binary counter Turing machine. After it has finished running, the binary expansion of is written on the Turing machine's work tape. In our construction, the binary expansion of contains the description of target . We will discuss the second computation in Section 3.4.
The binary counter QTM takes time to write out the binary expansion of on its work tape. Since target is encoded in the binary expansion of , this run time, as well as the size of the simulator system is exponential in the size of the target system. Moreover, since the runtime is exponential in the size of the target system, the spectral gap of the universal Hamiltonian closes exponentially fast. Therefore, the universal model constructed via this method is not efficient in terms of number of spins or the norm of the simulator system. See Theorem 4.6 for a full discussion of the scaling of this universal model.
In this case the interactions of the Hamiltonian are entirely fixed -they enforce that the ground state of the Hamiltonian is a history state encoding a QTM computation (for a detailed construction of the terms in the Hamiltonian we refer readers to [GI09]. There are two additional global parameters in the Hamiltonian which depend on the accuracy of the simulation -we defer discussion of those parameters to the technical proofs of Lemma 4.3 and Theorem 4.6. All the information about the target Hamiltonian (the Hamiltonian to be simulated) is entirely encoded in the binary expansion of -the number of spins in the simulator system.

Dovetailing for simulation
After the computation carried out by 1 has finished, the binary expansion of is written out on the work-tape shared by 1 and PE . We then construct (using standard techniques from [CPGW15b; GI09]) a Hamiltonian such that the two Turing machines 1 and PE share a work tape. At the beginning of its computation, PE reads in a description of the target Hamiltonian that we wish to simulate. PE then carries out phase estimation on some input state | (left unconstrained, just like a QMA witness) 3 with respect to the unitary generated by the target Hamiltonian, = e i for some such that < 2 . It then outputs the eigenphase in terms of a pair of natural numbers ( , ) such that = √ 2 − (which can be done efficiently via Remark 3.1). The ground space of the Hamiltonian which encodes the computation of 1 and PE has zero energy, and is spanned by history states in a superposition over all possible initial states | . In general the Hamiltonian we want to simulate doesn't have a highly degenerate zero energy ground state, so we need to break this degeneracy and construct the correct spectrum for target . In order to break the degeneracy and reconstruct the spectrum of target , we add one body projectors to the universal Hamiltonian, which are tailored such that the QPE output ( , ) identifies the correct energy penalty to inflict.
In order to ensure that the encoding of target in the universal Hamiltonian is local, we make use of an idea originally from [Aha+07] and used recently in [AZ18a; NVY18; Aha+14], which has bee called 'idling to enhance coherence'. Before carrying out the phase-estimation computation, the system "idles" in its initial state for time . By choosing appropriately large, we can ensure that with high probability the input spins (the spins which form the unconstrained input | to PE ) are found in their initial states. This means that (with high probability) there are a subset of spins on the simulator system whose state directly maps to the state which is being simulated in the target system. This ensures that the encoding is (approximately) local (see Lemma 4.3 for detailed analysis of how idling is used to achieve universality). 3 Although quantum phase estimation takes as input an eigenvector of the unitary, we show in the proof that this suffices, as the argument then extends to general input states by linearity.

Translationally-Invariant Universal Models in 1D
In this section we prove our main result: there exist translationally invariant, nearest neighbour Hamiltonians acting on a chain of qudits, which are universal quantum simulators.
All the 'circuit-to-Hamiltonian' mappings we make use of in this work are what are known as "Standard form Hamiltonians". Where "Standard form Hamiltonians" are a certain class of circuit-to-Hamiltonian constructions, defined in [Wat19]. We refer interested readers to [Wat19] for the full definition -and simply note that it encompasses the Turing-machine based mappings which we make use of in this work [CPGW15b;GI09]. In [Wat19], the following result was shown, which we will make use of in our proofs: We also require a digital quantum simulation algorithm, summarized in the following lemm: The polynomial time bound in Lemma 4.2 suffices for our purposes; a tighter (and more complicated) bound, also for the more general case of sparse Hamiltonians, can be found in [BCK15].
We can now start our main analysis by proving that "dovetailing" quantum computationsrigorously defined and constructed in [CPGW15b,Lem. 22  To prove this we show that the univ ( ) can satisfy the definition to be an approximate simulation of an arbitrary "target Hamiltonian" target , to any desired accuracy. We break up the proof into multiple parts. First we construct a history state Hamiltonian SF ( ), which encodes two Turing machine computations: 1 which extracts a description of target from a parameter of SF , and PE which carries out phase estimation on the unitary generated by target . Then we define the one-body projectors Π and Π which break up the ground space degeneracy of SF , and inflict just the right amount of penalty to approximately reconstruct the spectrum of target in its entirety.

Construction of H SF .
SF is a standard form history state Hamiltonian with a ground space laid out in Lemma 4.1. The local states of the spins on which SF acts are divided into multiple "tracks". There are a constant number of these, hence a constant local Hilbert space dimension. The exact number will depend on the standard form construction being used. Each track serves its own purpose, as outlined in Table 1. See [GI09; CPGW15b] for more detail.
The QTM PE reads in the description of target -provided as integer ∈ N output by the Turing machine 1 whose worktape it shares. PE further reads in the unconstrained input state | (see Table 1 for details of the local Hilbert space decomposition). But instead of proceeding immediately, PE idles for time-steps (where is specified in the input string , as explained in Section 3.2), before proceeding to carry out the quantum phase estimation algorithm.
The quantum phase estimation algorithm is carried out with respect to the unitary = e i target for some such that target < 2 . It takes as input an eigenvector | of , and calculates the eigenphase . The output of PE is then the pair of integers ( , ) (corresponding to the extracted phase = √ 2 − as explained in Remark 3.1), specified in binary on an output track. To calculate -the eigenvalue of target -to accuracy requires determining to accuracy O( / target ) which takes O( target / ) uses of = e i target . The unitary must thus be implemented to accuracy O( / target ), which is done using Lemma 4.2; the latter introduces an overhead poly( , , target , , 1/ ) in the system size , local dimension , locality , and target accuracy . The error overhead of size poly 1/ due to the digital simulation of the unitary is thus polynomial in the precision, as are the ∝ 1/ repetitions required for the QPE algorithm. The whole procedure takes time PE ≔ poly( , target / , ). (10) In our construction the input to PE is not restricted to be an eigenvector of | , but it can always be decomposed as | = | . By linearity, for input | = | the output of PE will be a superposition in which the output ( , ) occurs with amplitude . After PE has finished its computation, its head returns to the end of the chain. By Lemma 4.1, the ground space L of SF is spanned by computational history states as given in Definition 2.7, and is degenerate since any input state | yields a valid computation. Therefore: where ( ) denotes the state of the system at time step if the input state was | .

A Local Encoding.
In order to prove that univ ( ) can simulate all quantum Hamiltonians, we need to demonstrate that there exists a local encoding E ( ) such that the conditions of Definition 2.3 are satisfied. To this end, let where ′ = 1 + , and where 1 is the number of time steps in the 1 computation. This is the history state up until the point that PE begins its computation (i.e. the point at which the 'idling to enhance coherence' ends). So, throughout the computation encoded by this computation the spins which encode the information about the input state remain in their initial state, and we can write: The rest of the history state we capture in such that the total history state is We now define the encoding E ( ) = † via the isometry where | are the computational basis states (any complete basis will suffice). E is a local encoding, which can be verified by a direct calculation: where phys is the operator acting on the Hilbert space corresponding to the th qudit.
We now consider the encoding E ′ ( ) = ′ ′ † , defined via We have that By Lemma 4.1, E ′ is the ground space of SF .

Splitting the Ground Space Degeneracy of H SF .
What is left to show is that there exist one body-projectors Π and Π which add just the right amount of energy to states in the kernel L ( SF ) to reproduce the target Hamiltonian's spectrum. We first choose the one body terms in univ to be projectors onto local subspaces which contain the two states which are outputs of the PE computation -|Ω and |Ω : We have shown that if the input state is | , which is an eigenstate of with eigenphase = √ 2 − , then the history state will contain terms with one spin in the state |Ω and terms with one spin in the state |Ω (each term in the history state will have amplitude 1 ). If the input is a general state | = | then for each the history state will contain terms with one spin in the state |Ω and terms with one spin in the state |Ω , where now each of these terms has amplitude / . Let Π ≔ |Φ( ) Φ( )| for some complete basis | , and we define 1 ≔ ( √ 2Π − Π ), where is the total time in the computation. It thus follows that the energy of |Φ( ) with respect to the operator Π 1 Π is given by + O( ). Finally, we need the following technical lemma from [BH17].
Lemma 4.4 (First-order simulation [BH17] ). 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 has eigenvalue 0 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 1.
We can now prove our main theorem:

Theorem 4.5. There exists a two-body interaction depending on a single parameter ℎ( ) such that the family of translationally-invariant Hamiltonians on a chain of length ,
is a universal model, where Δ, and are parameters of the Hamiltonian, and the first sum is over adjacent site along the chain. Furthermore, the universal model is efficient in terms of the number of spins in the simulator system.
Proof. The two body interaction ℎ( ) makes up a standard form Hamiltonian which encodes a QTM, 1 dovetailed with the phase-estimation computation from Lemma 4.3.
The QTM 1 carries out phase estimation on the parameter in the Hamiltonian, and writes out the binary expansion of (which contains a description of the Hamiltonian to be simulated) on its work tape. There is a standard form Hamiltonian in [CPGW15b] which encodes this QTM, so by Lemma 4.3 we can construct a standard form Hamiltonian which simulates all quantum Hamiltonians by dovetailing 1 with PE . The space requirement for the computation is O(| |), where | | denotes the length of the binary expansion of , and the computation requires time 1 = O(| |2 | | ) [CPGW15a, Theorem 10] As we commented in Section 3.3.1, the standard form clock construction set out in [CPGW15a, Section 4.5] allows for computation time of O(| |2 | | ) using a Hamiltonian on | | spins. We therefore find that for a -local target Hamiltonian target acting on spins of local dimension , the number of spins required in the simulator system for a simulation that is close to target is given by = O(| |) = poly , , , 1/ , 1/ . Therefore, the universal model is efficient in terms of the number of spins in the simulator system as defined in Definition 2.4.
Note that this universal model is not efficient in terms of the norm univ . This is immediately obvious, since univ = Ω(Δ), and using the relations between Δ ′ , , , and and Δ from However if we only wish to simulate a translationally invariant -local Hamiltonian target , this can be specified to accuracy with just log( target 2 / ) bits of information. In this case (for , = O(1) and taking = ), the interaction strengths are then poly( , target , Δ ′ , 1 , 1 ), and the whole simulation is efficient. Lemma 4.3 also allows the construction of a universal quantum simulator with two free parameters.

Theorem 4.6. There exists a fixed two-body interaction ℎ such that the family of translationally-invariant Hamiltonians on a chain of length ,
is a universal model, where Δ and are parameters of the Hamiltonian, and the first sum is over adjacent sites along the chain.
Proof. As in Theorem 4.5, the two body interaction ℎ makes up a standard form Hamiltonian which encodes a QTM 1 dovetailed with the phase-estimation computation from Lemma 4.3. It is based on the construction from [GI09]. Take 1 to be a binary counter Turing machine which writes out -the length of the qudit chain-on its work tape. We will choose to contain a description of the Hamiltonian to be simulated, as per Section 3.2. There is a standard form Hamiltonian in [GI09] which encodes this QTM, so by Lemma 4.3 we can construct a standard form Hamiltonian which simulates all quantum Hamiltonians by dovetailing 1 with PE .
The standard form clock used in the construction allows for computation time polynomial in the length of the chain, so exp(poly)-time in the size of the target system. As before, by Eq. (10), we require According to the requirements of Definition 2.3, the universal simulator of the second theorem is not efficient in either the number of spins, nor in the norm. However-as was noted in [PB20]-this is unavoidable if there is no free parameter in the universal Hamiltonian which encodes the description of the target Hamiltonian: a translationally invariant Hamiltonian on spins can be described using only O(poly log( )) bits of information, whereas a -local Hamiltonian which breaks translational invariance in general requires poly( ) bits of information. So, by a simple counting argument, we can see that it is not possible to encode all the information about a -local Hamiltonian on spins in a fixed translationally invariant Hamiltonian acting on poly( ) spins.
We observe that the parameters Δ and are qualitatively different to , in that they do not depend on the Hamiltonian to be simulated, but only the parameters (Δ ′ , , ) determining the precision of the simulation.

No-Go for Parameterless Universality
Is an explicit Δ-dependence of a simulator Hamiltonian univ necessary to construct a universal model? Note that an implicit dependence of univ on Δ is possible via the chain length = (Δ) in Theorem 4.5. In the following, we prove that such an implicit dependence is insufficient, by giving a concrete counterexample for which an explicit Δ-dependence is necessary.
To this end, we note that it has previously been shown [AZ18b] that a degree-reducing Hamiltonian simulation (in a weaker sense of simulation, namely gap-simulation where only the ground state(s) and spectral gap are to be maintained) is only possible if the norm of the local terms is allowed to grow. In order to construct a concrete example in which an explicit Δ-dependence is necessary, we first quote Aharonov and Zhou's result, and then translate the terminology to our setting.
with a localized encoding, -incoherence, and energy spread˜ , for any number of Hamiltonian terms .

Corollary 4.8. Consider a universal family of Hamiltonians with local interactions and bounded-degree interaction graph. Hamiltonians in this family must have an explicit dependence on the energy cut-off (Δ) below which they are valid simulations of particular target Hamiltonians.
Proof. We first explain the notation used in Theorem 4.7. As mentioned, the notion of gap simulation is weaker than Definition 2.3. Only the (quasi-) ground space L of , rather than the full Hilbert space, needs to be represented -coherently: | L − | L < , where ·| L denotes the restriction to L). And only the spectral gap above the ground space, rather than the full spectrum, must be maintained:˜ = Δ(˜ ) ≥ = Δ( ). The rest of the spectrum in the simulation can be arbitrary. Energy spread in this context simply means the range of eigenvalues within L spreads out at most such that | 0 −˜ 0 | ≤˜ .
A [O(1), , O(1)] simulation with the above parameters then simply means ancoherent gap simulation, constant degree and local interaction strength, where -the number of local terms in the simulator-is left unconstrained, and the eigenvalues vary by at most˜ .
It is clear that this notion of simulation falls within our more generic framework of simulation (cf. [AZ18b, Sec. 1.1]): a simulation of also defines a valid gap simulation of . Since by Definition 2.4 this simulation can be made arbitrarily precise, with parameters ,˜ arbitrarily small, and has constant interaction degree by assumption, this contradicts Theorem 4.7.

Applications to Hamiltonian Complexity
As already informally stated, the L H problem is the question of approximating the ground state energy of a local Hamiltonian to a certain precision. Based on a history state embedding of a QMA verifier circuit and on Feynman's circuit-to-Hamiltonian construction [Fey85], Kitaev proved in 2002 that L H with a promise gap that closes inverse-polynomially in the system size is QMAcomplete [KSV02].
To be precise, let us start by defining the L H problem. We note that variants of this definition can be found throughout literature which commonly omit one or more of the constraints presented herein, in particular with regards to the bit precision to the input matrices. In order to be precise, we explicitly list the matrix entries' bit precision as extra parameter Σ in the following definition. Kitaev's QMA-completeness result was shown for a promise gap ( ) = poly [KSV02, Th. 14.1]. Following the proof construction therein reveals that this was done for a bit complexity of the matrix entries Σ( ) = O(1) (assuming a discrete fixed gateset for the encoded QMA verifier). Since his seminal result, the statement has been extended and generalized to ever-simpler many-body systems [OT05; HNN13; Aha+09]. Some of these results allow a coupling constant to scale in the system size, e.g. as poly -i.e. the matrix entries now feature a bit precision of Σ( ) = poly log .
We remark that despite the apparent relaxation in the bit precision, these results are not weaker than Kitaev, Shen, and Vyalyi's. Since the number of local terms = poly , a polynomial number of local terms of O(1) bit complexity acting on the same sites can already be combined to create -local interactions with polynomial precision (logarithmic bit-precision, Ω(1/poly) ∩ O(poly)). (Similar to how the encoding in Section 3.2 and Remark 3.1 works by adding up integers to approximate a number in the interval [0, 1].) We also emphasize that the overall bit complexity of the input is already poly , as there are that many local terms to specify in the first place. Indeed, many times in the literature, the matrix entries of the L H problem are simply restricted to bit precision Σ = poly (e.g. [CM14]).
However, translationally-invariant spin systems are common in condensed matter 1/poly promise gap. In short, the problem is easier for translationally-invariant systems, Hamiltonians in the bulk to local Hamiltonians in the boundary [KC19a]. The techniques in [KC19a] require at least a 2D boundary, and it was an open question whether a similar result could be obtained in lower dimensions.
Here we construct a HQECC from a 2D bulk to a 1D boundary which maps any (quasi-)local Hamiltonian in the bulk to a local Hamiltonian in the boundary. A quasi -local Hamiltonian is a generalisation of a -local Hamiltonian, where instead of requiring that each term in the Hamiltonian acts on only -spins, we require that each term in the Hamiltonian has Pauli rank at most , 6 along with some geometric restrictions on the interaction graph. More precisely: Definition 6.1 (Quasi-local hyperbolic Hamiltonians). Let H 2 denote -dimensional hyperbolic space, and let ( ) ⊂ H 2 denote a ball of radius centred at . Consider an arrangement of qudits in H 2 such that, for some fixed , at most qudits and at least one qudit are contained within any ( ). Let denote the minimum radius ball (0) containing all the qudits (which without loss of generality we can take to be centred at the origin). A quasi -local Hamiltonian acting on these qudits can be written as: where the sum is over the qudits, and each term can be written as: where: • ℎ ( ) local is a term acting non-trivially on at most qudits which are contained within some ( ) Wilson is a Pauli operator acting non trivially on at most O( − ) qudits which form a line between and the boundary of (0) The extension to quasi-local bulk Hamiltonians allows us to consider using the HQECC to construct toy models of AdS/CFT with gravitational Wilson lines in the bulk theory. 7 With this definition, we obtain the following result.
Theorem 6.2. Consider any arrangement of qudits in H 2 , such that for some fixed at most qudits and at least one qudit are contained within any ( ). Let denote the minimum radius ball (0) containing all the qudits. Let bulk = ℎ be any (quasi) -local Hamiltonian on these qudits.
Then we can construct a Hamiltonian boundary on a 1D boundary manifold M with the following properties: 1. M surrounds all the qudits and has diameter O (max (1, log( )/ ) + log log ). Proof. There are three steps to this simulation. The first two steps follow exactly the same procedure as in [KC19a].

Any local observable/measurement in the bulk has a set of corresponding observables/measurements
Step 1. Simulate bulk with a Hamiltonian which acts on the bulk indices of a HQECC in H 2 of radius = O (max (1, log( )/ ) ).
In order to do this, we embed a tensor network composed of perfect tensors in a tessellation of H 2 by a Coxeter polygon with associated Coxeter system ( , ), and growth rate . Note that in a tessellation of H 2 by Coxeter polytopes the number of polyhedral cells in a ball of radius ′ scales as O( ′ ), where we are measuring distances using the word metric, ( , ) = ( −1 ). (See [KC19a] for a detailed discussion.) If we want to embed a Hamiltonian bulk in a tessellation we will need to rescale distances between the qudits in bulk so that there is at most one qudit per polyhedral cell of the tessellation. If ′ = , then If log( )/ ≥ 1 then the qudits in bulk are more tightly packed than the polyhedral cells in the tessellation, and we need to rescale the distances between the qudits by a factor of O (log( )/ ). If log( )/ < 1 then the qudits in bulk are less tightly packed then the cells of the tessellation, and there is no need for rescaling. The radius of the tessellation needed to contain all the qudits in bulk is then given by (24) 8 The entanglement wedge, E is a bulk region constructed from the minimal area surface used in the Ryu-Takayanagi formula. It has been suggested that on a given boundary region, , it should be possible to reconstruct all operators which lie in E [Hea+14]. The greedy entanglement wedge is a discretised version defined in [Pas+15,Definition 8] After rescaling there is at most one qudit per cell of the tessellation. There will be some cells of the tessellation which do not contain any qudits. We can put "dummy" qudits in those cells which do not participate in any interactions, so their inclusion is just equivalent to tensoring the Hamiltonian with an identity operator. We can upper and lower bound the number of "real" qudits in the tessellation. If no cells contain dummy qudits then the number of real qudits in the tesselation is given by max is the number of cells in the tessellation. By assumption, there is at least one real qudit in a ball of radius ′ . Thus the minimum number of real qudits in the tessellation scales as min = O( / ′ ) = O( ) = O( ), and = Θ( ) = Θ( ).
If the tessellation of H 2 by Coxeter polytopes is going to form a HQECC, the Coxeter polytope must have at least 5 faces [KC19a, Theorem 6.1]. From the HQECC constructed in [Pas+15] it is clear that this bound is achievable, so we will without loss of generality assume the tessellation we are using is by a Coxeter polytope with 5 faces. The perfect tensor used in the HQECC must therefore have 6 indices.
It is known that there exist perfect tensors with 6 indices for all local dimensions [Rai97]. We will restrict ourselves to stabilizer perfect tensors with local dimension for some prime . These can be constructed for = 2 [Pas+15] and ≥ 7 [Hel13]. Qudits of general dimension can be incorporated by embedding qudits into adimensional subspace of the smallest prime which satisfies ≥ and = 2 or ≥ 7. We then add one-body projectors onto the orthogonal complement of these subspaces, multiplied by some Δ ′ ≥ | bulk | to the embedded bulk Hamiltonian. The Hamiltonian ′ bulk on the -dimensional qudits is then a perfect simulation of bulk . We can therefore simulate any bulk which meets the requirements stated in the theorem with a Hamiltonian which acts on the bulk indices of a HQECC in H 2 .
Step 2. Simulate bulk with a Hamiltonian on the boundary surface of the HQECC.
We first set ≔ ′ +Δ , where ′ satisfies ′ Π C = ( ′ bulk ⊗1 dummy ) † . Here is the encoding isometry of the HQECC, Π C is the projector onto the code-subspace of the HQECC, 1 dummy acts on the dummy qudits and is given by Π C ( ) is the projector onto the codespace of the quantum error correcting code defined by viewing the th tensor in the HQECC as an isometry from its input indices to its output indices (where input indices are the bulk logical index, plus legs connecting the tensor with those in previous layers of the tessellation). Provided Δ ≥ ′ bulk , [KC19a, Lemma 6.9] ensure that meets the conditions in Definition 2.2 to be a perfect simulation of ′ bulk below energy Δ , and hence-as simulations compose-a perfect simulation of bulk .
Naturally, there is freedom in this definition as there are many ′ which satisfy the condition stated. We will choose an ′ where every bulk operator has been pushed out to the boundary, so that a 1-local bulk operator at radius corresponds to a boundary operator of weight O( − ). We will also require that the Pauli rank of every bulk operator has been preserved (see [KC19a,Theorem D.4] for proof we can choose ′ satisfying this condition).
Step 3. Simulate with a local, nearest neighbour Hamiltonian using the technique from Theorem 4.5.
In order to achieve the scaling quoted we make use of the structure of due to the HQECC. It can be shown [KC19a]  In order to preserve entanglement wedge reconstruction [Pas+15], the location of the spins containing the input state on the Turing machine work tape has to match the location of the original boundary spins. So, instead of the input tape at the beginning of the PE computation containing the input state, followed by a string of |0 s, the two are interspersed. Information about which points on the input tape contain the input state can be included in the description of the Hamiltonian to be simulated.
It is immediate from the definition of the greedy entanglement wedge [Pas+15, Definition 8] that bulk local operators in E ( ) can be reconstructed on . The boundary observables/measurements { ′ } corresponding to bulk observables/measurements { } which have the same outcome, because by definition simulations preserve the outcome of all measurements. The claim follows.
It should be noted that the boundary model of the resulting HQECC does not have full rotational invariance. In order to use the universal Hamiltonian construction the spin chain must have a beginning and end, and the point in the boundary chosen to "break" the chain also breaks the rotational invariance. However, it is possible to construct a HQECC with full rotational symmetry by using a history state Hamiltonian construction with periodic boundary conditions, as in [GI09, Section 5.8.2].
In [GI09, Section 5.8.2] a Turing machine is encoded into a local Hamiltonian acting on a spin chain of length with periodic boundary conditions. The ground space of the resulting Hamiltonian is 2 fold degenerate. It consists of history states, where any two adjacent sites along the spin chain can act as boundary spins for the purpose of the Turing machine construction -giving rise to 2 distinct ground states. 9 We can apply this same idea to construct a rotationally invariant HQECC, which maps a (quasi-)local bulk Hamiltonian, bulk in H 2 to a local Hamiltonian boundary acting on a chain of qudits. The code-space of the HQECC is 2 -fold degnerate, and below the energy cut-off boundary has a direct sum structure: bulk → boundary | ≤ Δ We can choose that the bulk state maps into the 'unrotated' boundary Hilbert space, so that the geometric relationship between bulk and boundary spins is preserved. 11

Discussion
In this work we have presented a conceptually simple method for proving universality of spin models. The reliance of this novel method on the ability to encode computation into the low energy subspace of a Hamiltonian suggests that there is a deep connection between universality and complexity. This insight is made rigorous in [Koh+21], where we derive necessary and sufficient conditions for spin systems to be universal simulators (as was done in the classical case [CC16]).
This new, simpler proof approach is also stronger, allowing to prove that the simple setting of translationally invariant interactions on a 1D spin chain is sufficient to give universal quantum models. Furthermore, we have provided the first construction of 9 The factor of two arises because there is freedom about which of the two adjacent sites is assigned to be the 'left' boundary, and which is the 'right' boundary. 10 See [CMP18, Section 7.1] for a discussion of maps on states in simulations. 11 Although the bulk states maps into one factor of the direct sum structure, every state in the low-energy portion of the boundary does have a bulk interpretation. But most of these states are rotated with respect to the bulk geometry. translationally invariant universal model which is efficient in the number of qudits in the simulator system.
Translationally invariant interactions are more prevalent in condensed matter models than interactions which require fine tuning of individual interaction strengths. However, a serious impediment to experimentally engineering either of the universal constructions in this paper is the local qudit dimension, which is very large-a problem shared by the earlier 2d translationally invariant construction in [PB20].
An important open question is whether it is possible to reduce the local state dimension in these translationally invariant constructions, while preserving universality. One possible approach would be to apply the techniques from [BCO17], which were used to reduce the local dimension of qudits used in translationally invariant QMA-complete local Hamiltonian constructions.
It would also be interesting to explore what other symmetries universal models can exhibit. This is of particular interest for constructing HQECC, where we would like the boundary theory to exhibit (a discrete version of) conformal symmetry.