Distance measures to compare real and ideal quantum processes

0.29 MB·English

by  Alexei Gilchrist

Distance measures to compare real and ideal quantum processes
Alexei Gilchrist,1,∗ Nathan K. Langford,1,† and Michael A. Nielsen2,‡
1Centre for Quantum Computer Technology and Department of Physics,
The University of Queensland, Brisbane, Queensland 4072, Australia.
2School of Physical Sciences and School of Information Technology and Electrical Engineering,
The University of Queensland, Brisbane, Queensland 4072, Australia.
(Dated: January 27, 2009)
With growingsuccess inexperimentalimplementationsitis critical toidentifya“gold standard”
for quantuminformation processing, a single measure of distance that can beused to compare and
contrastdifferentexperiments. Weenumerateasetofcriteriasuchadistancemeasuremustsatisfy
to be both experimentally and theoretically meaningful. We then assess a wide range of possible
measures against thesecriteria, before making arecommendation as tothebest measures touse in
9 characterizing quantuminformation processing.
0
0 PACSnumbers: 03.67.Lx
2

n
I. INTRODUCTION basis states (i.e., verifying the truth table of the gate),
a
J and a few superposition states. Such demonstrations
are important, but it is clear that a figure of merit that
7 Many real-world imperfections arise when experimen-
2 tally performingaquantuminformationprocessingtask. is standardized, theoretically well motivated and experi-
mentally practicalwould be a considerable step forward.
These may arise either in the creation or measurement
2 Parenthetically, we note that such a measure would also
of a quantum state, or in the manipulation of the state
v be ofgreatuseinconcretelyconnectingrealexperiments
3 via some quantum process. It is important to quantita-
to results such as the fault-tolerance threshold for quan-
6 tively measure and characterize these imperfections in a
tum computation [8].
0 way that is theoretically meaningful and experimentally
8 practical. The purpose of this paper is to comprehensively ad-
0 dress the problem of developing such error measures.
How can this be done? Quantum states can be
4 Thereisasizeablepreviousliteratureonthissubject,but
completely determined using quantum state tomogra-
0 we believe that there has been a consistent gap between
/ phy [1, 2] and compared using a variety of well-known
h work motivated primarily by theoretical considerations,
measures[3]. Quantumprocessescanbe measuredusing
p and work constrainedby experimental realities. Our pa-
ananalogousprocedurecalledquantumprocesstomogra-
- per aims to address both theoretical and experimental
t phy [3, 4, 5]. However, the problem of developing quan-
n desiderata.
titativemeasurestocomparerealandidealizedquantum
a The key to our work is to introduce a list of six sim-
u processes has not been comprehensively addressed.
ple, physically motivated criteria that should be satis-
q Ideally there would be a single good measure, a “gold
fied by any good measure of distance between quantum
:
v standard” [6], enabling sensible comparison of different processes. These criteria enable us to eliminate many
i experimental implementations of quantum information
X approaches to the definition of an error measure that a
processing,andagreeduponbyexperimentalistsandthe-
priori appear highly plausible.
r orists alike. We will refer to candidates for such a gold
a The criteria are as follows. Suppose ∆ is a candidate
standard as “distance measures” for quantum processes,
measureofthedistancebetweentwoquantumprocesses.
or as “errormeasures”,when we wantto stress the com-
Suchprocessesaredescribedbymapsbetweeninputand
parison of real and idealized processes.
output quantum states, e.g., ρ = (ρ ), where the
out in
Suchanerrormeasurewouldbeextremelyusefulboth map is known as a quantum operatiEon [3, 9]. Physi-
when comparing experiments with the theoretical ideal, E
cally,∆( , ) may be thought ofin two ways: as a mea-
and in comparing different experiments that attempt to E F
sure of error in quantum information processing when
perform the same task. Existing experiments in quan-
one wants to do the ideal process but does instead;
tum information processing have typically been assessed F E
or of distinguishability between the two processes and
onaratheradhoc basis. Forexample,someimplementa- E
. We believe that any such measure must satisfy the
tionsofquantumlogicgateshavereliedondemonstrating F
following six properties, motivated by both physical and
thatthosegatesactinthe correctwayoncomputational
mathematical concerns.
(1) Metric: ∆ should be a metric. This requires three
properties: (i) ∆( , ) 0with ∆( , )=0 if andonly
E F ≥ E F
if = ; (ii) Symmetry: ∆( , ) = ∆( , ); and (iii)
∗Electronicaddress: [email protected] E F E F F E
the triangle inequality ∆( , ) ∆( , )+∆( , ).
†Electronicaddress: [email protected] E G ≤ E F F G
‡Electronic address: [email protected]; (2)Easy to calculate: itshouldbe possibleto evaluate
URL:www.qinfo.org/people/nielsen ∆ in a direct manner.
2
(3) Easy to measure: there should be a clear and sures for quantum information processing, within the
achievable experimental procedure for determining the broad framework of the criteria we have identified. So
value of ∆. far as we are aware,none of the prior work has surveyed
(4) Physical Interpretation: ∆ should have a well- andcomparederrormeasuresagainstsuchabroadarray
motivated physical interpretation. of theoretical and experimental concerns.
(5)Stability [10]: ∆( , )=∆( , ), where Error measures for quantum teleportation have re-
I⊗E I⊗F E F I
represents the identity operation on an additional quan- ceived particular attention in the prior literature, per-
tum system. Physically,this meansthatunrelatedancil- haps spurred by controversy over which experiments
lary quantum systems do not affect the value of ∆. shouldberegardedasdefinitivelydemonstratingthetele-
(6) Chaining: ∆( , ) ∆( , ) + portation effect [11]. Examples of this line of devel-
2 1 2 1 1 1
E ◦ E F ◦ F ≤ E F
∆( , ). Thus,foraprocesscomposedofmanysmaller opment include [12, 13, 14, 15, 16, 17], and references
2 2
E F
steps, the total error will be less than the sum of the er- therein. With the exception of Ref. [17] this work differs
rors in the individual steps. from ours in that it is focused primarily on the prob-
The chaining and stability criteria are key properties lem of teleportation. Reference[17] has a more general
for estimating the error in a complex quantum informa- focus, but is not primarily concerned with the develop-
tion processing task. Because quantum information pro- ment of error measures, but rather with the question of
cessingtasksaretypicallybrokendownintoasequenceof when quantum information processing can be modeled
simpler component operations, a conservative bound on classically.
thetotalerrorcanbefoundbysimplyanalyzingtheindi- More mathematical investigations of error measures
vidual components. This is critical for applications such have also been mounted, especially in the context of
asquantumcomputation,wherefullprocesstomography quantum communication and fault-tolerant quantum
on an n-qubit computation requires exponentially many computation. Examples of this work include [10, 18,
measurements, and is thus infeasible. Chaining and sta- 19, 20, 21, 22, 23, 24, 25, 26], and references therein.
bility enable one to instead benchmark the constituent This work(oftenembedded in somelargerinvestigation)
processes involved in the computation, which can then typically focuses on one or a few measures of specific in-
be used to infer that the entire computation is robust. terest for the problem at hand. These papers thus differ
Many other properties follow from these six criteria. from our work in that they don’t attempt a comprehen-
Forexample,fromthemetricandchainingcriteriawesee sivesurveyofpossibleerrormeasuresagainstsomesetof
that ∆( , ) ∆( , ), where is any quan- abstract criteria; nor, typically, do they address experi-
R◦E R◦F ≤ E F R
tumoperation. Thiscorrespondstotherequirementthat mental criteria such as ease of measurement. Nonethe-
post-processingby cannotincrease thedistinguishabil- less, while this prior work is different in character from
R
ity of two processes and . Another elementary con- ours, it has greatly informed our point of view, and we
E F
sequence of the metric and chaining criteria is unitary willhaveoccasiontociteitonspecificpointsthroughout
invariance, i.e., ∆( , )=∆( , ), where this paper. Ofparticular relevanceis Ref. [10], whichin-
U◦E◦V U◦F◦V E F
and are unitary operations. troduced one of the key measures we use, the stabilized
U For bVoth theoreticians and experimentalists, there are process distance, or S distance (referred to as the dia-
strong motivations to find a gold standard satisfying mond norm in Ref. [10]), and emphasized some of the
these criteria—the need for a physically sensible way of important properties satisfied by that measure.
evaluating the performance of a quantum process, and Structure of the paper: Secs. II and III summarize
the need to compare the success of a theoretical model background material on quantum operations and dis-
to the operation of a real, experimental system. For the tance measures for quantum states.
experimentalist,however,thereisalsoanotherimportant Section IV is the core of the paper, comprehensively
consideration. That is the need for diagnostic measures surveying possible approaches to the definition of error
which can be used to build insight into the source of im- measures. Ourstrategyistocastawidenet, considering
perfectionsinexperimentalimplementations. Diagnostic many different possible approaches to the definition of a
measuresmaynotnecessarilybegoodcandidatesforour distance measure, and then to use our list of criteria to
sought-after gold standard — they may fail to satisfy eliminate as many approaches as possible. This means
one or more of our criteria — but they still may be ex- a certain amount of tedium as we propose and then re-
tremely useful in the experimental context. Thus, some ject certain a priori plausible candidate error measures.
ofthemeasureswediscardasunsuitableforuseasagold Thebenefitofgoingthroughthisprocessofeliminationis
standardmaystillbeusefulasdiagnosticmeasures. Fur- considerable, however. First, it gives us confidence that
thermore, it is not difficult to construct other examples the few measures we identify as particularly promising
ofusefuldiagnosticmeasures,differenttoanyconsidered should be preferred over all other measures. Indeed, we
inthispaper. Thedetailedinvestigationofsuchdiagnos- quicklyeliminateallbutfourofthemeasureswedefineas
ticmeasuresis,however,beyondthescopeofthepresent follows: theJamiolkowski process fidelity (J fidelity),the
paper. Jamiolkowski process distance (J distance),thestabilized
Prior work: The principalcontributionofourpaperis process fidelity (S fidelity),andthestabilized process dis-
to comprehensively evaluate many plausible error mea- tance (S distance). Second, inseveralinstances we show
3
thaterrormeasuresproposedpreviouslyintheliterature where (χ ) a a∗ are the elements of the pro-
E mn ≡ j jm jn
(in one case, by one of the authors of this paper) should cess matrix, χE. PEquation (2) tells us that the process
be rejected as inadequate. matrix completely describes the action of the quantum
Section V applies the four promising measures identi- process. The big advantage of the process matrix repre-
fied in Sec. IV to the concrete problemof quantum com- sentationisthat,unliketheoperator-sumrepresentation,
putation, showing that each measure has a useful opera- once the basis A is chosen the process matrix can be
j
{ }
tional interpretation in terms of the success or failure of shown to be unique to the process [30]; i.e., it depends
a quantum computation. only on , not on the particular choice of operation el-
E
SectionVIconcludesthepaperwithasummaryofour ements E . We will not give an explicit proof of this
j
results, and the identification of the S distance and the facthere{,bu}tnotethatthisresultfollowseasilyfromthe
S fidelity as the two measures whose properties make discussion below.
them the most attractive candidates for use as a gold The process matrix gives a convenient way of repre-
standardin quantuminformationprocessing. We do not senting the operation . A closely-related but more ab-
make a final recommendation as to which of these two stractrepresentationisEprovidedbytheJamiolkowskiiso-
measuresshouldbe used,since they haveextremely sim- morphism [31], which relates a quantum operation to
ilar strengths and weaknesses. However, we do discuss a quantum state, ρ : E
E
and make definite recommendations regarding the re-
porting of quantum information processing experiments. ρ [ ](Φ Φ), (3)
E
≡ I⊗E | ih |
Furthermore, we sketch future research directions which
may ameliorate some of the weaknesses of one or both where Φ = j j /√disamaximallyentangledstate
measures, and which may therefore make it possible to | i j| i| i
ofthe(d-dimPensional)systemwithanothercopyofitself,
definitively choose a single measure as a gold standard. and j is some orthonormal basis set. The map
{| i} E →
ρ is invertible, that is, knowledge of ρ is equivalent to
E E
knowledge of [32]. This isomorphism thus allows us
II. DESCRIBING QUANTUM PROCESSES E
to treat quantum operations using the same tools as are
ordinarilyusedtotreatquantumstates. Forlaterusewe
Quantum operations describe the most general phys- note the useful property ρ =ρ ρ .
E⊗F E F
ical processes that may occur in a quantum system [3], The state ρ and the processmatr⊗ix χ areclosely re-
E E
includingunitaryevolution,measurement,noise,andde- lated. A direct calculation shows that if one chooses the
coherence. Any quantum operation may be given the operator basis sets A = m n , then χ = dρ , as
j E E
operator-sum representation relating input ρin and out- matrices. Thus we {shal}l refe{r| toihb|o}th χE and ρE as the
put ρout states, process matrix, and treat them interchangeably. This is
very convenient, as ρ is easy to work with mathemat-
ρ = (ρ )= E ρ E†, (1) E
out E in j in j ically, using the expression Eq. (3), while the elements
Xj ofχ haveanobviousphysicalsignificance,expressedby
E
wheretheoperatorsE areknownasoperation elements, Eq. (2).
j
and obey the condition that E†E I [27]. Note We conclude this section with a comment on our no-
j j j ≤ tationalconventions. We oftenuse notationlikeψ tode-
that the operation elements {PEj} completely describe note either a pure state ψ orthe correspondingdensity
the effect of the process. We will mostly be concerned | i
matrix ψ ψ , with the meaning to be determined from
with the case of trace-preserving operations, for which | ih |
context. Thus,forexample,wemaywriteψ =α0 +β 1
jEj†Ej = I. Physically, this corresponds to the re- toindicate a purestate ofa singlequbit, while a|lsiowr|iti-
qPuirement that represents a physical process without
E ing (ψ) to indicate a quantum operation acting on
post-selection [28]. Many of our results extend easily to E E
the density matrix corresponding to that pure state.
the case of non trace-preserving operations, but to ease
the exposition we assume processes are trace-preserving
unless otherwise noted.
III. DISTANCE MEASURES FOR QUANTUM
The operator-sum representation has the drawback
STATES
thatit is notunique,in the sensethat thereis a freedom
in the choice of operation elements [3]. This is inconve-
A natural starting place for an attempt to define a
nientifwearetryingtocomparetwoprocesses. Toallevi-
measure of distance for quantum processes is measures
atethis,letusfixabasis A forthespaceofoperators,
j
{ } of distance for quantum states. The quantum informa-
choosing for convenience a basis orthonormal under the
Hilbert-Schmidt inner product, i.e., tr(A†A )=δ [29]. tion science community has identified the trace distance
j k jk and the fidelity as particularly important approaches to
We can use this basis to expand the operation elements,
the definition of a distance measure for states [33], and
E = a A , and rewrite Eq. (1):
j m jm m these two measures will serve as the basis for our later
P
(ρ)= (χ ) A ρA† (2) definitions ofdistance measuresfor quantumoperations.
E E mn m n In keeping with the aims of the paper, we don’t make
Xmn
4
a choice between the trace distance and the fidelity at a fact that is not obvious from the definition we have
the outset. Instead,ourpreferenceis todevelopdistance given, but which follows from other equivalent defini-
measuresforquantumoperationsbasedonboth thetrace tions.
distance and the fidelity, and then assess them using the There is an ambiguity in the literature in the defini-
criteriadiscussedinthe introduction. We nowbrieflyre- tion of fidelity that is worth commenting on here. Both
view the basic properties of the trace distance and the thequantitydefinedaboveanditssquareroothavebeen
fidelity. referredtoasthe fidelity,andbothhavemanyappealing
Thetracedistance: Thetracedistance betweendensity properties [36].
matricesρandσisdefinedbyD(ρ,σ)≡ 21tr|ρ−σ|,where Nevertheless,westronglyadvocateusingthedefinition
X √X†X. From this definition it follows that the of Eq. (4), despite the other definition being used in ref-
| | ≡
trace distance is a genuine metric on quantum states, erencessuchas[3]. AswewillseeinSec.V,adoptingthe
with 0 D 1. The trace distance also has many definition of Eq. (4) gives rise to a measure of distance
≤ ≤
other attractive properties that make it a particularly betweenquantumprocesseswithaphysicallycompelling
good measure of distance between quantum states. We interpretation in terms of the probability of success of a
now briefly describe three of these. quantum computation. Adopting the other definition of
First, the trace distance has a compelling physical in- fidelitywouldmakeaboutasmuchsenseasreportingthe
terpretationasameasureofstatedistinguishability. Sup- square root of the probability that the quantum compu-
poseAlicepreparesaquantumsysteminthestateρwith tation succeeded.
probability 1,andinthe stateσ withprobability 1. She Althoughnotametric,thefidelitycaneasilybeturned
2 2
givesthesystemtoBob,whoperformsaPOVMmeasure- into a metric. Two common ways of doing this are the
ment [3] to distinguish the two states. It can be shown
Bures metric, defined by B(ρ,σ) 2 2 F(ρ,σ),
thatBob’sprobabilityofcorrectlyidentifyingwhichstate ≡ q −
Alice prepared is 1/2+D(ρ,σ)/2. That is, D(ρ,σ) can and the angle, defined by A(ρ,σ) arccospF(ρ,σ).
≡
be interpreted, up to the factor 1/2, as the optimal bias The origin of these metrics can be seen intupitively by
in favour of Bob correctly determining which of the two considering the case when ρ and σ are both pure states.
states wasprepared. This physicalinterpretationfollows The Bures metric is just the Euclideandistance between
from the identity D(ρ,σ) = max tr(E(ρ σ)) [34], the two pure states, with respect to the usual norm on
E≤I
−
where the maximum is over all positive operators E sat- statespace[37], whilethe angleis,asthenamesuggests,
isfying E I. justtheanglebetweenthetwostates,withrespecttothe
≤
Second, the trace distance possesses the contractivity usual inner product on state space.
property [35], that is, D( (ρ), (σ)) D(ρ,σ) whenever In addition to the angle and the Bures metric we will
E E ≤
is a trace-preserving quantum operation. This state- find it convenient to introduce a third metric based on
E
ment expresses the physicalfact that a quantum process the fidelity. This metric does not seemto havebeen pre-
acting on two quantum states cannot increase their dis- viously recognized in the literature, but arises naturally
tinguishability. Contractivity follows from the physical later in this paper in the context of quantum computa-
interpretation of D(ρ,σ) described above. tion. It is defined by C(ρ,σ) 1 F(ρ,σ). The only
≡ −
Third, the trace distance is doubly convex, i.e., difficult step in proving this is pa metric is the proof of
if p are probabilities then D( p ρ , p σ ) the triangle inequality [38].
j j j j j j j ≤
jpjD(ρj,σj). This inequality caPn be phyPsically inter- In later sections our discussion will sometimes focus
Ppreted as the statement that the distinguishability be- on the fidelity, and sometimes on metrics derived from
tween the states p ρ and p σ , where j is not the fidelity. We will say that a metric ∆F(ρ,σ) on state
j j j j j j
known, can neverPbe greater tPhan the average distin- space is a fidelity-based metric if it is a monotonically
guishability when j is known, but has been chosen at decreasingfunctionofthefidelityF(ρ,σ). Obviouslythe
random according to the distribution p . angle, the Bures metric and C(, ) are all fidelity-based
j
· ·
Fidelity: The fidelity between density matrices ρ and metrics. Itisoftenthecasethatthespecificdetailsofthe
σ is defined by metricusedarenotimportant,andwheneverpossiblewe
state results using the fidelity as a single unifying con-
2 cept. However, sometimes it will prove advantageous to
F(ρ,σ) tr √ρσ√ρ . (4)
≡ (cid:18)q (cid:19) usethefidelity-basedmetricsdirectly. Inparticular,they
have the advantage of satisfying the triangle inequality,
When ρ = ψ is a pure state, this reduces to F(ψ,σ) = which turns out to be useful proving the chaining crite-
ψ σ ψ , the overlapbetween ψ and σ. rion [property (6)].
h | | i
Thefidelityalsohasmanyattractiveproperties. Itcan Likethetracedistance,thefidelityanditsderivedmet-
be shown that 0 F(ρ,σ) 1, with equality in the sec- ricshavemanyotherniceproperties. Itcanbeshown[40]
≤ ≤
ond inequality if and only if ρ = σ. The fidelity is thus that F( (ρ), (σ)) F(ρ,σ) for any trace-preserving
E E ≥
not a metric as such, but serves rather as a generalized quantum operation . We call this the monotonicity
E
measureoftheoverlapbetweentwoquantumstates. The propertyofthe fidelity. Itfollows thatany fidelity-based
fidelityisalsosymmetricinitsinputs,F(ρ,σ)=F(σ,ρ), metricsatisfiesacontractivitypropertyanalogoustothat
5
satisfied by the trace distance. aprocess. Finally,inSec.IVCweinvestigateapproaches
The fidelity also satisfies a property analogous to the based on the worst-case behaviour of a process. In each
double convexity of the trace distance. Precisely, the caseweinvestigatemeasuresbasedonboththetracedis-
square root of the fidelity is doubly concave, that is, tance and the fidelity. We will describe connections be-
F( p ρ , p σ )1/2 p F(ρ ,σ )1/2. This dou- tweenthevariousmeasures,andidentifyfourmeasuresof
j j j j j j ≥ j j j j
blePconcavitPy can be usedPto prove double convexity of particular merit. The properties of these four measures
certain fidelity-based metrics. In particular, supposing will be discussed in more detail in the next section.
∆F is a fidelity-based metric which is convex in the Nomenclature: In the following treatment we shall
square root of the fidelity (the angle, the Bures metric use the unadorned symbol ∆ to mean a metric between
and C(, ) are all easily verified to have this property), states. Our approach is to use state-based metrics to
then it·is·easy to verify that ∆F is doubly convex. form metrics between processes, and these will also be
Onedrawbackofthefidelityisthatitisdifficulttofind represented by ∆ but with a subscript denoting the
a compelling physical interpretation. When ρ and σ are method used, e.g. ∆ave is a process metric based on the
mixed states, no completely satisfactory interpretation averageoverinputstates. Whereweneedtospecializeto
of the fidelity is known (but c.f. Refs [41, 42]). When a specific state-metric we will use a superscript with the
ρ = ψ is a pure state, we have F(ψ,σ) = ψ σ ψ , the symbol representing that metric (A, B, C, and D from
h | | i
overlap between ψ and σ. Physically, we might imagine section III), or use that symbol directly with a subscript
σ is anattempt to prepare the pure state ψ. In this case for the method, e.g. ∆Dave ≡ Dave is the process metric
the fidelity coincides with the probability that a perfect basedonthe averagetracedistance. The chiefdeparture
measurementtesting whether the state is ψ willsucceed. from these conventions will be due to the fidelity, which
It is this property of the fidelity that is used in Sec. V isnotametric. Wewillusethenotation∆F tomeanany
toconnectourfidelity-basederrormeasuresforquantum metric derived from the fidelity (e.g. A, B, and C) and
processestotheprobabilityofsuccessofaquantumcom- thesymbolF withasubscripttomeanaprocessmeasure
putation. basedonfidelity, for example Fave is the averagefidelity.
General comments: Thefidelityis,atpresent,perhaps
somewhatmorewidely used inthe quantuminformation
science community than is the trace distance. However, A. Error measures based on the process matrix
weshallseebelowthatthetracedistanceandthefidelity
havecomplementaryadvantagesasabasisfordeveloping Suppose∆(ρ,σ)isanymetriconthespaceofquantum
measures of distance for quantum operations, and so it states. A natural approach to defining a measure ∆
pro
is useful to investigate both. In any case, the two mea- of the distance between two quantum processes is
sures are, as one might expect, quite closely related. In
∆ ( , ) ∆(ρ ,ρ ). (6)
particular,it is possible to show thatthey are relatedby pro E F
E F ≡
the inequalities [43]:
Defining ∆ in this way automatically gives ∆ the
pro pro
metric property. Provided ∆(, ) is easy to calculate,
1 F(ρ,σ) D(ρ,σ) 1 F(ρ,σ). (5) · ·
− ≤ ≤ − ∆pro is also easy to calculate. Furthermore, since can
p p E
beexperimentallydeterminedusingquantumprocessto-
It is not difficult to construct examples of saturation
mography, it follows that ∆ can be experimentally
pro
for both inequalities. Note that the second inequal-
measured, at least in principle.
ity is always saturated for pure states, i.e., D(ψ,φ) =
What about the other properties? The properties of
1 F(ψ,φ) for pure states ψ and φ.
stability and chaining can be obtained by making some
−
p natural extra assumptions about the state metric ∆,
which we now describe. Suppose first that the metric
IV. ERROR MEASURES FOR QUANTUM ∆ is stable in the sense that ∆(ρ τ,σ τ) = ∆(ρ,σ).
PROCESSES This is easily seen to be the case⊗for the⊗trace distance
and for any fidelity-based metric, for example. The sta-
Our goal in this paper is to recommend a single er- bility property for ∆ follows immediately:
pro
ror measure enabling researchers to compare the perfor- ∆ (I ,I )=∆(ρ ρ ,ρ ρ )=∆(ρ ,ρ )=
pro I E I F E F
⊗E ⊗F ⊗ ⊗
mance of quantum information processing experiments ∆ ( , ).
pro
E F
againstthe theoreticalideal. As the basis for such a rec- The chaining property can be proved, with some
ommendation,in this sectionwecomprehensivelysurvey caveats to be described below, by assuming that ∆(, )
· ·
possibledefinitions ofsucherrormeasures,anddo apre- is contractive, i.e., ∆( (ρ), (σ)) ∆(ρ,σ), for trace-
E E ≤
liminary assessment of each measure against the criteria preserving operations . We have already seen that this
E
introduced earlier in this paper. is a natural physical assumption satisfied by the trace
We take three basic approaches to defining an error distance and any fidelity-based metric.
measure for processes. In Sec. IVA we investigate ap- Suppose then that ∆ is contractive with respect to
proachesbasedontheprocessmatrix,ρ . InSec.IVBwe trace-preservingoperations. Weclaimthat∆ satisfies
E pro
investigateapproachesbasedontheaverage behaviourof the chaining property,
6
∆ ( , ) ∆ ( , )+∆ ( , ), servation [44]
pro 2 1 2 1 pro 2 2 pro 1 1
E ◦E F ◦F ≤ E F E F
provided is doubly stochastic, i.e., is trace-
1 1
F F 1
preserving and satisfies 1(I) = I; this assumption is F ( ,U)= tr(UU†U† (U )), (10)
used at a certain point iFn our proof of chaining. This pro E d3 j E j
Xj
may seem like a significant assumption, since physical
processes such as relaxation to a finite temperature are where the U are a basis of unitary operatorsorthogo-
j
not doubly stochastic. However, in quantum informa- nal under {the}Hilbert-Schmidt inner product, satisfying
tion science we are typically interested in the case when tr(U†U ) = dδ . Up to scaling we saw an example of
j k jk
1 and 2 are ideal unitary processes, and we are using such a set in Sec. II, the n-qubit tensor products formed
F F
∆pro to compare the composition of these two ideal pro- from the Pauli matrices and the identity matrix. Equa-
cessestotheexperimentallyrealizedprocess 2 1. Since tion (10) does not provide a direct way of estimating
E ◦E
unitary processesare automatically doubly stochastic, it F . But suppose we expand the U in terms of a set
pro j
followsthatchainingholds inthis case,whichis the case of input states, ρ : U = a ρ . These input states
k j k jk k
of usual interest. mustspantheentireoperaPtorspace,andthustheremust
The proof of chaining begins by applying the triangle be d2 of them; we will see an explicit example below for
inequality to obtain two qubits. We also expand UU U† in terms of a set of
j
observables, σ : UU U† = b σ . These observables
l j l jl l
∆pro(E2◦E1,F2◦F1) = ∆(ρE2◦E1,ρF2◦F1) (7) must also span the entire oPperator space. Substitution
∆(ρ ,ρ ) into Eq. (10) gives
≤ E2◦E1 E2◦F1
+∆(ρ ,ρ ). (8)
E2◦F1 F2◦F1 1
F ( ,U)= M tr(σ (ρ )), (11)
pro E d3 kl lE k
Then note the easily-verified identity ρE◦F = ( T Xkl
F ⊗
)(Φ), where Φ is the maximally entangledstate defined
Eearlier, we define FT(ρ) ≡ jFjTρFj∗, and Fj are the wevhaelureatMekFl ≡:Pchjobojlsaejak.spTahninsienqgusaettioonfdgi2vienspaumtsettahtoedsρto
operation elements for [cP.f. Eq. (1)]. Applying this pro k
F which can be prepared experimentally, and a set of ob-
identity to both density matrices in the second term on
servables σ whose averageswe can reliably measure;de-
the right-hand side of Eq. (8) gives l
termine the matrix M = (M ), whose elements depend
kl
only on knownquantities (ρ ,σ , and the idealized oper-
∆ ( , ) k l
pro 2 1 2 1
E ◦E F ◦F ation U), not on the unknown . The non-zero matrix
∆(ρ ,ρ ) E
≤ E2◦E1 E2◦F1 elements in M will determine which observable averages
+∆((F1T ⊗E2)(Φ),(F1T ⊗F2)(Φ)). (9) enreaeld, dto4 boebseesrtvimabalteedavfeorragcaeslcuwlialltinngeeFdptroo(Eb,eUe)s.tiImnagteend-.
ThedoublestochasticityofF1impliesthatF1T isatrace- However, suppose we choose some fixed set of ρk, and
preserving quantum operation. We can therefore apply then define σ a UU U† [45]. In this case it is
l ≡ k kl k
contractivity to both the first and the second terms on easily verified thatPEq. (11) simplifies to:
the right-hand side of Eq. (9), giving the desired result.
1
Only one property of ∆ remains in question, and
pro F ( ,U)= tr(σ (ρ )), (12)
that is whether or not it has a good physical interpre- pro E d3 kE k
Xk
tation. We will see in Sec. V that D and F can
pro pro
bothberelatedinanaturalwaytotheaverageprobabil- which only requires between d2 and 2d2 measurements.
ity with whicha quantum computationfails or succeeds, The drawback is that in this method we are not free to
providing a good physical interpretation for these quan- choose the σ ; they are determined by U and the ρ .
l k
tities. In practical situations, certain input states and mea-
Although ∆ may be calculated easily in principle surements are easier to use than others. We envisage
pro
forboththetracedistanceandfidelity-basedapproaches, an experimentalist choosing the set of input states and
thefidelity-basedmeasureshavesomesubstantialadvan- measurements according to convenience and using the
tages. The reason is that, so far as we are aware, exper- prescription above to calculate which combinations are
imentally determining D requires doing full process necessary. This in general will be less than what is re-
pro
tomography, which for a d-dimensional quantum system quired to perform full process tomography. This direct
requires the estimation of d4 d2 observable averages. method has the additionaladvantage of making it easier
−
By contrast, when U is a unitary operation it turns out to estimate the experimental error in F .
pro
that the fidelity F ( ,U) (and related error measures) For example, consider an n-qubit process,U. Suppose
pro
E
canbe determined basedupon the estimation ofat most we selectthe U to rangeoverthe n-foldtensorproducts
j
2d2 observable averages,and inparticular,d2 observable of Pauli matrices (including the identity matrix). Sup-
averages for qubits. This makes F ( ,U) and related pose furthermore that for each qubit we select the input
pro
E
error measures substantially easier to determine experi- statesfromtheset I,I+X,I+Y,I+Z (whereX,Y,Z
{ }
mentally than D . The key to proving this is the ob- aretheusualPaulioperators),sothatwechooseρ from
pro k
7
the set of all possible tensor products of the single qubit for ∆ because ofthe lack ofany clear meaning for the
ave
input states. Now, choosing σ a UU U†, we see fidelity-based metrics.
l ≡ k kl k
that the akl will always be real, Pand since the Uk are Finally,completingthechecklistofcriteria,ournumer-
Hermitianthentheσ arealsoHermitian. ThusEq.(12) ical analysis shows that ∆ is not stable for any of the
l ave
tells us that we need to estimate only d2 observable av- four candidate state metrics we’ve investigated. Later
erages to evaluate F for any U, much fewer than the in the paper we describe in detail a method for “sta-
pro
d4 d2 observable averages necessary to do full process bilizing” measures which are not stable; we now briefly
−
tomography on n qubits. note the results that are obtained when this procedure
It is an interesting problem deserving further explo- is applied in the present context. The idea is to in-
rationto find the minimum number of measurementsre- troduce an ancillary system A, and consider the quan-
quired to estimate F when there are constraints on tity ∆ ( , ) lim∆ ( , ), where
pro stab−ave ave
E F ≡ I ⊗ E I ⊗ F
what input states and observables are available. For in- the limit is that of large ancilla dimension. Using the
stance, it would be useful to know the optimal number well-known result that a randomly chosen chosen state
for the case where we are restricted to separable inputs of a composite system AQ (dimA dimQ) has very
≫
andproductobservables,i.e.,inputsandobservablesthat close to maximal entanglement [46, 47], it follows that
can be given direct local implementations. ∆ ( , ) = ∆ ( , ), i.e., the stabilized aver-
stab−ave pro
E F E F
age distance reduces to the process distance considered
earlier.
B. Error measures based on the average case Thereisanalternativeapproach,availablebecausethe
fidelity-based metrics are nonlinear functions of the fi-
Another natural approach for defining error measures delity, whichis to createa measurebasedonthe average
for quantum operations is to compare output states and fidelity:
averageover all input state, where the output states can
be compared using the distance measures for states de- F ( , ) dψF( (ψ), (ψ)). (14)
ave
E F ≡Z E F
scribed in Section III. We define
When is a unitary operation, U, the average fidelity
F
∆ave( , ) dψ∆( (ψ), (ψ)), (13) has a physical interpretation that is at least plausible,
E F ≡Z E F
as the average overlap between U ψ and (ψ). It was
| i E
shown in Ref. [48] (see also Ref. [19]) that F and F
wheretheintegralisovertheuniform(Haar)measureon ave pro
are related by the equation
state space.
Whilethisapproachseemsintuitivelysensible,itturns
F ( ,U)d+1
pro
outthattheresultingmeasuressatisfyfewofourcriteria. Fave( ,U)= E , (15)
E d+1
Theonlytwopropertiesthesemeasuresappeartosatisfy
ingeneral,foranarbitrarystatemetric∆,arethemetric where d is the dimension of the quantum system, and
and chaining criteria, both of which follow immediately we are restricting ourselvesto the case where U is a uni-
from the metric property of ∆. tary operation. This relationship makes F ( ,U) easy
ave
E
The average-basedmetrics are less successful in meet- to calculate [19, 20] and also easy to measure experi-
ing the other criteria. Even when ∆ is easy to calculate, mentally, using the techniques described in the previous
it is not obvious that the integral in Eq. (13) will have a subsection for F ( ,U).
pro
E
simple form that enables easy calculation of ∆ . This, Although F has severaladvantages(ease of calcula-
ave ave
in turn, means that ∆ may not be so easy to deter- tion,easeofmeasurement,andaphysicalinterpretation),
ave
mine experimentally. So far as we are aware, no simple theoutlookfortheothercriteriaisnotsogood. Notonly
expressionsareknownfor ∆ for anyof the metrics we is F not a metric, it is not stable either, a fact that
ave ave
have discussed. followsfromEq.(15)andtheknowledgethatF issta-
pro
It is not surprising that the physical interpretations ble. The same argument shows that measures analogous
of these metrics rely heavily on the possible interpreta- toA,B,andC basedonF willalsonotbestable. We
ave
tions of the corresponding state metrics as discussed in do not know of any stable metrics that may be derived
section III. The earlier discussion of the trace distance, asafunctionofF ,andEq.(15)rendersanysuchmet-
ave
for example, follows onto give a meaning for D . Sup- rics equivalent in content to functions based on F so
ave pro
posewe areaskedtodistinguishbetween (ψ)and (ψ) the only reason to use them would be if they had better
E F
for some ψ which is known, but has been chosen uni- characteristics.
formly at random. On average, the optimal probability To summarize the results of this section, they show
of successfully distinguishing the two processes will be thatnoneofthe average-caseerrormeasureswehavede-
1/2+D ( , )/2. Thus,D ( , )maybeinterpreted finedareparticularlyattractive. However,thesenegative
ave ave
E F E F
as a measure of the average bias in favour of correctly results are vital because these approaches are all fairly
distinguishing which process was applied to a state ψ. natural solutions one might take to defining a plausible
Withregardtothefidelity-basedmetrics,however,there error measure. It was therefore important to consider
does not appear to be any clear physical interpretation them carefully before choosing to reject them.
8
C. Error measures based on the worst case thestabilizedtrace-distanceDstabhasanappealingphys-
ical interpretation—it is the worst-casebias in the prob-
Our finalapproachto defining errormeasuresis based ability of being able to distinguish ( )(ψ) from
I ⊗ E
on the worst case distance between (ψ) and (ψ). We ( )(ψ), where we allow an ancilla of arbitrary size.
define E F WIe⊗dFefer discussion of the physical interpretation of the
fidelity-based measures until the next section, where we
∆max( , ) max∆( (ψ), (ψ)), (16) willseethatboththeyandDstab canbegivenanelegant
E F ≡ ψ E F interpretation in the context of quantum computation.
Whatoftheremainingcriteria,easeofcalculationand
wherethemaximumisoverallpossiblepurestateinputs,
easeofmeasurement? Unfortunately,nopowerfulgeneral
ψ, and ∆ is a metric on quantum states.
formulae for calculating ∆ are known. Reference [10]
When ∆=∆F is a fidelity-based metric, we see ∆F stab
max gives a general formula for the distance D between
is a function of the minimal fidelity, defined by stab
two unitary operations, but the more interesting case of
the distance between an idealized unitary operation and
F ( , ) minF( (ψ), (ψ)). (17)
min
E F ≡ ψ E F a noisy quantum process has not been solved, even for
single-qubit operations.
In the definition of ∆ , we maximize over all pure
max The good news is that D and F (and thus
stab stab
state inputs. Is this maximum the same if all physical
A ,B and C ) are easy to calculate numerically,
stab stab stab
inputs, including mixed states, are considered? In fact,
because they can all be reduced to convex optimization
it is fairly simple to show that this is true, and therefore
problems [51]. For this special class of problem, where
thatitdoesnotmatterifweoptimizeoverpureormixed
the task is to minimize a convex function defined on a
states [49]. Suppose ∆ is a doubly convex metric, as are
convex set, extremely efficient numerical techniques are
all the metrics discussed in this paper (c.f. Sec. III). If
available. Among many other nice properties, it is pos-
themaximumisachievedatsomemixedstate,ρ,thenwe
sible to show that a local minimum of a convex opti-
have ∆ =∆( (ρ), (ρ)). Expanding ρ= p ψ as
max E F j j j mization problem is always a global minimum, and thus
a mixture of pure states, and applying doublePconvexity techniquessuchasgradientdescenttypicallyconvergeex-
we see that the maximummust also be attainedat some
tremely rapidly, with no danger of finding false minima.
pure state ψ . A similar argument holds for F , based
j min In Appendix A2, we prove explicitly that finding F
stab
on the double concavity of the fidelity.
belongstothis classofproblems,andthe proofforD
stab
To assess the suitability of these measures, it is useful
follows similar lines.
tofirstnotethatD hasalreadybeenshowningeneral
max We have seen that numerical calculation of D and
stab
nottobe stable[10],andsimilarargumentscanbe made
F can easily be carried out, and this enables a two-
stab
toextendthistothefidelity-basedmeasures. InRef.[10],
step procedure for experimental measurement of either
Aharonov et al. resolve this difficulty by constructing a
quantity—process tomography, followed by a numerical
variantofD whichis stable, but whichotherwisehas
max optimization. Of course, finding general formulae along
extremely similar properties to D . We now describe
max the lines of F ( ,U) or D is still a highly desirable
pro pro
how this procedure can be extended to define a stable E
goal. Aside from the intrinsic benefit, finding general
versionof∆ foranarbitrarystatemetric∆,anddefer
max formulae would simplify the experimental measurement
for the moment discussion of the other criteria.
and determination of error bars for D and F , and
stab stab
Suppose the original system Q on which and act
E F perhaps obviate the need for a full process tomography,
has state space dimension d. It will be convenientto use
as Eq. (10) did for F ( ,U).
pro
subscripts to indicate the system on which operations E
act (e.g. = , = ). We introduce a fictitious d-
Q Q
E E F F
dimensionalancillarysystemA, actedonby the identity
V. APPLICATION TO QUANTUM
operation , and define the stabilized quantity [50]
IA COMPUTING
∆ ( , ) ∆ ( , ). (18)
stab Q Q max A Q A Q
E F ≡ I ⊗E I ⊗F Can we find a good physical interpretation for any of
The proofthat ∆ is stable under addition of systems the error measures that we’ve identified? In this section
stab
is simple and has been included in Appendix A1. In wewillfocusoninterpretationsthatarisewithinthecon-
the same way, we can also define a stable form of the text of quantum computation and we will find that of
minimum fidelity, F ( , ) F ( , the errormeasures we have discussed, four have particu-
stab Q Q min A Q A
E F ≡ I ⊗E I ⊗
Q), with the proof of stability following similar lines. larlyoutstandingproperties: Dpro,Fpro,Dstab andFstab.
F
Note that the stabilized fidelity-based metrics ∆F are (Note that in the case of the fidelity, it will actually be
stab
functions of F in the obvious way (e.g. we define as moreconvenienttostateourresultsintermsoftheequiv-
stab
usual Astab, Bstab and Cstab). alent measures Cpro and Cstab.)
Which of the other criteria for an error measure does Assessed according to the criteria described in the in-
∆ satisfy? It is straightforward to show that ∆ troduction,these four measureshavealreadybeen found
stab stab
satisfies the metric and chaining criteria. Furthermore, to be superior to all the other measures we have stud-
9
ied. The additional fact that each arises naturally in operation is performed. A good measure of error in
E
the context of quantum computation strongly indicates the real computation is the actual probability p that
e
that these four measures are the most deserving of con- the measured output of the computation is not equal to
sideration as measures of error in quantum information f(x). In Appendix B1, we show that
processing. We will return in the conclusion, Sec. VI, to
the question of which of these four measures is the best pe ≤ pied+Dstab(E,F) (19)
possible measure of error. 2
Thereareavarietyofdifferentwaysofdescribingquan- pe ≤ (cid:20)qpied+Cstab(E,F)(cid:21) . (20)
tumcomputations,anditturnsoutthateachofthe four
error measures arises naturally in different contexts. We Which of these inequalities is better depends upon the
will discuss separately two broad divisions of quantum exact circumstances. For example, when pid =0, we see
e
computation,functioncomputation andsamplingcompu- that which inequality is better depends upon whether
tation looking at both worst-case and average-case per- D ( , ) is larger or smaller than C ( , )2. With
stab stab
E F E F
formance for each division. Eq.(5)inmind,itisnotdifficulttoconvinceoneselfthat
Most algorithms on classical computers are framed as either of these possibilities may occur.
function computations. We will see that our error mea- Function computation in the average case: Onceagain
surescanbegivenparticularlycompellinginterpretations our goal is to compute a function f(x) using an approx-
relating to the probability of error in a function compu- imation to some ideal operation . However, we now
E F
tation. However, in the context of simulating quantum look at the average-case error probability p that the
e
systems it is often more natural to consider sampling measuredoutput of (x x) is notequalto f(x), where
E | ih |
computations, where the goal is to reproduce the statis- the average is taken with respect to a uniform distribu-
tics obtained from a measurementof the system in some tionoverinstancesx. Correspondingly,weintroducepid,
e
specified configuration. Again, we will see that our error the averagecase error probability for the idealized oper-
measures can be given good interpretations in this con- ation . We show that (App. B2):
F
text,albeitsomewhatmorecomplexinterpretationsthan
for function computation. pe ≤ pied+Dpro(E,F). (21)
The reason for treating the two types of computation
Unfortunately,wehavebeenunabletodevelopafullnat-
separatelyis atleastpartiallyapracticalone,since both
uralanalogueofEq.(20) basedon the fidelity. However,
types of computation arise naturally in the context of
we have proved a partial analogue for when the ideal
quantum computation. However, a more fundamental
computation succeeds with probability one (pid =0). In
reason is that it does not appear to be known how to e
this case:
reduce sampling computation to function computation.
Rather remarkably, even when there is an efficient way
p C ( , )2 =1 F( , ). (22)
of computing a probability distribution, there does not e ≤ pro E F − E F
appear to be any general way to convert that into an
The proof uses very similar techniques to those used to
efficient way of sampling from that distribution.
establish Eqs. (21) and (20), and is therefore omitted.
A. Function computation B. Sampling computation
Infunctioncomputation,thegoalofthequantumcom- Insamplingquantumcomputation,thegoalistosam-
putation is to compute a function, f, exactly or with ple from some ideal distribution p (y) p on mea-
x x
{ } ≡
highprobabilityofsuccess. Moreprecisely,the goalis to surement outcomes y, with x representing input data
take as input an instance, x, of the problem, and to pro- for the problem. For instance, x might represent the
duce a final state ρ of the computer that is either equal coupling strengths and temperature of some spin glass
x
to f(x) , or sufficiently close that when a measurement model, with the goal being to sample from the thermal
| i
in the computational basis is performed, the outcome is distribution of configurations y for that spin glass. This
f(x)withhighprobability. Grover’salgorithmisusually type of computation is particularly useful for simulating
castinthisway,wherewewanttodeterminetheidentity the dynamics of another quantum system.
of the state marked by the oracle. Unlike Grover’s algorithm, Shor’s algorithm is usually
Function computation in the worst case: Suppose we describedas a sampling computation. The goalis not to
attempt to performa quantum computation represented directly produce a factor or list of factors, but rather to
by anideal operation thatacts on aninput x , where produce a distribution over measurement outcomes. By
F | i
x represents the instance of the problem to be solved, sampling from this distribution and doing classicalpost-
e.g., a number to be factored[52]. This process succeeds processingitispossibletoextractfactorsofsomenumber
in computing f(x) with an error probability of at most x. Ofcourse,asnotedinRef.[53],itispossibletomodify
pid, where ‘id’ indicates that this is the ideal worst-case Shor’salgorithmtobeafunctioncomputation,takingan
e
error probability. Of course, in reality some non-ideal instance x and producing a list of all the factors of x.
10
The desired result in sampling computation is that VI. SUMMARY, RECOMMENDATIONS, AND
the measurement outcomes y are distributed according CONCLUSION
to the ideal probabilities p (y), for a given problem in-
x
stance x. Suppose, however, that they are instead dis- Wehaveformulatedalistofcriteriathatmustbesatis-
tributed according to some nonideal set of real proba- fied by a good measure of error in quantum information
bilities qx(y). How should we compare these two distri- processing. These criteria provide a broad framework
butions? There are two widely-used classical measures that can be used to assess candidate error measures, in-
enabling comparison of probability distributions p and corporating both theoretical and experimental desider-
q. The first is the Kolmogorov or l1 distance, defined by ata.
D(p,q) ≡ y|p(y)−q(y)|/2. The second is the Bhat- We have used this framework to comprehensively sur-
tacharya oPverlap, defined by F(p,q) p(y)q(y). veypossibleapproachestothedefinitionofanerrormea-
≡ y
Since these measures are in fact commutaPtivepanalogues sure,rejecting many a priori plausible errormeasuresas
of the trace distance and fidelity, respectively, we rep- they fail to satisfy many of our criteria. Although many
resent them with the same symbols as their quantum oftheserejectederrormeasuresareofsomeinterestasdi-
analogues (D and F). As with the trace distance, the agnosticmeasures,nonearesuitableforuseasaprimary
Kolmogorov distance can be given an appealing inter- measureoftheerrorinaquantuminformationprocessing
pretation as the bias in probability when trying to dis- task.
tinguish the distributions p and q. No similarly simple Four error measures were identified which have par-
interpretation for the Bhattacharya overlap seems to be ticular merit, each of which satisfies most or all of the
known,althoughitisrelatedtotheKolmogorovdistance criteria we identified. These measures are the J distance
through inequalities analogous to Eq. (5). (Jamiolkowski process distance), the J fidelity (Jami-
olkowskiprocess fidelity), the S distance (stabilized pro-
The Kolmogorov distance and Bhattacharya overlap,
cess distance) and the S fidelity (stabilized process fi-
together with the quantum error measures we have in-
delity), denoted D ,F ,D andF , respectively.
troduced,canbeusedtorelateidealandrealprobability pro pro stab stab
All four measures either are metrics (in the case of
distributions obtained as the result of a quantum com-
the process distances) or give rise to a variety of as-
putation.
sociated metrics (for the process fidelities). Moreover,
Sampling computation in the worst case: Suppose we all of the metrics can be shown to satisfy stability and
attempt to performa quantum computation represented chainingpropertieswhichgreatlysimplifytheanalysisof
byanidealoperation thatactsonaninput x ,wherex multistage quantum information processingtasks, as de-
F | i
representsthe instance of the problemto be solved. The scribedintheintroduction. Themaindifferencesarisein
goal is to produce a final state (x x) which, when thecriteriaofeasycalculation,measurementandsensible
F | ih |
measured in the computational basis, gives rise to an physicalinterpretation. We now briefly summarize these
ideal distribution px. Instead, we perform the operation remaining properties for the four measures. Throughout
, giving rise to a distribution qx on measurement out- this section, we assume that the goal in each case is to
E
comes. In Appendix B3 we prove that: compare a quantum operation to an ideal unitary op-
E
eration U; the results vary somewhat when is being
E
compared to an arbitrary process .
maxD(q ,p ) D ( , ) (23) F
x x x ≤ stab E F (i) J distance: There is a straightforward formula en-
max[1 F(q ,p )] C ( , )2. (24) abling Dpro to be calculated directly from the process
x x stab
x − ≤ E F matrix, thus also allowing it to be experimentally deter-
mined using quantum process tomography. The J dis-
tance can be given an operational interpretation as a
Just as for function computation, which of these is the
boundonthe averageprobabilityoferrorp experienced
better inequality depends upon the details of the situa- e
duringquantumcomputationofafunction,orasabound
tion under study.
onthe distancebetweenthe realandidealjointdistribu-
Sampling computation in the average case: Given the tions of the computer in a sampling computation:
same situation as for the worst case, we now assume
that problem instances are chosen uniformly at ran- p pid+D ( ,U) (27)
e ≤ e pro E
dom. We will therefore use the Kolmogorov distance
D(q,p) D ( ,U). (28)
pro
andBhattacharyaoverlapbetweenthejoint distributions ≤ E
h{pa(sxa,pyp)}ro≡ximpaatnedd {q.(Ax,ryg)u}m≡enqtstaonamloeagsouursetohotwhawteulsleEd In the first expression pied is the average probability of
F errorin the ideal computation, representedby U. In the
in the worst case establish:
secondexpression,D(q,p)istheKolmogorovdistancebe-
tweentherealjointprobabilitydistribution p(x,y) p
{ }≡
onprobleminstancesxandmeasurementoutcomesyand
D(q,p) D ( , ) (25)
pro
≤ E F the ideal joint distribution q(x,y) q, for a uniform
1 F(q,p) Cpro( , )2. (26) distribution on problem inst{ances. } ≡
− ≤ E F