Strongly Universal Hamiltonian Simulators

Leo Zhou1, ∗ and Dorit Aharonov2, †

1Department of Physics, Harvard University, Cambridge, MA 02138, USA2School of Computer Science and Engineering, The Hebrew University, Jerusalem 91904, Israel

(Dated: Feb 4, 2021)

A universal family of Hamiltonians can be used to simulate any local Hamiltonian by encodingits full spectrum as the low-energy subspace of a Hamiltonian from the family. Many spin-latticemodel Hamiltonians—such as Heisenberg or XY interaction on the 2D square lattice—are knownto be universal. However, the known encodings can be very inefficient, requiring interaction energythat scales exponentially with system size if the original Hamiltonian has higher-dimensional, long-range, or even all-to-all interactions. In this work, we provide an efficient construction by which theseuniversal families are in fact “strongly” universal. This means that the required interaction energyand all other resources in the 2D simulator scale polynomially in the size of the target Hamiltonianand precision parameters, regardless of the target’s connectivity. This exponential improvement overprevious constructions is achieved by combining the tools of quantum phase estimation algorithmand circuit-to-Hamiltonian transformation in a non-perturbative way that only incurs polynomialoverhead. The simulator Hamiltonian also possess certain translation-invariance. Furthermore, weshow that even 1D Hamiltonians with nearest-neighbor interaction of 8-dimensional particles ona line are strongly universal Hamiltonian simulators, although without any translation-invariance.Our results establish that analog quantum simulations of general systems can be made efficient,greatly increasing their potential as applications for near-future quantum technologies.

I. INTRODUCTION

Building a simpler model of a quantum system whilereproducing all its physical properties has many appli-cations in physics, chemistry, and computation. This isthe task of analog quantum simulation, where one simu-lates a Hamiltonian H by another Hamiltonian H ′ that issimpler or more easily implemented. This goal has beenidentified as a main motivation for quantum computersas early as 1981 by Feynman [1]. Due to its less strin-gent requirements on error correction and controls, ana-log simulation is considered to be an important practicalapplication in the era of noisy intermediate-scale quan-tum technology [2, 3]. Efficient implementation of analogHamiltonian simulators allows one to probe new many-body physics, develop new materials and drugs[4], andimprove feasibility of Hamiltonian-based quantum com-putations such as adiabatic algorithms [5, 6]. In fact,coherent analog quantum simulation in systems as largeas hundreds of qubits have already been successfully re-alized to solve condensed matter physics problems [7–9].

When seeking analog simulators of Hamiltonians, it isnatural to consider families of such simulators that areuniversal, in the sense that they can simulate any lo-cal Hamiltonian. For any target Hamiltonian H, thereshould exists a Hamiltonian H ′ in the family that cansimulate H. The ability to implement these universalfamilies enables analog simulation of all local Hamilto-nians, much like how a universal set of quantum gatesallows implementation of any unitary quantum opera-tion. This notion of universal Hamiltonians was devel-

∗ leozhou@g.harvard.edu† dorit.aharonov@gmail.com

oped in Ref. [10], in which various simple families of quan-tum spin-lattice models in two dimensions with tunablenearest-neighbor interaction energy are shown to be uni-versal. More families were shown to be universal by laterworks [11–13]. These can reproduce all physical proper-ties of the target system—including time-evolution, ther-mal states, and effects of local noise processes—to anyprecision.

However, the constructions given by Refs. [10–13] arenot efficient in the general case. The efficiency in factdepends on the connectivity or spatial dimensionality ofthe target Hamiltonian. While any Hamiltonian H in 2Dcan be simulated by a Hamiltonian from the universalfamily with spatially local interactions in 2D with onlypolynomial overhead in both the number of particles andthe interaction energy, an exponential overhead in the in-teraction energy is required in general by these construc-tions if the target Hamiltonian H is embedded in a higherdimension (e.g., when H is 3D or has all-to-all interac-tions). We call such families, in which the efficiency ofthe simulation is not guaranteed, weakly universal. Notethat using some gadgets [14], one can maintain polyno-mial interaction energy if one is willing to make otherresources exponential: the number of particles and thedegree (connectivity) of interaction. In any case, whenusing the constructions of [10–13], either an overhead ofexponential-strength interaction or exponential numberof particles (along with exponential interaction degree)is required for simulating general Hamiltonians.

In this work, we overcome this exponential overheadand arrive at what we call strong universality. We pro-vide a constructive method to design an analog simulatorthat is efficient in both the number of particles as wellas the interaction energy, and allows simulation in 2D ofany target local Hamiltonian, regardless of the geometry

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of its interaction graph. In fact, we show that this can bedone by 2D spin-lattice models which include only a sin-gle type of nearest-neighbor interaction, with interactionenergy that vary for different neighboring pairs (we callthis semi-translation-invariant). The overheads in par-ticle number and energy both only grow polynomiallywith the target system size. Our results show that anyof the semi-translation-invariant 2D families of Hamil-tonians that have been found to be weakly universal inRef. [10] are in fact also strongly universal, when moreefficient constructions are applied.

We further show that a similar result holds even in onespatial dimension. Nevertheless, there is a caveat whenrestricting to simulators in 1D: the strongly universal 1Dfamily that we construct is no longer semi-translation-invariant. The interactions in the simulating Hamiltoni-ans take on a more complicated form that need to varyin space, but the simulation is still efficient.

We note that these results are tight in the followingsense: We cannot hope to bring the polynomial over-head in the interaction energy down to a constant whilestill requiring that the simulating Hamiltonian is embed-ded in 1D (or 2D). This is due to the existence of somecounterexamples [15] showing that general (i.e., univer-sal) Hamiltonian simulation is impossible if the interac-tion energy is required to not increase with the systemsize and the simulator is set on a lattice (or any geometrywith bounded degree of connectivity).

To achieve our results, we begin with a method similarto that used in our previous work [15] (and recently ap-plied in Ref. [13]) in which we convert the target Hamil-tonian to a quantum phase estimation circuit embeddedin 1D. We then map this circuit back to a low-degree sim-ulating Hamiltonian, using the Feynman-Kitaev circuit-to-Hamiltonian construction [16]. The reason for trans-forming Hamiltonians via circuits is that unlike Hamilto-nians, circuits can be straightforwardly made “sparse”—e.g. each qubit is only acted on by a few gates. This canbe done by swapping qubits to fresh ancilla qubits af-ter every computational gate. In this work, we extendthis method to simulate any target Hamiltonian witha 1D or 2D Hamiltonian, by embedding the circuit ina spatially local manner in 1D or 2D using techniquesfrom earlier Hamiltonian complexity literature [6, 17]. Toobtain a semi-translation-invariant simulator Hamilto-nian in 2D, we borrow additional gadgets from Ref. [10].To obtain a 1D Hamiltonian simulator, we employ amodified construction of QMA-complete 1D Hamiltoni-ans from [18, 19] to simulate the circuit using nearest-neighbor Hamiltonian interactions on a line of particleswith 8 internal dimensions. These combinations of tech-niques allow us to overcome the exponential overheadcommon to previous constructions that mostly rely onperturbative gadgets for simulations [10–12, 17].

II. BACKGROUND ON UNIVERSALHAMILTONIANS FOR ANALOG SIMULATION

We first define what it means for a Hamiltonian tosimulate another. We adopt the well-motivated definitionof Ref. [10], which posits that H ′ simulates H if the fullspectrum ofH can be encoded as the low-lying part of thespectrum of H ′, by an encoding that preserves localityof observables. More precisely,

Definition 1 (Local encoding, adapted from [10]). Con-sider an encoding map E taking Hermitian operators on nqudits (d-dimensional systems), into operators acting onn′ ≥ n particles (not necessarily of the same dimensiond). We say E is a local encoding if we can write

E(H) = V (H ⊗ P + H ⊗Q)V †, (1)

such that V is an isometry, and can be written as V =⊗i Vi, where each Vi is an isometry acting on at most 1

qudit of the original system. Furthermore, P and Q arelocally orthogonal projectors (i.e., ∀i ∃ orthogonal projec-tors Pi, Qi acting on the same subsystem as Vi such thatPiQi = 0, PiP = P and QiQ = Q). H denotes complexconjugation.

Definition 2 (Hamiltonian simulation, adapted from[10]). Given an n-qudit Hamiltonian H, we say a Hamil-tonian H ′ is a (∆, η, ε)-simulation of H if for some localencoding E of the form of Eq. (1), we have

1. There exists an isometry V and corresponding en-coding E(H) = V (H ⊗ P + H ⊗ Q)V † such that

‖V − V ‖ ≤ η and E(1) = P≤∆(H′).

2. ‖H ′≤∆ − E(H)‖ ≤ ε.

Here, ‖ · ‖ is the spectral norm, P≤∆(H′) is the projectoronto the subspace of eigenstates of H ′ with eigenvalue≤ ∆, and H ′≤∆ = H ′P≤∆(H′) is the restriction of H ′ ontothese states. We say the simulation is efficient if both thenumber of particles in H ′ and its maximum energy ‖H ′‖are at most O(poly(n, η−1, ε−1,∆)), and the descriptionof H ′ is efficiently computable.

Under this definition, Ref. [10] showed that implement-ing H ′ allows one to approximately reproduce all physi-cal properties of H, implying that the term “simulation”means essentially any aspect. Specifically, since E is alocal encoding, all local A observables with respect toH are mapped to local observables E(A) for H ′. Corre-spondingly, there is a local map Estate(ρ) that maps quan-tum states, satisfying Tr(Aρ) = Tr[E(A)Estate(ρ)]. Gibbsstates (thermal ensembles) of H are mapped to Gibbsstates of H ′, and errors are exponentially suppressed bythe energy cutoff ∆. Time-evolution e−iHt can be alsosimulated by e−iH

′t applied on the appropriately encodedstate, and the error in this simulation grows as O(tε+η).Additionally, under a reasonable physical assumption, lo-cal noise and errors in the simulator has been shown to

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correspond to local noise and errors in the original sys-tem. Since any real physical system is subject to (typ-ically local) noise, the simulator can be used to probemany of its properties without error-correction.

Our goal is to understand which families of Hamiltoni-ans can be used to (efficiently) simulate all other physicalHamiltonians, as characterized by the notion of “univer-sal Hamiltonians” that we define below:

Definition 3 (Weak and strong universality). A fam-ily of Hamiltonians F = {Hm} is weakly universal ifgiven any ∆, η, ε > 0, any O(1)-local n-particle Hamil-tonian can be (∆, η, ε)-simulated by some Hm ∈ F .Such a family is strongly universal if the simulation isalways efficient—i.e., Hm is efficiently computable inO(poly(n)) time, requires n′ = O(poly(n, η−1, ε−1,∆))particles, and ‖Hm‖ = O(poly(n, η−1, ε−1,∆)).

Following Ref. [10], we consider Hamiltonians that onlyinvolve up to 2-local interactions, which can be writtenin the following form:

H ′ =∑〈i,j〉∈E

Jijh(i,j)αij , where ‖hαij‖ ≤ 1 and Jij ∈ R.

(2)Here, E is some set of edges describing the connectiv-

ity of the qudits, h(i.j)αij is some two-body operator hαij

acting on qudit i and j, and Jij is the interaction en-ergy. Ref. [10] studied such Hamiltonians in the casewhen hαij is drawn from some set of two-body interac-tions S = {hα}, which could be highly restricted andsometimes even just contain a single term. In this set-ting, we say H ′ is an S-Hamiltonian.

It is shown in Ref. [10] that many families of S-Hamiltonians on qubits are weakly universal, in the sensethat any local Hamiltonian can be simulated by a Hamil-tonian drawn from such a family. In fact, even restrict-ing the connectivity E of the qubits to the 2D squarelattice, many such S-Hamiltonian remains weakly uni-versal. For example, models such as Heisenberg in-teraction (S = {X ⊗ X + Y ⊗ Y + Z ⊗ Z}) or XY-interaction (S = {X ⊗ X + Y ⊗ Y }) on the 2D squarelattice are weakly universal. Here and below, we de-note (X,Y, Z) = (σx, σy, σz) as the Pauli matrices. Thismeans that the terms in Eq. (2) are all equal up to theirrelative weights, so S contains only a single term; yet,universality can still be achieved. We will need anotherdefinition to state this more concisely:

Definition 4 (Full and Semi-Translation-Invariance).We say that a Hamiltonian H ′ has semi-translation-invariance (or semi-TI) if every two-body operators are

the same up to the scaling by Jij, i.e., h(i,j)αij = h(i,j). We

say that H ′ has full translation-invariance (or full-TI) ifit has semi-translation-invariance and all the interactionenergy are the same, i.e., Jij = J .

We note that more generally, Ref. [10] has shown thatany family of S-Hamiltonians (even when restricted to

the 2D lattice) is weakly universal as long as S is non-2SLD, which roughly means that the 2-local part of allthe interactions in S are not simultaneously and locally(i.e., by 1-local unitaries) diagonalizable. More precisely,the property of 2SLD is defined as:

Definition 5 (2SLD interactions [10]). Suppose S isa set of interaction on 2 qubits. We say S is 2SLDif there exists U ∈ SU(2) such that for each Hi ∈ S,U⊗2Hi(U

†)⊗2 = αiZ⊗Z+Ai⊗1+1⊗Bi, where αi ∈ Rand Ai, Bi are any 1-local operator. Otherwise, S is non-2SLD.

These results are summarized in the following theorem:

Theorem 1 ([10]). Any S-Hamiltonian set on a 2Dsquare lattice of qubits is weakly universal as long as Sis non-2SLD.

Since there are many non-2SLD set of interactionsS which contain a single interaction term, this impliesthat there are many semi-translation-invariant familiesof Hamiltonians in 2D that are weakly universal. Laterworks have extended these results to show weak univer-sality of various families that are using qudits [11], em-bedded in higher dimensions [12], or fully translation-invariant in 2D or 1D [12, 13]. A major question is ofcourse whether the simulation can be made efficient, sothat universality is achieved in the strong sense.

Unfortunately, the constructions used by Ref. [10] toprove Theorem 1 are only efficient if the target Hamilto-nian have the same or lower spatial dimensionality as thesimulator Hamiltonian. When attempting to simulate a3D (or worse, all-to-all interacting) target Hamiltonianby a 2D simulator, however, the simulation is no longerefficient and requires exponentially large interaction en-ergy Jij = 2O(poly(n)). Alternatively, one can circumventthe exponentially large interaction by using exponentiallymany particles and degree of interaction [14], which alsogives up spatial locality.

III. MAIN RESULTS: STRONGLY UNIVERSALHAMILTONIANS

We are motivated by the fact that in many importantsituations, one is interested in studying Hamiltonians em-bedded in large spatial dimensions, sometimes even all-to-all interactions such as the SYK model [20]. On theother hand, experimental implementations typically onlyhave access to simulator Hamiltonians that are restrictedin their interaction geometry.

In this work, we show how to use the families of Hamil-tonians proposed in Ref. [10] to achieve not only weakuniversality but also strong universality for analog sim-ulation. In order to accomplish this improved efficiency,we have applied a different constructive method for sim-ulation than that of Ref. [10].

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spatial dimension translation-invariance interaction energy particle number

Cubitt et al. [10] 2D semi exp(poly(n)) poly(n)

Piddock-Bausch [12] 2D full exp(poly(n)) exp(poly(n))

Kohler et al.[13] 1D full* exp(poly(n)) poly(n)

Kohler et al.[13] 1D full exp(poly(n)) exp(poly(n))

This work (Theorem 2) 2D semi poly(n) poly(n)

This work (Theorem 3) 1D none poly(n) poly(n)

TABLE I. The properties of currently known constructions of universal families of Hamiltonians that can simulate any O(1)-localn-qudit Hamiltonian. The 1D construction of Kohler et al. [13] with full* translation-invariance uses a Hamiltonian interactionthat changes depending on the target Hamiltonian.

Theorem 2. Any S-Hamiltonian on the 2D square lat-tice is strongly universal, as long as S is non-2SLD. Inparticular, it’s sufficient for S to contain only a single in-teraction (such Heisenberg or XY-interaction), implyingthat there are semi-translation-invariant Hamiltonians in2D that are strongly universal.

We further show that strongly universal simulationcan even be achieved by 1D Hamiltonians with nearest-neighbor interactions, acting on particles of 8 internaldimensions. However, we give up any form of translation-invariance in the process.

Theorem 3. There is a strongly universal family of 1DHamiltonians consisting of nearest-neighbor interactionsacting on a line of particles with 8 internal dimension.

This family of 1D Hamiltonians is based a circuit com-putation on a line of qubits. The interactions in theHamiltonian family are tailored to represent various (uni-versal) gates that make up the computations, and thusdo not have any translation-invariance.

In Table I, we summarize our results and compare themto previously known constructions of universal families ofHamiltonians, in terms of the resources required for simu-lating general local Hamiltonians. Importantly, our con-structions only require polynomial overhead in both in-teraction energy and particle number for simulating gen-eral Hamiltonians, which makes them much more feasi-ble than previous constructions that require exponentialoverhead in one or both resources. We can do this evenwhen restricting our simulator to a low-dimensional spa-tial geometry, and imposing semi-translation-invariancein the case of 2D.

IV. PROOF SKETCHES

A. Overview

The starting point of our proofs for both Theorem 2and 3 is as follows. We note that previous construc-tions [10–12, 17] relied purely on perturbative gadgets toreduce the connectivity of qudits. Specifically, in theseconstructions, in order to reduce degree in the interactiongraph from O(n) to O(1) so that the Hamiltonian can be

embedded on a finite-dimensional lattice, it is necessaryto apply O(log n) rounds of perturbative gadgets, eachof which roughly halves the degree. Since the requiredinteraction energy increase polynomially for each appli-cation of perturbation gadget (i.e., J → [J poly(n)]c forsome constant c > 1), the final Hamiltonian requires in-

teraction energy scaling as Jfinal = ncO(logn)

= 2O(poly(n))

in these constructions.To circumvent this problem, we reduce the connectiv-

ity of the qudits by first mapping the target HamiltonianH to a quantum circuit performing the phase estimationalgorithm with respect to eiHt. Using standard tech-niques including Trotter decomposition [21], we can em-bed this circuit in 1D, using only nearest-neighbor gateson a line of qubits, while still applying the desired phaseestimation with sufficient accuracy. This circuit writesdown the energy eigenvalue of H as a bit-string in someancilla qubits.

In the 2D case, we can utilize swap gates to make thecircuit spatially sparse non-perturbatively. Here, spatialsparsity means that the qubits can be placed on a 2Dplane where each qubit participates in a constant numberof spatially local gates, in a sequence that traverses spacein a local way (see Definition 6). Note that a circuit thatuses only nearest-neighbor gates in 1D is not spatiallysparse if each qubit participates in O(poly(n)) gates.

Applying the Feynman-Kitaev circuit-to-Hamiltonianmapping [16] to the spatially sparse circuit gives us aspatially sparse Hamiltonian Hcircuit which has an ex-ponentially large degeneracy of ground states: for eacheigenstate of H, we have a groundstate of Hcircuit corre-sponding to the computational history of the phase esti-mation circuit running on that eigenstate as input. Wethen restore the spectral features of H in Hcircuit by im-posing bit-wise energy penalties on the energy bit-stringancilla qubits to match the energy of the eigenstate. Theincoherence induced by different computational historiesof different eigenstates can be repaired by the tricks of“uncomputing” and “idling.” Finally, Hcircuit, which isspatially sparse in 2D, can then be converted to a semi-translation-invariant Hamiltonian H ′ on a 2D square lat-tice using gadgets from Ref. [10]. This means that such afamily of semi-TI 2D Hamiltonian is strongly universal,as all steps of our construction incur only polynomialoverhead in the interaction energy and the number of

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any O(1)-localHamiltonian

Phase estimation circuit using NN gates in 1D

1D NN Hamiltonian on8-dimensional particles

2D spatially sparseHamiltonian

2D semi-TI Hamiltonianon a square lattice

Proposition 1 Proposition 2

Theorem 3 Theorem 2

FIG. 1. Overview of our constructions of 1D and 2D universalfamilies of Hamiltonians. NN = nearest-neighbor.

qubits.Likewise, in the 1D case, we construct a family of

Hamiltonians that simulate the 1D phase estimation cir-cuits. We combine the tools of uncomputing and idlingwith previously known circuit-to-Hamiltonian construc-tions deriving QMA-complete 1D Hamiltonians [18, 19]to achieve this result. The interactions in this Hamilto-nian family are nearest-neighbor operators that enforcesa set of transition rules, so that the zero-energy eigen-states are those corresponding to performing the circuitcomputation correctly. The spectral properties of H aresimilarly recovered by imposing bit-wise energy penalties.We have to employ some additional tricks to modify thetransition rules so that the original qudits of H are en-coded locally in the new 1D Hamiltonian.

In both cases, our non-perturbative techniques avoidthe exponential blow-up in the required interaction en-ergy, which brings the scaling of the energy overheadJfinal down to only O(poly(n)).

Figure 1 explains the proof structure for constructingboth the 2D and 1D universal Hamiltonians. The proofsdiverge only after the first step, Proposition 1, in whichthe target Hamiltonian is replaces by a phase estimationcircuit in 1D. The 1D and 2D cases differ in the way wemap the resultant 1D circuit back to a Hamiltonian.

B. The 1D phase estimation circuit: Proposition 1

We first show—and this is fairly standard—that onecan construct a phase estimation circuit UNN

PE that wouldtake eigenstates of H as input and (approximately) writedown their energy on ancilla qubits, using only nearest-neighbor gates acting on a line of qubits. Note that if His an O(1)-local n-qudit Hamiltonian where d is a con-stant, we can easily convert it to an O(1)-local O(n)-qubit Hamiltonian by simply encoding each qudit in thesubspace of a group of dlog2 de qubits. We can separatethe extra states in this redundant encoding (when d isnot a power of 2) from the spectrum by adding to theHamiltonian a local energy penalty term on each groupwith ‖H‖ = O(poly(n)) magnitude. Hence, for simplic-

flow of tim

eFIG. 2. Illustration of converting a circuit to a spatially sparseone. (a) Adding swap gates to make a 2-qubit gate (red)acting on distant qubits nearest-neighbor. (b) Adding newqubits so that the execution of gates are in a spatially localsequence.

ity of the discussion that follows, we will assume that thisconversion is always performed first, and H is an O(1)-local n-qubit Hamiltonian whose interaction energy is atmost O(poly(n)).

Given such a Hamiltonian H, let us write H =∑µEµ|ψµ〉〈ψµ| in its energy eigenbasis, where 0 ≤ Eµ ≤

Emax without loss of generality. Note the upper boundEmax = O(poly(n)) can be computed without knowledgeof the energy eigenvalues, e.g. by adding up the spectralnorm of individual local terms of H.

Ideally, we want a phase estimation circuit U idealPE

that acts on any input state of the form |ψ〉 |0m〉 =∑µ cµ |ψµ〉 |0m〉 in the following way

U idealPE

∑µ

cµ |ψµ〉 |0m〉 =∑µ

cµ |ψµ〉 |Eµ〉 , (3)

where the first s = O(log(n)) qubits of the m-qubitancilla register encodes the energy Eµ as |Eµ〉 =|ϕ1ϕ2 . . . ϕs〉 ⊗ |restµ〉. Here, ϕµ = 0.ϕ1ϕ2 . . . is the bi-nary representation of the real number ϕµ = Eµ/Emax.

Nevertheless, we want to implement U idealPE with only

polynomial number of local gates, which can be doneusing the Trotter decomposition [21]. We also want touse a discrete set of 2-qubit universal gates, which can bedone by invoking the Solovay-Kitaev theorem [22]. Thisgives us U local

PE , which will have some error ζ = ‖(U localPE −

U idealPE ) |ψ〉 |0m〉 ‖. This error ζ can be made small using

only O(poly(n, ζ−1)) resources (see Appendix A). Then,as shown in Fig. 2(a), we make all gates to be nearest-neighbor on a line by adding swap gates, obtaining UNN

PE .This fact is summarized as Proposition 1:

Proposition 1. Given any n-qubit O(1)-local Hamilto-nian H =

∑µEµ|ψµ〉〈ψµ|, one can construct a circuit

UNNPE consisting of poly(n, ζ−1) gates acting on n + m

qubits with 1- or 2-qubit nearest-neighbor gates chosenfrom a universal gate set, where m = poly(n), such thatfor any normalized input state

∑µ cµ |ψµ〉∥∥∥UNN

PE

∑µ

cµ |ψµ〉 |0m〉 −∑µ

cµ |ψµ〉 |Eµ〉∥∥∥ ≤ ζ (4)

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C. Constructing 2D semi-TI strongly universalHamiltonians

To prove Theorem 2, we want to simulate any localHamiltonian H with a S-Hamiltonian on a 2D squarelattice with polynomial overhead, for any non-2SLD S.We start by first constructing a spatially sparse cir-cuit Hamiltonian Hcircuit that simulates our original tar-get Hamiltonian H (Proposition 2). Then the (semi-translation-invariant) S-Hamiltonian simulator on the2D lattice is obtained from Hcircuit by applying a series ofgadgets. The full technical proof in given in Appendix B,but we will sketch the essential ideas below. To that end,we borrow the notion of spatial sparsity from Ref. [10, 17]and generalize it to circuits:

Definition 6 (Spatial sparsity of Hamiltonians and cir-cuits (adapted from [17])). A Hamiltonian on n-qudits isspatially sparse if its interaction hypergraph is one where(i) every vertex participates in O(1) hyper-edges, and (ii)there is a straight-line drawing in the plane such that ev-ery hyper-edge overlaps with O(1) other hyper-edges, andthe surface covered by every hyper-edge is O(1). More-

over, we say a quantum circuit U =∏Tt=1 Ut is spatially

sparse if there is a placement of the qudits in the two-dimensional plane such that (i) each qudit participates inO(1) gates, and (ii) the spatial supports of Ut and Ut+1

are only O(1) distance apart for all t, each covering O(1)contiguous area.

With the definition of spatial sparsity in hand, we nowformally state the Proposition we want to show:

Proposition 2. Given any O(1)-local n-qudit Hamil-tonian H with ‖H‖ = O(poly(n)), one can constructa spatially sparse 5-local Hamiltonian Hcircuit that ef-ficiently simulates H to precision (∆, η, ε), with ∆ =O(ε−1‖H‖2 + η−1‖H‖). Hcircuit has O(poly(n, ε−1))terms and qubits, and interaction energy at mostO(poly(n, η−1, ε−1)).

To prove this Proposition, we start from the 1D phaseestimation circuit obtained using Proposition 1 and con-struct a spatially sparse circuit using swap gates. To dothis, we first make UNN

PE spatially sparse into U sparsePE by

moving it from the line to a 2D grid. As visualized inFig. 2(b), starting from the leftmost column of qubits,we apply just one nearest-neighbor gate before swappingall qubits to the next column and applying the next gate,getting us U sparse

PE . By ordering the gates in a snake-likefashion similar to Ref. [6, 17], we make sure that eachqubit participates in only a constant number of gates,and that the temporally proximate gates in the sequencehave spatially proximate support.

Before converting the circuit back to a Hamiltonian,we need to address the issue of the entanglement betweeneach energy eigenstate and the ancilla register that hasthe energy bit-string, as evident in Eq. (3). This inco-herence between different eigenstates causes a large error

for the simulation if left unchecked. We repair this er-ror by running the circuit backwards (“uncomputing”)and then adding L identity gates at the end (“idling”),so that most of the computational history of each eigen-state is simply the state itself. We then get a new circuitwhich Ucircuit = (1)L(U sparse

PE )†(1)sU sparsePE , which is spa-

tially sparse as long as U sparsePE is. Note we inserted s

identity gates before applying (U sparsePE )† so that we can

examine the energy bit-string bit-by-bit before it is un-computed.

Subsequently, we apply Kitaev’s circuit-to-Hamiltonian mapping [16] to convert the spatiallysparse, “uncomputed” circuit Ucircuit to a spatiallysparse Hamiltonian Hcircuit. The energy of the eigen-states of H are restored by adding an appropriatebit-wise energy penalty on the s qubits where the energyis written, while we idle between U sparse

PE and (U sparsePE )†

[see Eq. (B7)]. We then use perturbative argument(such as those shown in Ref. [23]) to show that Hcircuit

simulates H with polynomially small error, with onlypolynomial overheads. This proves Proposition 2.

Finally, to prove Theorem 2, we map the spatiallysparse Hamiltonian Hcircuit to a Hamiltonian in a uni-versal family on the 2D square lattice with additionalpolynomial overhead. This is done via a sequence of re-ductions, with known techniques [10, 17]: We first con-verted Hcircuit to a real-valued Hamiltonian by doublingthe number of qubits, and encoding any Pauli Y ’s intoa pair of Y ⊗ Y . We then remove all Pauli Y ’s, andreduce the locality of the Hamiltonian to 2-local via ap-plications of perturbative gadgets. This is subsequentlyconverted to a spatially sparse S0-Hamiltonian whereS0 = {XX + Y Y + ZZ} or {XX + Y Y }. Through-out this sequence of reductions, the spatial sparsity ofthe Hamiltonian is preserved, and both the interactionenergy and qubit number only increases polynomially asthe original system size and the target precision param-eters (∆, η, ε). This spatially sparse S0-Hamiltonian in-volving only Pauli-interactions without Y ’s can then bemapped to a S0-Hamiltonian on the 2D lattice. Since theinput Hamiltonian is spatially sparse, this mapping onlyincurs polynomial overhead in the interaction energy asshown in Ref. [10, 17]. Furthermore, Ref. [10] has shownthat S0-Hamiltonian on a 2D square lattice can be sim-ulated by any S-Hamiltonian on a 2D square lattice forany non-2SLD S, with polynomial overhead. We havethus provided a construction that allows any such familyof S-Hamiltonians on the 2D square lattice to efficientlysimulate any local Hamiltonian with arbitrary geometry.

D. Constructing 1D universal Hamiltonians

We now show how to construct a strongly univer-sal family of Hamiltonian simulators in 1D. To provethis result as stated in Theorem 3, we extend thecircuit-to-1D-Hamiltonian constructions in Ref. [18, 19].These constructions were originally used to show QMA-

7

completeness of Hamiltonians involving nearest-neighborinteraction on a 1D line of particles, by using them toencode the outcome of any circuit computation in theirground state energy.

The basic idea in these constructions is as follows.Suppose we are given any quantum circuit consistingof R rounds of nearest-neighbor gates on n qubits in1D such as UNN

PE , where each round is of the formUn−1,n · · ·U23U12 (some may be identity). We consideran equivalent circuit on a line of 2nR particles, dividedinto R blocks, where each block encodes the computa-tional state of the original n qubits. In this equivalentcircuit, a single round of nearest-neighbor gate is appliedin each block before the qubits are moved to the nextblock where subsequent gates can be performed. The8 internal dimensions of each particle are necessary tostore both the computational state of the original qubitand marker states that allows us to locally distinguishdifferent stages of the computation of each particle (e.g.,whether a gate has already been applied or needs to beapplied).

We want to apply these constructions to simulate gen-eral Hamiltonians. Following the same idea as in theproof of Theorem 2, we first convert the target Hamilto-nian H to a phase estimation circuit UNN

PE using nearest-neighbor gates on a line of qubits as in Proposition 1.Then, we use the method in Ref. [19] as well as our tricksof “uncomputing” and “idling” to map the circuit to a1D Hamiltonian of nearest neighbor-interactions, and pe-nalize the energy bit-string so that we simulate the fullspectral properties of H. However, a naıve applicationof this method would yield a highly non-local encoding,since the idling part of the circuit corresponds to sim-ply move the computational qubits down the line, caus-ing the encoded eigenstates of H to be delocalized overmany blocks of particles. To circumvent this issue, wemodified the method so that the idling step is done with-out moving the computational qubits, while maintainingthe consistency of all transition rules so that the legalcomputational history states are spectrally gapped fromthe rest. Consequently, the eigenstates of H are encodedin the qubit-subspace of some 8-dimensional qudits fromjust one block of the line, yielding a local encoding. Forthe full proof with technical details, see Appendix C.

In this construction, our 1D simulator does not haveany form of translation-invariance, since its nearest-neighbor terms vary from block to block where differentgates from UNN

PE are applied. Note the semi-translation-invariance in our 2D simulator is achieved (as done inRefs. [10–12]) using gadgets to simulate general inter-actions with a single type of interaction. Since thesegadgets require ancilla particles placed in more than onedimension, it appears we cannot apply them to make our1D construction semi-translation-invariant.

V. DISCUSSION

We have significantly improved the prospects of univer-sal analog quantum simulation by showing that a strongnotion of universality is possible with simple families ofHamiltonians embedded in constant dimensions. Unlikeprevious works [10–13], our results show that only poly-nomial overheads in both particle number and interactionenergy are sufficient to simulate any local Hamiltonianwith arbitrary connectivity by some universal Hamilto-nians embedded in 1D or 2D. Our results are tight inthe sense that the overhead in the interaction energycannot be brought down to a constant using constant-dimensional Hamiltonian simulators, due to an earlierresult [15] that gave counterexamples showing such simu-lations are impossible. We remark that even though thereare weakly universal, fully translation-invariant familiesof Hamiltonians [12, 13], the interaction energy in thoseHamiltonians has to scale exponentially with the size ofthe target Hamiltonian, since the target Hamiltonian’sspectrum is encoded in an exponentially vanishing frac-tion of the spectrum of the simulator.

An interesting open question is: Are there are stronglyuniversal semi-translation-invariant Hamiltonians in 1D?Since the known gadgets to simulate general interactionswith a single type of interaction seem to require ancillaparticles placed in more than one dimension, we wouldlikely need to invent new gadgetry.

We can also ask if there are strongly universal Hamil-tonians with full translation-invariance in any constantdimensions. This is impossible if translation-invarianceis required in the strongest sense, where the only freeparameter of the Hamiltonians is the number of particlen′; since such Hamiltonians can be described by O(log n′)bits of information, they cannot represent general Hamil-tonians on n qudits that are described by poly(n) bits un-less n′ = exp(poly(n)). Indeed, Refs. [12, 13] constructedsuch families of Hamiltonians that are weakly univer-sal, but cannot be strongly universal. This implies thatthere not all weakly universal families are strongly uni-versal. To move towards full-TI strong universality, wemay consider relaxing the notion of simulation: for exam-ple, Ref. [24] constructed a 1D fully translation-invariantfamily of Hamiltonians that can simulate any Hamilto-nian by allowing for polynomial-sized encoding and de-coding circuits. However, the desirable properties of ana-log Hamiltonian simulations such as preserving locality ofobservables and noise will no longer hold once we have en-coding circuits that induce non-local correlations. Alter-natively, one can consider relaxing translation-invarianceby letting Hamiltonian interactions have more free pa-rameters to encode the target Hamiltonian. Ref. [13] hasdone this to keep the number of particles in the simulatorO(poly(n)), but their construction still requires exponen-tial overhead in the interaction energy. Nevertheless, itis likely possible to efficiently simulate any full-TI Hamil-tonian by a family of full-TI Hamiltonian, where the de-scription of both Hamiltonians areO(log n) bits. It would

8

be worthwhile to investigate whether these constructionscan be improved, or show that strong universality of full-TI Hamiltonians is impossible.

While this work has established that efficient univer-sal analog quantum simulation using simple 1D or 2Dsystems is possible, the constructions we have providedhere is far from optimal. Although we have shown thatresources scaling only polynomially in the target sys-

tem size are sufficient for analog simulation of any localHamiltonian, there is much room for improvement in thescaling for practical applications. As experimental re-alizations of analog quantum simulators develop rapidly,we hope our work provides a starting point for researchersto develop methods to expand their scope to simulate allphysical systems and tackle classically intractable prob-lems.

[1] R. P. Feynman, International Journal of TheoreticalPhysics 21, 467 (1982).

[2] J. I. Cirac and P. Zoller, Nature Physics 8, 264 (2012).[3] J. Preskill, Quantum 2, 79 (2018), arXiv:1801.00862.[4] J. Arguello-Luengo, A. Gonzalez-Tudela, T. Shi,

P. Zoller, and J. I. Cirac, Nature 574, 215 (2019).[5] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser,

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S. Lloyd, and O. Regev, SIAM J. Comput. 37, 166(2007).

[7] J. Y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse,I. Bloch, and C. Gross, Science 352, 1547 (2016),arXiv:1604.04178.

[8] D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Se-meghini, S. Ebadi, T. T. Wang, A. A. Michailidis,N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner,V. Vuletic, and M. D. Lukin, (2020), arXiv:2012.12276.

[9] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Se-meghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pich-ler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner,V. Vuletic, and M. D. Lukin, (2020), arXiv:2012.12281.

[10] T. S. Cubitt, A. Montanaro, and S. Piddock, Proceed-ings of the National Academy of Sciences 115, 9497(2018).

[11] S. Piddock and A. Montanaro, (2018), arXiv:1802.07130.[12] S. Piddock and J. Bausch, (2020), arXiv:2001.08050.[13] T. Kohler, S. Piddock, J. Bausch, and T. Cubitt, (2020),

arXiv:2003.13753.[14] Y. Cao and D. Nagaj, Quantum Inf. Comput. 15, 1197

(2015).[15] D. Aharonov and L. Zhou, in Proceedings of the 2019

ACM Conference on Innovations in Theoretical Com-puter Science, ITCS ’19 (2019) arXiv:1804.11084.

[16] A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical andQuantum Computation (American Mathematical Soci-ety, 2002).

[17] R. Oliveira and B. M. Terhal, Quantum Inf. Comput. 8,900 (2008).

[18] D. Aharonov, D. Gottesman, S. Irani, and J. Kempe,Communications in Mathematical Physics 287, 41(2009).

[19] S. Hallgren, D. Nagaj, and S. Narayanaswami, QuantumInfo. Comput. 13, 721 (2013).

[20] S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993);A. Kitaev, A simple model of quantum holography (KITPstrings seminar and Entanglement 2015 program, 2015).

[21] H. F. Trotter, Proceedings of the American MathematicalSociety 10, 545 (1959).

[22] C. M. Dawson and M. A. Nielsen, Quantum Info. Com-put. 6, 81 (2006).

[23] S. Bravyi and M. Hastings, (2014), arXiv:1410.0703.[24] T. C. Bohdanowicz and F. G. Brandao, (2017),

1710.02625 [quant-ph].[25] M. A. Nielsen and I. L. Chuang, Quantum Computation

and Quantum Information, 10th ed. (Cambridge Univer-sity Press, New York, NY, USA, 2011).

[26] J. Kempe, A. Kitaev, and O. Regev, SIAM J. Comput.35, 1070 (2006).

Appendix A: A 1D nearest-neighbor implementation of phase estimation circuit

In this appendix, we show that given any local Hamiltonian H, how to construct a phase estimation circuit suchthat the energy of any input eigenstate of H can be written down as bits on some ancilla qubits to O(log n) bitprecision with O(1/poly(n)) error in the state. In particular, we show that this can be done with a circuit acting ona line of qubits with nearest-neighbor gates. This will serve as the backbone of our efficient construction of universalHamiltonian simulators.

Proposition 1 (formal). Consider any O(1)-local Hamiltonian H =∑aHa =

∑µEµ|ψµ〉〈ψµ| acting on n qubits,

where we assume w.l.o.g. that 0 ≤ Eµ ≤ Emax for some known number Emax = O(poly(n)). For any s = O(log n) andζ > 0, we can construct a phase estimation circuit UNN

PE consisting of O(poly(n, ζ−1)) 1- or 2-qubit nearest neighborgates drawn from any universal gate set, acting on a line of n+m qubits, where m = O(poly(n)). For any normalizedstate

∑µ cµ |ψµ〉, the circuit UNN

PE satisfies∥∥∥∥∥UNNPE

∑µ

cµ |ψµ〉 |0m〉 −∑µ

cµ |ψµ〉 |Eµ〉 |restµ〉

∥∥∥∥∥ ≤ ζ (A1)

9

where |Eµ〉 = |ϕµ,1ϕµ,2ϕµ,3 · · ·ϕµ,s〉 is the s-bit truncated representation of ϕµ = Eµ/Emax = 0.ϕµ,1ϕµ,2ϕµ,3 · · · withϕµ,j ∈ {0, 1}, and |restµ〉 is some unimportant state on the remaining ancilla qubits.

Proof. Let us first consider the standard implementation of quantum phase estimation algorithm circuit UPE. Here,

the circuit uses the evolution operator uj = eiHτ2j−1

under H, where τ = 2π/Emax, and writes phase of the eigenvaluesof u1 = eiHτ on some ancilla qubits. Note the eigenvalues of u1 are ei2πϕµ , where ϕµ = Eµτ/(2π). Since 0 ≤ ϕµ ≤ 1,we can write ϕµ = 0.ϕµ,1ϕµ,2ϕµ,3 · · · .

Ideally, the action of the phase estimation circuit on input states {|ψµ〉 |0m〉}2n

µ=1 is

U idealPE |ψµ〉 |0m〉 = |ψµ〉 |Eµ〉 |restµ〉 , (A2)

Correspondingly, let us denote Eµ = 2πϕµ/τ as approximate values of the energy Eµ, where ϕµ =

0.ϕµ,1ϕµ,2ϕµ,3 · · ·ϕµ,s. In the ideal case, Eµ = Eµ for some sufficiently large s.In reality, there are two sources of errors that cause the phase estimation circuit to deviate from U ideal

PE .Error 1: finite-bit-precision— The first is due to the fact that the energy eigenvalues don’t generally have finite-

bit-precision representation, i.e., |Eµ − Eµ| = O(2−s) is non-zero. In other words, since ϕµ 6= ϕµ, there’s additionalerror from imprecise phase estimation. Let us consider a phase estimation circuit UPE implemented to p-bit precision,where p > s. Let bµ be the integer in the range [0, 2p − 1] such that 0 ≤ ϕµ − bµ/2p ≤ 2−p. It is well-known [25] thatthe action of UPE on any input state |ψµ〉 |0〉 result in the following state

UPE |ψµ〉 |0m〉 = |ψµ〉 |rest′µ〉 ⊗1

2p

2p−1∑k,`=0

e−i2πk`/2p

ei2πϕµk |`〉 = |ψµ〉 |rest′µ〉2p−1∑`=0

αµ` |`〉 (A3)

where |`〉 = |`1 · · · `p〉 is the binary representation of `, and

αµ` =1

2p

2p−1∑k=0

[ei2π(ϕµ−`/2p)]k =1

2p

[1− ei2π(2sϕµ−`)

1− ei2π(ϕµ−`/2p)

](A4)

The analysis from Sec. 5.2.1 in Ref. [25] shows that the probability of getting a state that is a distance of e integeraway is

perrorµ (e) ≡

∑|`−bµ|>e

|αµ` |2 ≤ 1

2(e− 1)(A5)

Note that we only care about the first s < p bits, so we can choose e = 2p−s − 1.Hence,

UPE |ψµ〉 |0m〉 = |ψµ〉 |rest′µ〉 ⊗

∑|`−bµ|≤e

αµ` |`〉+∑

|`−bµ|>e

αµ` |`〉

= |ψµ〉 |rest′µ〉 ⊗

(√1− perror

µ |Eµ〉 |rest1µ〉+

√perrorµ |rest2

µ〉)

(A6)

Comparing this with the idealized output in Eq. (A2), we can identify |restµ〉 = |rest′µ〉 |rest1µ〉, and observe that

(UPE − U idealPE ) |ψµ〉 |0m〉 = |ψµ〉 |errorµ〉 , where ‖|errorµ〉‖2 ≤ 2perror

µ = O(2−(p−s)) (A7)

Thus, for any normalized state |ψ〉 =∑µ cµ |ψµ〉 |0m〉, we have

‖(UPE − U idealPE )

∑µ

cµ |ψµ〉 |0m〉 ‖2 = O(2−(p−s)) ≤ O(ζ) (A8)

where we chose, for example, p = 2s+O(log ζ−1) and s = O(log(n)), and thus make this first source of error due toimprecision to be smaller than any constant c1.

Error 2: local gate approximation of e−iHτj— The second source of error is due to the fact that we need toimplement the circuit UPE using only 1 or 2-qubit gates, in order to ensure the corresponding circuit-Hamiltonianis local, The only non-local gates in the p-bit precise phase estimation algorithm that we need to address are thecontrolled-application of Hamiltonian evolution, |0〉〈0| ⊗ 1 + |1〉〈1| ⊗ uj , where uj = e−iHτj , τj = 2j−1τ and j =1, 2, . . . , p. This can be implemented with local gates via Trotter decomposition [21].

10

Specifically, we write H =∑M0

a=1Ha, where Ha is a k-local term, and M0 = O(poly(n)) is the number of terms.

We can implement uj = (∏M0

a=1 e−iHaτj/rj )rj for some integer rj , so that ‖uj − uj‖ ≤ O(τ2

j /rj). Since p = O(log n+

log ζ−1), we have τj = O(2p/‖H‖) = O(poly(n, ζ−1)). We can then choose rj = O(τ2j poly(ζ−1)) = O(poly(n, ζ−1))

to ensure each such error is polynomially small. The error from Trotter decomposition is bounded by

‖UTrotPE − UPE‖ ≤

p∑j=1

O(τ2j /rj) ≤ O(ζ) (A9)

The total number of local gates in UTrotPE is RTrot = O(M0

∑j rj) = O(poly(n, ζ−1)), and the locality of each gate is

at most k + 1.We still need a circuit with only 1- or 2-qubit gates drawn from a universal set of gates. To that end, we can apply

the Solvay-Kitaev algorithm to approximate each (k + 1)-local gate with a sequence of 2-local gates. It is known[22]that to approximate any gate in SU(d) with 2-local gates to ε0-precision, we’ll need at most O(d2 poly log ε−1

0 ) 2-qubit gates from some universal gate set of finite size. Note d = 2k+1 when we are approximating (k + 1)-localgates with 2-qubit gates. As there are at most RTrot such gates, to keep the overall error to be below O(ζ), weonly need ε0 ≤ O(ζ/RTrot) = O(1/ poly(n, ζ−1))). This means we need to approximate each (k + 1)-local gates withRSK = O(4k poly(log n, log ζ−1)) 2-qubit gates. The final circuit is U local

PE consisting of only Rlocal = O(RTrotRSK) =O(poly(n, ζ−1)) 1- or 2-qubit gates, for k = O(1).

We can then ensure that this circuit U localPE only consists of nearest-neighbor gates by the following procedure:

1. Place all n qubits on a line with any pre-determined ordering.

2. Iterate over each gate Ut, t = 1, . . . Rlocal. If Ut acts on qubits that are not neighbors on the line, add a sequenceSt of swap gates on nearest neighbors in the circuit before Ut so that Ut acts on neighbors. Then add the sameswap gates in reversed order in the circuit after Ut so that the qudits returned to their original order on the

line. At the end of this step, the new circuit is of the form UNNPE =

∏R0

t=1(S†tUtSt), with R0 = O(RlocalN) gates,each only acting on a neighbors group of qubits. See Fig. 2(a) for an example.

Putting everything together, we have UNNPE = U local

PE ≈ UTrotPE ≈ UPE ≈ U ideal

PE . In conclusion, for any ζ > 0, we canconstruct a phase estimation circuit UNN

PE comprised of only O(poly(n, ζ−1)) 1-qubit or 2-qubit nearest-neighbor gateson a line, such that its action is ζ-close to U ideal

PE on any valid input state∑µ cµ |ψµ〉 |0m〉:

‖(UNNPE − U ideal

PE ) |ψ〉 |0m〉 ‖ ≤ ζ. (A10)

Appendix B: Proof that spin models on 2D lattice is strongly universal

In this section, we show the details of the construction for strongly universal Hamiltonian in 2D square lattice.

Theorem 2. Any S-Hamiltonian on the 2D square lattice is strongly universal, as long as S is non-2SLD. Inparticular, it’s sufficient for S to contain only a single interaction (such Heisenberg or XY-interaction), implying thatthere are semi-translation-invariant Hamiltonians in 2D that are strongly universal.

1. Efficient, spatially sparse Hamiltonian simulator

Before proving the strong universality of 2D spin-lattice models, we first prove the following result, where we showthat any local Hamiltonian can be simulated by a spatially sparse Hamiltonian.

Proposition 2. Given any O(1)-local n-qudit Hamiltonian H with ‖H‖ = O(poly(n)), one can construct a spatiallysparse 5-local Hamiltonian Hcircuit that efficiently simulates H to precision (∆, η, ε), with ∆ = O(ε−1‖H‖2 +η−1‖H‖).Hcircuit has O(poly(n, ε−1)) terms and qubits, and interaction energy at most O(poly(n, η−1, ε−1)).

To prove the above Proposition, we first prove two smaller Lemmas 1 and 2 about different aspects of using theFeynman-Kitaev circuit-to-Hamiltonian construction [16] for Hamiltonian simulation. The following concept of historystates will be useful in the discussion:

11

Definition 7 (history states). Let U = UT · · ·U2U1 be a quantum circuit acting on n+m qudits. Then for any inputstate |ψµ〉 ∈ Cd

n

, the history state with respect to U and |ψµ〉 is the following

|ηµ〉 =1√T + 1

T∑t=0

(Ut · · ·U2U1 |ψµ〉 |0m〉anc

)|1t0T−t〉clock

(B1)

We now prove the first of the two Lemmas, which describes a circuit-to-Hamiltonian transformation that can be usedfor analog Hamiltonian simulation, assuming an appropriate energy penalty Hamiltonian Hout can be constructed.

Lemma 1 (Circuit-Hamiltonian simulation). Consider an orthonormal basis of states {|ψµ〉}dn

µ=1 on n qudits. Let

U =∏Tt=1 Ut be a quantum circuit where each gate Ut is at most k-local. Let L = span{|ηµ〉}d

n

µ=1 be the subspace ofhistory states with respect to U and {|ψµ〉}, and let H be any Hamiltonian. Suppose there exists a Hamiltonian Hout

such that

‖V HV † −Hout|L‖ ≤ ε/2 (B2)

where E(H) = V HV † is a local encoding, and O|L means operator O restricted to subspace L. Then for any η > 0,we can construct a Hamiltonian Hcircuit from the description of U such that Hcircuit is a (∆, η, ε)-simulation of Hwith local encoding E, where ∆ ≥ O(ε−1‖Hout‖2 + η−1‖Hout‖), per Def. 2. The constructed Hcircuit is (k + 3)-local,has O(T ) terms and particles, and uses O(poly(n, T,∆)) interaction energy. Furthermore, Hcircuit is spatially sparseif the circuit U is spatially sparse.

Lemma 2 (Idling to enhance simulation precision). Consider an uncomputed quantum circuit UD · · ·U1 = 1. Supposewe add L identity gates to the end of the circuit, so that we obtain a new circuit U = 1LUD · · ·U1 with length T = D+L.Let |ηµ〉 be the history state with respect to U and |ψµ〉. Suppose H =

∑µEµ|ψµ〉〈ψµ| and Heff =

∑µEµ|ηµ〉〈ηµ|.

For any ε > 0, if we choose L = O(D‖H‖2

ε2 ), then there is an ancilla state |α〉 such that ‖H ⊗ |α〉〈α| −Heff‖ ≤ ε.

We are now ready to prove our main Proposition 2:

Proof of Proposition 2. Given any O(1)-local n-qudit Hamiltonian where d is a constant, we can easily convert itto an O(1)-local O(n)-qubit Hamiltonian by simply encoding each qudit in the subspace of a group of dlog2 de qubits.We can separate the extra states in this redundant encoding (when d is not a power of 2) from the relevant part ofspectrum by adding to the Hamiltonian a local energy penalty term on acting each group with ‖H‖ = O(poly(n))magnitude. Hence, we will call H the O(1)-local n-qubit Hamiltonian containing O(poly(n))-strength interactionsobtained after this conversion.

Let us denote the normalized eigenstates of H as |ψµ〉, with corresponding eigenvalues Eµ. We assume they areordered such that E1 ≤ E2 ≤ E3 ≤ · · · ≤ E2n .

From Proposition 1, for any s = O(log n), we can construct a ζ-approximate, s-bit precise phase estimation circuitUNN

PE such that it acts on a line of N = O(poly(n)) qubits with R0 = O(poly(n, ζ−1)) nearest-neighbor gates. Wewant to replace it with a spatially sparse circuit U sparse

PE with O(R0N) qubits and gates (see Definition 6). This canbe done with polynomial overhead in the same way as in Ref. [6, 17]. We begin by placing the N original qubits onthe first column of a N ×R0 grid of qubits. For column i = 1, 2, . . . , R0, we execute only the i-th gate from the circuitUNN

PE , and other uninvolved qubits are acted on by identity gates. After each column, we swap the state of the qubitof column i to i + 1. We order the execution of all the gates such that the gates from UNN

PE and identity gates areexecuted top-to-bottom, and the swap gates between column are executed from bottom-to-up [see Fig. 2(b)]. It isthus easy to see that in this new circuit U sparse

PE , each qubit participates in at most 3 gates (up to two swap gates anda non-trivial gate), and the gate are executed in a spatially local sequence. Note the action of U sparse

PE is equivalent toUNN

PE up to re-ordering of the qubits, since we’ve only added swap gates. Thus, Proposition 1 gives us∥∥∥∥∥U sparsePE

∑µ

cµ |ψµ〉 |0m〉 −∑µ

cµ |ψµ〉 |Eµ〉

∥∥∥∥∥ ≤ ζ (B3)

where |Eµ〉 = |ϕµ,1ϕµ,2ϕµ,3 . . . ϕµ,s〉 ⊗ |restµ〉 contains the s-bit truncated representation of ϕµ = Eµ/Emax =0.ϕµ,1ϕµ,2ϕµ,3 · · · , and Emax is the upper bound on the maximum energy of the target Hamiltonian H used inthe construction of UNN

PE . Let

Eµ = Emax × (0.ϕµ,1ϕµ,2ϕµ,3 . . . ϕµ,s) = Eµ +O(Emax2−s) (B4)

be the truncated-approximation to the energy eigenvalue Eµ.

12

The new spatially sparse circuit U sparsePE now has t0 = O(R0N) = O(poly(n, ζ−1)) gates. From this, we construct

the following uncomputed, spatially sparse circuit

Ucircuit = (1)LU sparse†PE (1)sU sparse

PE , (B5)

which we will transform into our spatially sparse Hamiltonian. Note we add U sparse†PE for uncomputing and s + L

idling identity gates, making the entire circuit gate count T = 2t0 + s + L. The s identity gates are used for localmeasurements of energy to s-bit precision, and L = O((2t0 +s)‖H‖2/ε2) = O(poly(n, ζ−1)/ε2) identity gates are usedto ensure O(ε) simulation precision as in Lemma 2. The history states with respect to eigenstate |ψµ〉 of H and thiscircuit are

|ηµ〉 =1√T + 1

T∑t=0

(Ut · · ·U2U1 |ψµ〉 |0m〉

)|1t0T−t〉 (B6)

We can convert the circuit to a Hamiltonian Hcircuit using the method described in Lemma 1, where Hout is chosento be

Hout = (T + 1)Emax

s∑b=1

2−b|1〉〈1|ancb ⊗ P clock(t = t0 + b). (B7)

We also denote P clock(t) = |110〉〈110|clockt−1,t,t+1, which projects onto legal clock states corresponding to time step t.

To show that Hcircuit simulates the original Hamiltonian H, we first show that Hout restricted to the subspace ofhistory states L = span{|ηµ〉 : 1 ≤ µ ≤ 2n} can be approximated by the following effective Hamiltonian

Heff =∑µ

Eµ|ηµ〉〈ηµ|. (B8)

Consider arbitrary states |η〉 ∈ L. We write |η〉 =∑µ aµ |ηµ〉, and observe

〈η|Hout|η〉 = Emax

s∑b=1

2−b

[∑ν

a∗ν 〈ψν | 〈0m|

]U sparse†

PE |1〉〈1|bU sparsePE

[∑µ

aµ |ψµ〉 |0m〉

](B9)

Then using (B3), we have

〈η|Hout|η〉 = Emax

s∑b=1

2−b

[∑ν

a∗ν 〈ψν | 〈Eν |+ 〈ζ|

]|1〉〈1|b

[∑µ

aµ |ψµ〉 |Eµ〉+ |ζ〉

](B10)

where |ζ〉 is some residual state vector with ‖ |ζ〉 ‖ ≤ ζ. Hence

|〈η|Hout −Heff|η〉| ≤∑µ

|aµ|2|Eµ − Eµ|+ 2sζEmax ≤ maxµ|Eµ − Eµ|+ 2sζEmax ≤ (2−s + 2sζ)Emax (B11)

We can ensure this is always less than ε/4 by choosing for example

s = log2(8Emax/ε) = O(log n+ log ε−1) and ζ = ε/(16sEmax) = O(1/ poly(n, ε−1)). (B12)

Hence,

|〈η|Hout −Heff|η〉| ≤ ε/4 ∀ |η〉 ∈ L =⇒ ‖Heff −Hout|L‖ ≤ ε/4 (B13)

Furthermore, since we have added L idling gates such that ‖H ⊗ |α〉〈α| −Heff‖ ≤ ε/4 for some ancilla state |α〉 byLemma 2, then together with (B13) we have

‖H ⊗ |α〉〈α| −Hout|L‖ ≤ ε/2. (B14)

Observe that we can rewrite H ⊗ |α〉〈α| = V HV †, where V |ψ〉 = |ψ〉 |α〉 ∀ |ψ〉 ∈ C2n is an isometry. Hence, byLemma 1, for any η > 0, the constructed Hcircuit simulates H to precision (∆, η, ε), where ∆ = O(ε−1‖Hout‖2 +η−1‖Hout‖) = O(ε−1‖H‖2 + η−1‖H‖). Note that Hcircuit is spatially sparse since U sparse

PE is spatially sparse. SinceU sparse

PE contains at most 2-local gates, which means Hcircuit is at most 5-local. Furthermore, Hcircuit contains O(T ) =O(poly(n, ζ−1)/ε2) = O(poly(n, ε−1)) terms (and qubits), with O(poly(n, T, ε−1, η−1, ‖Hout‖)) = O(poly(n, η−1, ε−1))interaction energy.

13

To finish the proof, we just need to prove Lemma 1 and 2. We start with the proof of Lemma 1.

Proof of Lemma 1. For a given circuit U = UT · · ·U2U1, the corresponding circuit-Hamiltonian is

Hcircuit = H0 +Hout (B15)

where H0 = JclockHclock + JpropHprop + JinHin (B16)

The role of H0 is to isolate L = span{|ηµ〉} as its zero-energy groundspace separated by a large spectral gap 2∆from the rest of the eigenstates. Then Hout is used recover the eigenvalue structrue of H in the subspace L, allowingHcircuit to simulate H.

Now we give the explicit form of the circuit-Hamiltonian. The first part of H0 is

Hclock =

T−1∑t=1

|01〉〈01|clockt,t+1, (B17)

which sets the legal state configurations in the clock register to be of the form |t〉clock ≡ |1t0T−t〉clock. Then, we

simulate the state propagation under the circuit using

Hprop =

T∑t=1

Hprop,t, (B18)

where Hprop,t = 1⊗ |100〉〈100|clockt−1,t,t+1 − Ut ⊗ |110〉〈100|clock

t−1,t,t+1

−U†t ⊗ |100〉〈110|clockt−1,t,t+1 + 1⊗ |110〉〈110|clock

t−1,t,t+1 for 1 < t < T,

Hprop,1 = 1⊗ |00〉〈00|clock12 − U1 ⊗ |10〉〈00|clock

12 − U†1 ⊗ |00〉〈10|clock12 + 1⊗ |10〉〈10|clock

12 ,

and Hprop,T = 1⊗ |10〉〈10|clockT−1,T − UT ⊗ |11〉〈10|clock

T−1,T − U†T ⊗ |10〉〈11|clock

T−1,T + 1⊗ |11〉〈11|clockT−1,T .

These terms check the propagation of states from time t − 1 to t is correct. Now, we also need to ensure that the

input states are valid, i.e. ancilla qudits are in the state |0m〉ancwhen t = 0 (i.e., the clock register is |0T 〉clock

). Thiscan be done using

Hin =

m∑i=1

(1− |0〉〈0|)anci ⊗ |0〉〈0|clock

tmin(i), (B19)

where tmin(i) = min{t : Ut acts nontrivially on ancilla qudit i}.

In other words, for each ancilla qudit i, Hin penalizes the ancilla if it’s not in the state |0〉 before it is first used bythe tmin(i)-th gate. Note that Hcircuit has O(T ) terms, each of which is most (k + 3)-local when Ut are k-local. If Uis spatially sparse, then it is easy to see that Hcircuit is also spatially sparse.

Note that H0L = 0. We then need to lower bound the spectral gap of H0, i.e. λ1(H0|L⊥), where λ1(H) denotesthe lowest eigenvalue of H. To that end, let us denote the following subspaces:

Sclock = span{|ψ〉 |y〉 |1t0T−t〉 : |ψ〉 ∈ Cdn

and |y〉 ∈ Cdm

, 0 ≤ t ≤ T}, (B20)

Sprop = span{|ηµ, y〉 ≡1√T + 1

T∑t=0

(Ut · · ·U2U1 |ψ〉 |y〉

)|1t0T−t〉 : 1 ≤ µ ≤ dn, 0 ≤ y ≤ dn−1}. (B21)

Note that L ⊂ Sprop ⊂ Sclock. Let us denote A = A∩L⊥ for any subspace A. Note HclockSclock = 0, HpropSprop = 0,HinL = 0. We will use the following Projection Lemma 3:

Lemma 3 (Projection Lemma, adapted from [26]). Let H = H1 +H2 be sum of two Hamiltonians operating on someHilbert space S0 = S ⊕ S⊥. Assuming that H2 has a zero-energy eigenspace S ⊆ S0 so that H2S = 0, and that theminimum eigenvalue λ1(H2|S⊥) ≥ J > 2‖H1‖, then

λ1(H1|S)− ‖H1‖2

J − 2‖H1‖≤ λ1(H) ≤ λ1(H1|S). (B22)

In particular, if J ≥ K‖H1‖2 + 2‖H1‖ = O(K‖H1‖2), we have λ1(H1|S)− 1K ≤ λ1(H) ≤ λ1(H1|S). �

14

Applying the above Lemma successively to H0, we obtain

λ1(H0|L⊥) ≥ λ1

[(JpropHprop + JinHin)|Sclock

]− 1

Kif Jclock = O(K‖JpropHprop + JinHin‖2) (B23)

≥ λ1

[(JinHin)|Sprop

]− 2

Kif Jprop/T

2 = O(K‖JinHin‖2) (B24)

where we used the fact that λ1(Hclock|S⊥clock) ≥ 1, and λ1(Hprop|S⊥prop) ≥ c/T 2 for some constant c. We now lower

bound (B24). Let us denote n = 1− |0〉〈0|. Then within Sclock, we can rewrite

Hin|Sclock =

m∑i=1

nanci ⊗

∑0≤t≤tmin(i)

|t〉〈t|clock =

maxi tmin(i)∑t=0

Hin,t (B25)

where Hin,t =∑

{i: t≤tmin(i)}

nanci ⊗ |t〉〈t|clock.

In particular, Hin,t=0 =∑mi=1 n

anci ⊗ |t = 0〉〈t = 0|. Thus, for any |ηµ, y〉 , |ην , y′〉 ∈ Sprop, where necessarily y, y′ > 0,

we have

〈ην , y′|Hin,t=0|ηµ, y〉 =1

T + 1〈ψν | 〈y′|Hin,t=0 |ψµ〉 |y〉

=1

T + 1δµν 〈y′|

m∑i=1

nanci |y〉 =

1

T + 1δµνδy,y′ × w(y), (B26)

where w(y) is the Hamming weight of y in d-ary representation, which is at least 1 for any y > 0. Hence, the minimumeigenvalue of Hin,t=0|L⊥ is 1/(T + 1). Since Hin consists of only positive semi-definite terms, we have

λ1(Hin|Sprop) ≥ λ1(Hin,t=0|Sprop) ≥ 1/(T + 1). (B27)

Thus, to ensure that H0 has spectral gap λ1(H0|L⊥) ≥ 2∆, we simply choose Jin = O(∆(T + 1)), Jprop =O(KT 2J2

inm2), and Jclock = O(KJ2

propT2) = O(poly(n, T,∆)).

Now that we have shown H0 has L as its groundspace with spectral gap 2∆, we are ready to show that Hcircuit

(∆, η, ε)-simulates H with only polynomial overhead in energy. To this end, we use the following result regardingperturbative reductions adapted from Lemma 4 of [23] (also Lemma 35 of [10]):

Lemma 4 (First-order reduction, adapted from [23]). Suppose H = H0 +Hout, defined on Hilbert space H = L⊕L⊥such that H0L = 0 and λ1(H0|L⊥) ≥ 2∆. Suppose H is a Hermitian operator and V is an isometry such that

‖V HV † − Hout|L‖ ≤ ε/2, then H is a (∆, η, ε)-simulation of H, as long as ∆ ≥ O(ε−1‖Hout‖2 + η−1‖Hout‖), per

Def. 2. In other words, ‖H≤∆ − V HV †‖ ≤ ε for some isometry V where ‖V − V ‖ ≤ η. �

Observe we are given in the premise of this Lemma 1 that

‖V HV † −Hout|L‖ ≤ ε/2. (B28)

Hence, Hcircuit = H0 +Hout simulates H to precision (∆, η, ε) where ∆ ≥ O(ε−1‖Hout‖2 +η−1‖Hout‖). The maximuminteraction energy in Hcircuit is Jclock = O(poly(n, T,∆)) = O(poly(n, T, ε−1, η−1, ‖Hout‖)). This concludes the proofof Lemma 1.

We now prove the second Lemma, which shows that in order to ensure the circuit-Hamiltonian simulates the originalHamiltonian with good precision with trivial encoding, we only need to add O(poly(n, ε−1)) “idling” identity gatesto the end of a polynomial-sized circuit before transforming the circuit back to a Hamiltonian.

Proof of Lemma 2. Note that we can write

|ηµ〉 =√

1− χ2 |ψµ〉 ⊗ |α〉+ χ |βµ〉 (B29)

15

where

|α〉 =1√L+ 1

|0m〉anc ⊗D+L∑t=D

|1t0T−t〉clock, (B30)

|βµ〉 =1√D

D−1∑t=0

(Ut · · ·U2U1 |ψµ〉 |0m〉anc

)|1t0T−t〉clock

, (B31)

and χ =√D/(D + L+ 1). (B32)

Observe that 〈βµ| (|ψν〉 |α〉) = 0 since the clock register are at different times, and

〈βµ|βν〉 =1

D

D−1∑t=0

〈ψµ|ψν〉 = δµν . (B33)

Let Panc = |α〉〈α|. Then

Heff −H ⊗ Panc =∑µ

[Eµ|ηµ〉〈ηµ| − Eµ|ψµ〉〈ψµ| ⊗ |α〉〈α|

]=⊕µ

(−Eµχ2 Eµχ

√1− χ2

Eµχ√

1− χ2 Eµχ2

)(B34)

And so

‖Heff −H ⊗ Panc‖ ≤ χmaxµ

Eµ ≤ χ‖H‖. (B35)

To ensure ‖Heff −H ⊗ Panc‖ ≤ ε, it’s sufficient to choose L so that χ‖H‖ = ε. Plugging in χ =√D/(D + L+ 1), we

find that it is sufficient to choose L = O(D‖H‖2

ε2 ).

2. Proof of Theorem 2 – Strongly Universal Hamiltonian on 2D Square Lattice

In this subsection, we show how to transform the spatially sparse Hamiltonian constructed previously into a Hamil-tonian from a universal family of 2D spin-lattice model, with only polynomial overhead, proving our main Theorem 2.This follows from our Proposition 2 and the following result from Ref. [10]:

Lemma 5 (Essentially Ref. [10]). Given any k-local n-qudit Hamiltonian H that is spatially sparse, we can constructH ′ from a family of S-Hamiltonian on the 2D square lattice that efficiently (∆, η, ε)-simulates H as long as S isnon-2SLD. Here, ∆ = O(poly(‖H‖, η−1, ε−1), and H ′ has O(poly(n, ε−1)) qubits and interaction energy O(poly(∆)).

Then our main Theorem is a simple consequence of the above Lemma and our Proposition 2:

Proof of Theorem 2. As we showed in Proposition 2, any O(1)-local qudit Hamiltonian H with ‖H‖ can be simu-lated by a spatially sparse 5-local Hamiltonian Hcircuit with O(poly(n)/ε2) terms and qubits, and interaction energyat most O(poly(n, η−1, ε−1)). By Lemma 5, we can simulate Hcircuit by a S-Hamiltonian on the 2D square latticewith polynomial overhead, as long as S is non-2SLD.

Although it was essentially shown in Ref. [10], we provide here a sketch of the proof of Lemma 5 for completeness.

Proof Sketch of Lemma 5. To show this, we use a sequence of reductions originally described in Ref. [10, 17], whichtogether performs the desired transformation. These reductions are enumerated in the following list of Lemmas:

Lemma 6 (Lemma 21 of [10]). Given any k-local Hamiltonian on n qudits H, we can construct a kdlog2 de-localHamiltonian H ′ on ndlog2 de qubits that (∆, 0, 0)-simulates H, for ∆ ≥ ‖H‖. For d = O(1), the constructionpreserves spatial sparsity, and uses terms of interaction energy O(∆).

16

The construction maps each qudit to dlog2 de qubits, and uses local terms of strength ∆ to penalize any redundantstates among the qubits. Specifically, consider any isometry W : Cd → (C2)⊗dlog2 de. The construction maps H toH ′ = W⊗nHW †⊗n + ∆′

∑ni=1 Pi, for any ∆′ > ∆, where P = 1−WW †. It is easy to see that H ′ is spatially sparse

if H is spatially sparse and d = O(1).

Lemma 7 (Lemma 22 of [10]). Given any k-local n-qubit Hamiltonian H, we can construct a real-valued 2k-local2n-qubit Hamiltonian H ′ that (∆, 0, 0)-simulates H, for any ∆ ≥ 2‖H‖. The construction preserves spatial sparsity,and uses terms of interaction energy O(∆).

The construction adds one additional qubit per original qubit, and map the individual Pauli operators in thefollowing way:

1 7→ 1⊗ 1, σx,z 7→ 1⊗ σx,z, σy 7→ σy ⊗ σy. (B36)

For each new pair of qubits (i, n+ i), an additional local term ∆′(YiYn+i +1) is added, where ∆′ > ∆. Note the newHamiltonian is real-valued, and spatially sparse if H is spatially sparse.

Lemma 8 (Lemma 39 of [10]). Real-valued k-local qubit Hamiltonian H with M terms can be (∆, η, ε)-simulated byreal (k+ 1)-local Hamiltonian with O(M +n) qubits and terms, whose Pauli-decomposition contains no any Y terms.The construction preserves spatial sparsity, and uses interaction energy at most ∆ = O(poly(‖H‖, η−1, ε−1)).

The construction here takes any terms in H with (necessarily) even number of Y ’s, and recreates it with a per-turbative gadget involving only X,Z terms and an additional mediator qubit a. Specifically, the gadget perform thefollowing mapping:

Y ⊗2m ⊗A 7→ ∆h0 +√

∆h2 (B37)

where h0 = (1 + Za)/2 = |0〉〈0|a, h2 = Xa(X⊗2m ⊗ 1+ (−1)m+1Z⊗2m ⊗A). (B38)

Every term is mapped in parallel with an independent mediator qubit gadget. Since there are M terms in H, we haveat most O(M + n) qubits in the end. The large interaction energy ∆ = O(poly(‖H‖, η−1, ε−1)) is required to ensuresmall errors from perturbation. It is easy to see that if H is spatially sparse, so is the new Hamiltonian.

Lemma 9 (Theorem 40 of [10]). Suppose H is any k-local qubit Hamiltonian with M terms whose Pauli-decompositioncontains no Y . Then H can be simulated by 2-local qubit Hamiltonians with O(M + n) terms and qubits to preci-sion (∆, η, ε) whose Pauli-decomposition contains no Y terms. The construction preserves spatial sparsity, and usesinteraction energy at most Θ(∆) = O(poly(‖H‖, η−1, ε−1)) assuming k = O(1).

This construction makes use of the subdivision and 3-to-2 local gadgets, which are described in details in Ref. [17].For each k-local term of the form A ⊗ B, one can map it to (dk/2e + 1)-local terms of the form A ⊗Xw + Xw ⊗ Busing a subdivision gadget that introduces an extra ancilla qubit w. Thus, O(log k) applications of the subdivisiongadget is sufficient to reduce the locality to 3-local. This is then reduced to 2-local terms using the 3-to-2 local gadget:this converts term of the form A ⊗ B ⊗ C to 2-local terms such as (A − B)Xw, C|1〉〈1|w, AB, and (A2 + B2)C. Inother words, the construction maps each k-local term to O(k) 2-local terms mediated by O(k) ancilla qubits. Thismapping clearly preserves spatial sparsity. The required interaction energy blows up exponentially in k; however,since k = O(1), the interaction energy required is at most O(poly(‖H‖, η−1, ε−1)).

Lemma 10 (Theorem 41 of [10]). Suppose H is a 2-local n-qubit Hamiltonian whose Pauli-decomposition containsno Y terms. Then it can be simulated by a 4n-qubit S-Hamiltonian to precision (∆, η, ε), where S = {XX + Y Y +ZZ} or {XX + Y Y }. The construction preserves spatial sparsity, and uses interaction energy at most Θ(∆) =O(poly(‖H‖, η−1, ε−1)).

Here, the construction uses a perturbative gadget that maps every logical qubit in H to a group of 4 physical qubitsthat interact only via terms from S. The 1-local and 2-local interaction on any two logical qubits can be implementedusing two-body terms from S coupling different pairs of physical qubits from the two groups. Hence, spatial sparsityis preserved by this construction. The required interaction energy scales as Θ(∆) = O(poly(‖H‖, η−1, ε−1)).

Finally, we restate the following two results from Ref. [17]:

Lemma 11 (Lemma 46 of [10]). Let S0 be either {XX + Y Y + ZZ} or {XX + Y Y }. Any spatially sparse S0-Hamiltonian on n qubits, whose largest interaction energy is Λ0, can be simulated by a S0-Hamiltonian on a 2Dsquare lattice of poly(n) qubits using interaction energy at most Jij = O(poly(nΛ0(1/ε+ 1/η))).

Lemma 12 (Theorem 42 of [10]). Suppose S be a set of interactions on 2-qubits that is non-2SLD. Then given an{XX +Y Y +ZZ}- or {XX +Y Y }-Hamiltonian on the 2D square lattice, we can simulate it with an S-Hamiltonianon the 2D square lattice.

17

In what follows, we denote S0 as either {XX + Y Y + ZZ} or {XX + Y Y }. For any S that is non-2SLD, we mapHcircuit to an S-Hamiltonian on the 2D square lattice in the following sequence:

1. By Lemma 6, we can simulate Hcircuit with H1 that is spatially sparse, O(1)-local on O(poly(n)/ε2) qubits andinteraction energy at most O(‖Hcircuit‖) = O(poly(n, η−1, ε−1)).

2. By Lemma 7, we can simulate H1 with H2 that is spatially sparse, real-valued, and O(1)-local, with onlypolynomial overhead in qubit-number of interaction energy.

3. By Lemma 8, we can simulate H2 with H3 that is spatially sparse and O(1)-local, and contains no Y terms inPauli-decomposition, with only polynomial overhead in qubit-number of interaction energy.

4. By Lemma 9, we can simulate H3 with H4 that is spatially sparse and 2-local, contains no Y terms in Pauli-decomposition, with polynomial overhead.

5. By Lemma 10, we can simulate H4 with H5 that is a spatially sparse S0-Hamiltonian, with polynomial overhead.

6. By Lemma 11, we can simulate H5 by H6 an S0-Hamiltonian on a 2D square lattice, with polynomial overhead.

7. By Lemma 12, we can simulate H6 by the broader class of S-Hamiltonian on the 2D square lattice, for any Sthat is non-2SLD, with polynomial overhead.

Since every step of the above sequence of reductions only incurs a polynomial overhead in the number of qubits and thestrength of interactions, we have shown that any O(1)-local, polynomial-sized qudit Hamiltonians can be efficientlysimulated by an S-Hamiltonian with polynomial qubits and interaction energy, assuming S is non-2SLD.

Appendix C: Proof that 1D nearest-neighbor Hamiltonian is strongly universal

Here we give our proof of Theorem 3, whose statement we reproduce below for convenience:

Theorem 3. There is a strongly universal family of 1D Hamiltonians consisting of nearest-neighbor interaction actingon a line of particles with 8 internal dimension.

The proof is based heavily on the framework in Ref. [19]. We will only try to provide a succinct and somewhatself-contained description of the most important elements of the construction here. For the full technical details ofthe construction, we encourage the reader to also examine Section 3 and 4 of Ref. [19].

1. Preliminaries

We first describe how the computation is encoded within a line of 8-dimensional particles.

Definition 8 (1D-encoded L-idling history state). Consider any quantum circuit U consisting of R = O(poly(n))rounds of 1-qubit or nearest-neighbor 2-qubit gates on a line of n qubits. This can be further converted to an encodedcircuit U with nearest-neighbor gates acting on a line 2nR + L qudits (d = 8), implicitly arranged in R blocks of

2n 8-dimensional particles, followed by L particles for idling. The Hilbert space of each particle is H8 = © ⊕ �© ⊕◦© ⊕ ש ⊕ ⊕ I , where and I are 2-dimensional subspaces designed to hold a qubit state, and the rest are

1-dimensional subspaces. For a given input state of the form |ψµ〉 |0m〉 ∈ C2n on the original n qubits, this is encodedas

|γµ0 〉 =

R blocks︷ ︸︸ ︷I ◦© ◦© · · · ◦© ©︸ ︷︷ ︸

the first block of length 2n

©© ©© · · · · · · ��CC ©©© · · · ©︸ ︷︷ ︸L idling qudits

(C1)

where the odd qudits in the first block of length 2n encodes the input state |ψµ〉 |0m〉. Here, the symbols , and ��CC aresimply boundary markers in space that help us identify the role of each particle and do not indicate anything about

the internal state of particles. In particular, the symbol ��CC marks a special boundary that separating the computationalpart of the line and the idling part.

The encoded circuit U acts on |γµ0 〉 with R rounds of computation, each corresponding to applying a round of gatesfrom U ′ to the currently active block of qudits, and then moving the block 2n positions to the right. This entails a total

18

of K = (R−1)(3n2 +2n−1)+2n steps of nearest-neighbor gates that map configuration to configuration, according tothe transition rule outlined in Table 1 of Ref. [19]. This is then followed by L steps of “idling” where the L rightmostqudits perform a trivial counting operation. To facilitate the idling, we add the following transition rules.

(α) I ��CC© ←→ ��CC ש unmarks the active qubit I once it hits the special boundary ��CC . The © changes to ש soas to signal that the idling is supposed to the start.

(β) ש© ←→ ש ש for any location to the right of the special boundary ��CC .

This results in a history of K + L + 1 configurations on the 2nR qudits {|γµt 〉}K+Lt=0 which is pairwise orthogonal:

〈γµt |γµt′〉 = δtt′ . The 1D-encoded L-idling history state with respect to U and |ψµ〉 |0m〉 is the following superposition:

|ηµ〉 =1√

K + L+ 1

K+L∑t=0

|γµt 〉 . (C2)

To help visualize the computational history, note the configuration after all R rounds of computation comes to haltis

|γµK〉 = ש 2n(R−1) ש ◦© · · · ◦© ◦© I ��CC©⊗L. (C3)

All ensuing configurations are of the form (for 1 ≤ ` ≤ L):

|γµK+`〉 = ש 2n(R−1) ש ◦© · · · ◦© ◦© ��CC ש⊗`©⊗(L−`). (C4)

Lemma 13 (1D circuit-Hamiltonian [19]). Given a circuit U consisting of poly(n) 1- or 2-qubit gates on n qubits.We can construct a Hamiltonian on O(poly(n)) + L qudits, d = 8, with only nearest-neighbor interaction of the form

Hhist = JinHin + JpropHprop + JpenHpen (C5)

such that the 1D-encoded L-idling history states with respect to U and |ψµ〉 |0m〉 are of zero-energy, and all otherstates have energy ≥ 1. The interaction energy of nearest-neighbor terms in Hhist are Jin, Jprop, Jpen = O(poly(n)).

Proof. The construction is essentially the same as described in Section 4 of Ref. [19], except for a few small changes:

a. Changes to legal configurations and penalty Hamiltonian — To the right of the special boundary ��CC , onlyש ש , ש© and ©© are allowed configurations in these locations. This can be addressed by tweaking the penalty

HamiltonianHpen to penalize all configurations using the term |XY 〉〈XY |i,i+1 whereXY ∈ H⊗28 \{ש ש , ש© ,©©}

for these locations.b. Changes to propagation Hamiltonian — We need to incorporate the two new rules (α) and (β) added above.

The Rule (α) is similar to Rule 4a I © ←→ �© from Table 1 of Ref. [19]. Since there’s only a unique locationwhere Rule (α) applies, at the special boundary, we can simply use the following propagation Hamiltonian for thatpair of sites

H(α)prop,i = |I ��CC©〉〈I ��CC©|i,i+1 + | ��CC ש〉〈 ��CC ש|i,i+1 − | ��CC ש〉〈I ��CC©|i,i+1 − |I ��CC©〉〈 ��CC ש|i,i+1 (C6)

where i = 2nR for this special pair of sites.Now for Rule (β). Note this is the only propagation rule that is applicable in the region to the right of the special

boundary ��CC . Thus, we only need the following propagation Hamiltonian for i > 2nR.

H(β)prop,i = | ש©〉〈ש©|i,i+1 + | ש ש〉〈ש ש|i,i+1 − |ש ש〉〈ש©|i,i+1 − |ש©〉〈ש ש|i,i+1 (C7)

We note there can be mis-timed transitions from H(α)prop,i, e.g.

◦© ��CC ש ש −→ − ◦© I ��CC© ש (C8)

and H(α)prop,i, e.g.,

ש ש ש −→ −ש© ש (C9)

However, they will all result in energy penalty from Hpen, because they have illegal configuration © ש that is locallydetectable.

19

c. Proof that the Hamiltonian has the 1D-encoded history states as the only ground states — This is essentiallygiven in Section 5 and 6 of [19].

2. Proof of Theorem 3

Proof of Theorem 3. As in the Proof of Theorem 2, we can always convert any input qudit Hamiltonian to a qubitHamiltonian by encoding each qudit in a group of dlog2 de-qudits with polynomial overhead (assuming d = O(1)).Hence, we will take the input H as an O(1)-local n-qubit Hamiltonian. We write H =

∑µEµ|ψµ〉〈ψµ| in its eigenbasis,

with 0 ≤ Eµ ≤ Emax. By Proposition 1, we can construct a circuit U localPE consisting on O(poly(n, ζ−1)) 1- or 2-qubit

nearest-neighbor gates on n + m qubits, m = O(poly(n)), such that its action on any normalized state∑µ cµ |ψµ〉

can be described as ∥∥∥∥∥UNNPE

∑µ

cµ |ψµ〉 |0m〉 −∑µ

cµ |ψµ〉 |Eµ〉 |restµ〉

∥∥∥∥∥ ≤ ζ (C10)

where |Eµ〉 = |ϕµ,1ϕµ,2ϕµ,3 . . . ϕµ,s〉 is the s-bit string representation of ϕµ = Eµ/Emax = 0.ϕµ,1ϕµ,2ϕµ,3 · · · . Fromnow on we will denote n = n+m as the number of qubits in the phase estimation circuit.

We then consider the identity circuit U = (UNNPE )†1n+mUNN

PE = 1 that corresponds to running UNNPE , applying

identity to every qubit, and then uncomputing. Let RPE be the number of rounds of nearest-neighbor gates in UPE,and R = 2RPE + 1 be the total number of rounds. Let K = (R− 1)(3n2 + 2n− 1) + 2n be the total number of stepswhen applying U . By Lemma 13, we can construct a 1D nearest-neighbor Hamiltonian Hhist on poly(n) +L particleswith poly(n) interaction energy such that all the 1D-encoded L-idling history states with respect to U and |ψµ〉 |0m〉are the only zero-energy states, and all other states have energy at least ∆.

We claim that the following 1D Hamiltonian (∆, η, ε)-simulates H

H1D = 2∆Hhist +Hout. (C11)

To describe Hout, we start by considering the configuration of all 2nR + L particles after the RPE-th round of gatescorresponding to the end of applying UNN

PE . This looks like

· · · ש ש I ◦© ◦© · · · ◦© © ©© · · · © ��CC© · · · (C12)

Note the next round of gates corresponds to 1n+m, i.e., applying the identity gate to every qubit. Without lossof generality, we assume the first s active particles corresponds (ζ-approximately) to the ancilla with |Eµ〉 on them.

Notice that every active qudit will at one point transition into their appropriate state in the I subspace, with identitygate applied. Then, the appropriate Hout is

Hout = (K + L+ 1)Emax

s∑b=1

2−b|I (1)〉〈I (1)|2nRPE+2b−1 (C13)

where |I (1)〉 corresponds to the |1〉 qubit state in the I subspace.

To prove that H1D simulates H to the desired precision, we first show that Hout restricted to the set of 1D-encodedhistory states L = span{|ηµ〉} can be approximated by the following effective Hamiltonian:

Heff =∑µ

Eµ|ηµ〉〈ηµ| (C14)

Consider arbitrary states |η〉 ∈ L. We write |η〉 =∑µ aµ |ηµ〉, and observe

〈η|Hout|η〉 = (K + L+ 1)Emax

s∑b=1

2−b∑ν,µ

a∗νaµ 〈ην |(|I (1)〉〈I (1)|2nRPE+2b−1

)|ηµ〉

= Emax

s∑b=1

2−b∑ν,µ

K∑t,t′=0

a∗νaµ 〈γνt |(|I (1)〉〈I (1)|2nRPE+2b−1

)|γµt′〉 (C15)

20

Note the 2nRPE + 2b−1-th particle is only in the I subspace at one point in time among the K+ 1 steps, regardlessof possible input state. Table 2 of [19] provides a clear illustration for the reason. Let’s call that time tb, and write

〈η|Hout|η〉 = Emax

s∑b=1

2−b∑ν,µ

a∗νaµ 〈γνtb |(|I (1)〉〈I (1)|2nRPE+2b−1

)|γµtb〉 (C16)

Note the configuration∑µ aµ |γ

µtb〉 is the encoded version of UNN

PE

∑µ aµ |ψµ〉 |0m〉, which is ζ-close to the encoded

version of∑µ aµ |ψµ〉 |Eµ〉 |restµ〉. Hence

〈η|Hout|η〉 = Emax

s∑b=1

2−b∑ν,µ

a∗νaµϕµ,b 〈ψν |ψµ〉 〈restν |restµ〉+ Θ(ζEmax)

=∑µ

|aµ|2s∑b=1

2−bϕµ,bEmax + Θ(ζEmax)

=∑µ

|aµ|2Eµ + Θ(ζEmax). (C17)

where we have denoted Eµ = Emaxϕµ = Emax(0.ϕµ,1ϕµ,2 · · ·ϕµ,s) as the s-bit representation of energy eigenvalue Eµof H. Then

|〈η|Hout −Heff|η〉| ≤∑µ

|aµ|2|Eµ − Eµ|+ Θ(sζEmax) ≤ [2−s + Θ(ζ)]Emax (C18)

By choosing s = Θ(logEmax/ε) = Θ(log n + log ε−1) and ζ = Θ(ε/Emax) = Θ(1/poly(n, ε−1)), just like we did inEq. (B12), we can ensure

|〈η|Hout −Heff|η〉| ≤ ε/4 ∀ |η〉 ∈ L =⇒ ‖Heff −Hout|L‖ ≤ ε/4 (C19)

Furthermore, looking at the 1D-encoded L-idling history state more carefully, it looks like

|ηµ〉 =1√

K + L+ 1

K∑t=0

|γµt 〉 =√

1− χ2 |αµ〉+ χ |βµ〉 (C20)

where

χ =

√K + 1

K + L+ 1(C21)

and

|βµ〉 =1√

K + 1

K∑t=0

|γµt 〉 (C22)

|αµ〉 =1√L

L∑t=1

|γµK+t〉 = ש 2n(R−1) ש ◦© · · · ◦© ◦© ��CC1√L

L∑`=1

ש⊗`©⊗(L−`)

= (V |ψµ〉)⊗ |α〉 (C23)

where V |ψµ〉 is simply the input state |ψµ〉 stored in the qubit subspace of a subset of the n qudits marked within the R-th block, |α〉 is the state of the remaining ancilla. Therefore,

|ηµ〉 =√

1− χ2(V |ψµ〉)⊗ |α〉+ χ |βµ〉 . (C24)

By choosing L = O(K/ε2), we can ensure that ‖|ηµ〉〈ηµ| − V |ψµ〉〈ψµ|V † ⊗ |α〉〈α|‖ ≤ O(ε). With the same argumentin Lemma 2, we can show that

‖Heff − V HV † ⊗ |α〉〈α|‖ ≤ ε/4 (C25)

Therefore, together with Eq. (C19) we have∥∥V HV † ⊗ |α〉〈α| −Hout|L∥∥ ≤ ε/2 (C26)

To finish proving our Theorem 3, we again use Lemma 4 which we restate below for the reader’s convenience:

21

Lemma 4 (First-order reduction, adapted from [23]). Suppose H = H0 +H1, defined on Hilbert space H = L ⊕ L⊥such that H0L = 0 and λ1(H0|L⊥) ≥ 2∆. Suppose H is a Hermitian operator and V is an isometry such that

‖V HV † −H1|L‖ ≤ ε/2, then H (∆, η, ε)-simulates H, as long as ∆ ≥ O(ε−1‖H1‖2 + η−1‖H1‖), per Def. 2. In other

words, ‖H≤∆ − V HV †‖ ≤ ε for some isometry V where ‖V − V ‖ ≤ η.

We apply the above Lemma with H0 = 2∆Hhist and H1 = Hout. Thus, H1D = H0 + H1 simulates H to precision(∆, η, ε) by choosing ∆ ≥ O((ε−1 + η−1) poly(n, ‖H‖)). Note that H1D contains O(poly(n, ε−1)) nearest-neighborterms, with interaction energy O(poly(n, η−1, ε−1,∆)), which proves our Theorem.