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overhead that precludes efficient universal quantum
computation.
We consider generalizations of quantum walk
with many interacting walkers and show that multiparticle
quantum walk is universal for quantum
computation using a graph of polynomial size.
We present efficient universal constructions based
on the Bose-Hubbard model, fermions with nearestneighbor
interactions, and distinguishable particles
with nearest-neighbor interactions. We also
show that almost any interaction gives rise to universality
when using indistinguishable particles.
Because our graphs are exponentially smaller
than those in reference (7), our construction is
amenable to experimental implementation using
an architecture where vertices of the graph occupy
different spatial locations (although we leave
issues of fault tolerance for future work). Current
two-particle experiments involve only noninteracting
bosons (11–14), but other experiments
could realize interactions. Specifically, a Bose-
Hubbard model of the type we consider could
naturally be realized in a variety of systems, including
traditional nonlinear optics (15), neutral
atoms in optical lattices (16, 17), or photons in arrays
of superconducting qubits (18).
Multiparticle quantum walk has been considered
as an algorithmic tool for solving graph isomorphism
(19), although this technique has known
limitations (20). Other previous work has focused
on two-particle quantum walk (11–14, 21–25)and
multiparticle quantum walk without interactions
(11–14, 21, 22, 26). Here, we consider multiparticle
quantum walks with interactions, which
seem to be required to achieve computational universality
(27, 28).
It has been shown that systems of interacting
particles on a lattice can perform universal computation
with a time-dependent Hamiltonian
(29, 30). However, because of the time dependence,
such a system should not be considered a
quantum walk. In our scheme, the Hamiltonian
is time-independent and the computation is encoded
entirely in the unweighted graph on which
the particles interact.
In a multiparticle quantum walk, particles (either
distinguishable or indistinguishable) interact
in a local manner on a simple graph G with vertex
set V(G) and edge set E(G). The Hilbert space
for m distinguishable particles on G is spanned
by the basis
fji1,…,im〉 : i1,…,im ∈ VðGÞg ð1Þ
where iw is the location of the wth particle. A
continuous-time multiparticle quantum walk of m
distinguishable particles on G is generated by a
time-independent Hamiltonian
H ðmÞ
G
¼ ∑m
w¼1
∑
ði, jÞ∈EðGÞ
ðji〉〈 jj w þjj〉〈ij w Þþ
∑ Uijðn% i; n% jÞ ð2Þ
i, j∈V ðGÞ
where the subscript w indicates that the operator
acts nontrivially only on the location register for
the wth particle. Here, the operators n% i and n% j
count the numbers of particles at vertices i and j,
respectively (explicitly, n% i ¼ ∑ m w¼1 ji〉〈ij w ).
The first term of Eq. 2 moves particles between
adjacent vertices, and the second term is an
interaction between particles. We assume that Uij
is zero whenever one of its arguments evaluates
to zero and that Uii is zero if there is only one
particle at vertex i (thus, the Hamiltonian for a
single particle reduces to that of a standard quantum
walk). We also assume that the interaction
Uij only depends on the distance between i and j
and has a constant range. Finally, we assume that
the norm of each term Uij is upper-bounded by a
polynomial in m.
Indistinguishable particles can be represented
in the basis specified in Eq. 1 as states that are
either symmetric (for bosons) or antisymmetric
(for fermions) under the interchange of any two
particles. The Hamiltonian preserves both the symmetric
and antisymmetric subspaces and, restricted
to the appropriate subspace, generates a quantum
walk of m bosons or m fermions on G.
This framework includes well-known interacting
many-body systems defined on graphs. For example,
it includes the Bose-Hubbard model, with
interaction Uijðn% i,n% jÞ¼ðU=2Þdi, jn% iðn% i − 1Þ. It
also includes systems with nearest-neighbor interactions,
such as with Uijðn% i,n% jÞ¼Ud ði, jÞ∈EðGÞn% in% j.
The dynamics of our scheme can be understood
by considering scattering events involving
only one or two particles and can be analyzed
using a discrete version of scattering theory (31).
We first discuss single-particle scattering.
Consider a single-particle quantum walk on
an infinite graph G obtained by attaching a semiinfinite
path to each of N chosen vertices of an
arbitrary (N + m)–vertex graph G % (Fig. 1A). A
particle is initially prepared in a state that moves
(under Schrödinger time evolution) toward the
A (4, 1)
(3, 1)
(2, 1)
(4, 2)
(1, 1) (3, 2)
G (1, 2)
(2, 2)
(1,N)
(2,N)
(3,N)
(4,N)
B
subgraph G % along one of the semi-infinite paths.
After scattering through the subgraph, the particle
moves away from G % in superposition along the
semi-infinite paths. To understand this process, we
discuss the scattering eigenstates of the Hamil-
tonian H ð1Þ
G .
Given the graph G % , for each momentum
k ∈ ð−p,0Þ and path j ∈ f1,2…,Ng, there is a
scattering state jscjðkÞ〉 with amplitudes
〈x,qjscjðkÞ〉 ¼ e −ikx dqj þ e ikx SqjðkÞ ð3Þ
on the semi-infinite paths [with the labeling (x,q)
of vertices on the paths as in Fig. 1A], where S(k)
is an N by N unitary matrix called the S-matrix
[see section S1 of (32)]. The state jscjðkÞ〉 is an
eigenstate of H ð1Þ
G with energy 2 cos k.
A wave packet is a normalized state with
most of its amplitude on scattering states with
momenta close to some particular value. The scattering
state jscjðkÞ〉 gives us information about
how a wave packet with momenta near k located
on path j scatters from G % . The wave packet
initially moves toward the graph with speed
j dE
dk j¼j2sinkj. After scattering, the amplitude
associated with finding the wave packet on path q
is Sqj(k). This picture of the scattering process
remains valid when the finite extent of the wave
packets is taken into account and when the infinite
paths are truncated to be long but finite
[section S5 of (32)].
We now consider scattering of two indistinguishable
particles on an infinite path. Translation
symmetry makes this system easier to analyze
than more general two-particle quantum walks
(33). Consider two indistinguishable particles
initially prepared in spatially separated wave packets
moving toward each other along a path with
momenta k1 ∈ ð−p; 0Þ and k2 ∈ ð0; pÞ. After
scattering, the particles move apart with momenta
k
Fig. 1. (A) The infinite graph G. The vertices labeled (1, j) belongtoG%. (B) Encodedj0〉 (left) andj1〉
(right).
A
0in
0out
B
0in
1in 1out 1in
1out 1in
Fig. 2. (A)Phasegateat–p/4. (B) Basis-changing gate at –p/4. (C) Hadamard gate at –p/2.
0out
15 FEBRUARY 2013 VOL 339 SCIENCE www.sciencemag.org
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0in
k
0out
1out
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