Tapering off qubits to simulate fermionic Hamiltonians

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by  Sergey Bravyi

Tapering off qubits to simulate fermionic Hamiltonians
Sergey Bravyi,1 Jay M. Gambetta,1 Antonio Mezzacapo,1 and Kristan Temme1
1IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
(Dated: January 31, 2017)
We discuss encodings of fermionic many-body systems by qubits in the presence of symmetries.
Such encodings eliminate redundant degrees of freedom in a way that preserves a simple structure
ofthesystemHamiltonianenablingquantumsimulationswithfewerqubits. FirstweconsiderU(1)
symmetry describing the particle number conservation. Using a previously known encoding based
on the first quantization method a system of M fermi modes with N particles can be simulated
on a quantum computer with Q = Nlog (M) qubits. We propose a new version of this encoding
2
tailoredtovariationalquantumalgorithms. Alsoweshowhowtoimprovesparsityofthesimulator
Hamiltonianusingorthogonalarrays. Nextweconsiderencodingsbasedonthesecondquantization
method. It is shown that encodings with a given filling fraction ν = N/M and a qubit-per-mode
7
ratio η = Q/M < 1 can be constructed from efficiently decodable classical LDPC codes with the
1
relative distance 2ν and the encoding rate 1−η. A family of codes based on high-girth bipartite
0
2 graphsisdiscussed. Graph-basedencodingseliminateroughlyM/N qubits. FinallyweconsiderZ2
symmetries, and show how to eliminate qubits using previously known encodings, illustrating the
n
technique for simple molecular-type Hamiltonians.
a
J

7 I. INTRODUCTION simulation without loss of information [15, 17]. Impor-
2
tantly, the removal of qubits in these examples preserves

a simple structure of the encoded Hamiltonian enabling
] Quantum information processing holds the promise of
h efficient simulations with fewer qubits. This motivates
solving some of the problems that are deemed too chal-
p thequestionofhowtogeneralizethesemethodsandhow
lengingforconventionalcomputers. Oneimportantprob-
- to eliminate redundant qubits in a computationally effi-
t leminthiscategoryisthesimulationofstronglyinteract-
n cient manner for larger systems.
ing fermionic systems, in the context of quantum chem-
a
u istry or material science. A natural application for a To address these questions let us first give a more pre-
q quantumcomputerwouldbepreparinglow-energystates cisenotionofasimulation. Weshalldescribeafermionic
[ and estimating the ground energy of a fermionic Hamil- system to be simulated by a target Hamiltonian Htgt

tonian. Severalmethodshavebeenproposedintheliter- composed of one- and two-body operators such as those
1
v ature to accomplish this task, for example, preparation describing hopping, chemical potential, and two-particle
3 of a good trial state followed by the quantum phase es- interactions,
1 timation [1, 2] or state preparation by the adiabatic evo-
2 lution [3, 4]. These methods however require a univer- (cid:88)M (cid:88)M
8 salquantumcomputercapableofimplementingverylong Htgt = tαβa†αaβ + uαβγδa†αa†βaγaδ. (1)
0
circuits,exceedingthestate-of-the-artdemonstrationsby α,β=1 α,β,γ,δ=1
.
1 many orders of magnitude [5]. Alternative methods that
0 could be more viable in the near future are variational Here M is the number of fermi modes, a†α and aα are
7 quantumalgorithms[6–9]. Suchalgorithmsminimizethe creation and annihilation operators for a mode α, and
v:1 etrniearlgsytaotfeas ttahragtetcafenrmbeionpircepHaarmediltoonniaangoivveenr aqucalanstsumof atαnβd,uuααββγγδδa=re cuo∗δmγβpαl.exLceoaevffiincgienatssidseucshuptheractontβdαuc=tivti∗αtβy
Xi hardware by varying control parameters. and relativistic effects, all natural fermionic Hamilto-
nians have the above form. Since each term in H
Sincethebasicunitsofaquantumcomputerarequbits tgt
r has equal number of creation and annihilation opera-
a rather than fermi modes, any simulation method relies
tors, H commutes with the particle number operator
on a certain encoding of fermionic degrees of freedom tgt
by qubits [10–16]. A choice of a good encoding is im- Nˆ ≡(cid:80)Mα=1a†αaα. We shall assume that the system con-
portant as it may affect both the number of qubits and tains a fixed number of particles N. For example, N
the running time of a simulation algorithm. Here we could be the number of valence electrons in a molecule.
propose encodings tailored to variational quantum algo- DefineatargetHilbertspaceHtgt astheN-particlesub-
rithms and fermionic Hamiltonians that possess symme- spacespannedbyallstates|φ(cid:105)ofM fermimodessatisfy-
triessuchastheparticlenumberconservation. Thepres- ing Nˆ|φ(cid:105) = N|φ(cid:105). Without loss of generality, N ≤ M/2
ence of symmetries allows one to restrict the simulation (otherwise consider holes instead of particles). Our goal
to an eigenspace of the symmetry operator whereby re- is to estimate the minimum energy of H restricted to
tgt
ducingthenumberofqubitsthatencodeafermionicsys- theN-particlesubspaceH . Belowweshallofteniden-
tgt
tem. In several specific examples, such as the hydrogen tify H and the restriction of H onto the subspace
tgt tgt
moleculeandtheFermi-Hubbdardmodelithasbeenob- H . We shall choose the energy scale such that all co-
tgt
served that some qubits can indeed be removed from the efficients in Eq. (1) have magnitude at most one.
2
Let us now formally define an encoding of fermionic We also give a poly(M) upper bound on the norm of the
degreesoffreedombyqubits. FollowingRef.[18],weshall terms D although we do not expect this bound to be
i
describesuchencodingasanisometryE : H →H , tight. ThisencodingmostlyfollowsideasofRefs.[19,20]
tgt sim
where H = (C2)⊗Q is the Hilbert space of Q qubits. and relies on the first-quantization method. We extend
sim
A state of the target system |φ(cid:105)∈H is identified with the results of Refs. [19, 20] in two respects. First we
tgt
astateE|φ(cid:105)ofthesimulatorsystem. EncodedstatesE|φ(cid:105) showhowtoimprovethesparsityofH usingorthogo-
sim
span a codespace Im(E)≡E ·H . nalarrays[21]. Sucharrayshavebeenpreviouslyusedfor
tgt
A Hamiltonian H describing a system of Q qubits quantum simulations and dynamical decoupling [22–24]
sim
is called a simulator of H if it satisfies two conditions. but their application in the context of variational quan-
tgt
First, we require that tum algorithms appears to be new. Secondly, we intro-
duce a sparse Hamiltonian enforcing the anti-symmetric
H E =EH . (2)
sim tgt structureofencodedstatesandcomputethespectralgap
In words, H must preserve the codespace and the re- of this Hamiltonian using arguments based on the Schur
sim
striction of Hsim onto the codespace must be unitarily duality [25–27]. This allows us to bound the norm (cid:107)Di(cid:107).
equivalent to H . The action of H on the orthogo- Next consider encodings based on the second quanti-
tgt sim
nal complement of the codespace may be arbitrary. Sec- zation method. Let ν =N/M be the filling fraction and
ondly, we require that the codespace contains a ground η =Q/M be the desired qubit-per-mode ratio. We show
state of H . This guarantees that H and H have that sparse encodings with prescribed ν, η can be con-
sim tgt sim
thesamegroundenergyandtheirgroundstatescoincide structedfromclassicalerrorcorrectingcodeswithcertain
modulo the encoding E. properties. Namely, we need binary linear codes that
OurgoalistoconstructencodingsthatrequireQ<M have a column-sparse parity check matrix, relative dis-
qubits and, at the same time, all target Hamiltonians tance 2ν, and the encoding rate 1−η. It is known that
Eq. (1) have sufficiently simple simulators. We shall say the requisite codes can be constructed whenever
that a simulator Hamiltonian is r-sparse if
ν <1/4 and η >h(2ν), (5)
r
(cid:88)
H = D , (3) where h(x)=−xlog (x)−(1−x)log (1−x) is the bi-
sim i 2 2
nary Shannon entropy. For example, one can use good
i=1
LDPCcodes[28]whoseparametersapproachtheGilbert-
where each operator D is diagonal in some tensor prod-
i Varshamov bound [29, 30]. Thus a constant fraction of
uct basis of Q qubits. This basis may depend on i. We
qubits can be eliminated if the target system has a fill-
require that matrix elements of D are efficiently com-
i ingfractionν <1/4. ThesimulatorHamiltonianEq.(3)
putable. A family of encodings E as above is called
has sparsity r proportional to the number of non-zero
sparse if there exist small constants c,d such that any
coefficients t , u in the target Hamiltonian. Fur-
target Hamiltonian Eq. (1) has an r-sparse simulator αβ αβγδ
thermore, (cid:107)D (cid:107)≤1 for all i.
Eq. (3) with r ≤ Mc and (cid:107)D (cid:107) ≤ Md for all M. Sparse i
i TheboundEq.(5)isworsethanwhatonecouldexpect
encodings are well-suited for applications in variational naively. Indeed, since H has dimension (cid:0)M(cid:1)≈2Mh(ν),
quantum algorithms [6–9]. Indeed, a basic subroutine of tgt N
the information-theoretic bound on the qubit-per-mode
variational algorithms is estimating energy (cid:104)ψ|H |ψ(cid:105)
sim ratio is η ≥h(ν). We leave as an open question whether
of a given trial state ψ ∈ H that can be prepared
sim sparse encodings can achieve this bound.
on the available quantum hardware. One can estimate
The above result has one important caveat. Namely,
the expectation value e ≡ (cid:104)ψ|D |ψ(cid:105) by preparing the
i i weshallseethatmatrixelementsofthesimulatorHamil-
trial state ψ, performing a local change of basis in each
tonian can be computed efficiently only if the chosen
qubitsuchthatD becomesdiagonalinthestandard|0(cid:105),
i code is efficiently decodable. In Appendix C we describe
|1(cid:105) basis, and then measuring each qubit. Performing a
a brute force implementation of the decoding algorithm
sequence of such measurements with a freshly prepared
thatmaybepracticalforsmallnumberofmodesM ≤50.
trial state ψ for each term D gives an estimate of the
energy (cid:104)ψ|H |ψ(cid:105)=(cid:80)r e .i Simulating larger systems may require LDPC codes that
sim i=1 i are both good and efficiently decodable. Designing such
codes is an active research area, see Refs. [31–33].
Toenableefficientdecodingandimprovesparsenessof
Summary of results
the simulator Hamiltonian we consider a special class of
LDPCcodesassociatedwithhigh-girthbipartitegraphs.
Firstassumethatthenumberofparticlesissmallsuch
For such encodings any two-body operator a†a ±a†a
that Nlog (M) < M. We expect that this regime may α β β α
2 has a 2-sparse simulator, while any four-body opera-
be relevant for high-accuracy quantum chemistry calcu-
tor has 32-sparse simulator. Furthermore, matrix ele-
lations with large basis sets. We construct a sparse en-
mentsofthesimulatorscanbecomputedintimeO(M3).
coding with Q=Nlog (M) qubits such that any target
2 Graph-based encodings can eliminate M/N qubits for
HamiltonianEq.(1)hasasimulatorEq.(3)withsparsity
N ≤ M1/2. Figure 1 shows a numerically computed
r ≤9M3.17. (4) lower bound on the number of modes M that can be
3
qubits Q hardware. Weleaveasopenquestionswhethersparseen-
codingcanbeconstructedforthefillingfractionν ≥1/4,
20 22 24 26 28 30 32 34 36 38 40 whatisthetradeoffbetweentheparametersM,N,Qand
2 34 39 43 49 53 59 64 69 74 79 85 the encoding sparseness, and how to generalize our tech-
N
s 3 25 28 31 35 38 40 46 47 51 54 58 niques to other types of symmetries.
e
cl 4 22 25 27 30 34 36 38 41 44 48 49 Therestofthepaperisorganizedasfollows. SectionII
arti 5 21 24 26 28 30 33 36 39 41 43 45 summarizes our notations. Encodings based on the first
p
quantization method are described in Section III. These
6 20 22 25 27 29 32 34 36 39 42 43
encoding are applicable if the number of particles N is
sufficiently small. Section IV shows how to construct
FIG.1. AlowerboundonthenumberoffermionicmodesM sparse encodings for a constant filling fraction N/M us-
that can be simulated for fixed values of N and Q using the ing classical LDPC codes. Encodings based on high-
graph-based encodings. girth bipartite graphs are described in Sections V,VI.
Discretesymmetriesandapplicationsofourtechniquesto
small molecular-type Hamiltonians are discussed in Sec-
simulated for fixed values of Q,N using the graph-based tions VII,VIII. Appendix A summarizes the previously
encodings. known encodings of fermions by qubits. Appendix B il-
Our third result concerns Z -symmetries such as the lustratesourmethodsusingthehydrogenmoleculeasan
2
example. AppendixCshowshowtocomputematrixele-
fermionic parity conservation. For concreteness, we con-
siderasymmetrygroupZ ×Z describingparityconser- mentsofsimulatorHamiltoniansconstructedfromLDPC
2 2
codes.
vation for electrons with a fixed spin orientation. Such
symmetryispresent,forexample,inthestandardmolec-
ular electron Hamiltonians that neglect spin-spin and
II. NOTATIONS
spin-orbit interactions. The target Hilbert space H
tgt
is chosen as a subspace in which the number of electrons
with a given spin is fixed modulo two. Using the second A system of M fermi modes is described by the Fock
quantizationmethodandboththeparityandthebinary space FM of dimension 2M equipped with the standard
treeencodingofRef.[11]weconstructasimulatorHamil- basis |x(cid:105) ≡ |x1,...,xM(cid:105), where xα = 0,1 is the occupa-
tonian Eq. (3) that acts on Q = M −2 qubits. More tion number of the mode α such that a†αaα|x(cid:105) = xα|x(cid:105).
generally, we give a systematic method of detecting Z OurtargetHilbertspaceisdefinedastheN-particlesub-
2
symmetries in a given target Hamiltonian and show how space of FM,
to construct sparse encodings that eliminate one qubit
foreachZ2 symmetry. Ourtechniquesareillustratedus- Htgt =span(|x(cid:105)∈FM : |x|=N). (6)
ing quantum chemistry Hamiltonians describing simple
Here|x|denotestheHammingweightofx. Thesimulator
molecules.
system consists of Q qubits, where Q satisfies
To summarize, we observed that the encoding based
on the first quantization method achieves the best per- (cid:18) (cid:19)
M
formance in terms of the number of qubits that can be dim(Htgt)= N ≤2Q ≤2M.
eliminated. However, it is applicable only if the number
of particles N is relatively small. Furthermore, the en-
The Hilbert space H = (C2)⊗Q is equipped with the
coding does not take advantage of any structure present sim
standard basis |s(cid:105), where s ∈ {0,1}Q. We shall reserve
inthecoefficientsofH . Forexample,H mighthave
tgt sim letters s,t for qubit basis vectors and letters x,y for the
sparseness 9M3.17 even if H has only O(M) non-zero
tgt Fock basis vectors. For any integer K ≥ 1 let [K] ≡
coefficients. In contrast, encodings based on the sec-
{1,2,...,K}.
ondquantizationmethodeliminatefewerqubitsbuthave
SupposeO isatwo-bodyorfour-bodyfermionicob-
broader applicability and produce more sparse simulator tgt
servable (hermitian operator) listed below
Hamiltonian such that the number of terms in H and
tgt
H are roughly the same (up to a constant factor).
sim i(cid:15)(a†a ±a†a ), i(cid:15)(a†a†a a ±a†a†a a ), (7)
Which encoding should be preferred may depend on de- α β β α α β γ δ δ γ β α
tails of the target system.
where (cid:15)=0,1 is chosen to make the operator hermitian.
We expect our results to find applications in the near- WeshallsaythataqubitobservableO actingonH
sim sim
future experimental demonstrations of variational quan- is a simulator of O if
tgt
tum algorithms. In this context the number of qubits Q
isfixedbythehardwareconstraintsandmaynotbelarge O E =EO . (8)
sim tgt
enough to simulate interesting molecules directly. Com-
biningthestandardvariationalalgorithms[6–8]withthe AdirectconsequenceofEq.(8)isthatO preservesthe
sim
encodings described in this paper may extend the range codespaceandtherestrictionofO ontothecodespace
sim
of molecules that can be simulated on a given quantum is unitarily equivalent to O . The action of O on
tgt sim
4
the orthogonal complement of the codespace may be ar- (cid:88) (cid:88)M
U =− u |α,β(cid:105)(cid:104)γ,δ| (13)
bitrary. Let us say that the simulator O is r-sparse if αβγδ i,j
sim
it can be written as 1≤i(cid:54)=j≤N α,β,γ,δ=1
r Here and below the subscripts i,j indicate the registers
(cid:88)
Osim = Di, (9) Qi,Qj acteduponbyanoperator. Finally,H⊥ penalizes
i=1 states orthogonal to the codespace,
whereDiarehermitianoperatorssuchthateachoperator (cid:88) 1
Di is diagonal in some tensor product basis of Q qubits. H⊥ = 2(I+(↔)i,j). (14)
This basis may depend on i. The maximum sparsity r 1≤i<j≤N
oftwo-bodyandfour-bodysimulatorswillbedenotedr
2 Here(↔) istheSWAPoperatorthatexchangesQ and
and r respectively. i,j i
4 Q . NotethatH⊥ haszerogroundenergyanditsground
j
subspace coincides with the codespace Im(E). The coef-
ficient g > 0 in Eq. (11) will be chosen large enough so
III. SPARSE ENCODINGS FOR SMALL
that the ground state of H belongs to the codespace.
NUMBER OF PARTICLES sim
Note that [T,P ] = [U,P ] = 0 for any permutation
π π
π ∈S . Thus H preserves the codespace Im(E). The
Here we discuss encodings based on the first quantiza- N sim
standard correspondence between the first and the sec-
tion method. Assume for simplicity that the number of
ond quantized Hamiltonians implies that the restriction
modes M is a power of two, M =2m. Otherwise, round
ofH ontothecodespaceisunitarilyequivalenttoH ,
M up to the nearest power of two. Given a fermi mode sim tgt
so that Eq. (2) is satisfied. Thus H is indeed a simu-
α ∈ {1,...,M}, let α ∈ {0,1}m be the binary represen- sim
lator of H (for large enough g to be chosen later).
tation of the integer α−1. tgt
Let us show that H is r-sparse, where
The simulator system consists of Q=mN qubits par- sim
titioned into N consecutive registers Q1,...,QN of m r =9m ≤M3.17. (15)
qubits each. For any N-tuple of modes α ,...,α let
1 N
|α1,α2,··· ,αN(cid:105) ∈ Hsim be a basis vector such that the Furthermore, Hsim = (cid:80)ri=1Di, where each term Di is
register Qi is in the state |αi(cid:105). diagonal in a tensor product of Pauli bases
Consider a basis vector |x(cid:105)∈Htgt and let √ √
X ≡{(|0(cid:105)±|1(cid:105))/ 2}, Y ≡{(|0(cid:105)±i|1(cid:105)) 2},
1≤α <α <...<α ≤M
1 2 N
and Z ≡{|0(cid:105),|1(cid:105)}. Let
be the subset of N modes that are occupied in the state
|x(cid:105), that is, x = 1 iff i ∈ {α ,...,α }. Define the Pauli(m)={σ ,σ ,...,σ }
i 1 N 1 2 M2
encoding as
be the set of all m-qubit Pauli operators (ignoring the
E|x(cid:105)= √1 (cid:88) (−1)πP |α ,α ,··· ,α (cid:105), (10) overall phase). By definition, each operator σa is a ten-
N! π 1 2 N sor product of single-qubit Pauli operators I,σx,σy,σz.
π∈SN Note that there are 4m = M2 such operators. We note
where S is the group of permutations of N objects, thattheSWAPoperatorontwoqubitscanbewrittenas
N
(−1)π isthesignofapermutationπ,andP isaunitary
π 1
operator that applies a permutation π to the registers (I⊗I+σx⊗σx+σy⊗σy+σz⊗σz).
Q ,...,Q such that 2
1 N
SincetheSWAPoperator(↔) exchangingm-qubitreg-
P |α ,α ,··· ,α (cid:105)=|α ,α ,··· ,α (cid:105). i,j
π 1 2 N π(1) π(2) π(N) istersQ ,Q isatensorproductofmtwo-qubitSWAPS,
i j
TherighthandsideofEq.(10)canbeviewedasthefirst- one gets
quantized version of the state |x(cid:105). The codespace Im(E)
M2
is spanned by antisymmetric states ψ ∈H such that (cid:88)
sim (↔) =M−1 (σ ⊗σ ) . (16)
i,j a a i,j
P |ψ(cid:105)=(−1)π|ψ(cid:105), for all π ∈S . a=1
π N
ExpandingeachterminEqs.(12,13)inthebasisofPauli
We choose the simulator Hamiltonian as
operators and using Eq. (16) to expand H⊥ one gets
H =T +U +gH⊥, (11)
sim
M2
(cid:88) (cid:88)
whereT+U isthefirst-quantizedversionofHtgt,namely Hsim = ca,b(σa⊗σb)i,j (17)
1≤i<j≤N a,b=1
N M
(cid:88) (cid:88)
T = tαβ|α(cid:105)(cid:104)β|i (12) forsomerealcoefficientsca,b. WeshallgroupPaulioper-
ators that appear in Eq. (17) into r bins such that Pauli
i=1α,β=1
5
operators from the same bin are diagonal in the same quantum algorithms one can start from g =0 and grad-
tensor product basis. First consider a single register Q . uallyincreaseg untilthebestvariationalstateψ satisfies
i
Obviously, any Pauli operator acting on Q is diagonal (cid:104)ψ|H⊥|ψ(cid:105)=0.
i
in a tensor product of the bases X,Y,Z. Such tensor LetusnowproveEq.(19). Recallthat∆⊥isthesmall-
product bases can be labeled by letters in the alphabet est non-zero eigenvalue of the Hamiltonain H⊥ defined
inEq.(14). Weshalluseasymmetry-basedargumentto
A≡{X,Y,Z}m. (18) compute all eigenvalues of H⊥. Consider the permuta-
tiongroupS andtheunitarygroupU(M). TheHilbert
N
Recallthatanorthogonalarray[21]overanalphabetAis space H ∼= (CM)⊗N defines a unitary representation
sim
amatrixRofsizen×kwithentriesfromAsuchthatany of these groups such that a permutation π ∈ S acts
N
pairofcolumnsofRcontainseachtwo-letterwordinthe on H as P and a unitary matrix V ∈ U(M) acts
sim π
alphabet A the same number of times. (More precisely, on H as V⊗N. The standard result from the group
sim
theabovedefinesanorthogonalarraywithstrengthtwo). representation theory known as Schur duality [25] gives
Orthogonal arrays have been previously used for quan- a decomposition
tum simulations and dynamical decoupling [22–24]. We
(cid:77)
shall use a family of orthogonal arrays based on the (CM)⊗N = P ⊗Q , (20)
λ λ
Galois field GF(3m) known as Rao-Hamming construc- λ
tion [21]. It gives an orthogonal n×k array R over an
where the sum runs over all Young diagrams with N
alphabetAofsize3m withn=9m andk =3m+1. Note
boxes, P is the irreducible representation (irrep) of the
that the equality n = 9m is possible only if any pair of λ
permutation group S and Q is the irrep of the uni-
columnsofR containseachtwo-letterwordinthealpha- N λ
tarygroupU(M). TheoperatorsP andV⊗N areblock-
bet A exactly one time. Also note that the number of π
diagonal with respect to Schur decomposition Eq. (20).
particles N obeys N ≤M =2m <k =3m+1. We shall
Furthermore, within each sector λ the operator P acts
use only the first N columns of R. Let f =1,...,9m be π
non-trivially only on the subsystem P , while V⊗N acts
somerowofR. ItdefinesatensorproductofPaulibases λ
non-trivially only on subsystem Q .
λ
Importantly, the Hamiltonian H⊥ commutes with the
R ≡R ⊗R ⊗···⊗R
f f,1 f,2 f,N action of both groups S and U(M), that is,
N
for the system of Q qubits. By construction, each Pauli [H⊥,P ]=[H⊥,V⊗N]=0
π
term c (σ ⊗σ ) that appears in Eq. (17) is diagonal
a,b a b i,j
inatleastonebasisR . Thuswecanchooseadecompo- for all π ∈ S and for all V ∈ U(M). Since the groups
f N
sitionHsim =(cid:80)9fm=1Df whereDf isdiagonalinthebasis SN and U(M) generate the full operator algebra in each
sectorλinthedecompositionEq.(20), weconcludethat
R . This shows that any target Hamiltonian Eq. (1) has
f
a simulator with sparsity r = 9m = M2log2(3) ≈ M3.17. H⊥ =(cid:77)e Π , (21)
RoundingM uptothenearestpoweroftwogivesEq.(5). λ λ
λ
The coefficient g in Eq. (11) must be large enough so
that the ground state of H belongs to the codespace. where Π is the projector onto the sector λ in Eq. (20)
sim λ
This is always the case if g∆⊥ > 2(cid:107)U +T(cid:107), where ∆⊥ and e are eigenvalues of H⊥. Thus we can compute e
λ λ
is the smallest non-zero eigenvalue of H⊥, see Eq. (14). by picking an arbitrary state ψ from the sector λ and
λ
Indeed, suppose ψ is an eigenvector of H orthogonal computing e =(cid:104)ψ |H⊥|ψ (cid:105).
sim λ λ λ
to the codespace. Then the corresponding eigenvalue is We shall consider a Young diagram λ with d columns
as a partition of the integer N, that is,
(cid:104)ψ|H |ψ(cid:105)≥(cid:104)ψ|U +T|ψ(cid:105)+g∆⊥ >(cid:107)U +T(cid:107).
sim
N =λ +...+λ , λ ≥...≥λ ≥1.
1 d 1 d
Such eigenvalue cannot be the ground energy of H
tgt Namely, λ is the number of boxes in the p-th column of
since the latter is unitarily equivalent to a submatrix of p
λ. Foranyintegerudefineastate|φ(u)(cid:105)∈(CM)⊗u such
U +T. Thus the ground state of H must belong to
sim that
the codespace. Recall that we assume the coefficients
t ,u to have magnitude at most one. This gives a 1 (cid:88)
αβ αβγδ |φ(u)(cid:105)= √ (−1)π|π(1),π(2),...,π(u)(cid:105).
conservative estimate (cid:107)U +T(cid:107) = O(N2M4). Below we u!
show that π∈Su
√
For example, |φ(1)(cid:105)=|1(cid:105), |φ(2)(cid:105)=(|1,2(cid:105)−|2,1(cid:105))/ 2,
N
∆⊥ = for all N ≤M. (19)
2 |φ(3)(cid:105) = √1 ( |1,2,3(cid:105)+|3,1,2(cid:105)+|2,3,1(cid:105)−
6
|2,1,3(cid:105)−|3,2,1(cid:105)−|1,3,2(cid:105))
Thus it suffices to choose g >4N−1(cid:107)U+T(cid:107)=O(NM4)
andalltermsD inthesimulatorHamiltonianhavenorm
i etc. We claim that a state
poly(M). We do not expect the bound g > O(NM4)
to be tight. In practical implementation of variational |ψ (cid:105)≡|φ(λ )(cid:105)⊗···⊗|φ(λ )(cid:105) (22)
λ 1 d
6
belongs to the sector λ of the decomposition Eq. (20). IV. SPARSE ENCODING BASED ON LDPC
Namely, such state can be obtained by applying a suit- CODES
able Young symmetrizer [26] to a basis vector. Indeed,
defineaYoungtableau(λ,T)obtainedbyfillingcolumns In this section we use the second quantization method
of λ one by one with consecutive integers 1,...,N start- and classical LDPC codes to construct sparse encodings
ing from the first column, see Figure 2 for an example. forthecasewhenthetargetsystemhasaconstantfilling
Let Scol ⊆ SN and Srow ⊆ SN be the subgroups that fraction ν =N/M.
permute integers from the same column and from the LetAbeabinarymatrixwithQrowsandM columns.
same row of (λ,T) respectively. The Young symmetrizer Given a binary vector x of length M, we shall write Ax
corresponding to tableau (λ,T) is defined as for the matrix-vector multiplication modulo two, that is,
(cid:32) (cid:33) (cid:32) (cid:33) M
(cid:88)
Π ∼ (cid:88) (−1)πP · (cid:88) P . (23) (Ax)i = Ai,αxα (mod 2) (26)
λ,T π τ
α=1
π∈Scol τ∈Srow
We consider encodings E : H →H defined by
tgt sim
It is well-known [25, 26] that Π is proportional to a
λ,T
E|x(cid:105)=|Ax(cid:105) (27)
(non-orthogonal) projector onto a subspace of the sector
λ in the Schur decomposition. In particular, Π maps
λ,T wherex∈{0,1}M and|x|=N. Letussaythatamatrix
any state to some state that belongs to the sector λ.
A is N-injective if it maps distinct M-bit vectors x with
Let s(λ) be a sequence of N integers obtained by filling
theHammingweightN todistinctQ-bitvectorss=Ax.
columns of λ one by one with consecutive integers such
It follows directly from the definitions that E is an isom-
that the j-th column is filled with integers 1,...,λ , see
j etry iff A is N-injective. The N-injectivity condition is
Figure 2 for an example. Let |s(λ)(cid:105) ∈ (CM)⊗N be the
satisfied if A is chosen as a parity check matrix describ-
basis vector corresponding to s(λ). We observe that the
ing a binary linear code of length M with the minimum
distance2N+1. Indeed,inthiscaseallerrorsxofweight
up to N must have different syndromes s=Ax and thus
a syndrome s uniquely identifies a weight-N error x.
How sparse is the encoding defined in Eq. (27)? Let
columns of A be A1,...,AM ∈{0,1}Q and let
|𝑠 𝜆 ⟩ = |1234121⟩
c(A)=max|Aα|
α
bethemaximumcolumnweight. Weclaimthatfermionic
(𝜆,𝑇)
observables defined in Eq. (7) have simulators Eq. (9)
with sparsity
FIG. 2. Example of a Young tableau (λ,T) and the basis
vector |s(λ)(cid:105). Here N =7 and (λ1,λ2,λ3)=(4,2,1). r2 ≤22c(A)−1 and r4 ≤24c(A)−1 (28)
for two-body and four-body observables respectively.
secondfactorinEq.(23)hastrivialactionon|s(λ)(cid:105)since Furthermore, (cid:107)D (cid:107) ≤ 1 for all i. Thus the encoding de-
i
Pτ|s(λ)(cid:105)=|s(λ)(cid:105) for all τ ∈Srow. It follows that finedbyEq.(27)issparsewheneverAisacolumn-sparse
matrix.
Π |s(λ)(cid:105)∼|ψ (cid:105) (24) First, let us introduce some notations. Let A−1s be a
λ,T λ
set of all weight-N vectors x satisfying Ax=s,
andthus|ψ (cid:105)indeedbelongstothesectorλoftheSchur
λ A−1s≡{x∈{0,1}M : Ax=s and |x|=N}. (29)
decomposition. One can easily check that |ψ (cid:105) is an
λ
eigenvector of H⊥ with the eigenvalue The set A−1s may be empty for some s. By definition,
A is N-injective iff the set A−1s contains at most one
(cid:18) (cid:19) d (cid:18) (cid:19)  element for any s ∈ {0,1}Q. Below eα = (0...010...0)
eλ = 21 N2 −(cid:88) λ2a + (cid:88) λb. (25) denotesastring withasinglenon-zeroat thepositionα.
We use the notation ⊕ for the bitwise XOR.
a=1 1≤a<b≤d
For concreteness, consider a pair of modes α < β and
a target observable
FromEq.(21)oneinfersthatanyeigenvalueofH⊥ must
haveaformeλ. Notethateλ =0iffλisasinglecolumn, O =a†a +a†a . (30)
that is, d=1, λ =N. One can check that the smallest tgt α β β α
1
non-zero value of eλ is achieved when λ has two columns We have Otgt|x(cid:105)=0 if xαxβ =00,11 and
with length N−1 and 1, that is, d=2, λ =N−1, and
1
λ =1. In this case e =N/2 which proves Eq. (19). O |x(cid:105)=S (x)|x⊕eα⊕eβ(cid:105)
2 λ tgt αβ
7
if x x = 01,10 where S (x) = ±1 is the parity of all g(u,s(cid:48))byapplyingtheWalsh-Hadamardtransformwith
α β αβ
bits of x located between α and β, that is, respect to the first argument:
(cid:88)
β−1 h(t,s(cid:48))≡2−k (−1)t·ug(u,s(cid:48)). (35)
(cid:89)
S (x)= (−1)xγ.
αβ u∈{0,1}k
γ=α+1
Here t·u≡(cid:80)k t u . Substituting the identity
Since A(x⊕eα⊕eβ)=Ax⊕Aα⊕Aβ, we have i=1 i i
(cid:88)
EO |x(cid:105)=S (x)|Ax⊕Aα⊕Aβ(cid:105) if x x =01,10, |s(cid:105)(cid:104)s|≡|u,s(cid:48)(cid:105)(cid:104)u,s(cid:48)|=2−k (−1)t·uZ(t)⊗|s(cid:48)(cid:105)(cid:104)s(cid:48)|
tgt αβ α β
t∈{0,1}k
EO |x(cid:105)=0 if x x =00,11. (31)
tgt α β
into Eq. (34) gives
Letussaythatabasisvectors∈{0,1}Qisαβ-flippableif
s=Ax for some weight-N string x such that x x =01 (cid:88) (cid:88)
α β Γ = h(t,s(cid:48))Z(t)⊗|s(cid:48)(cid:105)(cid:104)s(cid:48)|. (36)
orx x =10. NotethatA−1sisasinglestringwhenever αβ
α β
sisαβ-flippable. DefineanoperatorΓ actingonH t∈{0,1}k s(cid:48)∈{0,1}Q−k
αβ sim
such that
For each t∈{0,1}k define an operator
Γαβ|s(cid:105)=Sαβ(A−1s)|s(cid:105) if s is αβ-flippable (cid:88)
Γ (t)= h(t,s)|s(cid:105)(cid:104)s|. (37)
Γ |s(cid:105)=0 otherwise. (32) αβ
αβ
s∈{0,1}Q−k
Given a bit string s, let X(s) be the product of Pauli σx
acting on the last Q−k qubits. Then
operators over all qubits i such that s = 1. We claim
i
that the observable O has a simulator (cid:88)
tgt Γ = Z(t)⊗Γ (t). (38)
αβ αβ
O =X(Aα⊕Aβ)Γ =Γ X(Aα⊕Aβ). (33) t∈{0,1}k
sim αβ αβ
Aswasshownabove,X(Aα⊕Aβ)=X⊗k commuteswith
First let us check that X(Aα⊕Aβ) commutes with Γ .
αβ Γ . Since X⊗k commutes (anti-commutes) with Z(t)
Suppose s is αβ-flippable and let t = s⊕Aα ⊕Aβ. By αβ
for even (odd) t, we infer from Eq. (38) that Γ (t)=0
assumption, s = Ax for some x such that |x| = N and, αβ
whenever t has odd weight. Combining Eqs. (33,38) we
say, x x = 01. It follows that y ≡ x ⊕ eα ⊕ eβ has
α β arrive at
weight N and Ay = t. Furthermore, y y = 10. Thus
α β
t is αβ-flippable and A−1t = y. Since Sαβ(x) = Sαβ(y), O = (cid:88) X⊗kZ(t)⊗Γ (t), (39)
we have shown that S (A−1s)=S (A−1t) and thus sim αβ
αβ αβ t∈{0,1}k
|t|even
X(Aα⊕Aβ)Γ |s(cid:105)=S (x)|t(cid:105)=Γ X(Aα⊕Aβ)|s(cid:105).
αβ αβ αβ
This gives a 2k−1-sparse decomposition of O as de-
sim
If s is not αβ-flippable then so is t, so that fined in Eq. (9) where
X(Aα⊕Aβ)Γαβ|s(cid:105)=ΓαβX(Aα⊕Aβ)|s(cid:105)=0. Di ≡X⊗kZ(t)⊗Γαβ(t). (40)
We have shown that X(Aα⊕Aβ) commutes with Γ . Thisoperatorcanbemadediagonalinthestandardbasis
αβ
Next, let us check the simulation condition Eq. (2). by applying a Clifford operator exchanging Pauli Y and
Suppose x has weight N and let s=Ax. Using the first Z for all qubits i=1,...,k such that t =1. It remains
i
equality in Eq. (33) one infers that to note that
O E|x(cid:105)=X(Aα⊕Aβ)Γ |s(cid:105)=S (x)|s⊕Aα⊕Aβ(cid:105) k =|Aα⊕Aβ|≤|Aα|+|Aβ|≤2c(A).
sim αβ αβ
if x x = 01,10 and O E|x(cid:105) = 0 otherwise. Compar- Thus the simulator Eq. (39) has sparsity 22c(A)−1. Fur-
α β sim
ing this and Eq. (31) shows that OsimE =EOtgt. thermore, all matrix elements of Di are contained in
Let us show that O is r-sparse with r ≤ 22c(A)−1. the interval [−1,1], see Eqs. (34,35,37,40). We omit the
sim
By construction, Γ is diagonal in the standard basis, derivation of simulators for other observables defined in
αβ
that is, Eq. (7) since it follows exactly the same steps as above.
ConsideratargetHamiltonianH definedinEq.(1).
tgt
(cid:88)
Γαβ = g(s)|s(cid:105)(cid:104)s|, g(s)=0,±1. (34) DecomposeHtgt asalinearcombinationoftwo-bodyand
four-body observables O defined in Eq. (7). Replac-
s∈{0,1}M tgt
ing each observable O by a qubit simulator O con-
tgt sim
Let k ≡|Aα⊕Aβ|. To simplify notations, let us reorder structed above gives a simulator Hamiltonian
thequbitssuchthatX(Aα⊕Aβ)=X⊗k actsonthefirst
r
k qubits. Decompose s = (u,s(cid:48)), where u ∈ {0,1}k and H =gEE†+(cid:88)D . (41)
sim i
s(cid:48) ∈ {0,1}Q−k. Define a function h(t,s(cid:48)) obtained from
i=1
8
HerewecombinedthetermsD fromeachsimulatorO V. IMPROVING THE SPARSITY
i sim
intoasinglesum. ThetermgEE†penalizesstatesorthog-
onaltothecodespace. Notethatrisupperboundedbya Here we show how to improve the sparsity bounds in
constant O(1) times the number of non-zero coefficients Eq. (28) if the parity check matrix A has a certain addi-
tαβ,uαβγδ in the target Hamiltonian. Since (cid:107)Di(cid:107) ≤ 1, tional structure. At this point we shall exploit the fact
we can guarantee that the ground state of Hsim belongs that simulators only need to reproduce the action of tar-
to the codespace provided that g ∼M4. get observables within the codespace and can act arbi-
Let us choose A as a parity check matrix of a binary trarily on the orthogonal complement to the codespace.
linearcodethatencodesK bitsintoM bitswiththemin- Let A be a binary matrix of size Q×M with columns
imum distance 2N +1<M. As was argued above, such A1,...,AM. We shall say that A is bipartite if the set
matrixAisN-injective. Itisknownthatcertainfamilies of rows [Q] ≡ {1,...,Q} can be partitioned into two
of codes described by sparse parity check matrices can disjoint subsets, [Q]=L∪R, such that each column Aα
approachtheGilbert-Varshamovbound[29,30],namely, intersects both L and R on odd number of rows,
K =M(1−h(2N/M)−(cid:15)), (42)
|Aα∩L| (mod 2)=|Aα∩R| (mod 2)=1 (44)
where h(x)=−xlog (x)−(1−x)log (1−x) is the bi-
2 2
for all 1≤α≤M. We claim that the encoding Eq. (27)
naryShannonentropyfunctionand(cid:15)>0canbemadear-
based on a bipartite matrix A has sparsity parameters
bitrarilysmallbychoosinglargeenoughc(A). Thisclaim
follows from the existence of good LDPC codes [34], see
r ≤22c(A)−3 and r ≤24c(A)−3. (45)
for instance Theorem A.3 of [28]. We can assume wlog 2 4
that all rows of A are linearly independent in which case
Indeed, consider any weight-N string x and let s = Ax
A has Q = M −K rows. Then a family of good LDPC
be its syndrome. Let s(L),s(R)∈{0,1} be the parity of
codes as above gives a family of sparse encodings with
s restricted to L and R,
the filling fraction ν =N/M and the qubit-per-mode ra-
tio η = Q/M = 1−K/M that satisfy η = h(2ν)+(cid:15), (cid:88) (cid:88)
s(L)≡ s (mod 2) and s(R)≡ s (mod 2).
i i
as claimed in Eq. (5). Unfortunately, the constant c(A)
i∈L i∈R
grows quickly as (cid:15) approaches 0, see [28]. Since the spar-
sityofsimulatorsconstructedforfew-bodyfermionicob- From Eq. (44) one infers that flipping any bit of x flips
servables is exponential in c(A), see Eq. (28), encodings the values of s(L) and s(R). Therefore
based on good LDPC codes are not quite practical. We
show how to overcome this problem using “bad” LDPC s(L)=s(R)=(−1)N (46)
codes in Sections V,VI.
are constants independent of x. Thus the codespace
Nextletusdiscusshowtocomputematrixelementsof
Im(E) is stabilized by the products of Pauli Z operators
the simulator Hamiltonian Eq. (41). Note that all steps
over L and R,
in the definition of H are computationally efficient
sim
except for inverting the action of A, that is, computing
Z(L)|s(cid:105)=Z(R)|s(cid:105)=(−1)N|s(cid:105) for all |s(cid:105)∈Im(E).
the set A−1s defined in Eq. (29). Define a function
(47)
x (s)=arg min |x|. (43) BelowweusenotationsandterminologyofSectionIV.
min
x∈{0,1}M Consideratwo-bodyfermionicobservableO actingon
tgt
Ax=s a pair of modes α,β, see Eq. (30). It has a 2k−1-sparse
It returns an error x∈{0,1}M of minimum weight con- simulator defined by Eq. (39) where k = |Aα ⊕Aβ|. If
sistent with a given syndrome s∈{0,1}Q. Suppose A is Aα ∩ Aβ (cid:54)= ∅ then k ≤ |Aα| + |Aβ| − 2 ≤ 2c(A) − 2
aparitycheckmatrixofalinearcodewiththeminimum and thus the simulator has sparsity 22c(A)−3, as claimed.
distance 2N +1. It follows easily from the definitions FromnowonweassumethatAα∩Aβ =∅. Theassump-
that A−1s = {x (s)} if x (s) has weight N and tion Eq. (44) implies that X(Aα) anti-commutes with
min min
A−1s = ∅ otherwise. Thus it suffices to give an efficient Z(L) and Z(R) for all α. Therefore X(Aα ⊕ Aβ) =
algorithmforcomputingx (s). Thelatterisknownas X(Aα)X(Aβ) commutes with Z(L) and Z(R). We
min
a minimum weight decoding problem. Although in gen- shall modify the simulator O defined in Eq. (39) by
sim
eralthisproblemisNP-hard[35],therearespecialclasses multiplying some terms in Eq. (39) by (−1)NZ(L), or
ofLDPCcodesthatadmitalineartimedecoder[31,32]. (−1)NZ(R), or Z(L)Z(R). As was argued above, these
These codes have a non-zero encoding rate and relative operators commute with each term in Eq. (39) and have
distance, but they are not good in the sense of Eq. (42). trivial action on the codespace due to Eq. (47). Thus we
In Section VI we discuss a special class of LDPC codes themodifiedsimulatorO(cid:48) hasexactlythesameaction
sim
basedonhigh-girthbipartitegraphsthatcanbedecoded on the codespace as O , that is, O(cid:48) is a simulator of
sim sim
in time O(M3). Appendix C gives a simple algorithm O .
tgt
that computes the set A−1s for any N-injective matrix. Fix some pair of qubits i ∈ Aα ∩L and j ∈ Aα ∩R.
Althoughthisalgorithmisnotefficientasymptotically,it Multiply each term in Eq. (39) with t = 1 and t = 0
i j
can be implemented for small system sizes M ≤50. by (−1)NZ(L). Multiply each term in Eq. (39) with
9
t = 0 and t = 1 by (−1)NZ(R). Multiply each term for instance [36] and the references therein. In particu-
i j
in Eq. (39) with t = 1 and t = 1 by Z(L)Z(R). This lar, nearly maximal bipartite graphs with a given girth
i j
cancelstheactionofPauliZ(t)inEq.(39)onthechosen can be constructed by greedy algorithms [37, 38]. Such
pair of qubits i,j. Thus we can write algorithms start from an empty graph and sequentially
add random edges drawn from a suitable (time depen-
(cid:88)
O(cid:48) = X⊗kZ(t)⊗Γ(cid:48) (t) (48) dent) probability distribution. The process terminates
sim αβ
once there is no edge that can be added without reduc-
t∈{0,1}k
|t|even ing the girth below the specified value, see Refs. [37, 38]
ti=tj=0 for details. The data shown on Figure 1 was generated
usingthegreedyalgorithmofRef.[38]with103 trialsfor
for some new operators Γ(cid:48) (t) diagonal in the standard
αβ each pair Q,N and selecting the maximum graph with
basis. ThusO(cid:48) hassparsity2k−3 ≤22c(A)−3asclaimed
sim girth at least 2N +2. The number of edges M in the
inEq.(45). Weomitthederivationforotherobservables
maximum graph gives a lower bound on the number of
definedinEq.(7)sinceitfollowsthesamestepsasabove.
fermimodesthatcanbesimulatedforagivenpairQ,N,
see Figure 1.
Asasimpleexampleconsideragirth-6bipartitegraph
VI. GRAPH-BASED ENCODINGS
shown onFigure3. It has12 verticesand 16edges. This
graph encodes a system of M = 16 fermi modes with
Suppose G is a bipartite graph with Q vertices and M N =2particlesintoasystemofQ=12qubits. Onecan
edges. We assume that vertices of G are partitioned into generalize this example as follows. Start from a cycle of
twodisjointsubsetsL,RsuchthatonlyedgesbetweenL even length L such that L ≥ 2N +2 and connect each
and R are allowed. Let A be the incidence matrix of G. pair of vertices j and j+L/2 (mod L) by a chord. Each
Bydefinition,AhasQrows,M columns,andAi,α =1if chord contains N −1 vertices and N edges. This defines
a vertex i is an endpoint of an edge α. Otherwise Ai,α = abipartitegraphwithgirthg =2N+2similartotheone
0. Consider the encoding E|x(cid:105) = |Ax(cid:105). Since c(A) = 2 shownonFigure3. ThegraphhasM =L+NL/2edges
and A is bipartite, few-body fermionic observables have andQ=L+(N−1)L/2vertices. ConditionL≥2N+2
simulatorswithsparsityr2 =2andr4 =32,seeEq.(45). is equivalent to (1+N)(2+N)≤M. Thus
Suppose the number of qubits Q and the number of
particles N are fixed. What is the maximum value of M M
Q=M − for N ≤M1/2−O(1).
that can be achieved using encodings based on bipartite 2+N
graphs? First let us rephrase the N-injectivity condition
in terms of the girth of the graph G. Recall that a graph ThisencodingcaneliminateapproximatelyM/N qubits.
Ghasgirthg ifanyclosedloopinGhasatleastg edges. WeobservethatsomeentriesinthetableofFigure1can
WeclaimthatthematrixAisN-injectiveiffthegraph be obtained using the above construction. For example,
G has girth g ≥ 2N +2. Indeed, assume that A is not theencodingwithQ=20,N =3,M =25correspondto
N-injective. Then Ax = Ay for some pair of weight-N the graph of Figure 3 where the cycle has length L=10
stringsx(cid:54)=y. Letz =x⊕ysothatAz =0and|z|≤2N. and each chord contains N = 3 edges. Such graph has
WecanconsiderzasasubsetofedgesinG. FromAz =0 girth g =8 and M =25 edges.
one infers that z is a cycle, that is, each vertex has even
number of incident edges from z. However, each cycle
contains at least one closed loop. If z(cid:48) ⊆ z is such a
loop then |z(cid:48)| ≤ |z| ≤ 2N, that is, g ≤ 2N. Conversely,
assume that g ≤ 2N. Let z be any loop of length at
most 2N. Note that z must have even length since G
is bipartite. Choose any partition z = x⊕y such that
|x|=|y|=|z|/2 and x∩y =∅. Then Ax=Ay. Choose
any subset of edges u such that |u|=N−|z|/2 and such
that x,y,u are pairwise disjoint. This is always possible
since
FIG.3. Exampleofagirth-6bipartitegraphwith12vertices
and 16 edges. Each empty or filled circle is a vertex.
M −|x|−|y|=M −|z|≥M −2N ≥0.
Let x(cid:48) =x⊕u and y(cid:48) =y⊕u. Then x(cid:48) (cid:54)=y(cid:48), Ax(cid:48) =Ay(cid:48) IncontrasttoencodingsbasedongeneralLDPCcodes,
and |x(cid:48)| = |y(cid:48)| = N, that is, A is not N-injective. This graph-basedencodingsgivesimulatorHamiltonianswith
proves the claim. efficiently computable matrix elements. Indeed, suppose
TheaboveshowsthatmaximizingM forfixedN andQ x is a minimum weight solution of the equation Ax=s,
is equivalent to finding the largest bipartite graph with see Eq. (43). If A is an incidence matrix of a graph,
a fixed number of vertices Q and a girth g ≥ 2N +2. one can view x and s as subsets of edges and vertices re-
This problem has a long history in the graph theory, see spectively. Obviously, x is minimal if it consists of edge
10
disjointpathsconnectingpairsofverticesins. Moreover, Finally, the parity encoding [13] is obtained by choosing
x defines a perfect matching on the set s such that each A as a lower-triangular M ×M matrix,
matched pair of vertices in s is connected by a short-
(cid:26)
1 if i≥j
est path. Thus computing a minimum weight solution of A = (51)
i,j 0 otherwise
Ax = s is equivalent to (a) computing a shortest path
between each pair of vertices in s and (b) finding a min-
For example, choosing M =4 one gets
imum weight perfect matching of vertices of s. These
steps can be done in time O(M3) using the Dijkstra’s 1 0 0 0
algorithm to compute the shortest paths and Edmonds 1 1 0 0
A= .
blossom algorithm to find the minimum weight perfect 1 1 1 0
matching, see [39] for more details. 1 1 1 1
The main advantage of the binary tree encoding is that
anyfew-bodyfermionicobservableO definedinEq.(7)
tgt
VII. DICRETE Z2 SYMMETRIES – PARTICLE hasaqubitsimulatorOsim =EOtgtE†suchthatOsimisa
AND SPIN CONSERVATION Pauli-likeoperatoractingnon-triviallyonlyonO(logM)
qubits[11]. Incontrast,thestandardJordan-Wignerand
In the next two sections we discuss encodings based the parity encodings can map a few-body fermionic ob-
on the Jordan-Wigner transformation [10] and its recent servable to a Pauli-like operator acting on all M qubits,
generalizations [11, 13]. Such encodings are well suited see Ref. [13] and Appendix A for more details.
for the removal of qubits in the presence of discrete Z ConsideratargetHamiltonianEq.(1)thatdescribesa
2
symmetries such as those describing the fermionic par- molecule with M spin-orbitals. Accordingly, each fermi
ityconservation. RemovalofqubitsfortheH molecular mode α is a pair α = (i,ω), where i = 1,...,M/2 is a
2
Hamiltonian and a two-site Hubbard model were consid- spatial orbital and ω ∈ {↑,↓} is the spin orientation. It
ered in [15]. We generalize the approach and consider is well-known that molecular Hamiltonians based on the
a system of M fermi modes and assume that our target non-relativistic Schr¨odinger equation conserve the num-
Hilbert space is the full Fock space F . The simula- ber of particles with a fixed spin orientation [40]. Let
M
tor system consist of M qubits. We consider encodings us order the M modes such that the first (the last) M/2
E : F →(C2)⊗M such that modes describe orbitals with spin up (spin down). Then
M
E|x(cid:105)=|Ax(cid:105) for all x∈{0,1}M, (49) [Htgt,Nˆ↑]=[Htgt,Nˆ↓]=0, (52)
where
whereAissomeM×M invertiblebinarymatrixandAx
stands for the matrix-vector multiplication modulo two, Nˆ =M(cid:88)/2a†a and Nˆ = (cid:88)M a†a (53)
cf. Eqs. (26,27). The standard Jordan-Winger transfor- ↑ α α ↓ α α
mation is obtained by choosing A as the identity ma- α=1 α=M/2+1
trix, A = I. A binary tree and the parity encodings
are particle number operators for spin-up and spin-down
introduced in Refs. [11, 13] can be viewed as general-
modes. We claim that the symmetry Eq. (52) can be ex-
izations of the Jordan-Wigner transformation. A binary
ploited to remove two qubits from the simulator Hamil-
tree encoding is defined for M being a power of two,
tonian obtained via the binary tree encoding. For a spe-
M = 2m. If M (cid:54)= 2m, one refers to the definition for
cial case when M = 4 and H describes the hydrogen
a number of modes 2m > M, using only the correspon- tgt
molecule see Ref. [17]. Indeed, suppose M =2m and let
dences for the first M modes. The binary tree encoding
A=A bethematrixdefinedbyEq.(50). Wenotethat
is obtained by choosing A ≡ A , where a sequence of m
m the M-th row of A has a form 1M (the all-ones string).
matrices A ,A ,...,A is defined recursively [13] as
0 1 m Furthermore, the row M/2 has a form 1M/20M/2. It fol-
lows that
(cid:20) (cid:21) (cid:20) (cid:21)
1 0 A 0
A =1, A = , A = k−1 , (50)
0 1 1 1 k Bk−1 Ak−1 (cid:88)M
(Ax) = x (mod 2) (54)
M α
α=1
where 0 is the all-zeros matrix and B is a matrix of
k
size 2k ×2k that has the last row filled by ones and all and
remaining rows filled by zeros. For example, choosing
M/2
M =4 one gets (cid:88)
(Ax) = x (mod 2). (55)
M/2 α
1 0 0 0 α=1
1 1 0 0 Comparing Eqs. (53,54,55) one concludes that
A= .
0 0 1 0
1 1 1 1 E(−1)Nˆ↑E† =σz and E(−1)Nˆ↑+Nˆ↓E† =σz . (56)
M/2 M