Perturbative gadgets for gate-based quantum computing Non-recursive constructions without subspace restrictions Simon Cichy12Paul K. Faehrmann1Sumeet Khatri1and Jens Eisert134

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Perturbative gadgets for gate-based quantum computing:

Non-recursive constructions without subspace restrictions

Simon Cichy,

1, 2

Paul K. Faehrmann,

1, ∗

Sumeet Khatri,

and Jens Eisert

1, 3, 4

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland

Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany

Fraunhofer Heinrich Hertz Institute, 10587 Berlin, Germany

(Dated: August 28, 2024)

Perturbative gadgets are a tool to encode part of a Hamiltonian, usually the low-energy subspace, into a

diﬀerent Hamiltonian with favorable properties, for instance, reduced locality. Many constructions of perturbative

gadgets have been proposed over the years. Still, all of them are restricted in some ways: Either they apply to

some speciﬁc classes of Hamiltonians, they involve recursion to reduce locality, or they are limited to studying

time evolution under the gadget Hamiltonian, e.g., in the context of adiabatic quantum computing, and thus

involve subspace restrictions. In this work, we ﬁll the gap by introducing a versatile universal, non-recursive,

non-adiabatic perturbative gadget construction without subspace restrictions, that encodes an arbitrary many-body

Hamiltonian into the low-energy subspace of a three-body Hamiltonian and is therefore applicable to gate-based

quantum computing. Our construction requires

𝑟 𝑘

additional qubits for a

𝑘

-body Hamiltonian comprising

𝑟

terms.

Besides a speciﬁc gadget construction, we also provide a recipe for constructing similar gadgets, which can be

tailored to diﬀerent properties, which we discuss.

I. INTRODUCTION

The study of many-body Hamiltonians is a well-researched

ﬁeld in condensed-matter physics and quantum information

theory. While the presence of Hamiltonian terms acting on

many qubits simultaneously can result in exciting phenomena,

in many situations, it also carries additional diﬃculties. Be it

due to the hardness of generating them experimentally or other

limitations, local Hamiltonians are usually simpler to deal with.

Born in the context of complexity theory for proving the

QMA-completeness of the local Hamiltonian problem [

a problem located at the interface of the theory of quantum

many-body physics and Hamiltonian complexity, so-called

perturbative gadgets can be used to reduce the locality of a

given many-body Hamiltonian. This is done by embedding

it in the low-energy subspace of a tailored, local, i.e., few-

body, Hamiltonian acting on a larger Hilbert space [

Among the best-known constructions is the subdivision gadget

proposed by Oliveira and Terhal [

]. It employs mediator

qubits to halve the support of a given Hamiltonian term and

can be recursively applied to construct an at best three-body

Hamiltonian mimicking the original

𝑘

-body Hamiltonian. As is

typical for perturbative gadgets, such a procedure, unfortunately,

leads to an exponential increase of the interaction strengths

required for the gadget Hamiltonian to accurately mimic the

low-energy subspace of the original Hamiltonian.

A diﬀerent gadget, proposed by Jordan and Farhi [

], can

avoid the need for recursion and oﬀers a direct reduction from

𝑘

-body to two-body Hamiltonians, albeit with an exponential

suppression of energies. However, their direct construction is

only applicable when one can restrict the evolution of chosen

initial states to predeﬁned subspaces of the Hilbert space, as

can be done in adiabatic quantum computing, where commuta-

∗SC and PKF have contributed equally.

tion relations guarantee the system remains in the initialized

subspace.

Since such a limitation restricts the applicability of this direct

𝑘

-to-two-local gadget, the question arises whether there exists a

direct, non-recursive

𝑘

-to-

𝑘′

-local gadget without such restric-

tions and whether such a gadget could have any applications

besides being a tool for proving complexity theoretic results.

After all, perturbative gadgets have so far mostly found their

application in proving hardness results for the general local

Hamiltonian problem [

] or certain classes of Hamiltoni-

ans [6–10] equipped with further constraints and properties.

In this work, we address these questions by proposing a

direct

𝑘

-to-three-local gadget without the need for subspace

restrictions, that is inspired by Ref. [

] but that avoids initial-

ization and commutation properties from adiabatic quantum

computing to achieve the same locality reduction as the re-

peated subdivision procedure [

], without requiring recursive

applications.

As expected of gadget constructions [

], our construction

is also not free from unfavorable energy scalings, but we still

explore the use of such a gadget within gate-based quantum

computing and, in particular, as a method for reducing the

locality of the cost function in variational quantum algorithms,

a setting that does not comply with the “adiabatic” restriction,

or the grouping of measurement terms.

While most likely unfavorable for situations where the lo-

cality of a Hamiltonian scales with the system size, there are

plenty of situations where the locality is constant but still high

enough to cause problems. Examples include measurement

noise scaling with the locality of the measurement operator in-

ﬂuencing noise-induced barren plateaus [

] or time evolution

and quantum phase estimation for restricted hardware settings.

The proposed methods do not rely on subspace restrictions and

can thus be implemented in gate-based quantum computing. By

providing a more general recipe for constructing perturbative

gadgets with similar performance guarantees and suggesting

plausible heuristics, we hope to start a discussion about how

arXiv:2210.03099v4 [quant-ph] 27 Aug 2024

the trade-oﬀs of perturbative gadgets could still be used for

practical problems.

We begin by introducing the main ideas and a historical

overview of perturbative gadgets in Section II, before focusing

on a speciﬁc

𝑘

-to-three-local gadget and discussing guarantees

on its performance in Section III. We then generalize the

gadget construction to provide a recipe for creating custom

perturbative gadgets in Section IV and discuss the nuances

of applying such gadgets in gate-based quantum computing

in Section V, before concluding with some remarks and open

questions in Section VI.

II. PERTURBATIVE GADGETS IN A NUTSHELL

Perturbative gadgets [

–

] encode the low-energy sub-

space of a many-body Hamiltonian into the low-energy subspace

of a fewer-body Hamiltonian. They make use of perturbation

theory in the opposite direction than what is more common:

instead of considering the eﬀect of some perturbation

𝜆𝑉

a well-understood Hamiltonian

𝐻

according to

𝐻′=𝐻+𝜆𝑉

perturbative gadgets provide a suitable

𝐻

and perturbation

𝑉

such that

𝐻′

is a few-body Hamiltonian that approximates the

low-energy subspace of a given, target Hamiltonian

𝐻target

, as

visualized in Figure 1. They are often accompanied by an

increase in the number of required qubits and either suppress

the energy of the desired low-energy subspace or an increase

in the norm of 𝐻′compared with 𝐻target.

In most cases, a local and simple Hamiltonian with de-

generate ground space is used as the so-called “unperturbed

Hamiltonian”

𝐻

. The degeneracy is chosen to correspond to the

dimension of the space acted upon by the target Hamiltonian.

Then, a perturbation is designed to split said degeneracy in a

way that the resulting low-energy subspace emulates the target

spectrum, so roughly

Π𝐻′Π∼𝐻target,(1)

where

denotes a projector on the low-energy subspace. This

statement will later be made formal in Theorem 1.

To be explicit and clear, we use interaction weight to refer

to the number of particles acted upon by a given operator and

interaction strength to refer to the coeﬃcient of that operator.

When talking about qubits and expanding operators in the

Pauli basis, the weight of a Pauli string (a tensor product

of Pauli operators) is deﬁned as the number of non-trivial

Pauli operations in the string. Throughout this work, we use

“locality” to refer to the maximum weight of all terms in a

Hamiltonian, independently of geometry. We use the term

local for Hamiltonians with ﬁxed small weights, that is weights

that do not scale with problem size, while global is reserved

for Hamiltonians with all-to-all interactions.

An overview of works related to perturbative gadgets is

presented in Table I. Historically, perturbative gadgets were

developed to prove the QMA-hardness of the local Hamiltonian

problem. The

-local Hamiltonian problem is known to be

QMA-complete [

] and Kempe, Kitaev, and Regev have shown

that so is the

-local problem by proposing the ﬁrst perturbative

gadget, which reduces the locality from

[

]. Oliviera

⋮

2 -1

=(0)+

(0)

  ∼target

(0)

dim =2

FIG. 1. Sketch of the main idea behind perturbative gadgets. Starting

from an unperturbed Hamiltonian

𝐻(0)

with gap

𝛾

and degenerate

ground space, add a perturbation

𝑉

such that the lowest energy states

resulting from the perturbation of the ground space mimic the target

Hamiltonian 𝐻target.

and Terhal have added to this by showing that the

-local

Hamiltonian problem is still QMA-complete when restricting

the interaction to a 2D square lattice, and in doing so introduced

a new perturbative gadget. This subdivision gadget reduces the

locality of a

𝑘

-local Hamiltonian to

⌈𝑘/2⌉+1

-local interactions

by introducing a single auxiliary qubit per interaction term.

These kinds of gadgets have also been considered to be used

outside of the world of complexity-theoretic analysis, but they

are inherently not practical, as discussed further in Section V.

Furthermore, the recursive use of gadgets to arrive at two-

local Hamiltonians can be avoided by a single gadget introduced

by Jordan and Farhi in Ref. [

], which encodes

𝑘

-local Hamil-

tonians in

𝑘th

order of perturbation theory of a two-local gadget

Hamiltonian. However, just like the three-to-two local gadget,

their construction relies on certain properties of the evolu-

tion under a Hamiltonian where the state is restricted to a

predetermined subspace , such as provided in the adiabatic

setting.

In the following years, several works proposed improvements

in resource requirements, convergence bounds, or coupling

strengths [

–

], as well as introduced highly specialized

gadget constructions for restricted classes of Hamiltonians, such

as for topological quantum codes and for realizing certain parent

Hamiltonians featuring intrinsic topological order [20,21].

There are two important things to note at this point: While all

of these gadget constructions rely on perturbation theory, they

use diﬀerent techniques, ranging from perturbative expansion

due to Bloch [

], which underlies the Jordan-Farhi gadget, the

Feynman-Dyson series employed in Ref. [

], to the Schrieﬀer-

Wolf transformation [

], discussed in Ref. [

]. Furthermore,

except for the so-called subdivision gadget due to Oliveira

and Terhal [

], the other gadgets are restricted to the setting

of time evolution under the Hamiltonian, which guarantees

the evolution to remain in the initialized subspaces, rendering

them inapplicable for gate-based quantum computing. A non-

recursive

𝑘

-to-two-local gadget without these restrictions is

currently unknown.

Authors Main statement

Kempe, Kitaev, Regev

(KKR) [3]

3- to 2-local gadget to prove QMA completeness of the local Hamiltonian problem. Requires

restriction of the state evolution to a subspace of the Hilbert space.

Oliveira, Terhal (OT) [4]

Subdivision gadget (

𝑘

- to

𝑘/2

-local) and proof of QMA completeness of the Hamiltonian problem on

a square grid. Has to be applied recursively resulting in exponential coupling strengths.

Biamonte, Love [13] “Realizable Hamiltonians” with a focus on using gadgets for adiabatic computing and adapting to

hardware restrictions, not targeting a reduction of locality.

Bravyi, DiVicenzo, Loss,

Terhal [14]

Improvements on the OT gadget through extension to non-converging perturbative expansion, thus

reducing the gap in coupling strengths

Jordan, Farhi [5]

Direct

𝑘

- to

-local gadget through higher-order perturbation theory. Requires restriction of the state

evolution to a subspace of the Hilbert space.

Cao, Babbush, Biamonte,

Kais [15]

Resource requirement improvements of some of the OT and KKR previous gadgets.

Cao [16] Overview of existing gadgets and improvements of the OT and KKR gadgets.

Cao, Kais [17] Improvement of the convergence bound for the JF gadget.

Subasi, Jarzynski [18] Nonperturbative gadgets for adiabatic Hamiltonian evolution.

Bausch [19] Augmentation of other gadget constructions and changes their energy scales by (potentially)

increasing their locality by one.

Harley, Datta, Klausen,

Bluhm, França, Werner,

Christandl [11]

A general framework for analog quantum simulation and unavoidable unfavorable size-dependent

scalings of gadget constructions.

This work Direct 𝑘- to 3-local gadget without restriction to some subspace of the Hilbert space.

TABLE I. Overview of some relevant works related to perturbative gadgets. Not included are those that apply only to a restricted subset of

Hamiltonians, such as topological Hamiltonians.

III. A DIRECT 𝒌-TO-THREE-BODY GADGET WITHOUT

SUBSPACE RESTRICTION

One of the main results of this work is the proposal of a

direct, i.e., non-recursive, three-body gadget Hamiltonian

𝐻gad

derived from an arbitrary

𝑘

-body Hamiltonian

𝐻target

that does

not require any restrictions to a subspace of the Hilbert space,

inspired by the work of Jordan and Farhi [

]. We then ﬁnd

that the low-energy subspace of

𝐻gad

mimics the low-energy

subspace of

𝐻target

(Theorem 1), implying a guarantee on the

closeness of their ground states (Corollary 1).

We start by deﬁning the speciﬁc gadget Hamiltonian that is

the focus of our study.

Deﬁnition 1 (Gadget Hamiltonian).Let

𝐻target

be a

𝑘

-body

Hamiltonian acting on 𝑛qubits, given by

𝐻target B

𝑟



𝑠=1

𝑐𝑠ℎ𝑠,with ℎ𝑠=𝜎𝑠,1⊗𝜎𝑠,2⊗···⊗𝜎𝑠, 𝑘 ,(2)

where each term is a tensor product of at most

𝑘≤𝑛

single qubit

operators, with

𝜎𝑠, 𝑗 =𝑛𝑋𝑋+𝑛𝑌𝑌+𝑛𝑍𝑍

and

(𝑛𝑋, 𝑛𝑌, 𝑛𝑍) ∈ R3

a unit vector. We deﬁne the three-body gadget Hamiltonian

𝐻gad corresponding to 𝐻target, acting on 𝑛+𝑟 𝑘 qubits, as

𝐻gad B

𝑟



𝑠=1

𝐻aux

𝑠+𝜆

𝑟



𝑠=1

𝑉𝑠,(3)

h1h2

Htarget

aux + V2

aux

FIG. 2. Representation of a target Hamiltonian (left) and its associated

gadget Hamiltonian, as deﬁned in Eq.

(3)

, for the case of

𝑛=8

𝑘=4

, and

𝑟=3

. Qubits that interact according to the terms of

the Hamiltonian are placed in the same colored region. Shown in

the central and right-hand panels are the contributions to the gadget

Hamiltonian from two of the three four-body terms, with the auxiliary

qubits in gray and the target qubits in black.

where

𝐻aux

𝑠B

𝑘



𝑗=1

21𝑠, 𝑗 −𝑍aux

𝑠, 𝑗 =

𝑘



𝑗=1|1⟩⟨1|aux

𝑠, 𝑗 ,(4)

𝑉𝑠B

𝑘



𝑗=1

˜𝑐𝑠, 𝑗 𝜎target

𝑠, 𝑗 ⊗𝑋aux

𝑠, (𝑗+1)mod 𝑘(5)

and

˜𝑐𝑠, 𝑗 =−(−1)𝑘𝑐𝑠

𝑗=1

˜𝑐𝑠, 𝑗 =1

otherwise. We refer

to the

𝑛

qubits on which

𝐻target

acts as target qubits, and the

𝑟 𝑘 additional qubits as auxiliary qubits.

In our gadget Hamiltonian, each term

𝐻aux

𝑠

acts only on the

𝑠th

group of auxiliary qubits, while each term

𝑉𝑠

acts on a

target qubit and the

𝑠th

group of auxiliary qubits; see Fig. 2for

a graphical depiction for a toy model.

The proposed gadget is heavily inspired by the gadget intro-

duced by Jordan and Farhi in Ref. [

], in which they rely on

properties of adiabatic quantum computing. Both their gadget

and ours take advantage of a larger Hilbert space to encode

the low-energy subspace of the target Hamiltonian, so the

dimension of the ground space of the unperturbed Hamiltonian

must be the same as the space on which the target Hamiltonian

acts. This point is made more explicit in Appendix B. Jordan

and Farhi start from a larger ground space of the unperturbed

Hamiltonian and reduce the dimension by restricting the region

of the Hilbert space that can be explored by initializing the

system in a ﬁxed subspace. Remaining in this chosen subspace

is then ensured by using properties of adiabatic evolution,

something that cannot be done straightforwardly outside of the

regime of adiabatic quantum computing. This requirement,

therefore, places a strong limitation on the applicability of their

construction. On the other hand, our gadget construction does

not require such a restriction to a subspace, therefore avoids

this limitation, and can be applied even in the context of digital,

gate-based quantum computing. We refer to Appendix Ffor

a more detailed discussion of why a direct application of the

gadget introduced by Jordan and Farhi is not possible in the

realm of non-adiabatic quantum computing.

Similar to the results of Jordan and Farhi, we can show that the

subspace of the

2𝑛

lowest energy eigenstates of our three-body

gadget Hamiltonian in Eq.

(3)

mimics that of the

𝑘

-body target

Hamiltonian. For the formal statement, we deﬁne the following

notation: Let

𝐻

be a Hamiltonian acting on

𝑛

qubits, with

spectral decomposition

𝐻=Í2𝑛−1

𝑗=0𝐸𝑗|𝜓𝑗⟩⟨𝜓𝑗|

such that

𝐸0≤

𝐸1≤ ··· ≤ 𝐸2𝑛−1

, we deﬁne

𝐻eﬀ (𝐻, 𝑑)=Í𝑑−1

𝑗=0𝐸𝑗|𝜓𝑗⟩⟨𝜓𝑗|

to be the eﬀective Hamiltonian corresponding to the

𝑑

lowest

eigenvalues of 𝐻.

Theorem 1 (Main result).Let

𝐻target

and

𝐻gad

be as

in Deﬁnition 1, and let

𝜆≤𝜆max

, with

𝜆max =

4Í𝑟

𝑠=1|𝑐𝑠| +𝑟(𝑘−1)−1

. Then, there exists an

𝑓(𝜆)=

O(poly 𝜆)and Ξ = O(poly 𝑘)such that

𝐻eﬀ (𝐻gad,2𝑛)=𝜆𝑘

Ξ𝐻target ⊗(|0⟩⟨0|)⊗𝑟 𝑘

+𝑓(𝜆)Π+ O(𝜆𝑘+1),(6)

where Πis the projector onto the support of 𝐻eﬀ (𝐻gad,2𝑛).

Furthermore, we can provide a guarantee on the closeness

of the ground states of the target and gadget Hamiltonians.

Corollary 1 (Guarantees on the closeness of the ground states).

Let

𝐻target

and

𝐻gad

be as in Theorem 1. Then, there exists a

perturbation strength 𝜆∗, with

𝜆∗≤𝜆max and 𝜆∗≤𝐸target

1−𝐸target

Ξ∥𝑂err ∥,(7)

such that for all 𝜆≤𝜆∗it holds that

∥𝜓0−Traux [𝜙0]∥2≤ O(𝜆),(8)

where

𝜓0=|𝜓0⟩⟨𝜓0|

and

𝜙0=|𝜙0⟩⟨𝜙0|

are in the ground spaces

of 𝐻target and 𝐻gad, respectively.

In Fig. 3, we provide a visualization of the main steps in the

proof of Theorem 1, focusing on the main ideas behind these

results and referring the mathematically interested reader to the

appendices. Speciﬁcally, after a recap in Appendix Aof the

perturbation theory on which our results are based, we prove

Theorem 1in Appendix Band Corollary 1in Appendix C.

For clarity, the construction is stated for target Hamiltonians

where each term has the same Pauli weight, and for the maximal

reduction: to a three-local gadget Hamiltonian. This gadget

can be generalized to both of these aspects and we show how

to construct a gadget for mixed Pauli weights in Appendix D 1

and also how to trade oﬀ target locality for reduced resources

in Appendix D 2.

IV. A RECIPE FOR CREATING YOUR OWN

PERTURBATIVE GADGET

In the previous section, we presented a speciﬁc perturbative

gadget construction. However, that construction is by no

means unique, because there is some freedom in designing

such gadgets. For instance, the choice of Pauli operators in

the unperturbed Hamiltonian acting on the auxiliary qubits or

in the perturbation could have been diﬀerent. More generally,

the general construction of the penalization in

𝐻aux

𝑠

and of the

perturbation

𝑉𝑠

leaves many knobs to tweak. Especially for

practical implementations, it might be interesting to derive

slightly diﬀerent gadgets, which result in similar low-energy

spectra but are constructed from a diﬀerent set of operators, for

instance, some that are easier to implement on the hardware of

choice.

Here, we present a more general framework for the construc-

tion of non-recursive, non-adiabatic perturbative gadgets based

on higher-order perturbation theory that all lead to similar

results as Theorem 1, with the only notable diﬀerence being

the exact form of Ξin Eq. (6).

The core idea is again to start from a well-known, unperturbed

Hamiltonian

𝐻aux

and to perturb it with a perturbation

𝜆𝑉

, such

that we recover the original,

𝑘

-body target Hamiltonian at

𝑘th

order in perturbation theory. Since the

𝑘th

order in perturbation

theory comes with

𝑘

-fold applications of the perturbation

𝑉

, we

engineer the perturbation such that only cross-terms leading to

the desired result contribute. That is if the target Hamiltonian

target

−aux

aux

(aux

2)2

aux

target

1𝕀aux

target

1𝕀aux

(c)(b)(a) (d)

target gad

aux 2nd

order perturbation

5th

order perturbation

FIG. 3. Visualization of the key steps in the proof of Theorem 1. (a) For illustrative purposes, we take as the

𝑘

-body target Hamiltonian

𝐻target =Ë5

𝑗=1𝜎𝑗

, where

𝜎𝑗∈ {𝑋 , 𝑌, 𝑍 }

are Pauli operators, so that

𝑘=5

. (b) The unperturbed auxiliary Hamiltonian, given in Eq.

(4)

, acts

only on the auxiliary qubits. (c) Acting with the perturbation

𝑉

in Eq.

(5)

, at second order, one term could result in ﬂipping the second and

last auxiliary qubits, hence ejecting the state out of the ground space and not contributing to the perturbative expansion. (d) At

𝑘th

order in

perturbation theory, we obtain a non-trivial contribution acting on all target qubits simultaneously, mimicking the target Hamiltonian, while

acting trivially on the auxiliary qubits.

has been cut into

𝑘

pieces

𝑂target =𝑂1⊗. . . ⊗𝑂𝑘

, we ensure

that only the correct 𝑘-fold products of 𝑉survive.

To construct a perturbative gadget following a similar recipe

as the one presented in this work, we require a penalization

Hamiltonian

𝐻

and a perturbation operator

𝐴

, both acting on a

single auxiliary register of

𝑘

qubits. Although those are not the

unperturbed Hamiltonian

𝐻aux

and the perturbation

𝑉

directly,

they are closely related.

𝐻aux

will be built using instances of

𝐻

, while the perturbation operator will be a key component

in the construction of the perturbation

𝑉

. We further require

𝐻aux

to be composed of few-body terms and to have a non-

degenerate ground space to avoid the limitations of prior gadget

constructions [3,5,18]. That is, we need

𝐻=

𝑖

ℎ𝑖,(9)

for some ﬁnite sum of operators

ℎ𝑖

, each of which is few-body,

such that

𝐻

has a unique ground state vector, which we denote

|𝐺𝑆⟩

. We note that the restriction on few-body terms is

not strictly required for the construction to yield the desired

low-energy subspace, but for decreasing the locality of the

Hamiltonian.

Let us now argue that the operator

𝐴

should exhibit a product

form of 𝑘operators 𝑎1, 𝑎2, . . . , 𝑎𝑘, such that

𝐴=

𝑘

𝑗=1

𝑎𝑗,(10)

with the following properties:

𝐴|𝐺𝑆⟩∝|𝐺𝑆⟩,(11)

⟨𝐺𝑆|

𝑚

ℓ=1

𝑎𝑗ℓ|𝐺𝑆⟩=0∀𝑚 < 𝑘, ∀ {𝑗ℓ}⊂{1,2, . . . , 𝑘},

(12)

𝑎2

𝑗=1∀𝑗∈ {1,2, . . . , 𝑘}.(13)

The ﬁrst condition states that the ground state of

𝐻

should also

be an eigenstate of

𝐴

, while the second enforces that any partial

application of the operators

𝑎𝑗

results in a state orthogonal to

|𝐺𝑆⟩

. These are vital to ensure that at orders lower than

𝑘

perturbation theory, the perturbation will always expel the state

out of the ground space. The ﬁnal property ensures that when

the same perturbation is applied twice, it results in a constant

energy shift.

With the above properties, we can construct 𝐻aux and 𝑉as

𝐻aux

𝑠=1⊗𝑛

|{z}

target register

⊗1⊗(𝑠−1)𝑘⊗𝐻⊗1⊗(𝑟−𝑠)𝑘

| {z }

auxiliary registers

,(14)

𝑉𝑠=

𝑘



𝑗=1

˜𝑐𝑠, 𝑗 𝜎𝑠, 𝑗 ⊗1⊗(𝑠−1)𝑘⊗𝑎𝑗⊗1⊗(𝑟−𝑠)𝑘,(15)

where the deﬁnition of the coeﬃcients

˜𝑐𝑠, 𝑗

provides another

knob to tweak. They could be deﬁned similarly to Deﬁnition 1

or in an arbitrary fashion, as long as

𝑘

𝑗=1

˜𝑐𝑠, 𝑗 =−(−1)𝑘𝑐𝑠(16)

still holds. Given these conditions, similar results as in Theo-

rem 1will hold due to the arguments laid out in Appendix B.

Indeed, the gadget we propose is simply a special case of

ℎ𝑖=1

2(1−𝑍𝑖)=|1⟩⟨1|𝑖, 𝑖 ∈ {1,2, . . . , 𝑘},(17)

𝑎𝑗=𝑋𝑗⊗𝑋(𝑗+1)mod 𝑘, 𝑗 ∈ {1,2, . . . , 𝑘}.(18)

Note that the Hamiltonian in Eq.

(14)

has a non-degenerate

ground space when considering only the auxiliary registers but

has a

2𝑛

degenerate ground space when additionally considering

the target qubits which it acts trivially upon.

Furthermore, we would like to stress that any gadget con-

structed in this fashion will be accompanied by a suppression of

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Perturbativegadgetsforgate-basedquantumcomputing:Non-recursiveconstructionswithoutsubspacerestrictionsSimonCichy,1,2PaulK.Faehrmann,1,∗SumeetKhatri,1andJensEisert1,3,41DahlemCenterforComplexQuantumSystems,FreieUniversitätBerlin,14195Berlin,Germany2InstituteforTheoreticalPhysics,ETHZürich,8093Zürich,...

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Perturbative gadgets for gate-based quantum computing Non-recursive constructions without subspace restrictions Simon Cichy12Paul K. Faehrmann1Sumeet Khatri1and Jens Eisert134.pdf

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Perturbative gadgets for gate-based quantum computing Non-recursive constructions without subspace restrictions Simon Cichy12Paul K. Faehrmann1Sumeet Khatri1and Jens Eisert134

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