How robust is a quantum gate in the presence of noise?

14 Pages·0.31 MB·

by  Aram W. Harrow

How robust is a quantum gate in the presence of noise?
Aram W. Harrow1,2,∗ and Michael A. Nielsen2,†
1MIT Physics, 77 Massachusetts Ave., Cambridge MA 02139, USA
2School of Physical Sciences, University of Queensland, Queensland 4072, Australia
(Dated: February 1, 2008)
We define several quantitative measures of the robustness of a quantum gate against noise. Ex-
act analytic expressions for the robustness against depolarizing noise are obtained for all unitary
quantumgates,anditisfoundthatthecontrolled-notisthemostrobusttwo-qubitquantumgate,
in the sense that it is the quantum gate which can tolerate the most depolarizing noise and still
generate entanglement. Our results enable us to place several analytic upper bounds on the value
ofthethresholdforquantumcomputation,withthebestboundinthemostpessimisticerrormodel
being pth≤0.5.
3
0 PACSnumbers: 03.67.-a,03.65.Ud,03.67.Lx
0
2
I. INTRODUCTION below,entanglementgenerationisnecessaryforquantum
n
a computationto be possible,evenifthe methods offault-
J tolerantcomputationareused,thisprogramallowsusto
An idealquantumcomputer[1]is usually describedas
1 determine upper bounds on the value of the threshold.
asequenceofunitaryquantumgatesappliedtothequbits
2 Our work is different from most other work on es-
makingupthecomputer. Atypicaluniversalsetofquan-
tum gates is the controlled-not gate, and single-qubit timating thresholds, which usually aims to determine
1
lower bounds. The interest in lower bounds stems from
v unitary operations [2]. A crucial element in a universal
their more immediate practical interest: if we know that
8 gate set is that it be capable of generating entanglement
p > 10−6, for example, then that gives experimental-
0 between the qubits making up the computer. th
1 ists a target to shoot for in pursuit of a working quan-
In the real world quantum gates suffer from noise [3],
1 tum computer. Nonetheless, as emphasized in [11], from
which can inhibit the creation of entanglement. This
0 a fundamental point of view it would be extremely in-
3 problem led to the development of fault-tolerant meth- teresting to have exact values for the threshold, and this
0 ods for quantum computation (see the discussion and
requires techniques for obtaining upper bounds.
/ references in [1]) based on quantum error-correcting
h Our work is based upon the results of Vidal and Tar-
codes [4, 5]. One of the outstanding achievements of
p rach [12], who investigated the robustness of entangled
- workonfault-toleranceisthethresholdtheorem forquan-
quantum states, that is, how much noise can be added
nt tum computation [6, 7, 8, 9, 10]. The threshold theorem to a quantum state before it becomes unentangled, i.e.,
a statesthat,underreasonablephysicalassumptionsabout
separable. Our work also naturally extends and comple-
u noise in the computer, it is possible to correctfor the ef-
ments the work of Aharonov and Ben-Or [11], who, to
q fects of that noise, provided the strength of the noise is
: our knowledge, have done the only prior work obtaining
v below some constant threshold, pth. (Roughly speaking, upper bounds on the value of the threshold.
i p canbethoughtofasthemaximalprobabilityoferror
X th Another interesting context in which our measures of
duringasinglequantumgatethatcanbecorrectedusing
r the methods of fault-tolerance.) The exact value of the gate robustness may be placed is the program of defin-
a ing“dynamicstrengthmeasures”forquantumdynamical
threshold depends on what assumptions are made about
operations[13]. Dynamicstrengthmeasuresquantifythe
the noiseinthe quantumcomputer,andestimatesofthe
intrinsic power or strength of a quantum dynamical op-
value ofthe thresholdthereforevary quite a bit. Typical
eration as a physical resource, much as an entanglement
current estimates place it in the range 10−4 to 10−6.
measurequantifiestheentanglementinaquantumstate.
Motivated by the practical problem of noise, and the
[13] developed a framework for the analysis of dynamic
theoryoffault-tolerantquantumcomputation,inthispa-
strength measures, and we will see that gate robustness
per we considerthe problemofquantifying howrobusta
can be regarded as a measure of dynamic strength, and
quantum gate is to the effects of noise. More precisely,
analyzed within this framework.
for a given gate U we attempt to quantify how much
The structure of the paper is as follows. Sec. II re-
noisethegatecantoleratewhilepreservingtheabilityto
views background material on the Schmidt decomposi-
generateentanglement. Since,inasensewemakeprecise
tion for operators. This decomposition is central to our
later work on the robustness of quantum gates. Sec. III
reviews the notion of separable quantum gates, which
may be defined as the class of gates that cannot gener-
∗Electronicaddress: [e-mail];
ate entanglement in a quantum computer. Furthermore,
URL:http://web.mit.edu/aram/
†Electronicaddress: [e-mail]; thissectionprovesthataquantumcircuitcontainingonly
URL:http://www.qinfo.org/people/nielsen/ separablegatescanbeefficiently simulatedonaclassical
2
computer. Sec. IV reviews Vidal and Tarrach’s work on β j j / d (4)
| i ≡ | B RBi B
the robustness of quantum states. This section also in-
j
X p
troduces a novel measure of the robustness of quantum
states useful in our later work on gate robustness, and denote normalized, maximally entangled states of RAA
proves some elementary properties of the new measure. andBRB, respectively. Now let be a generalquantum
E
Sec.Vgivesourdefinitionsandresultsonthe robustness operation[35];wewillshortlyspecializetothecasewhen
ofquantumgates,andrelatestheresultstothetheoryof corresponds to the action of U. We define ρ( ) to
E E
fault-tolerant quantum computation. Sec. VI concludes. be the density operator resulting when acts on α β .
E | i| i
Writing this out explicitly, with subscripts to make it
clear which operations are acting on which systems:
II. THE OPERATOR-SCHMIDT
DECOMPOSITION ρ(E) ≡ (IRA ⊗EAB⊗IRB)◦(|αihα|⊗|βihβ|), (5)
where denotes the identity quantum operation on a
The operator-Schmidt decomposition is an operator IS
systemS. Inthespecialcasewhen representsaunitary
analogue of the well-known Schmidt decomposition for E
operation,U,onAB, we define ψ(U) to be the quantum
pure quantum states [1]. The present treatment of the
state obtained when U acts on α β , and let ρ(U) be
operator-Schmidt decomposition is based on the discus- | i| i
the corresponding density operator. Note that we will
sion in [13, 14], with the addition of a result on the con-
interchange notations like ψ(U) and ψ(U) , depending
tinuity of the Schmidt coefficients of a unitary operator. | i
on which is more convenient in a particular context.
We begin by introducing the Hilbert-Schmidt inner
TheSchmidtcoefficientsofψ(U)arecloselyconnected
product on d d operators, (Q,P) tr(Q†P), for any
× ≡ to the operator-Schmidt coefficients of U, which we de-
operators Q and P. We define an orthonormal opera-
note u . Letting U = u A B be an operator-
tor basis to be a set {Qj} which satisfies the condition Schmidjt decomposition, wejseje tjha⊗t j
(Q ,Q ) = tr(Q†Q ) = δ . For example, an orthonor- P
j k j k jk
mal basis for the space of single-qubit operators is the ψ(U) = (I U I )α β (6)
set I/√2,X/√2,Y/√2,Z/√2 , where X,Y, andZ are RA ⊗ ⊗ RB | i| i
{ } = u (I A )α (B I )β . (7)
the Pauli sigma operators, and I is the identity. j RA ⊗ j | i j ⊗ RB | i
The operator-Schmidt decomposition states that any Xj
operator Q acting on systems A and B may be writ-
Direct calculation shows that √d (I A )α and
ten [14]: A RA ⊗ j | i
√d (B I )β formorthonormalbases for R Aand
B j⊗ RB | i A
BR , respectively. Thus, we obtain the useful result
Q= qA B, (1) B
l l⊗ l that the quantum state ψ(U) has Schmidt coefficients
Xl u /√d d equal, up to the factor 1/√d d , to the
j A B A B
where q 0, and A and B are orthonormal oper- Schmidt coefficients of U.
l l l
ator bases≥for A and B, respectively. To prove the The following proposition shows that the Schmidt co-
operator-Schmidt decomposition, expand Q in the form efficientsofU arecontinuousfunctionsofU. Inthestate-
tQho=normjaklMopjkeCrajto⊗rDbaks,eswfhoerreACajndanBd, rDeskpaecretivfiexlye,daonrd- nmoetnets othfethuesuparlooppoesritaitoonr,nkoMrmk.= maxkψk=1kM|ψik de-
P
M are complex coefficients. The singularvalue decom-
jk
Proposition 1 Let U and V be operators on AB, with
positionstatesthatthematrixM with(j,k)thentryM
jk
respective Schmidt coefficients u and v , ordered into
may be written M = UqV, where U and V are unitary j j
decreasing order, u u ..., and v v ....
matrices and q is a diagonal matrix with non-negative 1 2 1 2
≥ ≥ ≥ ≥
Then
entries. We thus obtain
u v
Q= UjlqlVlkCj Dk, (2) 2 1 j j j U V 2 (8)
⊗ − d d ≤k − k
Xjkl (cid:18) PA B (cid:19)
whereq isthelthdiagonalentryofq. Definingorthonor- To understand why Eq. (8) can be interpreted as
l
mal operator bases A U C and B V D , a statement about continuity requires a little thought.
l ≡ j jl j l ≡ k lk k Note that tr(U†U) = tr(V†V) = d d , and thus
we obtain the operator-Schmidt decomposition, Eq. (1). A B
P P u2 = v2 = d d . It follows that we can think
Tobetterunderstandthecoefficientsqlintheoperator- j j j j A B
Schmidt decomposition, imagine that associated with Pof u2j/dAdPB and vj2/dAdB as probability distributions.
each system, A and B, there are reference systems, RA With this interpretation, the quantity jujvj/dAdB is
and R , with the same state space dimensionalities, d just the fidelity of these two probability distributions,
B A
P
and dB, as A and B. Let and it follows from Eq. (8) that if U ≈ V then uj ≈ vj
for all j.
α j j / d , and (3) Proof: The key is to observe that the norm is
| i ≡ | RA Ai A k·k
stable when extended trivially to an ancilla system, i.e.,
j
X p
3
M = M I . Using this observation we have theoremofCiracet al [20]linkingseparabilityofaquan-
k k k ⊗ k
tum operation to separability of the quantum state
U V = I (U V) I (9) E
k − k k RA ⊗ − ⊗ RBk ρ( ) introduced in Eq. (5).
E
(I (U V) I ) α β (10)
≥ k RA ⊗ − ⊗ RB | i| ik
Theorem 1 (Operation-separability theorem [20])
= ψ(U) ψ(V) . (11)
k − k A trace-preserving quantum operation is separable if
E
Squaring both sides of the inequality, and interchanging and only if ρ( ) is a separable quantum state, that is,
E
the roles of the two sides, we obtain: ρ( ) can be written in the form
E
kψ(U)k2+kψ(V)k2−2Re(hψ(U)|ψ(V)i)≤kU −Vk2(.12) ρ(E)= pjρRjAA⊗ρBj RB, (15)
j
X
Since ψ(U) 2 = ψ(V) 2 =1, this implies
k k k k where the pj are probabilities, ρRjAA are quantum states
2(1 ψ(U)ψ(V) ) U V 2. (13) of system R A, and ρBRB are quantum states of system
−|h | i| ≤k − k A j
BR .
B
Since ψ(U) and ψ(V) have Schmidt coefficients
u /√d d and v /√d d , respectively, it follows
j A B j A B When we say in the statement of the theorem that
from the results of [15, 16] that ψ(U)ψ(V)
ρ( ) is separable there is initially some ambiguity, due
|h | i| ≤
u v /d d . Combining this inequality with Eq. (13) E
givjesjthje dAesiBred result. 2 to the multiple ways the system RAABRB can be de-
composed into subsystems. To avoid this ambiguity, it
P
is convenient to introduce notational conventions as fol-
lows. Let σ be a state of a composite system CD. We
III. SEPARABLE AND
say σ is separable with respect to the C : D cut if σ can
SEPARABILITY-PRESERVING QUANTUM
be written σ = p ρC ρD for probabilities p , and
GATES j j j ⊗ j j
quantumstatesρC,ρD ofsystemsC andD,respectively.
Pj j
The advantage of this notation comes when more sys-
We now formally introduce the notion of separable
tems areintroduced. Forexample,whenσ isa state ofa
quantum gates, and study their basic properties, in
tripartite system, CDE, it is immediately clear what we
Sec. IIIA. Sec. IIIB states and proves a theorem show-
mean by separability with respect to the C :DE cut, or
ing that quantum circuits built entirely out of separable
with respectto the C :D :E cut, or other possible cuts.
quantum gates can be efficiently simulated on a classi-
Thus, in the operation-separability theorem, the asser-
cal computer. Finally, Sec. IIIC notes that the classi-
tion is that is separableif and only if ρ( ) is separable
cal simulation theorem of the previous subsection can E E
with respect to the R A:BR cut.
be extended to a somewhat larger class of gates, the A B
We have stated the operation-separabilitytheorem for
“separability-preserving” gates, and considers some of
the case of trace-preserving quantum operations, but a
the implications of this fact.
similar result also holds for non trace-preserving quan-
tum operations . The only change is that the p are no
j
E
longer probabilities, but instead can be any set of non-
A. Definition and basic properties
negativerealnumbers. Wehavealsorestrictedouratten-
tiontobipartitequantumoperations,thatis, whichact
Suppose is a quantum operation acting on a com- E
E on quantum systems with just two components, A and
posite quantum system with two components labeled A
B. Itisnotdifficulttoshowthatananalogousstatement
and B. is said to be separable if it can be given an
E holds for k-party quantum operations . We do this by
operator-sum representationof the form E
endowingeachpartywithanassociatedreferencesystem
(ρ)= (A B )ρ(A† B†) (14) withwhichitisinitiallymaximallyentangled,anddefin-
E j ⊗ j j ⊗ j ingρ( )tobetheresultofallowing toactonthisinitial
j E E
X state. isthenseparableifandonlyifρ( )isseparable.
E E
Separablequantumoperationswereindependentlyintro- An interesting corollary of the operation-separability
duced in [17, 18], where it was speculated that trace- theorem is that a quantum operation is separable if and
preserving separable quantum operations might corre- onlyifitisincapableofproducingentangledstates. Fur-
spond to the class of quantum operations that can be thermore,byconnectinggateseparabilitytostatesepara-
implemented on a bipartite system using local opera- bility,theoperation-separabilitytheoremallowsustoap-
tions and classicalcommunication. This speculationwas ply results from the theory of state separability to prove
false [19]. However, a related conjecture is true, namely, that certain gates are separable, and thus incapable of
that trace-preservingseparable quantum operations cor- producing entanglement.
respond to the class of trace-preserving quantum opera- The operation-separability theorem tells us that a
tions which cannot be used to generate quantum entan- trace-preserving quantum operation is separable pre-
E
glement. This follows from an elegant characterization ciselywhenρ( )isseparable. However,itdoesnotfollow
E
4
thatallseparablestatesofR A:BR canbewrittenas What does it mean to simulate this computation ef-
A B
ρ( ) for some trace-preserving quantum operation. To ficiently on a classical computer? Suppose we have a
E
understand why this is the case, observe that when classical computer that, on input of x, produces an out-
E
is trace-preserving, tr (ρ( )) must be the completely put y with probability distribution p˜ (y). A good mea-
AB x
E
mixed state of R R . In general, however, it is not dif- sure of how well this simulates the quantum computa-
A B
ficult to find separablestates σ ofR A:BR such that tion is provided by the L distance. For probability dis-
A B 1
tr (σ) is not completely mixed. tributions r(y) and s(y) the L distance is defined by
AB 1
An elegant result of M., P., and R. Horodecki [21] can D(r(y),s(y)) r(y) s(y)/2. Thus,werequirethat
≡ y| − |
be used to characterize precisely which separable states the L distance D(p (y),p˜ (y)) = p (y) p˜ (y)/2
1 P x x y| x − x |
canbewrittenintheformρ( )fortrace-preserving,sep- satisfies
arable . Their result, whicEh we have restated in the P
E
context of multipartite systems, is as follows: D(px(y),p˜x(y)) ǫ (16)
≤
for some parameter ǫ > 0. We will show that the com-
Theorem 2 Thesetofdensitymatrices,σ,ofR ABR
A B
putational resources required to achieve this accuracy
such that σ = ρ( ) for some trace-preserving quantum
E on a classical computer scale as O(poly(p(n)/ǫ)), where
operation is precisely the set such that tr (σ) is the
AB
E poly()is somepolynomialoffixeddegreenotdepending
completely mixed state of RARB. ·
on the circuit family C . Thus, high accuracies in the
n
{ }
simulation can be achieved with modest computational
Combining this theorem with the operation-
cost.
separability theorem we obtain the following result:
As an example of the practical implications of this re-
sult, suppose C is a uniform family of quantum cir-
n
{ }
cuits solving a decision problem, outputting the correct
Theorem 3 Thesetofdensitymatrices,σ,ofR ABR
A B
answer to an instance, x, of the decision problem with
such that σ =ρ( ) for some trace-preserving and separa-
E probability at least 3/4. Our result implies that there
ble quantum operation is precisely the set such that (a)
E is a classical simulation using O(poly(p(n))) gates, and
σ is separable with respect to the R A : BR cut; and
A B
outputting the correct solution to the decision problem
(b) tr (σ) is the completely mixed state of R R .
AB A B
with probability 2/3. (The probability of obtaining the
correct answer may easily be boosted up beyond 3/4 by
a constant number of repetitions.)
B. Separable gates and quantum computation
To analyze the method described below for classical
simulation, we need the notion of the trace distance, a
Having discussed the basic properties of separable quantum generalization of the L distance. The trace
1
quantum operations, we now turn to their utility for distance, D(ρ,σ), between density matrices ρ and σ is
quantum computation. Imagine a quantum circuit is defined by [1] D(ρ,σ) trρ σ /2. Note that we use
built entirely outofseparablequantumgatesandsingle- thesamenotationD(, ≡)for|th−etra|cedistanceandtheL
1
qubit gates. It is intuitively plausible that such a quan- distance, with the m·ea·ning to be determined from con-
tum circuit can be efficiently simulated on a classical text. The properties of the trace distance are discussed
computer,andwenowprovethisresult. Themajortech- in detail in [1], and we need only a few properties here:
nical difficulty is issues involving the accuracy required
inthesimulation,andtheassociatedcomputationalover- The trace distance satisfies the triangle inequality,
•
head. D(ρ,τ) D(ρ,σ)+D(σ,τ).
≤
Our model of quantum computation is as follows. Let
Thetracedistanceisdoublyconvex,meaningthatif
be a fixed set of one- and two-qubit quantum gates. •
GBy “quantum gate” we mean a trace-preserving quan- pj are probabilities, and ρj and σj are correspond-
ing density matrices, then
tum operation. We assume that all the two-qubit gates
in are separable. We let C be a uniform family of
n
G { }
quantum circuits [1, 22] containing p(n) gates, and act-
D p ρ , p σ p D(ρ ,σ ). (17)
ing on q(n) qubits, where p(n) and q(n) are polynomials  j j j j≤ j j j
j j j
in some parameter n. The initial state of the computer X X X
 
is assumed to be a computational basis state, x . The
| i The trace distance is contractive. That is, if
computation is concluded by performing a measurement •
is a trace-preserving quantum operation, then
in the computational basis, yielding a probability dis- E
D( (ρ), (σ)) D(ρ,σ).
tribution px(y) over possible measurement outcomes y. E E ≤
The measurement may be either on all the qubits, or on
Thetracedistancehasthestabilityproperty D(ρ
1
some prespecified subset. For instance, if one is solving • ⊗
σ,ρ σ)=D(ρ ,ρ ).
2 1 2
a decision problem, it is only necessary to measure the ⊗
first qubit ofthe computer,to get a single zero or one as Suppose E are POVM elements describing the
y
•
output. statisticsfromanarbitraryquantummeasurement.
5
Let r(y) tr(ρE ) and s(y) tr(σE ) be the cor- I+~sj ~σ I +~sj ~σ
≡ y ≡ y p A· B· c2−l, (20)
responding probability distributions for ρ and σ. j 2 ⊗ 2 !≤
Then the L1 distance and the trace distance are Xj
related by the inequality
for some constant c that does not depend on ,A or B.
E
D(r(y),s(y)) D(ρ,σ). (18) To see that this is possible, we make use of the fact that
≤
We now describe how the classical simulation is per- I+~sA ~σ I+~sB ~σ
· · (21)
formed,followedbyananalysistodeterminetheaccuracy E 2 ⊗ 2
(cid:18) (cid:19)
of the simulation.
Variables used in the classical simulation: For is a separable, two-qubit state, and therefore, by
eachj =1,...,q(n)welet~s beathree-dimensionalreal Carath´eodory’stheorem [23], can be written in the form
j
vector. Each vector ~s is valid, meaning that it has the
j
following three properties: (a) Each component of ~sj is q I +~tjA·~σ I+~tjB ·~σ, (22)
in the range[ 1,1];(b) Eachcomponentis specified to l j
2 ⊗ 2
−
bitsofprecision,wherelisanumberthatwillbefixedby Xj
the lateranalysis,inorderto ensurethe overallaccuracy
is at least ǫ; and (c) ~sj 1. where the qj are probabilities, ~tjA,~tjB are real-three vec-
We use the notatiokn ~sk≤ (~s ,...,~s ) to denote the tors satisfying ~tj , ~tj 1, and there are at most 16
≡ 1 q(n) k Ak k Bk ≤
3q(n)-dimensional real vector containing all the ~s s as terms in the sum. Choosing the p to be probabilities
j j
subvectors. We say that ~s is valid if each ~sj is valid. It which are l-bit approximations to the qj, and the~sjA,~sjB
will also be convenient to introduce the notation to be valid vectors which approximate ~tj ,~tj also to l
A B
I +~s ~σ I +~s σ bits, we obtain the result.
ρ(~s) 1· ... q(n)· . (19)
≡ 2 ⊗ ⊗ 2 Note that while Carath´eodory’s theorem ensures that
such probabilities and vectors exist, finding them may
Note that ρ(~s) is a legitimate density operator of q(n)
be a non-trivial task. The obvious technique, a brute
qubits, whenever ~s is valid. The idea of the classical
force search over probability distributions and valid vec-
simulationisthatthevariables~swillbeusedtorepresent
tors, requires poly(2l) operations, where poly() is some
thestateρ(~s). Notethatρ(~s)isnot avariableusedinthe ·
fixed polynomial function. Although we believe it likely
classicalsimulation;itissimplyamathematicalnotation
that better techniques — perhaps even polynomial in l
convenient in the analysis of the simulation.
—arepossible,forthepurposesofthepresentsimulation
Initial state of the classical variables: Suppose
poly(2l) turns out to be sufficient.
the initial stateofthe quantumcomputeris x , where x
| i Output of the procedure: For k =A,B we define ~sj
h(0a,s0b,1in)airnyiteiaxlplya,nswiohnilex1if..x.jx=q(n)1.wIfexsjet=~sj0 =we(s0e,t0,~sj 1=) ~sk. Set ~sj = (~sj1,...,~sjq(n)). Not6 e that ~sj is valid,kb≡y
−
initially. construction. With probability p , output~s′ =~sj.
j
Simulatingasingle-qubitgate: Asingle-qubitgate Simulating the final measurement in the com-
can be regarded as a two-qubit separable gate in which putational basis: Let S be the subset of qubits that
oneofthequbitsisactedontrivially. Thus,weneedonly is measured at the output of the quantum computation.
consider the case of two-qubit separable gates. For each k S, let s3 be the third component of ~s .
∈ k k
Simulating a two-qubit separable gate: Suppose The measurement result for that qubit is 0 with proba-
is a two-qubit separable gate, and it acts on qubits bility (1+s3)/2,and1with probability(1 s3)/2. Note
E k − k
A and B. We simulate this gate by using ~s as input to that,bydefinition,p˜ (y)isthedistributionoverpossible
x
thefollowingstochasticgatesimulationprocedure,which outcomes, y, produced by following this procedure.
producesavalid3q(n)-dimensionalvector,~s′,asoutput. Analysis: The key to the analysis of the classical
We then set~s=~s′, and repeat over,going through each simulation is a simple equivalence between the classical
gate, 1,..., p(n), in the computation, until a final out- simulationandcertainmeasurementsonquantumstates.
E E
put value of ~s is produced, at which point we proceed Suppose we define p˜m(~s) to be the probability distribu-
to the simulation of the final measurement, as described tionon validvectorsafter m steps ofthe simulationpro-
below. cedure,thatis,after ,..., havebeensimulated. For
1 m
Gate simulation procedure: m=0,...,p(n) definEe E
Input to the procedure: A valid vector,~s.
Body of the procedure: Findvalidthree-vectors~sjA and σ˜m p˜m(~s)ρ(~s). (23)
~sj , a probability distribution, p , containing at most 16 ≡
B j X~s
elements,andwitheachp specifiedtolbitsofprecision,
j
such that Itis notdifficult to see thatthe distribution obtainedby
measuringσ˜p(n) inthe computationalbasisofthe subset
I +~s ~σ I+~s ~σ
D A· B · , S is exactly the same as the output distribution p˜x(y)
E(cid:18) 2 ⊗ 2 (cid:19) produced by the classical simulation.
6
For m = 0,...,p(n) define σm to be the state of the By definition σm+1 = (σm), so this equation may
m+1
E
actual quantum computer after m gates have been ap- be rewritten
plied. Thus σ0 = x x, σ1 = (σ0), and so on. The
1
idea of the proof th|atihth|e classicEalsimulation works well D(σm+1,σ˜m+1) D( m+1(σm), m+1(σ˜m))
≤ E E
istoboundthedistancebetweenσm andσ˜m. Wedothis +D( (σ˜m),σ˜m+1). (32)
m+1
using the following lemma. E
Applying the contractivity of the trace distance to the
Lemma 1 Suppose a valid vector ~s is used as input to
first term, and Lemma 1 to the second term, we obtain
the gate simulation procedure with probability p(~s), and
letp(~s′)bethecorresponding outputdistribution on valid D(σm+1,σ˜m+1) D(σm,σ˜m)+c2−l. (33)
vectors. Define ≤
Applying the inductive hypothesis to the firsttermgives
σ p(~s)ρ(~s) (24)
≡
X~s D(σm+1,σ˜m+1) cm2−l+c2−l =c(m+1)2−l, (34)
≤
σ′ p(~s′)ρ(~s′) (25)
≡ which completes the induction. 2
~s′
X
We conclude from the proposition that
Ifthegatesimulationproceduresimulatesthegate ,then D(σp(n),σ˜p(n)) cp(n)2−l. It follows from Eq. (18)
E
we have ≤
that the simulated distribution p˜ (y) and the actual
x
distribution p (y) are related by the inequality
D( (σ),σ′) c2−l, (26) x
E ≤
where c is the constant introduced earlier in the discus- D(px(y),p˜x(y)) cp(n)2−l. (35)
≤
sion of the gate simulation procedure.
Choosing l log (cp(n)/ǫ) we therefore have
≡⌈ 2 ⌉
Proof: Letp(~s′~s)betheprobabilitythat~s′ isoutput
|
by the gate simulation procedure, given that ~s is input. D(px(y),p˜x(y)) ǫ. (36)
≤
Then we have p(~s′)= p(~s′~s)p(~s), so
~s | The total number of times the gate simulation proce-
σ′ = Pp(~s) p(~s′~s)ρ(~s′). (27) dure is performed is p(n), and the number of operations
~s ~s′ | performed in one iteration of the gate simulation proce-
X X durescalesaspoly(2l),sothetotalnumberofoperations
Applyingthedoubleconvexityofthetracedistancegives
in the classical simulation is O(poly(p(n)/ǫ)), where we
abuse notation by letting poly() be a (new) polynomial
·
D( (σ),σ′) p(~s)D (ρ(~s)), p(~s′~s)ρ(~s′) . function. We have proved the following theorem:
E ≤ ~s E ~s′ | !
X X
Theorem 4 Let be a fixed set of one- and two-qubit
(28)
G
gates. Suppose all two-qubit gates in are separable.
G
By inspection of the construction used in the gate sim- Let C be a uniform family of quantum circuits of size
n
{ }
ulation procedure, notably Eq. (20), and the stability p(n), acting on q(n) qubits, where both p(n) and q(n) are
property for trace distance, we have polynomials. Theinitialstateofthecomputerisassumed
to be a computational basis state, x . The computation
| i
is concluded by performing a measurement in the com-
D (ρ(~s)), p(~s′~s)ρ(~s′) c2−l. (29)
E X~s′ | !≤ qpuubtaittsi,onyaielldbiansgisaopnrosboambielitpyredsipsterciibfiuetdionsupbsxe(ty,)Sov,eorfptohse-
Combining this observation with Eq. (28) gives sible measurement outcomes y. Then for any ǫ > 0 it
is possible to sample from a distribution p˜ (y) satisfying
x
D( (σ),σ′) c2−l, (30) D(p (y),p˜ (y)) < ǫ using a classical algorithm taking
E ≤ x x
O(poly(p(n)/ǫ)) steps, where poly() is some fixed poly-
which was the desired result. 2 ·
nomial.
Proposition 2 For m = 0,...,p(n), D(σm,σ˜m)
cm2−l. ≤ Results related to Theorem 4 have been obtained in
the past, but, so far as we have determined, no proof
Proof: We induct on m. For m = 0 the result fol- of this result has previously been published. In particu-
lows from the fact that σ0 = σ˜0. Assuming the result is lar,AharonovandBen-Or [11]studied the roleof entan-
true for m, we now prove it for m+1. By the triangle glement in quantum computation, proving that many-
inequality party entanglementmust be presentin order for a quan-
tum computation to be difficult to simulate classically.
D(σm+1,σ˜m+1) D(σm+1, m+1(σ˜m)) This conclusion was subsequently clarified and extended
≤ E
+D( (σ˜m),σ˜m+1). (31) by Jozsa and Linden [24]. However, the conclusions of
m+1
E
7
both [11] and [24] are not applicable in the present con- IV. ROBUSTNESS OF QUANTUM STATES
text, since they apply inthe contextofpure state entan-
glement of a quantum computer, rather than the mixed Tounderstandhowrobustquantumgatesaretonoise,
state case considered in this paper. itisusefultofirstreviewpriorworkonthe robustnessof
Theissueofmixedstatequantuminformationprocess- entangled quantum states. This section describes Vidal
ing was considered by Braunstein et al [25], who raised, and Tarrach’s [12] definitions and results on the robust-
withoutanswering,thequestionofwhatrolemixed-state ness of quantum states, introduces a novel measure of
entanglement can play in quantum computation. This robustness, and relates that measure to Vidal and Tar-
lineofthoughthasbeencarriedfurtherbymanyauthors, rach’smeasure. Thenovelmeasureanditspropertieswill
without completely answering the question. See [26, 27] be of especial interestin applications to gate robustness.
for recent work and further references. Let ρ be a quantum state of a bipartite system, AB,
and let σ be a state of AB. Vidal and Tarrach [12] de-
fine the robustness of ρ relative to σ, R(ρ σ), to be the
C. Separability-preserving gates smallest non-negative number t such thatkthe state
1 t
It is straightforwardto extend the proofof Theorem4
ρ+ σ (38)
inavarietyofways,withoutchangingtheconclusionthat 1+t 1+t
a classical simulation of the quantum circuit is possible.
is separable. Equivalently, we can define R(ρ σ) to be
In particular, we can change the gates in so they can k
G the smallest non-negative number t such that ρ+tσ is
actonanybounded number ofqudit systems,ratherthan
separable;this latter definition in terms ofunnormalized
two-qubit systems.
quantum states will frequently be useful in later work.
Furthermore, the proof relies on properties of gates in
Note that [12] specify that σ be separable; however, we
that are weaker than separability. In particular, the
G will find it convenient to extend the definition to non-
gates in need only be separability-preserving, that is,
G separable σ also, specifying that R(ρ σ) + if no
(ρ)isseparableforanyseparablestateρ. Wedenotethe k ≡ ∞
E value oft exists suchthatthe statein Eq.(38)is separa-
class of separability-preservinggates by SP. To see that
this is a weakerproperty,note thatswapis separability- ble. At first sight one is tempted to ask why we choose
this definition for the robustness, and not the related
preservingsinceitmapsproductstatestoproductstates,
but swap is not separable, since it can generate entan- quantity
glement with the aid of local ancilla systems. More gen-
min p:p 0,(1 p)ρ+pσ is separable . (39)
erally, note that AB is separable with respect to A : B { ≥ − }
E
if andonly if EAB⊗IA′B′ is separability-preservingwith This latter definition has a more obvious physical inter-
respect to AA′ :BB′.
pretationastheminimalprobabilitywithwhichσcanbe
SincetheproofofTheorem4onlyreliedonthestatein
mixedwithρ to obtainaseparablestate. It followsfrom
Eq.(21)beingseparable,itstillholdswhenthe available
the definitions that the quantity of Eq. (39) is equal to
gates are all separability-preserving. However, no sim-
R(ρ σ)/(1+R(ρ σ)). The reason we do not work with
ple and easy-to-use characterization of the separability- k k
thequantityofEq.(39),despiteitsapparentlymorecom-
preserving gates is known, which is why we prefer, for
pellingphysicalinterpretation,isthattherobustnessde-
most of the remainder of this paper, to work with the
fined in Eq. (38) has useful and easy-to-prove convexity
separable gates. We do make occasional later use of
properties not satisfied by Eq. (39), namely, R(ρ σ) is
separability-preserving gates, so it is convenient to note k
convex in both the first and the second entry.
here a few of their properties. Note that all separable
A special case of R(ρ σ) of particular interest is the
gates are in SP, and for gates operating on multiple qu- k
random robustness, defined to be the robustness ofρ rel-
dits, any permutation of the qudits (for example swap)
ative to the maximally mixed state I/d d . We de-
A B
is in SP. Furthermore, SP is convex and is closed under
note the random robustness of a state ρ by R (ρ)
r
composition, so ≡
R(ρ I/d d ). VidalandTarrach[12]foundausefulfor-
A B
k
mula for the random robustness of a pure state ψ of AB
SP Hull : separable and a permutation .
⊇ {E ◦P E P }(37) iwnittheromrdseroefdaSSchchmmididttcodeefficocmienptossiψtion ψψ = ...jψj0|j:i|ji
1 2
≥ ≥P ≥
However,it is unclear whether this convexhull describes R (ψ)=ψ ψ d d . (40)
r 1 2 A B
all of SP. For example, the operation which measures a
pairofaqubitsintheBellbasisandstoresthe answerin So far we have discussed the robustness of a state ρ
the computational basis (i.e. (00 + 11 )/√2 becomes relative to another fixed state σ. We now define the ro-
| i | i
00 ,(00 11 )/√2becomes 01 ,etc...) iscertainlyin bustness of ρ, R(ρ), to be the minimum relative robust-
| i | i−| i | i
SPthoughisdoesnotseemasthoughitcanbeexpressed ness R(ρ σ) over all separable σ. Thus, the robustness
k
as a convex combination of for separable and of ρ is a measure of how much local noise can be mixed
k k k
E ◦P E
permutations [36]. with ρ before it becomes separable.
k
P
8
We have defined three notions of robustness for quan- so we have
tum states, R(ρ σ),R (ρ), and R(ρ). All three of these
r
k 2
definitions have assumed that ρ is a state of a bipartite
quantum system, AB. However, robustness is easily ex- R (ψ)=R(ψ)= ψ 1. (43)
g j
  −
tended to more than two parties, and it is convenient to j
X
haveanotationtoexpresstheextendednotion. Suppose,  
forexample,thatρandσarestatesofatripartitesystem We do not know whether R (ρ) = R(ρ) in general. To
g
ABC. Then RA:B:C(ρ σ) is defined to be the minimal complete the proof of Eq. (43), we show that if there
k
value of t such that ρ+tσ is separable with respect to exists a density operator σ such that
A:B :C.
Of course, a many-party quantum system can be de- ψ+tσ (44)
composed in many different ways, by grouping subsys-
tems together. So, for example, we can define a notion is separable, then t ( ψ )2 1. (Our proof both
of robustness, RA:BC(ρ σ), when system B and C are extends and simplifie≥s a simjilajr p−roof in [12] for the ro-
grouped together. Morke explicitly, RA:BC(ρ σ) is de- bustness R(ρ).) P
k
fined to be the minimal value of t such that ρ +tσ is The proof is based on the positive partial transpose
separable with respect to A:BC. criterionofPeres[28]. Letusdenotethepartialtranspose
These examples may be extended in a natural way to on systems A and B by T and T , respectively. Then
A B
the random robustness and robustness, as well as to the thepositivepartialtransposecriterionimpliesthatifthe
case where more systems are present, and to more com- state of Eq. (44) is separable, then
plicatedgroupingsofsubsystems. Mostofourworkcon-
cerns two-partyrobustness,andso we usually do notex- 0 ψTB +tσTB, (45)
plicitly include superscripts in expressions like RA:B(ρ). ≤
The robustness has many useful properties, which are where indicates an operator inequality, that is, we are
≤
explored in detail in [12]. We mention just a few of the saying that the operator on the right-hand side is posi-
more striking properties here. The robustness is invari- tive.
ant under local unitary operations. Moreover, it is an WenowuseEq.(45)todeducealowerboundont. To
entanglementmonotone,thatis,cannotbeincreasedun- do this we introduce an operator, M, defined by M
der local operations and classical communication. It is I swap, where swap j k k j is the linea≡r
also a convex function of ρ. As for the random robust- op−eratorinterchanging≡statejsko|fishys|te⊗m|Aihan|dsystemB.
ness, Vidal and Tarrach [12] have obtained an elegant Note that M is positive,Psince swap2 = I implies that
formula for the robustness in the special case of a pure swaphaseigenvalues 1,andthusM isadiagonalizable
state, ψ, of a bipartite system, AB, operator with eigenva±lues 0 and 2.
Since the trace of a product of two positive operators
2
is non-negative, it follows from Eq. (45) that
R(ψ)= ψ 1, (41)
j
 j  − 0 tr MψTB +ttr MσTB . (46)
X ≤
 
(cid:0) (cid:1) (cid:0) (cid:1)
whereψ aretheSchmidtcoefficientsforψ. Inthecourse Usingalittlealgebraandtheobservationthatforanytwo
j
of their proof, Vidal and Tarrach explicitly construct a operators,K andL,tr(KLTB)=tr(KTAL),theprevious
state, σ , such that ψ ψ +R(ψ)σ is separable. σ equation may be rewritten
ψ ψ ψ
| ih |
maybeexpressedintermsoftheSchmidtdecomposition
ψ = ψ j j by tr MTAψ ttr MTAσ . (47)
j j| i| i − ≤
P 1 Direct calculation(cid:0)shows(cid:1)that (cid:0) (cid:1)
σ = ψ ψ k k l l . (42)
ψ k l
R(ψ) | ih |⊗| ih |
Xk6=l MTA =I α α, (48)
−| ih |
In the definition of robustness we mixed ρ with a
where α j j is the (unnormalized) maxi-
separable quantum state, σ, trying to determine what | i ≡ j| i| i
mally entangled state. Using Eq. (48) it follows that
minimal level of mixing will produce separability. An- P
tr MTAψ = 1 ( ψ )2 and tr MTAσ tr(σ) = 1.
other natural definition of robustness would allow σ to − j j ≤
range over arbitrary density matrices, not just separa- Substituting these results into Eq. (47) gives
(cid:0) (cid:1) P (cid:0) (cid:1)
ble density matrices. That is, we can define R (ρ)
g
≡ 2
min R(ρ σ),wheretheg subscriptindicatesthatweare
σ
k
minimizing globally over all possible density matrices σ. ψ 1 t, (49)
j
  − ≤
How are Rg(ρ) and R(ρ) related? It is clear from the Xj
definitions that Rg(ρ) R(ρ). We will prove that the  
≤
reverseinequalityisalsotruewhenρ=ψ isapurestate, which was the desired bound.
9
V. ROBUSTNESS OF QUANTUM GATES and B with probability p. Then the threshold probabil-
ity at which this gate crosses the separable-inseparable
We now extend state robustness to quantum gates. threshold is:
Suppose and are trace-preserving quantum opera- R ( )
tions on Ea compFosite system AB. Then we define the r E . (53)
1+R ( )
r
robustness of relative to , R( ), to be the mini- E
E F EkF
mum value of t such that From Eq. (51) we see that the random robustness for an
operation is related to the random robustness of a state
1 t
+ (50) by
1+tE 1+tF
R ( )=RRAA:BRB(ρ( )). (54)
r E r E
is separable. Equivalently, R( ) can be defined to be
the minimal value of t such EthkaFt + t is separable. Specializing to the case where is a unitary quantum
Applyingtheoperation-separabilityEtheorFemweimmedi- operation, U, we see that Rr(UE) = RrRAA:BRB(ρ(U)).
ately find the useful formula However, ρ(U) is a pure state. We showed earlier that
ρ(U) has Schmidt coefficients u /√d d , where u are
j A B j
R( )=RRAA:BRB(ρ( ) ρ( )). (51) theSchmidtcoefficientsofU. Thisobservation,together
EkF E k F with Eqs. (54) and (40) implies the formula
Just as for quantum states, the notion of gate robust-
R (U)=d d u u , (55)
ness extends in a natural way to systems of more than r A B 1 2
two parties, and we use notations analogous to those in- where we order the Schmidt coefficients of U so that
troducedearlier,suchasRA:B:C( )andRA:BC( ), u u ... 0. (Note that in deriving this equation,
to describe this scenario. Note thEaktFthese notationEskwFill w1e≥hav2e≥replac≥ed d by d2, and d by d2 in Eq. (40),
A A B B
alsobe extendedin a naturalwayto the randomrobust- sinceweareworkingwithrobustnessfortheR A:BR
A B
nessandrobustnessofaquantumgate,asdefinedbelow. system.)
As forquantumstates,whenidentifying superscriptsare Itis,perhaps,notimmediatelyclearwhatthephysical
omitted we assume that the quantum gate in question relevance of the random robustness is. After all, in real
acts on a bipartite system, AB. physical systems, the effects of noise on a quantum gate
Motivated by several different classes of noise com- willnotusuallybe tosimplymixinsomedepolarization,
monly occurring in physical systems, we now use the together with the gate. Despite this, there is still a very
notionofrelativegaterobustnesstodefineandstudysev- good physical reason to be interested in the random ro-
eral different measures of robustness for quantum gates. bustness. Thereasonisthat,asweshowinmoreexplicit
Firstistherandomrobustness,whichwedefineandstudy detail below, the randomrobustness canbe used to ana-
in Sec. VA. Also in this subsection, we use results on lyze the particularnoisemodels whichhavebeenusedin
the random robustness to place bounds on the thresh- estimatingbounds onthe thresholdforquantumcompu-
old for quantum computation. Two other measures of tation. Inturn,ithasbeenargued[6,7,8,9,10]thatby
robustness are the separable robustness and the global analyzing and correcting for the effects of noise in those
robustness,whichwedefineinSec.VB,andusetoprove particular models, it is possible to make general state-
bounds on the threshold for quantum computation. Our ments about a wide class of physically reasonable noise
resultsonthesemeasuresofrobustnessarelesscomplete, models. Thus, althoughthe physicalscenarioconsidered
and so our discussion is more limited. inthedefinitionoftherandomrobustnessappearsrather
specialized,it willenable insightinto muchmore general
physical situations.
A. Random robustness of quantum gates Asanexample,wemayaskhowrobustthecontrolled-
not is against the effects of depolarizing noise?
1. Definition and basic properties The controlled-not has Schmidt decomposition [13]
√20 0 I/√2+√21 1 X/√2, so Eq. (55) implies
The random robustness of , R ( ), is defined to be tha|tiRh (|c⊗not)=8. In|teirhes|t⊗ingly,we canalsoshow that
r r
equal to the robustness of ErelativEe to the completely the controlled-not is the most robust two-qubit gate.
E
depolarizing channel, (ρ) = I/d d for all states ρ of To see this, note that unitarity of U implies that the
A B
system AB: D Schmidt coefficients u satisfy u2 = d d , and thus
j j j A B
u2 +u2 d d . It follows from this observation and
Rr(E)≡R(EkD). (52) E1q. (552)≤thatARBr(U) ≤ d2Ad2B/2P, and thus no two-qubit
unitary gate can have random robustness greater than
The random robustness is especially interesting because 8,whichisthe randomrobustnessofthe controlled-not.
it measures the robustness of against complete ran-
E These results are worth highlighting as a proposition:
domization of systems A and B. Another way of stating
this is to imagine that we are applying the operation Proposition 3 For any quantum operation , R ( )
r
with probability 1 p, and randomizing the systems AE d2d2/2. If d =d =2 then R ( ) R (CENOT)E=8≤.
− A B A B r E ≤ r
10
The random robustness has many physically interest- Proof: Let u and v be the ordered Schmidt coeffi-
j j
ingproperties. Belowwelistsixeasily-provedproperties, cients of U and V, respectively. From Eq. (55),
before discussing in more depth two less easily-proved
R (U) R (V) = d d u u v v
properties. Our discussion of these properties is, in r r A B 1 2 1 2
| − | | − |
part,motivatedbytheframeworkof“dynamicstrength” = dAdB (u1 v1)u2+v1(u2 v2)
| − − |
measures introduced in [13], although the properties we d d u v u + v u v
A B 1 1 2 1 2 2
discuss are interesting independent of that motivation. ≤ | − || | | || − |
d2d2 (u v + u v )
In [13] it was argued that these properties, especially ≤ A B | 1− 1| | 2− 2|
the property of chaining, discussed below, are essential ≤ d2Ad2B |uj −vj|. (59)
if a measure can be said to quantify the strength of a j
X
quantumdynamicaloperationasaphysicalresource. By
The second part of the proof is to observe that by the
showingthatthese propertiesaresatisfied,wethus show
Cauchy-Schwartzinequality,
thattherandomrobustnessisagoodmeasureofdynamic
strength. u v d (u2+v2 2u v ) (60)
| j − j| ≤ M j j − j j
1. Non-negativity and locality: Rr( ) 0 with Xj Xj
eoqpueraalittiyoni.f and only if E is a separabEle≥quantum = 2dMdAdB 1− djudjvj . (61)
(cid:18) PA B (cid:19)
2. Local unitary invariance: If , , , are Applying Proposition1 we obtain
A B A B
U U V V
alllocalunitary quantumoperations,with the sys-
tembeingactedonindicatedbythesubscript,then uj vj dMdAdB U V 2 (62)
| − |≤ k − k
j
X
R (( ) ( ))=R ( ). (56)
r UA⊗UB ◦E ◦ VA⊗VB r E Combining with Eq. (59) gives the result. 2
Another physically interesting question is to ask how
3. Exchange symmetry: R ( ) = R (swap
swap), that is, the randomr rEobustnerss is no◦tEaf◦- therandomrobustnessofagateE1◦E2composedofquan-
tum gates and relates to the random robustness of
fected if we interchange the role of the systems. E1 E2
the individual gates. The following proposition bounds
4. Time-reversal invariance: For a unitary, U, the random robustness of the combined operation:
R (U)=R (U†).
r r Proposition 5 (Chaining for random robustness)
Let be a doubly stochastic quantum operation, that
5. Convexity: The randomrobustness R ( ) is con- 1
r E
E is, a quantum operation which is both trace-preserving
vex in .
E and unital (i.e. (I) = I), and let be an arbitrary
1 2
E E
6. Reduction: Suppose a trace-preserving quantum trace-preserving quantum operation. Then
operation actingonAB isobtainedfromatrace-
E R ( ) R ( )+R ( )+R ( )R ( ). (63)
preservingquantumoperation actingonABC as r 1 2 r 1 r 2 r 1 r 2
E ◦E ≤ E E E E
F
follows:
Note that unitary operations are trace-preserving and
unital, so the proposition is true when and are
E(ρAB)=trC[F(ρAB⊗σC)], (57) unitary. There is an equivalent way of phEr1asing PEr2opo-
sition 5 that is physically more intuitive. Suppose we
for some fixed state σ of system C. Then the
C
define
randomrobustnesssatisfiesthereductionproperty,
namely, RA:B( ) RA:BC( ).
r E ≤ r F Cr( ) ln(1+Rr( )). (64)
E ≡ E
The random robustness satisfies two other physically Then C ( ) is monotonically related to the random ro-
r
interesting properties that are more difficult to prove. bustness oEf , and thus can be thought of as carrying
First of all, the random robustness is continuous in . thesamequaElitativeinformationabouttherobustnessof
E
Physically,this is self-evident: making a small change in the gate. Simple algebra shows that the conclusion of
should not too drastically affect its robustness against Proposition 5 may be recast in the form
E
the effects ofnoise. We nowprovea quantitativeformof
this statement for unitary gates. Cr( 1 2) Cr( 1)+Cr( 2). (65)
E ◦E ≤ E E
The simplicity and clarity of this form may, perhaps,
Proposition 4 (Continuity of random robustness)
make it more useful in some circumstances.
Let U and V be unitary gates acting on a system A of
Proof: By definition of the random robustness, the
dimension d , and a system B of dimension d . Then
A B
quantum operations
R (U) R (V) d d3d3 U V 2, (58)
| r − r |≤ M A Bk − k +R ( ) , and (66)
1 r 1
E E D
where d min(d ,d ). +R ( ) (67)
M A B 2 r 2
≡ E E D