We present a scheme to efficiently simulate, with a classical computer, the
dynamics of multipartite quantum systems on which the amount of entanglement
(or of correlations in the case of mixed-state dynamics) is conveniently
restricted. The evolution of a pure state of n qubits can be simulated by using
computational resources that grow linearly in n and exponentially in the
entanglement. We show that a pure-state quantum computation can only yield an
exponential speed-up with respect to classical computations if the entanglement
increases with the size n of the computation, and gives a lower bound on the
required growth.Comment: 4 pages. Major changes. Significantly improved simulation schem

A. Ekert

A. Peres

D. Gottesman

E. Schmidt

Guifré Vidal

L. Valiant

M. A. Nielsen

English

arXiv

Crossref

Efficient Classical Simulation of Slightly Entangled Quantum Computations

We present a classical protocol to efficiently simulate any pure-state quantum computation that involves only a restricted amount of entanglement. More generally, we show how to classically simulate pure-state quantum computations on n qubits by using computational resources that grow linearly in n and exponentially in the amount of entanglement in the quantum computer. Our results imply that a necessary condition for an exponential computational speedup (with respect to classical computations) is that the amount of entanglement increases with the size n of the computation, and provide an explicit lower bound on the required growth

Vidal, Guifré

University of Queensland eSpace

P H Y S I C A L R E V I E W L E T T E R S week ending3 OCTOBER 2003VOLUME 91, NUMBER 14Efficient Classical Simulation of Slightly Entangled Quantum Computations
Guifre´ Vidal
Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
(Received 25 February 2003; published 1 October 2003)147902-1We present a classical protocol to efficiently simulate any pure-state quantum computation that
involves only a restricted amount of entanglement. More generally, we show how to classically simulate
pure-state quantum computations on n qubits by using computational resources that grow linearly in n
and exponentially in the amount of entanglement in the quantum computer. Our results imply that a
necessary condition for an exponential computational speedup (with respect to classical computations)
is that the amount of entanglement increases with the size n of the computation, and provide an explicit
lower bound on the required growth.
DOI: 10.1103/PhysRevLett.91.147902 PACS numbers: 03.67.Lx, 03.65.Ud, 03.67.Hk, 03.67.Mnment in the system scales with the number n of qubits. standard measure entropy of entanglement [12].Identifying nontrivial quantum dynamics that are effi-
ciently simulatable on a classical computer is important
in quantum many-body physics and in quantum informa-
tion science. On the one hand, our understanding of
quantum many-body dynamics is severely hindered by
the fact that, generically, the description of the state of n
interacting quantum systems requires O expn parame-
ters, rendering the simulation of such systems intractable.
Consequently, those exceptional cases where an efficient
simulation is possible are a precious source of insight into
the physics of quantum many-body systems.
In quantum information [1], on the other hand, uncov-
ering simulatable quantum dynamics is of interest in
order to characterize the essential resources required for
genuine quantum computation. Clearly, if a quantum
device is to offer an exponential speedup with respect
to classical computations, then it must involve dynamics
that cannot be efficiently simulated classically. Thus by
detecting simulatable quantum dynamics we learn about
which systems may be used as a quantum computer.
Examples of quantum evolutions that can be efficiently
simulated are given by n qubits prepared in a computa-
tional-basis state and acted upon by gates from the
Clifford group, as established by the Gottesman-Knill
theorem [2], and by n bosonic (fermionic) modes evolv-
ing in a Gaussian state [3] (respectively, [4]). Also, Jozsa
and Linden [5] have recently shown how to efficiently
simulate any quantum computation on n qubits such that
their state factors, at all times, into a direct product of
states of at most a constant (i.e., independent of n) number
of qubits.
In this Letter we show that any quantum computation
with pure states can be efficiently simulated with a clas-
sical computer provided the amount of entanglement in-
volved is sufficiently restricted. Here entanglement is
quantified by a most natural (yet not standard) measure
and may involve all n qubits of the quantum computer in
a single, nonfactorizable state. More generally, we give an
upper bound on the computational speedup achievable by
a quantum computation, in terms of how the entangle-0031-9007=03=91(14)=147902(4)$20.00 Thereby we (i) establish that entanglement is a necessary
(but in general not sufficient) resource for genuine quan-
tum computation with pure states and (ii) provide an
explicit lower bound for the entanglement required in
quantum computational speedups. (See also [6–8] for
related discussions on the case of mixed-state quantum
computations.) In addition, the simulation protocol pre-
sented here can be adapted as to apply to a class of
quantum many-body systems, including spin chains, as
explored in depth in a future contribution.
Measure of entanglement.—Let ji 2 H n2 denote a
pure state of n qubits, A a subset of the n qubits, and B the
rest of them. The Schmidt decomposition (SD) [9] of ji
with respect to the partition A:B reads
ji 
XA
1
jA	 i  jB	 i; (1)
where jA	 i (jB	 i) is an eigenvector of the reduced
density matrix A	 (B	) with eigenvalue jj2 > 0,
and the Schmidt coefficient  follows from the relation
hA	 ji  jB	 i. The rank A of A is a natural
measure of the entanglement between the qubits in A
and those in B [10]. Therefore, we could use ,
  max
A
A; (2)
i.e., the maximal value of A over all possible bipartite
splittings A:B of the n qubits, to quantify the entangle-
ment of state ji. Here we mainly use the related entan-
glement measure E,
E  log2; (3)
which fulfills the following appealing properties: (i) 0 
E  nlog2d=2 for ji 2 H nd , with E  0 if and
only if ji is a product (i.e., completely unentangled)
state [13]; (ii) E is an entanglement monotone [11] that
decreases both under deterministic and stochastic local
manipulations of the system; (iii) E is additive under
tensor products, E 0  E  E0; and
(iv) in the bipartite setting E upper bounds the more2003 The American Physical Society 147902-1
P H Y S I C A L R E V I E W L E T T E R S week ending3 OCTOBER 2003VOLUME 91, NUMBER 14State decomposition.—Let us now consider the expan-
sion of ji 2 H n2 in the computational basis,
ji 
X1
i10
  
X1
in0
ci1in ji1i      jini: (4)
The key ingredient of our simulation protocol is a par-
ticular decomposition of coefficients ci1i2in , namely,
ci1i2in 
X
1;;n1
1	i11 
1	
1 
2	i2
12
2	
2 
3	i3
23   n	inn1 : (5)
This decomposition employs n tensors f1	;    ;n	g
and n 1 vectors fl	;    ; n1	g, whose indices il
and l take values in f0; 1g and f1;    ; g, respectively.
Therefore, (5) reexpresses the 2n coefficients ci1i2in of(4) in terms of about 22  n parameters, a number
that is linear in n and exponential in E,
n qubit
state
$ n expEparameters: (6)
For a generic state ji, E is of the order n and the de-
composition in terms of  ’s and ’s is uninteresting, for it
employs On expn parameters. However, notice that if
E scales as O logn, then only polyn parameters are
required, leading to an efficient description of ji.
Decomposition (5) depends on the particular way qu-
bits have been ordered from 1 to n, and essentially con-
sists of a concatenation of n 1 SDs. More explicitly, we
first compute the SD of ji according to the bipartite
splitting of the systems into qubit 1 and the n 1 remain-
ing qubits (from now on we omit the tensor product
symbol),
ji 
X
1
1	1 j1	1 ij2n	1 i (7)

X
i1;1
1	i11 
1	
1 ji1ij2n	1 i; (8)
where in the last line we have expanded each Schmidt
vector j1	1 i 
P
i1
1	i1
1 ji1i in terms of the basis vectors
fj0i; j1ig for qubit 1. We then proceed according to the
following three steps: (i) we expand each Schmidt vector
j2n	 i in a local basis for qubit 2,
j2n	1 i 
X
i2
ji2ij3n	1i2 i; (9)
(ii) then we write each (possibly unnormalized) vector
j3n	1i2 i in terms of the at most  Schmidt vectors
fj3n	2 ig21 (i.e., the eigenvectors of 3n	) and the
corresponding Schmidt coefficients 2	2 ,
j3n	1i2 i 
X
2
2	i212
2	
2 j3n	2 i; (10)
(iii) finally we substitute Eq. (10) in Eq. (9) and the latter
in Eq. (8) to obtain147902-2ji 
X
i1;1;i2;2
1	i11 
1	
1 
2	i2
12
2	
2 ji1i2ij3n	2 i: (11)
Iterating steps (i)–(iii) for the Schmidt vectors
j3n	2 i; j4n	3 i;    ; jn	n1i, one can express stateji as in (5).
A useful feature of the description of ji in terms of
the ’s and ’s of Eq. (5) is that it readily gives the SD of
ji according to 1    l	:l 1    n	, i.e., the bipartite
splitting A:B such that A contains the first l qubits and B
the rest of them,
ji 
X
l
l	l j1l	l ijl1n	l i: (12)
Indeed, it can be checked by induction over l that
j1l	l i 
X
1;;l1
1	i11 
1	
1   l	ill1l ji1    ili; (13)
whereas by construction we already had that
jl1n	l i
X
l1;;n
l1	il1ll1 n1	n1 n	inn1 jil1 ini:
(14)
Finally, insight into the meaning of decomposition (5)
may be gained by defining 2 unnormalized states
j’l	0 i 
P
i
l	i
0 jii for qubit l and expanding ji as a
linear combination of n1 product states,
ji 
X
1;;n1
j’1	1 i1	1 j’2	12i   n1	n1 j’n	n1i; (15)
where the sum over ’s is what accounts for the correla-
tions between qubits [14].
Update of the decomposition.—The following lemmas
explain how to update the description of state ji when a
one-qubit gate or a two-qubit gate (acting on consecutive
qubits) is applied to the system. Remarkably, the compu-
tational cost of the updating is independent of the total
number n of qubits and grows in  only as a polynomial of
low degree.
Lemma 1. Updating the ’s and ’s of state ji after a
unitary operation Uacts on qubit l involves transforming
only l	. The incurred computational cost is of O2
basic operations.
Proof. In the SD according to the splitting
1    l 1	:l    n	, a unitary operation U on qu-
bit l does not modify the Schmidt vectors for part
1    l 1	 and therefore j	 and j	 1  j  l 1
remain the same. Similarly, by considering the SD for the
splitting 1    l	:l 1    n	, we conclude that also
j	 and j1	 l 1  j  n remain unaffected.
Instead, l	 changes according to
0l	i 
X
j0;1
Uij
l	j
8;  1;    ; : (16)
That is, for each value of  and , a small matrix147902-2
P H Y S I C A L R E V I E W L E T T E R S week ending3 OCTOBER 2003VOLUME 91, NUMBER 14multiplication is required, leading to a total of O2
basic operations.
Lemma 2. Updating the ’s and ’s of state ji after a
unitary operation V acts on qubits l and l 1 involves
transforming only l	, l	, and l1	. This can be
achieved with O3 basic operations.
Proof. In order to ease the notation we regard ji as
belonging to only four subsystems,
H  J H C HD K: (17)
Here, J is spanned by the  eigenvectors of the reduced
density matrix
1l1	 
X

jihj; ji  l1	 j1l1	 i; (18)
and, similarly, K is spanned by the  eigenvectors of the
reduced density matrix
l2n	 
X

jihj; ji  l1	 jl2n	 i; (19)
whereas H C and H D correspond, respectively, to qubits
l and l 1. In this notation we have
ji 
X
;;1
X1
i;j0
C	i 
D	j
 jiji; (20)
and, reasoning as in the proof of Lemma 1, when applying
unitary V to qubits C and D we need update only
C	; ;D	. We can expand j0i  Vji as
j0i 
X
;1
X1
i;j0
ijjiji; (21)
where
ij 
X

X
kl
Vijkl
C	k
 
D	l
 : (22)
By diagonalizing 0DK	,
0DK	  trJCj0ih0j

X
j;j0;;0
X
;i
hjiijij00 

jjihj00j; (23)
we obtain its eigenvectors fj0DK	 ig, which we can ex-
pand in terms of fjjig to obtain 0D	 ,
j0DK	 i 
X
j;
0D	j jji: (24)
The eigenvectors of 0JC	 and 0 follow then from
0j0JC	 i  h0DK	 j0i (25)

X
i;j;;
0D	j ijhjijii; (26)
and by expanding each j0JC	 i,147902-3j0JC	 i 
X
i
0C	i jii; (27)
we also obtain 0C	. In the above manipulations, the
largest tensors contain O2 components. The most ex-
pensive manipulations are the following: in Eq. (22), for
each value of  and  the sum over  requires summing
up O terms, leading to a total of O3 basic opera-
tions; similarly O3 operations are required in Eq. (23)
and in Eq. (26); finally, the same cost applies to the
diagonalization of 0JC	 , a matrix of 2 elements, so
that the overall update can be done with O3 basic
operations.
Simulation protocol.—We consider a pure-state quan-
tum computation using n qubits, initially in state j0in,
and consisting of polyn one- and two-qubit gates and a
final measurement in the computational basis. The simu-
lation protocol works as follows. We use tensors ’s and
’s to store the initial state j0in and update its description
as the gates are applied [15]. Recall that decomposition
(5) assumes a specific ordering of the qubits. In order to
update ji according to a two-qubit gate between non-
consecutive qubits C and D, we first simulate On swap
gates between adjacent qubits to bring C and D together.
Notice that a swap gate does not modify . The update of
the decomposition after an arbitrary two-qubit gate has
been applied can thus be always achieved with at most
Onpoly	 basic operations. From the ’s and ’s it is
possible to compute, at a cost growing as poly, the
outcome probabilities of a measurement on qubit l and the
new ’s and ’s for the postselected states of the remain-
ing n 1 qubits. Thus simulating the outcome probabili-
ties of the final measurement on the computational basis
requires npoly basic operations. We can therefore state
the main results of the Letter.
Theorem. A pure-state quantum computation on n qu-
bits and consisting of polyn elementary gates can be
classically simulated with a cost in computational time
and memory space given by polyn expE, where E
denotes the maximal value of E achieved during the
computation,
n qubit
computation $
polyn expE
time and memory: (28)
Corollary. If E scales at most as logn, then a clas-
sical simulation can be accomplished with polyn com-
putational resources.
Discussion.—The above results establish a clear con-
nection between the cost of classically simulating a
pure-state quantum computation and the amount of en-
tanglement involved in the computation. The amount of
entanglement determines the computational cost incurred
when using the specific protocol described in this Letter.
In particular, the corollary states that any slightly en-
tangled quantum computation, such that E is upper
bounded by k logn for some k > 0, can be efficiently147902-3
P H Y S I C A L R E V I E W L E T T E R S week ending3 OCTOBER 2003VOLUME 91, NUMBER 14simulated in a classical computer. On the other hand, also
some highly entangled quantum computations can be
simulated efficiently [2–4] through other simulation
schemes. In other words, a small amount of entanglement
is a sufficient condition for an inexpensive classical simu-
lation, but not a necessary one.
The above considerations translate into a lower bound
on the amount of entanglement that is produced dur-
ing a pure-state quantum computation leading to a com-
putational speedup. Indeed, it follows from the theorem
that if a strictly exponential speedup is to occur, then E
must grow linearly in n. Similarly, a computational
speedup of, say, exp np , requires that E grow at least
as

n
p
, and so on.
The tools presented in this Letter can also be applied to
simulate continuous-time dynamics in certain quantum
many-body systems, as further discussed in a future
contribution. Roughly, time is divided into small steps
so as to approximate the continuous-time evolution by a
sequence of small gates, and then the present simulation
protocol is used to simulate the sequence of gates. As
above, the computational cost of a given simulation de-
pends on the amount of entanglement in the system.
Efficient simulations correspond to a very restricted sub-
set of states [16] and one could have expected not to find
physical systems of interest where the simulation protocol
is efficient. However, it turns out that the amount of
entanglement in many one-dimensional many-body sys-
tems, such as quantum spin chains at zero temperature
[17], is typically sufficiently small so that an efficient
classical simulation is possible.
Thus, the uncovered connection between entanglement
and the cost of classical simulations does not only con-
tribute to our understanding of the essential ingredients
of quantum computation, but it also stimulates the devel-
opment of new techniques for the study of realistic quan-
tum many-body phenomena.
The author thanks Dave Bacon, Ignacio Cirac, Ann
Harvey, Richard Jozsa, and Debbie Leung for valuable
advice. Support from the U.S. National Science Foun-
dation under Grant No. EIA-0086038 is acknowledged.14790[1] M. A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information (Cambridge University Press,
Cambridge, 2000).
[2] D. Gottesman, in Proceedings of the 23rd International
Colloquium on Group Theoretical Methods in Physics
(International Press, Cambridge, 1999), p. 23.
[3] S. D. Barlett et al., Phys. Rev. Lett. 89, 207903 (2002);
S. D. Barlett and B. C. Sanders, ibid. 89, 097904 (2002).
[4] L. Valiant, in Proceedings of the 33rd ACM Symposium
on the Theory of Computation (ACM Press, New York,
2001), pp. 114–123; B. M. Terhal and D. P. DiVincenzo,
Phys. Rev. A 65, 032325 (2002); E. Knill, quant-ph/
0108033.2-4[5] R. Jozsa and N. Linden, quant-ph/0201143.
[6] E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672
(1998).
[7] N. Linden and S. Popescu, Phys. Rev. Lett. 87, 047901
(2001).
[8] G. Vidal, ‘‘Efficient Classical Simulation of Barely
Correlated Mixed-State Quantum Dynamics’’ (to be
published).
[9] E. Schmidt, Math. Ann. 63, 433 (1906); A. Ekert and
P. L. Knight, Am. J. Phys. 63, 415 (1995); A. Peres,
Quantum Theory: Concepts and Methods (Kluwer
Academic Publishers, Dordrecht, 1995).
[10] The use of A [equivalently, of log2A] as a measure
of entanglement can be justified by considering how a
series of nonlocal resources can be interconverted under
(nonasymptotic) local operations and classical com-
munication (LOCC). For instance, about log2A EPR
pairs shared between A and B [or log2A Control-NOT
gates involving A and B] are necessary and sufficient
to prepare ji with the additional help of LOCC.
Also, log2A is the maximal number of EPR pairs
that, with finite probability, can be extracted from ji
by LOCC. Both A and its logarithm can be shown
not to increase (not even probabilistically) under
LOCC [11], as required of any entanglement mea-
sure, and they are related to the more popular measure
entropy of entanglement E trAlog2A [12]
through E log2A. Notice that using E to
quantify the entanglement of ji in the present context
may not be an appropriate choice, since E refers to
asymptotic properties of ji, i.e., to properties of jiN
in the limit N!1, whereas here we are concerned with
the case N1.
[11] G. Vidal, J. Mod. Opt. 47, 355 (2000).
[12] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu-
macher, Phys. Rev. A 53, 2046 (1996).
[13] We also note that E is not a continuous function of
ji with respect any reasonable distance. This implies
that a good approximation j ~i to ji may exist with a
significantly lower value of E, a fact that can be
exploited to improve the efficiency of the simulation
protocol described in this Letter.
[14] It is also insightful to derive the explicit decomposition
for simple n-qubit states with small , such as a product
state j0in, with   1; a cat state, j0in  j1in with
  2; and a W state j00    01i  j00    10i     
j10    00i, also with   2.
[15] A digital computer allows only for an approximate
description of gates and states, since real coefficients
are truncated. See, e.g., Ref. [5] for a discussion on
how to obtain efficient approximations by using rational
numbers.
[16] A pure state of n qubits picked up at random from H n2
(according, say, to the unitarily invariant probability
density) will have E  n=2 with certainty, as im-
plied by a simple parameter counting argument. From
this perspective, states with E  O logn are ex-
tremely rare.
[17] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev.
Lett. 90, 227902 (2003).147902-4


Efficient classical simulation of slightly entangled quantum computations

Efficient classical simulation of slightly entangled quantum
  computations

Efficient classical simulation of slightly entangled quantum computations

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