Nonperturbative Gadget for Topological Quantum Codes
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Citation
Ocko, Samuel, and Beni Yoshida. “Nonperturbative Gadget for
Topological Quantum Codes.” Physical Review Letters 107.25
(2011): n. pag. Web. 2 Mar. 2012. © 2011 American Physical
Society
As Published
http://dx.doi.org/10.1103/PhysRevLett.107.250502
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American Physical Society (APS)
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Final published version
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Thu May 26 20:55:36 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/69580
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Detailed Terms
PRL 107, 250502 (2011)
week ending
16 DECEMBER 2011
PHYSICAL REVIEW LETTERS
Nonperturbative Gadget for Topological Quantum Codes
Samuel A. Ocko1 and Beni Yoshida2
1
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 21 July 2011; published 14 December 2011)
2
Many-body entangled systems, in particular topologically ordered spin systems proposed as resources
for quantum information processing tasks, often involve highly nonlocal interaction terms. While one may
approximate such systems through two-body interactions perturbatively, these approaches have a number
of drawbacks in practice. In this Letter, we propose a scheme to simulate many-body spin Hamiltonians
with two-body Hamiltonians nonperturbatively. Unlike previous approaches, our Hamiltonians are not
only exactly solvable with exact ground state degeneracy, but also support completely localized
quasiparticle excitations, which are ideal for quantum information processing tasks. Our construction
is limited to simulating the toric code and quantum double models, but generalizations to other nonlocal
spin Hamiltonians may be possible.
DOI: 10.1103/PhysRevLett.107.250502
PACS numbers: 03.67.Ac, 03.67.Pp, 05.30.Pr, 05.50.+q
Many-body entanglement arising in strongly correlated
systems is a very promising resource for realizing various
ideas in quantum information, such as quantum communication and quantum computation. In particular, topologically ordered spin systems can be employed for reliable
storage of quantum information inside the degenerate
ground space [1] and for fault-tolerant quantum computation with non-Abelian anyonic excitations [2]. These topological approaches may resolve many problems in
quantum information science; qubits are encoded in
many-body entangled states and are thus naturally protected from decoherence.
Unfortunately, topologically ordered spin systems
capable of quantum information processing are very difficult to realize physically. Many proposed topologically
ordered spin Hamiltonians, such as the toric code, quantum
double model [2], and string-net model [3], involve
highly nonlocal interaction terms; this is a stark contrast
to Hamiltonians which occur in nature, which have only
geometrically local two-body interactions. Moreover, the
resource systems above are known not to be supported by
any two-body Hamiltonian [4].
Many efforts have been made to construct two-body
Hamiltonians which ‘‘approximate’’ nonlocal resource
Hamiltonians. The most commonly used approach is to
approximate target Hamiltonians through so-called ‘‘perturbative gadgets’’ [5–10]. The central idea of perturbative
gadgets is to design a two-body Hamiltonian whose
leading perturbative contribution gives rise to the desired
many-body Hamiltonian; unfortunately, most obtained
two-body Hamiltonians are not exactly solvable, and
their properties are hard to determine except for a few
exactly solvable examples [11,12]. In addition, the
perturbative Hamiltonian only approximates the target
Hamiltonian, and may give a very weak effective
Hamiltonian with a rather small energy gap. Furthermore, quasiparticle excitations (energy eigenstates) arising
0031-9007=11=107(25)=250502(5)
in perturbative Hamiltonians cannot be created through
completely localized manipulations of spins; excitations are always delocalized and the ground state degeneracy might be split for finite system sizes, resulting in
fatal errors in practice. While a nonperturbative approach based on the PEPS formalism was developed for
simulating the cluster state for measurement-based
quantum computation [13], such an approach may
not be applicable to degenerate systems with topological
order.
Here, we propose a scheme to simulate topologically
ordered Hamiltonians through two-body interactions nonperturbatively. Our scheme builds on previously established ideas in perturbative gadget studies, such as the
use of hopping particles proposed by König [8], and the
encoding of single particles into multiple particles used by
Brell et al. [10]. Combining these remarkable insights, we
are able to construct the first topologically ordered spin
system which satisfies the following. (1) The Hamiltonian
has at most two-body, geometrically local interactions.
(2) The Hamiltonian has exactly solvable ground states
and low-energy excitations, and is provably gapped for all
system sizes. (3) The ground space of the Hamiltonian is
exactly connected to that of the target Hamiltonian through
local unitary transformations, and anyonic excitations are
completely localized.
For clarity of presentation, we illustrate the gadget construction for the toric code. A generalization to the quantum double model is also possible [14].
Modified toric code.—We begin by defining a modified
version of the toric code, also known as the Z2 lattice gauge
model, that we will simulate through a two-body
Hamiltonian. Consider a system of qubits defined on edges
of a square lattice with periodic boundary conditions.
Unlike the conventional toric code, two qubits live on
each edge in our construction [see Fig. 1(a)], governed
by the following Hamiltonian:
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Ó 2011 American Physical Society
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PHYSICAL REVIEW LETTERS
PRL 107, 250502 (2011)
Gadget Hamiltonian.—The central idea behind our construction is to add a ‘‘gadget particle’’ at each star [see
Fig. 2(a)]. The gadget particle has four possible spin values
ms ¼ 0, 1, 2, 3. We replace the four-body star term As with
two-body terms Hhop and Hshield which involve the gadget
particles:
Hgadget ¼ Hp þ He þ Hhop þ Hshield
X
Hp ¼ J Bp ;
p
X
He ¼ J Ce :
e
The hopping term is Hhop ¼
s
p
Bp ¼
Y
j2p
Zj ;
e
Ce ¼
Y
Y
As ¼ Xj ;
j2s
Zj ;
j2e
where s, p, and e represent ‘‘star,’’ ‘‘plaquette,’’ and
‘‘edge,’’ respectively, as defined in Figs. 1(b)–1(d). Xj
and Zj are Pauli X and Z operators on qubit j, and J is
some positive constant. The model is exactly solvable since
interaction terms commute with each other, and it can be
considered to be a stabilizer code. The model has four
degenerate ground states, as in the toric code. Inside of
the ground space, As ¼ Bp ¼ Ce ¼ 1, meaning As j c i ¼
Bp j c i ¼ Ce j c i ¼ j c i for all s, p, and e when j c i is a
ground state. Notice that one can create the toric code from
this model by applying controlled-NOT gates between pairs
of qubits on each edge. Since the toric code and the
modified model are connected through local unitary transformations, they are considered to be in the same quantum
phase [15,16]. The ground space of the modified toric code
has a fourfold degeneracy, as seen by writing down two
pairs of ‘‘logical operators’’ which commute with the
Hamiltonian but anticommute with each other [see
Fig. 1(e)]. The nonlocality of logical operators makes the
model of great interest for robust storage of quantum
information.
As a first step towards obtaining a two-body
Hamiltonian simulating this modified toric code, we group
the four qubits in each plaquette into a single composite
particle with a sixteen-dimensional space. While Bp becomes one body, and Ce is two body through this grouping,
the star term As is only reduced to four body. Below, we
provide a scheme to simulate As through only two-body
terms.
s hs
where
hs ¼ Ujms ¼ 0ihms ¼ 0j tðDys þ Ds Þ
X
jms þ 1ihms j As ðms Þ
ðmod 4Þ;
Dys ¼
FIG. 1 (color online). (a) Construction of the modified toric
code. Dots represent qubits. (b) A star term As (red online). (c) A
plaquette term Bp (blue online). (d) An edge term Ce (green
online). (e) Two pairs of logical operators.
X
X
X
H ¼ J As J Bp J Ce ;
P
(1)
ms ¼0;1;2;3
where U and t are some positive constants, and ms represents the spin value of the gadget particle at s. Terms As ðmÞ
are products of two Pauli X operators as depicted in
Fig. 2(b). Since As ðmÞ are one-body operators when qubits
in a plaquette are viewed as a composite particle, hopping
terms are two body. These hopping terms will effectively
induce star terms As since As ¼ ðDys Þ4 . The shielding term
Hshield consists of two-body interactions between gadget
particles:
X
Hshield ¼ J T‘ ðms ÞTr ðmsþx^ Þ þ Td ðms ÞTu ðmsþy^ Þ;
s
where s þ x^ and s þ y^ are unit translations of a star s in the
horizontal and vertical directions, and
T‘ ðmÞ ¼ 1 2m;2 ;
Tr ðmÞ ¼ 2m;0 1
Td ðmÞ ¼ 1 2m;3 ;
Tu ðmÞ ¼ 1 2m;3
with m;m0 ¼ 1 for m ¼ m0 and 0 otherwise. As we will see
below, this choice of the shielding term decouples effective
FIG. 2 (color online). Construction of the hopping term Dys .
(a) Gadget particles at stars. (b) Terms As ðmÞ that are tensor
products of two Pauli X operators. Each term acts on two qubits
(red online), depending on spin values of gadget particles.
250502-2
interactions between neighboring gadget particles, and
makes the model exactly solvable.
Decomposition into subspaces.—Now, we solve the
gadget Hamiltonian in Eq. (1). It is convenient to decompose the entire Hilbert space into subspaces. Let us denote
computational basis states whose gadget values are all j0is:
~ ¼ j0i
~ qubit where jdi
~ qubit represents spin
~ gadget jdi
j c ðdÞi
~ qubit means that all the qubits
values jdj i for qubits, and j0i
~ such that it is
are j0is. We define the subspace MðdÞ
spanned by all the states which can be reached from
~ by applying Dys :
j c ðdÞi
Y
y s
~
~
M ðdÞ ¼
ðDs Þ j c ðdÞi; for all s :
s
~ is an invariant subspace of
We can verify that MðdÞ
Hgadget . Then, one can solve the gadget Hamiltonian inside
~ independently.
each subspace MðdÞ
Ground state subspace.—We will first solve for the
~ and then will show its lowest
ground state inside Mð0Þ,
~
energy state to be a ground state. We note that inside Mð0Þ
Bp ¼ 1, and thus plaquette terms need not be considered.
Denoting the total number of stars as N, we may view
~ as the Hilbert space of N particles.
Mð0Þ
Y
O
~
~ ¼
js i ¼ ðDys Þs j c ð0Þi:
(2)
ji
s
ðDys Þ4
s
As , ðDys Þ8
¼
¼ I, these particles can be
Noting that
considered to have eight-dimensional Hilbert spaces, s ¼
0; . . . ; 7 [17]. In this ‘‘ representation,’’ the hopping term
Hhop can be written as a one-body Hamiltonian: Hhop ¼
P
s hs where
hs ¼ Uðjs ¼ 0ihs ¼ 0j þ js ¼ 4ihs ¼ 4jÞ
t
7
X
s ¼0
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PHYSICAL REVIEW LETTERS
PRL 107, 250502 (2011)
ðjs þ 1ihs j þ H:c:Þ
ðmod 8Þ:
~
However, edge terms Ce are not one body inside Mð0Þ.
A key idea behind our gadget arises from the fact that
these two-body interactions arising from Ce can be exactly
~
canceled by adding the shielding term Hshield . Inside Mð0Þ,
edge terms have the same action as the following two-body
terms involving gadget particles: Ce ¼ T‘ ðms ÞTr ðmsþx^ Þ
^ and
for a horizontal edge e connecting s and s þ x,
Ce ¼ Td ðms ÞTu ðmsþy^ Þ for a vertical edge e connecting s
^ as one can verify from direct calculations [14].
and s þ y,
Then, the edge terms are exactly canceled: He þ Hshield ¼ 0
~ This means the gadget Hamiltonian is one
inside Mð0Þ.
P
body in the ‘‘ representation’’: Hgadget ¼ const þ s hs .
~
Because of this, all energy eigenstates inside
N Mð0Þ can
be written
~ ¼ s js i where
P in the tensor product form ji
js i ¼ s s ðs Þjs i. The lowest energy state is
~ ¼ Ns j0 i, where 0 ðÞ ¼ 0 ð þ 4Þ for all .
j c GS ð0Þi
Therefore, returning from the representation, we can
write the ground state as
~ ¼
j c GS ð0Þi
7
YX
~
0 ðÞðDys Þ j c ð0Þi
s ¼0
¼
Y
ðI þ As Þ
s
3
X
~
0 ðÞðDys Þ j c ð0Þi:
¼0
~ since
We see that there is a finite energy gap inside Mð0Þ,
Hgadget acts as a one-body Hamiltonian.
~
Unitary connection.—This lowest energy state j c GS ð0Þi
is connected to the ground state of the modified toric code
through the following local unitary transformation:
U¼
Y
Us ;
s
Us 3
X
ms ¼0
jms ihms j
Y
As ðmÞ:
(3)
m<ms
~ ¼ j
~ 0 iN
particular, we have Uj c GS ð0Þi
gadget P
3
~ qubit where j
~ 0 i ¼ m¼0 0 ðmÞjmi, and
j c Toric ð0Þi
Q
~
~ is a ground state of the
j c Toric ð0Þiqubit ¼ s ðI þ As Þj0i
modified toric code. We may verify that the gadget
Hamiltonian has three other ground states j c GS ðd~i Þi, i ¼
1, 2, 3, inside Mðd~i Þ, connected in the same way to the
ground states j c Toric ðd~i Þi of the modified toric code.
It is then simple to find the logical operators for the
gadget Hamiltonian; they are those of the modified toric
code conjugated by U: Uy X 1 U, Uy X 2 U, Uy Z 1 U, and
Uy Z 2 U. The ground space is topologically ordered since
it meets the criteria for the stability against local perturbations proposed in [18].
Anyonic excitations, which are also energy eigenstates,
can be created by applying ‘‘segments’’ of logical operators combined with local operations on gadget particles in a
similar way to the conventional toric code. As a result,
excitations can be created only through completely localized manipulations of spins in small regions. This is in
striking contrast to perturbative Hamiltonians where anyonic excitations are delocalized, and cannot be created
through completely localized manipulations of spins.
Energy gap.—Finally, we show that j c GS ðd~i Þi are the
ground states of the gadget Hamiltonian. To do so, we
prove that the lowest energy states within other non~ have a finite higher energy
ground-state subspaces MðdÞ
~
than the lowest energy state within Mð0Þ.
~ defined by j c ðdÞi
~ ¼
We first consider a subspace MðdÞ
~
~
As ð0Þj c ð0Þi where d has nonzero components for two
qubits acted on by As ð0Þ, as shown in Fig. 3. We notice
that As ð0Þ commutes with all terms except two edge terms
~ is
Ce1 and Ce2 . Therefore, solving Hgadget inside MðdÞ
equivalent to solving
In
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PHYSICAL REVIEW LETTERS
PRL 107, 250502 (2011)
~
FIG. 3 (color online). A non-ground-state subspace MðdÞ.
As ð0Þ anticommutes with two edge terms Ce1 and Ce2 .
As ð0ÞHgadget As ð0Þy ¼ Hgadget þ V
~ where V ¼ 2JðCe þ Ce Þ.
inside Mð0Þ,
1
2
Below, we show that the lowest energy states for
~ have finite higher energy than
Hgadget þ V inside Mð0Þ
those of Hgadget for appropriate choices of parameters U, t,
and J. For simplicity of discussion, we neglect a constant
correction resulting from plaquette term Hp by writing
P
~ Then, one may write
Hgadget ¼ Hhop ¼ s hs inside Mð0Þ.
P
0
Hgadget ¼ sfs ;s1 ;s2 g hs þ H with
H ¼
X
hs þ 2JðCe1 þ Ce2 Þ;
s¼fs ;s1 ;s2 g
where s1 and s are connected by e1 , and e2 connects s2 , s
(Fig. 3).
Returning to the representation, we note that all particles except s , s1 , s2 are noninteracting and are governed
under the same Hamiltonian hs as before. Let us denote
the lowest energy eigenvalue of hs as E0 . Noting that E0 is
upper bounded by U, it suffices to show that
H > 3U > 3E0 for the existence of an energy gap.
P
Let H ¼ H1 þ H2 where H1 ¼ t s¼fs ;s1 ;s2 g ðDys þ Ds Þ
P
and H2 ¼ U s¼fs ;s1 ;s2 g jms ¼ 0ihms ¼ 0j þ 2JðCe1 þ Ce2 Þ.
Since one cannot minimize H1 and H2 simultaneously, we
obtain a lower bound for H by finding minimal energy
eigenvalues for H1 and H2 individually. One can verify that
H1 6t by directly finding energy eigenvalues of
H1 . Similarly, one can verify that H2 minð3U þ
4J; 2U 4JÞ. Here, let us choose U and J such that J ¼
0
U=8, and H2 52 U. Hgadget
has a provably higher ground
state energy than Hgadget when H > 6t 5U
2 > 3U >
3E0 , so we simply set U > 12t. This proof may be easily
~ when U > 16t.
generalized to arbitrary MðdÞ
A drawback of this proof is that a small value of
t ¼ U=16 gives a weak constant gap for hs and thus the
~ is 104 U. Tighter analysis presented in
gap inside Mð0Þ
[14] shows that when J ¼ 0:09U, t ¼ 0:375U, the system
has a quite reasonable energy gap of >0:075U both inside
~
and outside Mð0Þ.
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Particle dimension.—In this construction, a gadget particle is four dimensional, and a composite particle is
eight dimensional after removing the internal degree of
freedom for Bp . This can be improved through defining a
similar construction on a triangular lattice, leading to sixdimensional gadget particles and four-dimensional composite particles. In addition, a more elaborate construction
reduces the dimension of the gadget particles to three while
keeping the dimension of composite particles at four [14].
Discussion.—Our gadget construction can be generalized to the quantum double model, which may be universal
for topological quantum computation, in a rather straightforward way [14]. We expect that similar generalizations
are possible for other interesting, but highly nonlocal topologically ordered Hamiltonians. In addition, our nonperturbative gadget may find use in adiabatic quantum
computation and Hamiltonian complexity problems.
In our construction, we have heavily taken advantage
of the fact that the terms being simulated commute.
Whether noncommuting terms can be simulated in this
way remains an open question. Perhaps insights from
related problems in theoretical computer science will prove
fruitful, opening new connections.
Conclusion.—In this Letter, we argue the necessity of
simulating topological quantum codes nonperturbatively,
and propose a model which is two body, exactly solvable,
and supports completely localized quasiparticle excitations. While our construction involves large particle dimensions and ‘‘gadgety’’ interaction terms which put it
beyond modern experimental capabilities, there has recently been remarkable theoretical and experimental
progress in engineering custom interactions between particles [19–21]. We hope our construction will provide a
stepping-stone towards physical realizability of topological quantum codes.
S. A. O. is supported by NSF Grant No. DGE-0801525,
IGERT: Interdisciplinary Quantum Information Science
and Engineering. B. Y. thanks Eddie Farhi and Peter Shor
for support. B. Y. is supported in part by DOE Grant
No. DE-FG02-05ER41360 and by Nakajima Foundation.
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