OptimizingFermionicEncodingsforbothHamiltonianandHardware

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Optimizing Fermionic Encodings for both Hamiltonian and

Hardware

Riley W. Chien1,2 and Joel Klassen1

1Phasecraft Ltd.

2Dartmouth College

October 12, 2022

Abstract

In this work we present a method for generating a fermionic encoding tailored to a set of target

fermionic operators and to a target hardware connectivity. Our method uses brute force search,

over the space of all encodings which map from Majorana monomials to Pauli operators, to ﬁnd

an encoding which optimizes a target cost function. In contrast to earlier works in this direction,

our method searches over an extremely broad class of encodings which subsumes all known second

quantized encodings that constitute algebra homomorphisms. In order to search over this class,

we give a clear mathematical explanation of how precisely it is characterized, and how to translate

this characterization into constructive search criteria. A beneﬁt of searching over this class is that

our method is able to supply fairly general optimality guarantees on solutions. A second beneﬁt

is that our method is, in principal, capable of ﬁnding more eﬃcient representations of fermionic

systems when the set of fermionic operators under consideration are faithfully represented by a

smaller quotient algebra. Given the high algorithmic cost of performing the search, we adapt our

method to handle translationally invariant systems that can be described by a small unit cell that

is less costly. We demonstrate our method on various pairings of target fermionic operators and

hardware connectivities. We additionally show how our method can be extended to ﬁnd error

detecting fermionic encodings in this class.

1 Introduction

An important application of quantum comput-

ing is the modelling of systems where quantum

physics dominates. Many such systems consist of

fermions – which are one of the two fundamental

types of particle. Any simulation of fermions on

a quantum computer requires a speciﬁcation of

how the fermions are represented on the mem-

ory of the device – an encoding of fermions into

qubits. These encodings must replicate the non-

local anti-commutation relations of the fermionic

operator algebra on the qubit system. This is a

non-trivial task, with many possible solutions of

which there is no a priori correct choice.

Many encodings have been invented ([1,2,3,

4,5,6,7,8,9,10,11]), each with their own

upsides and downsides which depend on the de-

tails of the model and of the hardware being

used. Historically, encodings have not been de-

signed for a particular model or piece of comput-

ing hardware. Instead they are often designed

for a generic advantageous trait, such as low op-

erator weight representations of certain interac-

tions1, eﬃcient use of qubits, ease of state prepa-

ration, the ability to detect or correct errors, or

1Here by operator weight we mean the number of

qubits on which operator has support

arXiv:2210.05652v1 [quant-ph] 11 Oct 2022

merely simplicity. When one wishes to imple-

ment these encodings on speciﬁc hardware, one

generally needs to think carefully about how the

encoding can be made to “ﬁt” into the device,

given the particular algorithm – generally with

the aim of minimizing circuit depth and quan-

tum memory use. This can be fairly non-trivial

and it suggests that instead of using an out of

the box encoding, it may be more fruitful to au-

tomatically generate a new encoding tailored to

the speciﬁcs of the hardware and of the model.

Some work has been done in this direction.

Ref. [12] explores ﬁnding optimal arrangements

of the Jordan-Wigner transform onto a speciﬁc

hardware geometry. This has the beneﬁt of min-

imizing the operator weights of the terms in

a given Hamiltonian, while not increasing the

qubit overhead. However it may be desirable

to make such a trade-oﬀ, in which case diﬀer-

ent methods are required in order to search over

the space of fermionic encodings that employ an-

cillary qubits to reduce the operator weight. In

Ref. [13] custom codes for the purposes of par-

allelization are considered. Here the optimiza-

tion is over the class of encodings described in

[14]. However there are many potential encod-

ings which may not fall within this class.

In this work we present a method for gener-

ating fermionic encodings tailored to the given

hardware connectivity of a device, and to a given

fermionic Hamiltonian. This method takes as

input a set of fermionic operators, constituting

the terms in the Hamiltonian, a graph specifying

the possible two-qubit interactions of the hard-

ware, and a maximum cost. The method ﬁnds

the fermionic encoding with the least possible

cost, or otherwise determines that no encoding

exists with cost less than the speciﬁed maximum.

Unlike earlier methods, ours searches in an ex-

haustive fashion over an extremely broad family

of encodings, namely all encodings which would

map Majorana monomials to Pauli operators. To

our knowledge this family subsumes nearly all

existing second quantized fermionic encodings –

with the notable exception of Ref. [15]2.

2This exception diﬀers from the other encodings in

The cost model we consider is as follows.

Given an encoding, every term in the Hamilto-

nian is mapped to a qubit operator, which has

support on a subset of qubits. In order to simu-

late the term, these qubits must be made to mu-

tually interact. This requires interactions across

at least all edges of a minimal Steiner tree (on

the hardware graph) that contains these qubits.

Thus an eﬀective proxy for the circuit depth of

simulating this term is the number of edges in

this Steiner tree, and the cost of a given encod-

ing is the maximum of this metric over all terms

in the Hamiltonian.

The method is brute force and employs a

branch-and-bound algorithm to search the space

of possible encodings. The size of the search

space grows very rapidly with the number of

terms in the Hamiltonian and the size of the

hardware graph. We introduce a number of

strategies to reduce the size of this search space

and exit branches early, which makes it possible

to solve small problem sizes. For large problem

sizes, it is likely that this method rapidly be-

comes too costly.

However there is a speciﬁc use case where the

size of the input speciﬁcation remains small and

where our method is likely to be useful. This

is where both the system being modelled and

the hardware graph each possess translational

symmetry, and may be described concisely by

a unit cell. Translational symmetry is ubiqui-

tous in physics, but the domain where we ex-

pect this will ﬁnd most use is in material sci-

ence and condensed matter, where the major-

ity of systems of interest possess translational

symmetry. Furthermore, we have already ob-

served that as superconducting quantum com-

puters scale up, qubits are being arranged in a

tiled pattern. We expect this trend to continue,

as it lends itself well to large scale manufactur-

ing. Since translational invariance is so impor-

that it constitutes a representation of the unitary group

of Majorana monomials, instead of an algebra homomor-

phism on the group algebra of the Majorana monomials,

and leverages block encodings and linear combination of

unitaries methods to simulate a unitary generated by the

required element of the algebra.

tant to this method, we incorporate it into all

aspects of the work presented here. We employ

the polynomial formalism popularized by Haah

[16,17] to describe translationally invariant fam-

ilies of fermionic and qubit operators.

All existing encodings that employ additional

qubits to reduce operator weights do so by way of

projecting into a code space via the stabilizer for-

malism. The encodings produced by our method

use the same strategy. An extra beneﬁt of this

strategy is that some encodings may also detect

or even correct errors [7,8], due to their high

code distance. We show how our method can be

made to restrict its search to fermionic encodings

with a minimum code distance, and thus ﬁnd

low cost encodings with good error mitigating

features. However high code distance is funda-

mentally at odds with the cost that we are trying

to minimize in our method, and so for practical

reasons we primarily concern ourselves with er-

ror detecting codes.

We have employed our method to ﬁnd optimal

encodings for fermionic models on various ge-

ometries, mapped to a number of diﬀerent trans-

lationally invariant hardware graphs, including

many existing superconducting architectures. In

some cases these optimal encodings are new, and

in other cases they were already known. Impor-

tantly, even in the case where an encoding was

known, our method certiﬁes that the encoding

chosen is the best choice for the given pairing

(under our cost model).

2 Preliminaries

2.1 Mathematical Formalism of

Fermionic Encodings

We now review the mathematical formalism un-

derpinning the kinds of fermionic encodings that

our method is able to ﬁnd. Nearly all known

fermionic encodings fall within this formalism,

with [15] being the only exception we are aware

of. Employing the Majorana operators

γj:= aj+a†

j(1)

γj:= i(a†

j−aj),(2)

where a†

jand ajare the standard fermionic cre-

ation and annihilation operators, we may con-

sider the group of Majorana monomials

M:= 





M(b) :=

m−1

j=0

γbj

jγbj+m

j

b∈ {0,1}2m





{±1,±i}(3)

Generally physicists are interested in an alge-

bra of observables, i.e. the group algebra C[M]

on M, or otherwise the group algebra C[F] on

some subgroup F≤ M of M. For example

fermionic parity superselection restricts the alge-

bra of observables of natural systems to be the

group algebra on the even monomials

M+:= Mwith |b|= 0,(4)

where |b|:= Pibimod 2. One may also be in-

terested in restricting to smaller subgroups, for

example if one is interested in conservation of a

symmetry.

To construct an encoding of these observables

into qubits, one must ﬁnd an algebra isomor-

phism between the given group algebra and an

algebra of observables on the qubit system. Since

these are group algebras it suﬃces to ﬁnd a faith-

ful group representation that acts trivially on el-

ements of the complex ﬁeld (i.e. maps −1 to −1

and ito i), which can then be extended to the

group algebra by linearity of the representation.

We are often interested in engineering our en-

coding to satisfy speciﬁc features on some privi-

leged subset F ⊆ Fof the group. So we want a

way to specify the form of the elements in F, and

then to build a faithful group representation of

the rest of the group Fwhich is consistent with

that form. The procedure for doing this is as

follows. For the sake of simplicity Fis assumed

to be generated by F.

Given a subset F ⊆ M of Majorana monomi-

als, the strategy is to ﬁnd a mapping σ:F → P

from Fto the group of Pauli operators on n

qubits :

P:= 





P(b) :=

n−1

j=0

Xbj

jZbj+n

j

b∈ {0,1}2n





{±1,±i}(5)

such that σpreserves the group commutation re-

lations

∀fi, fj∈ F : [fi, fj] := f−1

if−1

jfifj=±1 (6)

and inverse/hermiticity relations

∀f∈ F :f−1=f†=±f(7)

among elements of F. Note however, that σ

need not, and indeed typically does not, preserve

any other product relations. Next we show that

once a mapping of this form has been found, it

uniquely speciﬁes a group representation of the

subgroup F=hFi generated by F, subject to

one caveat.

Consider the ﬁnitely presented group

Γ =hF × {±1,±i}|(6),(7)i(8)

=





Γ(b) :=

|F|−1

j=0

fbj

j

b∈ {0,1}|F|





×(9)

{±1,±i}(10)

consisting of the free group on the set F ×

{±1,±i}, subject to relations (6) and (7), but

not any of the other product relations of M.

Most intuitively, one should think of the ele-

ments of Γ as having forgotten that they were

once products of Majorana operators, remem-

bering only commutation/anticommutation rela-

tions and their inverse. The mapping σadmits a

natural extension from the set Fto the group Γ

via the group structure of P, and by construction

is a group representation of Γ.

There is a natural group homomorphism τ:

Γ→Fwhich is deﬁned by its action on the

generating elements: τ(fi∈Γ) := fi∈Fand

then extended to the full group Γ via existing

product relations in F, i.e. τ(fifj) := τ(fi)τ(fj).

Furthermore, we have that F'Γ/ker(τ).

In what follows we will make use of the notion

of a “descended” representation. Given a group

A, a representation φand a subgroup B < A, if

Bis in the kernel of φthen φcan be descended to

a representation φ0of the quotient group A/B:

φ0(aB) := φ(a)

which is consistently deﬁned since φ(B) = 1. For

ease of exposition, we will drop the distinction

between φand its descended representation φ0,

which can be inferred implicitly from the partic-

ular group it acts upon. So for example we may

say φis a representation of A/B.

In order to construct a group representation of

F, we must identify the abelian subgroup:

S:= σ(ker(τ)) ⊆ P (11)

which, as long as −16∈ S, is a stabilizer group

with stabilizer code space V. This allows us to

deﬁne the projection of σonto the subspace V

σV:= ΠV◦σ(12)

which remains a representation of Γ, since ΠV

commutes with the group σ(Γ). Furthermore,

since σV(ker(τ)) = 1, it is also a descended rep-

resentation of Γ/ker(τ)'F.

Intuitively, the group ker τdescribes which

products of elements in Fshould be yielding the

identity once they remember that are were con-

structed from Majorana operators. By project-

ing into the code space Vwe ensure that this

product structure is preserved.

Finally, we need to ensure our representation

is faithful – i.e. there is a one-to-one corre-

spondence between qubit operations on the code

space and logical fermionic operations. It can be

shown (see Appendix B) that the representation

σVis a faithful representation of Fif and only

if ker(σ)⊆ker(τ). If this is not the case, then

there is a group of observables

G:= τ(ker(σ)) ⊆F(13)

whose value becomes ﬁxed to +1 when project-

ing into V. We refer to such a group Gas a

superselection group. In this case, σVis instead

a faithful representation of the quotient group

F/G (see Appendix B).

In summary, if we can map a set of fermionic

observables Fto Paulis which satisfy the group

commutation relations (6) and inverse relations

(7) among the elements of F, then we can iden-

tify a stabilizer group S⊆ P and a superselec-

tion group G⊆F:= hFi such that we can ex-

tend this mapping σto an algebra isomorphism

on the group algebra C[F/G] by projecting into

the stabilizer code space Vof S, ﬁxing the su-

perselection group Gto be +1. The one caveat

is that the stabilizer group Smay not contain

−1.

2.1.1 Binary Representation

A nice feature of both the Pauli group Pand

the Majorana monomial group Mis that up to

a phase their elements may be represented by

a binary vector (as made explicit in deﬁnitions

(3) and (5)). Furthermore, the product relations

among elements in these groups correspond di-

rectly to addition of the corresponding binary

vectors – again up to a phase.

Thus it is useful to specify the projective por-

tion of the maps τand σas matrices over a bi-

nary ﬁeld, and postpone the speciﬁcation of the

phase e.g.:

ˆτ=

f1f2f3· · · f|F|













γ0011· · · 0

γ1100· · · 0

.....

γm−1111· · · 1

¯γ0010· · · 0

¯γ1111· · · 1

.....

¯γm−1111· · · 1

(14)

ˆσ=

f1f2f3· · · f|F|













X0100· · · 0

X1101· · · 0

.....

Xn−1001· · · 0

Z0101· · · 1

Z1010· · · 0

.....

Zn−1000· · · 1

(15)

Where the columns ˆτiand ˆσiare given by

τ(fi) = δτ(fi)M(ˆτi), σ(fi) = δσ(fi)P(ˆσi)

where δτand δσare as yet unspeciﬁed phases.

Writing the maps in this way allows us to eas-

ily represent the group commutation relations as

a matrix equation, and also to easily identify the

stabilizer group Sand superselection group G.

For two Pauli operators P(a), P (b) speciﬁed

by a, b ∈F2n

2, their group commutation relation

is given by,

[P(a), P (b)] = (−1)ωq(a,b)(16)

where ωqis a binary symplectic form

ωq(a, b) = a†Λqb, Λq=0 1

1 0⊗In×n.(17)

In the case of Majoranas, if we restrict Fto be

a subset of the even Majorana monomials M+,

then for two Majorana operators M(a), M(b)

speciﬁed by a, b ∈F2m

2, their group commuta-

tion relation is given by

[M(a), M(b)] = (−1)ωf(a,b)(18)

Using the quadratic form ωf(a, b) = a†bworks

for even parity Majorana operators, however it

fails for odd parity operators. To correct this, the

products of the Hamming weights mod 2 should

be added.

In order to have quadratic form that behaves

properly also for odd operators, we will use a

matrix in the quadratic form

ωf(a, b) = a†Λfb, Λf=I+C1(19)

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OptimizingFermionicEncodingsforbothHamiltonianandHardwareRileyW.Chien1,2andJoelKlassen11PhasecraftLtd.2DartmouthCollegeOctober12,2022AbstractInthisworkwepresentamethodforgeneratingafermionicencodingtailoredtoasetoftargetfermionicoperatorsandtoatargethardwareconnectivity.Ourmethodusesbruteforcesearch...

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