Operator Locality in Quantum Simulation of Fermionic Models

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by  Vojtěch Havlíček

Operator Locality in Quantum Simulation of Fermionic Models
Vojtěch Havlíček
Institute for Theoretical Physics and Station Q Zurich, ETH Zurich, 8093 Zurich, Switzerland and
Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
Matthias Troyer
Institute for Theoretical Physics and Station Q Zurich, ETH Zurich, 8093 Zurich, Switzerland and
Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA
James D. Whitfield
Department of Physics and Astronomy, Dartmouth College,
7
6127 Wilder Laboratory, Hanover, NH 03755, USA
1
0 Simulating fermionic lattice models with qubits requires mapping fermionic degrees of freedom
2
to qubits. The simplest method for this task, the Jordan-Wigner transformation, yields strings of

n Pauli operators acting on an extensive number of qubits. This overhead can be a hindrance to
a implementation of qubit-based quantumsimulators, especially in the analog context. Here we thus
J reviewand analyzealternative fermion-to-qubitmappings, includingthetwoapproachesbyBravyi

4 andKitaevandtheAuxiliaryFermiontransformation. TheBravyi-Kitaevtransformisreformulated
2 in terms of a classical data structure and generalized to achieve a further locality improvement for
local fermionic models on a rectangular lattice. We conclude that the most compact encoding

] of the fermionic operators can be done using ancilla qubits with the Auxiliary Fermion scheme.
h
Without introducing ancillas, a variant of theBravyi-Kitaev transform provides the most compact
p
fermion-to-qubit mappingfor Hubbard-likemodels.
-
t
n
a I. INTRODUCTION systemshasrecentlybeenafocusofalgorithmicdevelop-
u
ments [6,10,12,13]. Besidesdirectsimulation[6],it has
q
been pointed out that quantum simulation of a strongly
[ Among the various applications of quantum comput-
correlated region can act as an impurity solver for dy-
ing,quantumsimulationhaslongstoodoutasaprimary
1 namical mean field theories [9, 10, 12]. In these recent
motivation [1, 2]. Classical computers can often perform
v
investigations,theauthorshavechosentousetheJordan-
2 rapid electronic structure calculations without explicit
Wigner fermionic encoding scheme [14–16] for their spe-
7 electron-electron interaction and obtain relatively accu-
cific simulations. However, under this transformation,
0 rate results [3]. However, systems where the electron-
7 electron interaction cannot be integrated out are called local fermionic operators become spin operators acting
0 on an extensive number of qubits, which may be prob-
strongly correlatedand representa new frontier for elec-
. lematic especially in the analog context. This can be
1 tronic structure in both theoretical chemistry [4] and
0 strongly correlated materials, such as high-temperature avoidedusing other encoding schemes andwe contribute
7 superconductors [5]. It is in this regime that quantum to the ongoing line of research by investigating various
1 fermion-to-qubit mappings.
simulation is a promising route forward [6–10].
:
v Quantumsimulationcomesintwodistinctflavors: dig- The Hubbard model has served as a paradigmatic ex-
i ital and analog, are each subject to different mindset ample for strongly correlated problems [5, 6, 8]. We
X
and constraints. In the digital context, the hardware is will continue this trend and use the Hubbard model as
ar thoughtofasauniversalquantumcomputerwhereanar- testbed for our ideas. Its Hamiltonian, on a graph with
bitraryquantumcircuitcanbe implementedandusedto edges E and vertices V, is given by:
approximatethesystemofinterest[2]. Sincehigh-quality
qubits are required in this context, the simulation qubit H =−t (a† a +a† a )+U n n , (1)
iσ jσ jσ iσ i↑ i↓
count can be thought of as an important constraint.
(i,Xj)∈EσX=↑,↓ iX∈V
An analog quantum simulator on the other hand ap-
proximatesthesystemwithanother,easiertoimplement,
where t and U are parameters of the model, n =
jσ
control and measure. Such a simulator or emulator is
a† a , and the fermion creation operators {a } satisfy
usually tailored to a specific problem and it is therefore jσ jσ iσ
argued to be technologically more viable to build such a†iσajτ +ajτa†iσ = δijδστ. We will consider t and U to
a chip rather than a generalpurpose quantum computer be fixed and assume, for now, that we are on a square
[8]. These analog simulators are typically restricted to lattice.
2-qubitcouplingsandalimited setofglobaloperations- The paper is organized into two parts. The first part
examples being both the trapped ions [11] or the super- reviews and extends mappings from fermionic Hamilto-
conducting qubits [8]. nians to qubits and the second part studies operator lo-
Quantum simulation of strongly correlated fermionic cality of the various fermion encoding methods.
2
II. MAPPING FERMIONIC HAMILTONIANS could therefore cancel, as is the case in 1D Hubbard
TO QUBITS model. Ina generalcasehowever(specifically fora Hub-
bard model on higher dimensional lattices), the hopping
The following section reviews and expands on a set of operator locality scales with the size of the lattice. We
localityimprovingtransformationsformappingfermionic therefore proceed by introducing an alternative scheme
Hamiltonians to qubits. Subsection IIA briefly summa- which improves locality of the resulting qubit Hamilto-
rizes the Jordan-Wigner transformation which will be nian.
used as a baseline for locality overhead comparison. In
subsectionIIB,wereviewthefirstmethodoriginallyout-
lined in Ref. [17] which has been referred to as Bravyi-
B. The Bravyi-Kitaev Transform
Kitaev transformation in the literature [18, 19]. We
reformulate the transform in terms of a classical data
The first of the two fermionic transformations intro-
structure (different perspective on the construction can
duced in [17], the Bravyi-Kitaev(BK) transform,can be
be found in [18, 19]). This allows for its generaliza-
described by a classical data structure, the Fenwick tree
tion outlined in subsection IIC. The generalized Bravyi-
[23], which we will introduce below. The BK transform
Kitaev transformation corresponds to a whole class of
has been previously reviewed in [18, 19] and formulated
fermion-to-spintransformations characterizedby a tran-
interms of recursiveprescriptionfor transformationma-
sition from linear to logarithmic operator locality. In
trices. This carriedanimplicit constraintonthe number
part IIE, we review the second method outlined in
of qubits being a power of 2. Our approach defines the
Ref. [17] and provide an example of 2D Hubbard model
scheme for an arbitrary number of qubits.
mapping. Lastly, subsection IIF reviews the Auxiliary
Fermionmethod introducedin [20, 21]with focus onop-
erator locality analysis. We have previously worked out
construction details of this transformation in Ref. [22]. 1. Fenwick Trees
Incontextofclassicalcomputation,aFenwicktreecan
A. Jordan-Wigner Transform
be used to map binary strings n n ...n , n ∈ {0,1}
0 1 N i
to binary strings x x ...x , x ∈{0,1} such that both
0 1 N i
The usual way to map fermionic operators to qubits
the prefix sum k−1 n and bit-flip operations have
is the Jordan-Wigner (JW) transformation [14]. This m=0 m
encoding stores information about the occupancy of N O(logN) acces(cid:16)sPcosts in(cid:17)the encoded representation.
fermionicsitesinN qubits. Thefermionraising/lowering This optimization is achieved by storing partial occu-
operatorsonk-thsite aremappedto qubitoperatorsby: pancy sums (xi) rather than occupancies/bits (ni) in
a way we now describe. The partial occupancy sums
k−1 k−1 xiaredictatedbythetreeconstructedusingAlgorithm1:
a 7→ Z |0ih1| , a† 7→ Z |1ih0| ,
k  j k k  j k
jY=0 jY=0 DefineFenwick(L,R):
    IF L6=R:
where Zj stands for an N-qubit operator corresponding Connect R to ⌊R+L⌋;
to a Pauli Z operator applied to the j-th qubit and 1 2
Fenwick(L,⌊R+L⌋);
to the rest of the qubit register. The above operators 2
Fenwick(⌊R+L⌋+1,R);
obey fermionic anti-commutationrelationsandtherefore 2
ELSE:
generate fermionic algebra on qubits. The
Terminate.
k−1 ALGORITHM1: Fenwick Tree Generation
Z =Z⊗ Z... Z ⊗1⊗...1,
j
jY=0 k N−k A tree generated by Fenwick(0, N − 1) has depth
string of Pauli Z ope|rator{szcoun}ts t|he {ezxcit}ation parity. d = ⌈log N⌉ and number of root-children equal to n =
2
The action of raising/loweringoperators canbe there- ⌊log N⌋. AnexampleofaFenwicktreeforN =7, d=3
2
fore thought of as a composition of two operations on is shown in Fig. 1. The partial sums x of the en-
j
qubit states: (1) counting the parity and (2) updating coded representationare givenby a (mod 2) sum of j-th
thefermionicsiteoccupancy. Thenumberofsingle-qubit fermionic occupancy n with the descendants of j in the
j
PauliZ operatorsusedforparitycountingscalesasymp- Fenwick tree. For example the zeroth bit encoded by a
totically as O(N), while the update is implemented with Fenwick tree in Fig. 1 stores only the occupancy of the
asinglequbitoperator|0ih1| oritsconjugate. Thecom- zeroth fermionic site as it has no descendants, while the
k
posite operation hence costs O(N) in operator locality. first bit stores x = n +x = n +n . Likewise, the
1 1 0 1 0
Thefermionicraising/loweringoperatorsoccuronlyin sixth bit has {3,5} as its children and therefore stores
pairs in any physical Hamiltonian. The Pauli strings x =n +(x +x )=n +n +n +n +n +n +n .
6 6 3 5 0 1 2 3 4 5 6
3
on the encoded states, as one needs to count the ex-
citation parity of 0 and 1 by applying Z , change the
1
0 1 2 3 4 5 6 occupancy of the second node by applying |1ih0| and
2
ensure consistency of the encoding by updating sites 3
n0 n1+ n2 n3+ n4 n5+ n6+ and 6 (ancestors of 2) with X3 and X6.
x0 x2+ x4 x5+ Mapping a† for a general j is more complicated, as
x x j
1 3
one has to condition application of |0ih1| or |1ih0| on
j j
FIG. 1: Fenwick treeof depth3 for N =7. The structure content of children of j in the Fenwick tree. This is the
can beconstructed bytaking thefirst nodeand making it caseforj =3inFig.1forexample. Ifthethirdfermionic
dependenton contents of thenode half way (rounded down) site is initially unoccupied (n3 =0), the raising operator
in the lattice and proceeding recursively for halves of the changes n from 0 to 1. In the encoded representation,
3
site array. The example hereis illustrated for N =7. Odd the third qubit stores x = n +n +n +n = (x +
3 0 1 2 3 1
N has been chosen in order to show a construction of the x )+n . So if (x +x ) = 1, a qubit lowering operator
2 3 1 2
mapping for N not being apower of 2, a restriction
|0ih1| should be applied in the encoded representation
implicitly imposed in [18]. Content of thewhite boxes 3
insteadof|1ih0| . Itfollowsthatonehastoconditionthis
corresponds to the information stored in each node. 3
operation on the children’s parity. The operator hence
maps to:
7 111
6 3 110 011 a†3 →−(|1ih1|1|0ih0|2+|0ih0|1|1ih1|2)|0ih1|3X6
5 1 101 001 +(|0ih0|1|0ih0|2+|1ih1|1|1ih1|2)|1ih0|3X6.
2 010
Theabovedescriptionconsiderablysimplifiesbywork-
4 0 100 000 ing in the Majorana basis:
FIG.2: Fenwick treeof depth3 for N =8. Fenwick treesfor c =a† +a →Z Z X X .
N =2d can be also described bya partial ordering on tree 3 3 3 1 2 3 6
node indices. Supposewe write theindices in binary as in
Consider now the set of children with indices less than j
the treeon theright. Then a bitstring with h>0 zeroes
ofallancestorsofj. WelabelthissetasC(j). Forexam-
labels a child of another bitstring with h−1 zeroes given by
ple,thesetofchildrenofallancestorsofqubit9inFig.3
flipping thelast 0 of thestring to 1. For example, 101, 011
isgivenby{7, 9, 10, 11, 13, 14}andoutofthis,only7is
and 110 are all children of 111. This construction manifests
apossibleconnection toalgebraic coding,aseverypathfrom less than 9 and hence C(9) = {7}. For consistency with
theroot toa leaf gives a Gray code[23,24]. Other refs.[18,19],wedenotethesetofchildrenofthej-thsite
definitions can befound, but working out examples is the byF(j)andworkwithasetP(j)=C(j)∪F(j). IfU(j)
fastest way to familiarize oneself with theconstruction. labels the set of all ancestors of j, then:
c =a +a† →Z X X , (2)
j j j P(j) j U(j)
The remaining bits are given by:
d =i a†−a →Z Y X =Z Y X ,
j j j P(j)/F(j) j U(j) C(j) j U(j)
x =n , x =n +x , x =n ,
0 0 1 1 0 2 2
(cid:16) (cid:17) (3)
x =n +x +x , x =n , x =n +n .
3 3 2 1 4 4 5 5 4
where Z implies Pauli Z operators applied to qubits
As a specific example, n n ...n = 0111010 is en- P(j)
0 1 6 in a set P(j).
coded as x x ...x =0111010.
0 1 6 Note that P(j) ∩ U(j) = ∅, since all nodes in U(j)
have indices greater than j while P(j) have all indices
less than j. Also note that the d operator acts trivially
2. Bravyi-Kitaev Transformation j
ontheF(j)qubits. Localityofthec Majoranaonqubits
j
is hence never better than d , since no operators are ap-
TheBKtransformusesFenwicktreestoimprovequbit j
pliedtochildrenofthej-thnode(Eq.3). Theworst-case
operator locality of the fermionic parity counting string
locality for the c Majorana operator is therefore given
to O(logN), while increasing fermionic occupancy up- j
by |U(j)∪P(j)|+1=|U(j)|+|P(j)|+1. In fact, for a
date cost to O(logN). The raising/lowering operators
Fenwick tree of N = 2d sites, the locality of c becomes
are hence mapped with O(logN) operator locality over- j
exactly log N +1 as we now show:
head, which is substantially better than O(N) for JW. 2
Starting with the simplest example, consider a† ap- Proof. Let d = 0. Then N = 1 and the c locality is
2 j
plied to the second fermionic site in a qubit register log 1+1 = 1 as there is only single node in the tree.
2
|x x ...x i encoding an occupancy state |n n ...n i Now suppose the locality is log N +1 for a tree with
0 1 6 0 1 6 2
of 7 fermionic sites as in Fig. 1. This operator acts as: N = 2d nodes. The 2N = 2d+1 tree is constructed by
connecting roots of two N trees - compare for example
a† →Z |1ih0| X X ,
2 1 2 3 6 the descendants of 7 to the tree of the remaining nodes
4
15 Every iteration of the algorithm defines new rep-
resentation of fermionic algebra with qubit operators.
14 7
13 With increasing recursion depth, the occupancy update
11 6 3
cost worsens, while the parity counting costs improve.
12 5 1 Asymptotically, the representation transitions from op-
10 9 2 erator locality overheads of O(1) → O(logN) for occu-
pancy update and O(N) → O(logN) for parity compu-
4 0
tation.
8
FIG. 3: An example of c Majorana operator mapped with
9
theBravyi-Kitaev method. The white colored nodes of the D. Segmented Bravyi-Kitaev (SBK) transform
Fenwick tree correspond to P(9)={7,8} and are thequbits
to which Pauli Z operators are applied. The black nodes are Itispossibletousethesegmentedtransformforopera-
in U(9)={11,15} to which X is applied. torlocalityimprovementofspecificfermionicHamiltoni-
ansonrectangularlattices. Asalreadydiscussed,theBK
transform optimizes update and lookup costs simultane-
in Fig. 10. Any node in the right subtree has to update ously. Composition of these operations describes the ac-
the new root, which worsens the locality by 1 - hence tionoffermionicraising/loweringoperators,butdoesnot
the terms are now log22N +1 local. Operators on the strictlycorrespondtooperatorsoccurringintheHubbard
restofthe tree willhaveto lookupanadditionalnode to model - or in fact any other physical fermionic Hamilto-
obtainthe parity,whichalsoimplies log 2N+1locality. nian. For closed systems, physical fermionic Hamiltoni-
2
So every cj on a tree with N = 2d nodes is log2N +1 ans only contain pairs of raising and lowering operators
local. [17],whichplacesfurtherconstraintsonthesetofopera-
torsweneedtomaptoqubits. Ifweadditionallyrestrict
We now provide a unifying framework for JW and ourattentiontooperatorswhicharelocal(asistheHub-
BK encodings and introduce an optimized variant of the bardmodel),thereisalotofredundancyonecanexploit
method suitable for rectangular lattice geometries. for further qubit operator locality optimization.
We focus on the case of a w×h, w ≤ h rectangular
lattice and build a Fenwick tree for every row - this is
C. FenwickTrees as a Class of FermionicEncodings a specific example of segmented Fenwick tree as defined
in the previous section. At first sight, this appears to
The recursive Algorithm 1 from the previous section worsen locality of the qubit operators as the locality of
gives rise to a class of fermionic encodings with JW and single raising/lowering operators now scales asymptoti-
BK schemes as limiting cases. Instead of using a single cally as O(hlogw). If we howeverrestrictthe setof pos-
Fenwick tree to encode allfermionic modes, we partition sible operations to on-site and nearest-neighbor terms,
the fermionic sites into Fenwick trees of varying depth the single-qubit operators on segmented tree roots can-
and include the set of all roots of segmented trees less celand the locality becomes O(logw) - a substantialim-
than j to P(j). The definition of P(j) in this context provement, as the operator locality is now independent
becomes: of the lattice height.
It turns out to be slightly more optimal to store two
P(j)=F(j)∪C(j)∪{set of all roots i, i<j}. disconnected trees per lattice row, as shown in Fig. 5.
Thisisbecausethe fullparityoftherowisnotnecessary
Inparticular,wecouldchoosetoputeachnodeinitsown for the vertical nor horizontal hopping terms.
Fenwicktreeofdepth0,whichwouldcorresponddirectly
to the JW transformation (Fig. 4).
E. Loop-Stabilized Fermion Simulation (LSFS)
0 1 2 3 4 5 6 7
We now shift our attention to the second method in-
troduced in Ref. [17] which finds an alternative repre-
sentation of the local fermionic Hamiltonians on qubits
0 1 2 3 4 5 6 7 in a line graph (i.e. on graph edges - see Fig. 6).
The method, referred to as a “Superfast Simulation of
Fermions” in [17] improves qubit operator locality to a
0 1 2 3 4 5 6 7 constant for any bounded-degree graph. The newly pro-
posedname“Loop-StabilizedFermionSimulation” ismo-
FIG.4: Recursion stepsoftheFenwicktreealgorithm shown tivated by the fermionic algebra being represented in a
for N =8. Thebottom case of corresponds tothe JW
subspace defined by a stabilizer condition on the set of
transformation.
all possible loops in the line graph.
5
0 1 2 3 0 1 2 3
70
y
t
ali 60 4 5 6 7 4 5 X 6 Z 7
c
o
g L 50 Y Y
n 8 9 10 11 8 Z 9 X 10 11
pi 40
p
o
al H 30 12 13 14 15 12 13 14 15
c
ti 20 FIG. 6: (left) Fermionic site ordering used with theBK
r
e
V scheme. Qubits( ) are on edges of the lattice. (right) A
10
0 10 20 30 40 50 60 70 stabilizer on (5,9,6,10) plaquette. There are 9 such
Tree width stabilizers for4×4latticecorrespondingtothe9plaquettes.
If a state is a simultaneous eigenstate of all stabilizers, it
encodes a physical fermionic state. Ifa qubit operator is to
FIG. 5: Locality of vertical hoppingterms as a function of
beapplied to theedge outside thelattice, ignore it.
thesegment tree size for W =64. The optimal tree size is
W/2. The hopping term localities of segments of W =64
and W =32 sizes are given by14 and 13 in this lattice.
as an additional rule to the above set. This motivates
the following choice for the qubit operator B˜ :
k
1. Loop-Stabilized Fermionic Simulation
B˜ = Z ,
k (jk)
As discussed in the construction of the SBK method,
j∈n(k)
Y
physicalfermionic Hamiltoniansaresumsorproductsof
fermionic raising/lowering operator pairs. Equivalently, where n(k) is the set of nearest neighbors of k. The
allfermionicHamiltonianscanbeobtainedbycombining B˜ = 1 condition is then trivially satisfied for
k∈V k
the following operators [17]: theoperatorsinceeachedgesharespreciselytwovertices
Q
(this is colloquiallyknownas the “handshakinglemma”).
B =1−2a†a for a vertex k,
k k k We now derive the form of A˜ . Firstly assume that
(jk)
A(jk) =−i(aj +a†j)(ak+a†k) for an edge (j,k). A˜(jk) is a tensor productof Paulioperatorsand/oriden-
tity on edges adjacent to vertices j, k. In order to sat-
Here, the subscript on B corresponds to a vertex and
the subscript on A labels a graph edge. Specifically, the isfy A(jk)Bl = (−1)δjl+δklBlA(jk), we first address the
case for which k 6= l, j 6= l, so that the fermionic oper-
fermionic site to site hopping is expressed by:
ators commute. If A˜ and B˜ are to obey the same rela-
a†a +a†a =−i A B +B A /2. tion,allsinglequbitoperatorsofA(jk) onedgesadjacent
k j j k (jk) k j (jk) to j, k, except for the edge (j,k), must be Hermitian
More formally, the A (cid:0), B operators gene(cid:1)rate the al- and identity-squaring operators in a subspace spanned
(jk) j
gebra of physical fermionic Hamiltonians. The algebra by {1,Z}. There are only two options - either Z or 1.
defining rules are [17]: Focusing on the case l ∈ {j,k}, the A(jk)Bl operators
anti-commute, implying that the operator on the (j,k)
A(jk)Bl =(−1)δjl+δklBlA(jk), [Bk,Bl]=0, edgequbitisaHermitianandidentity-squaringoperator
A A =(−1)δjl+δjs+δkl+δksA A , in the subspace spanned by {X,Y} - again X or Y are
(jk) (ls) (ls) (jk)
the only possibilities.
In other words, A anticommutes with any other gen- Itremainstosatisfytheconditionimposedbythegen-
(jk)
eratoraslongastheysharepreciselyonevertex. Further- eralizedcommutatoroftwoA ,A operators. These
(jk) (lm)
more note that B† = B , A† = A , B2 = A2 = 1 anticommute iff they share a vertex. An example of
and A(ij) =−A(jik). Addkition(iaj)lly: (ij) k (ij) qubit operators A˜(jk) satisfying this and the previous
constraints on a square lattice is given in Fig. 7. An
(i)pA A ...A =1, (4) example of operators satisfying these constraints on a
(j0j1) (j1j2) (jpj0)
general lattice is given by:
for any closed path j j j ... j .
0 1 2 p
Bravyi and Kitaev found in [17] qubit operators
n(j) n(k)
B˜l, A˜(jk) obeying the above algebra, subject to further A˜(jk) ∝X(jk) Z(lj) Z(sk),
constraints on excitation parity sector that we now im-
l<k s<j
Y Y
pose. For an even number N of fermions, one has that:
f
andisaspecificcaseoftheprescriptionfoundinRef.[17].
Bk = (1k−2a†kak)=(−1)Nf1=1, Lastly, we impose anticommutation of A(jk) using anti-
k∈V k∈V symmetric tensor ǫ which is +1 when j > k and −1
Y Y jk
6
0 1 2 3 0 1 2 3 for any plaquette (αβγδ) and an arbitrary closed loop
Z of Eq. 4 can be obtained by taking product of such
plaquette operators, encoded physical fermionic states
4 5 6 7 4 5 Z 6 Z 7
correspond to qubit states which are +1 eigenstates of
Z Z X A˜ A˜ A˜ A˜ . Formally, we restrict the qubit
(αβ) (βγ) (γδ) (δα)
8 Z 9 X 10 11 8 9 10 11 states by a set of stabilizer operators:
C = A˜ A˜ A˜ A˜ ,
12 13 14 15 12 13 14 15 (αβγδ) (αβ) (βγ) (γδ) (δα)
FIG. 7: Example of LSFS generators A and A . where (αβγδ) labels vertices of the plaquette. An exam-
(9,10) (6,10)
pleofsuchstabilizerisworkedoutinFigure6foratrivial
lattice ordering. Since C is a Kröneckerproduct of
(αβγδ)
when j <k. Thus, Paulimatrices,itseigenvaluesare±1,implyingthatvio-
lation of a single stabilizer condition costs 2 units. Note
n(j) n(k)
that if the simulation starts in the subspace encoding
A˜ =ǫ X Z Z .
(jk) jk (jk) (lj) (sk) physicalfermionic states, it stays in it, as the generating
l<k s<j
Y Y operators B , A commute with the stabilizers [17].
k (jk)
Even though this implies the generator itself is multi- Inspired by the toric code construction [25], we note
ply defined for each edge, we will see that all interesting thatonecanrestrictthesystemtothephysicalcodespace
physical operators will be independent of ǫ . by including a penalty term H = −∆ C ,
jk penalty 2 k k
The loop condition of Eq. 4 will be imposed by a set where ∆ ≫ t,U,ǫ corresponds to the “energy gap” of
of stabilizer operators which can be concisely presented the system and k runs over all plaquettes. P
with a specific lattice geometry in mind. We therefore The above representation is now applied to the
postponeitsdiscussionuntilafterthe followingexample. fermionic nearest-neighbor hopping operator. The
nearest-neighbor couplings for horizontal edges maps to
5-qubit-local operators:
2. Example: 2D Hubbard Model
1
a† a +a†a → Y→(Z↓Z↑ −Z↑Z←Z→ Z↓ ).
We illustrate this method by mapping the Hubbard k+1 k k k+1 2 k k k+1 k k k+1 k+1
Hamiltonian on rectangular lattice to qubits. Let Zk↑ Analogously, the vertical nearest-neighbor couplings are
denote a Pauli Z operator applied to the qubit on the encoded by 7-local operators:
verticaledgeadjacenttothevertexk -ifthereisnosuch
edge, substitute the term with an identity operator (for a†a +a†a → 1Y↑ Z←Z→Z↑Z←Z→Z↓−1 .
example, Z↑ = 1 in the diagram in Fig. 6). Operators j k k j 2 j k k k j j j
1 (cid:16) (cid:17)
onotheradjacentedgesaredefinedanalogouslyby using
It remainsto accountfor the Hubbard repulsionterm.
{→,←,↑,↓} superscripts.
The simplest way to implement it is by simulating the
The B˜ operator is represented with a “cross” of Pauli
k above on two lattices labeled with ↑,↓, coupled by the
Z operators:
density-density interaction term. We obtain that:
B˜ = Z =Z←Z↑Z→Z↓, for a vertexk.
k jk k k k k H =−t (a† a +a† a )+ǫ a† a ,
j∈Yn(k) ↑ i↑ j↑ j↑ i↑ i↑ i↑
(i,Xj)∈E Xi∈V
The form of A˜(jk) differs for horizontal and vertical H =−t (a† a +a† a )+ǫ a† a ,
edges, which we denote by E and E respectively. It ↓ i↓ j↓ j↓ i↓ i↓ i↓
H V
also depends on a specific lattice indexing - we choose (i,Xj)∈E Xi∈V
the simplest one shown in Fig. 6 and leave it an open whichleadstothefollowingexpressionforthefullHamil-
question whether a better ordering exists. The operator tonian:
A˜ is then given by:
(jk)
H =H +H +U n n .
↑ ↓ i↑ i↓
ǫ X Z←Z↑Z→ for(j,k) ∈ E ,
A˜(jk) =(ǫjjkkXjjkkZjj↑Zk↑jZj←j for(j,k) ∈ EHV . The H and H Hamiltonian teiX∈rmVs are decoupled and
↑ ↓
It remains to satisfy the loop condition of Eq. 4, since havebeen already mapped to spins. The density-density
in order to represent the fermionic algebra, the qubit interaction term maps to:
operators must obey the same relation. Because the
1
fermionic operators obey: nk↑nk↓ → 4(1−Zk←Zk↑Zk→Zk↓)(1−Zk←′Zk↑′Zk→′Zk↓′),
A A A A =(−i)4c c c c c c c c =1,
(αβ) (βγ) (γδ) (δα) α β β γ γ δ δ α
where the primed indices correspond to fermions in spin
c = a +a† ,
α α α ↓ lattice and the unprimed ones to the sites in the spin
(cid:0) (cid:1)
7
2 3 8 9 14 15 0 3 6 9 12 15 2 3 2 1 2 2 1 1 0
1 4 7 10 13 16 1 4 7 10 13 16 3 4 3 2 2 2 1 1 1
0 5 6 11 12 17 2 5 8 11 14 17 2 3 2 2 2 1 0 1 1
FIG. 8: (left) Snakepattern used for G in aw=l=3 in FIG. 9: (left) Degrees of fermionic sites in the3×3 2D
1
theauxiliary fermion method. (right) Optimal JW ordering Hubbardlattice. (middle) Degrees along the linear path G
1
for geometrically local fermionic models. through each sublattice (right). The nonlocal degree of each
nodedetermines thenumberof auxiliary modes to beused.
↑ lattice. It hence follows that in order to implement
this inter-lattice coupling on spins, the density-density no auxiliarysites since the edge setofG is sufficient for
1
Hamiltonianterm acts on8 qubits simultaneouslyin the coupling them to their nearest neighbors.
worst case. This presents an upper bound on the spin Asanexample,weillustratetheseideasusingasquare
Hamiltonian locality in this setting. lattice of width w = 3 and height h = 3. In the general
case,thereare(w−2)(h−2)siteswithdegreefourinthe
interiorofthe graphG. Eachofthese siteshasnon-local
F. The Auxiliary Fermion Scheme degree two and hence each needs just a single auxiliary
fermionic site. Then there are 2(h−2)+2(w−2) sites
The auxiliary fermion (AF) scheme [20–22] was in- alongtheboundarywithdegreethree. Eachofthesesites
troduced in [20] and also (independently) by [21]. The havenon-localdegreeof1andwillalsorequireoneauxil-
schemeusesauxiliaryfermionicmodestoallowfermionic iaryfermionicsite each. Finally,the foursitesoccupying
models on general lattice geometries to be simulated lo- the corners of the lattice to be simulated have degree
cally. We recall the relevant details from [22] to al- two. Presuming that the snake-like pattern is used, two
low comparison against the other fermionic encoding of the corners will have non-local degree zero while the
schemes. othertwoeachhavenon-localdegree1. Summing allthe
Agivensetofsite-to-sitehoppingtermscanbecharac- cost together we find that the total number of qubits is
terizedbyagraphG=(E, V),whereverticescorrespond 4wh−4.
tositesandtheedgestopairsofsitesparticipatinginthe Thelocalityofsimulationoperatorswasdetailedinour
hopping. In one dimension, this graph is a linear path, previous publication [22]. For a hopping term from site
G ,wherethedegreeoftheendpointsisoneandallother i to a non-consecutive site j, the qubit operator acts on
1
siteshavedegreetwo. Inhigherdimensionalsettingssuch site i and on site j as well as auxiliary modes associated
asthe 2DHubbardmodelstudiedhere,thedegreeofthe with site j and with site i. Hence, the qubit operator
sites may be greater than two. The non-local degree d is either 4-local or 3-local in the Hubbard model. The
nl
of a site is the number of edges that are not included in density-density operators are 2-local [22].
the linear path G . Therefore, it is important to choose
1
a path which overlaps maximally with the desired in-
teraction graph G. In the case of 2D Hubbard, this is III. COMPARISON
accomplished by a snake-like pattern (Fig. 8). Next, we
must account for spin in the Hubbard model. One re-
In the following we assess the various methods in con-
quires2whspinlessfermionicsitesifthemodelisdefined
text of the 2D Hubbard Hamiltonian. We analyze local-
on w×h rectangular lattice. If we take the first half of
ity of the nearest-neighbor fermionic hopping and den-
thesesitestobespindownandthesecondhalftobespin
sityoperators. Thefirstsubsectiongoesthroughlocality
up, there is no need to track phase factors between the
analysisfor 2D Hubbardmodel. We then presentresults
spinupandspindownsubsets. ThisisbecausetheHub-
in higher dimensions.
bard model preserves spin; thus, there are no hopping
terms between the first half of the sites and the second
half. The density-density terms are each a product of
A. 2D Hubbard Model
two one-pointcoupling terms which does not require the
tracking of antisymmetric phase factors.
The nonlocal degree of each node (when G is a sub- WeconsideraHubbardmodeldefinedonaw×h, w <
1
graphofG)isd =d(G)−d(G )and,aswe’vepreviously h rectangular lattice. For JW, we order the fermions as
nl 1
shown, each auxiliary fermionic site can facilitate up to in Fig. 8. The longest string of Z operators introduced
twonon-localcouplings [22]. Hence the number ofauxil- by the mapping has then length (w+1).
iary sites required per fermionic site is given by ⌈d /2⌉. For LSFS, we have already shown that the density-
nl
Since each non-local degree in the 2D Hubbard model is densitytermsintheHubbardHamiltonianareall8-local
less than or equal to two, only a single auxiliary mode is for lattices with w, h≥3. It is therefore only sensible to
needed for each site with non-local degree greater than use LSFS for locality reduction if w ≥ 8. Likewise AF
zero. The(1,w)and(h,1)corners(foreachspin)require only improves locality compared to JW if w ≥ 4, since
8
15 depth) and correspondsto hopping between two deepest
leafnodes. Thiscanbereducedto2⌈log w⌉+1=2d+1
14 7 2
13 by splitting the segmented trees in half, which however
11 6 3
increases the worst case locality for horizontal hoppings
12 5 1 to 2⌈log w⌉. The scheme hence provides locality advan-
2
10 9 2 tage compared to JW for lattices of width w > 2. The
worst case localities for 2D are sumarized in Tab. I and
4 0
in Fig. 11.
8
FIG. 10: The worst case for hopping operator locality, if
16
all-to-all couplings are allowed is thehopping from thefirst
to thelast node. In thediagram above, this corresponds to 14 LSFS
0↔15 - thenodes with raising/lowering operators are AUX
colored white. The set F(15) of all children of node 15 12 JW
corresponds to theblack nodes. Theupdateset U(0) of 0 is y 10 SBK
theset of squarenodes. This implies that XZ =iY will be alit 8
applied at node7. c
o
L 6
4
the most nonlocal term of the qubit Hamiltonian is the
4-local vertical hopping term. 2
The AF transform is hence superior to LSFS for the
0
Hubbard model defined on a rectangular lattice. Both 5 10 15 20
methodsusemorequbitsthanfermionicsitesoftheorig- Lattice width
inalmodel. Foraw×hrectangularlattice,LSFSrequires
4(w−1)(h−1)qubits,whiletheauxiliaryfermionmethod
FIG. 11: Worst case locality of qubit operators with lattice
needs 4wh−4, compared to a minimum of 2wh. size in the2D Hubbardmodel.
We now compare these methods to the BK transform
and its optimized SBK variant introduced in subsec-
tion IID. The density operator can be written in Ma-
jorana basis and converted by BK transform as:
B. Hypercubic Lattices
1+ic d 1
n = j j → 1−Z . (5)
j 2 2 F(j)∪{j} We now extendthe previous analysisto higher dimen-
Its locality therefore only dep(cid:0)ends on |F(j(cid:1))|, i.e. the sionsandconsidermappingofthehoppingterminagen-
number of children of a node j in the Fenwick tree. eral hypercubic lattice of D >0 dimensions with side w.
The root has the largest number of children, which im- Withthe simplestordering,the hoppingterminJordan-
plies that the worst-case locality is (⌊log N⌋+1). The Wigner transformation becomes wd−1 + 1 local, where
2
density-density term is hence (2⌊log N⌋+2)-local. w is the number of sites along a side of the hypercubic
2
Using the expression for Majorana operators in the lattice.
Bravyi-Kitaevmapping (Eq. 3), we haveforthe hopping In the case of AF, the number of auxiliaries per site
operator that: scalesasD−1,sinceinD-dimensionalhypercubiclattice,
eachvertexhas2D nearestneighbors. This translatesto
i
a†kaj +a†jak = 2(ckdj +cjdk) . (6) sdintles=p2eDr e−a2chnosint-eloocfatlhdeegorreigeiannadl lDat−tic1ea.uTxhileiawryofresrtmcaiosne
Ifall-to-allcouplingswereallowed,theworst-caselocality hopping term locality hence goes as 2D−2, whereas the
would be ⌊log N⌋+⌈log N⌉ - the sum of tree depth d number of qubits scales as D×wD.
2 2
and the number of root children n. This corresponds to Since each vertex has 2D neighbors, locality of the
hopping from the first to the last site of the lattice (see LSFS hopping term goes as 4D−1 in a bulk of the hy-
Fig. 10). percube (one neighbor is shared), while locality of the
Focusingourattentiontothenearest-neighborhopping density-density term becomes 4D. The number of edges
intheSBKtransformation,wehavetoanalyzetheverti- in a hypercubic lattice of side w in dimension D is given
cal and horizontal hoppings independently. For hopping by E(D,w) =D(w−1)wD−1, which is also the number
along horizontal edges, a loose bound on worst case lo- of qubits required for the mapping.
cality is givenby 2⌈log w⌉−1=2d−1andcorresponds
2
to hopping to the root child with smallest index j from Proof. One has that E(2,w) = 2(w−1)w, as there are
the (j+1)-th node. 2 edges per vertex with the exception of the bound-
The worstcase locality for hopping along the verticals ary, where we over-count by 2w. In 3 dimensions, we
is given by 2⌈log w⌉+2 = 2d+2 (where d is the tree construct a w × w × w cube by connecting vertices of
2
9
Method Density-density Horizontal Vertical Qubits
JW 2 2 w+1 2wh
BK 2⌊log2(wh)⌋+2 ⌊log2(wh)⌋+⌈log2(wh)⌉ ⌊log2(wh)⌋+⌈log2(wh)⌉ 2wh
SBK 2⌊log2w⌋+2 ⌊log2w⌋+⌈log2w⌉ 2⌊log2w⌋+1 2wh
AF 2 2 4 4(wh−1)
LSFS 8 7 7 4wh−2h−2w
TABLE I:Operator locality and qubit resource overheads for transformation of a 2D rectangular lattice Hubbardmodel.
Method Worst-caselocality Qubits N. Onsuchdevicesthe mainlimitationwillbethenum-
JW wD−1+1 2wD ber of available logical qubits and then our proposed
BK 2⌊log2wD⌋ 2wD modificationofthe Bravyi-Kitaevtransformleads to the
SABFK 2⌊log2w2DD−1⌋+1 22DwwDD best locality improvement of O(log2w)-local spin opera-
LSFS 4D 2D(w−1)wD−1 tors, where w is the lattice width.
Foranalogsimulations,ontheotherhand,operatorlo-
TABLE II: Worst-case locality of thehopping term as a cality will be the decisive factor. On such quantum sim-
function of thehypercubiclattice dimension. ulators natively only few-qubit couplings are available,
typically only two-qubit terms. In that setting multi-
qubittermshavetobegeneratedusingperturbativegad-
w × w squares by (w − 1)w2 edges. It follows that gets[26,27]. These,however,requirelargeenergypenal-
E(3,w) = wE(2,w) + w2(w − 1) = 4w2(w − 1). To tiestobe sufficientlydeepinsidetheperturbativeregime
count the number of edges in a w×D hypercube, one wheretheeffectivehigher-orderinteractionsappear. One
can take w (D − 1)-dimensional hypercubes and con- hencewantstooptimizethelocalityofthetermsandaim
nect them by (w − 1)wD−1 edges. This implies that for transformations with the most local terms. The pre-
E(D,w)=wE(D−1,w)+wD−1(w−1). It follows that sented analysis shows that it is possible to map the 2D
E(D,w)=D(w−1)wD−1. Hubbardtoa4-localqubitHamiltonianbytheAuxiliary
Fermion method, at the expense of using number of an-
The BK method requires N = wD qubits in gen- cillaryqubits. Thisistheoptimalfermionrepresentation
eral, while its worst-case locality scales as ⌈log (wD)⌉+ for 2D lattices of width w ≥4.
2
⌊log (wD)⌋=d+n. Fornearestneighbourhopping,this
2
can be further optimized to 2⌈log (wD−1)⌉+1=2d+1
2
by using the segmentation trick of the SBK method at V. ACKNOWLEDGEMENTS
the highest level of the lattice. Hopping term locality as
afunctionofhypercubiclatticedimensionissummarized We would like to thank to Peter D. Johnson, Peter
in Tab. II. Winkler, Amit Chakrabarti, Thomas H. Cormen and
Alexey Soluyanov for useful discussions. V. H. would
like to thank Dartmouth College for support and hos-
IV. CONCLUSION pitality while finishing the work and the Clarendon and
Keble de Breyne scholarships for support. This project
Whichschemeissuperiordependsonthetypeofquan- wassupportedbytheSwissNationalScienceFoundation
tum simulation. For digital quantum simulation on a through the National Competence Center in Research
universalquantumcomputerwithlogicalqubits,N-qubit NCCR QSIT and by the European Research Council
operators can be simulated with just linear overhead in through ERC Advanced Grant SIMCOFE.
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