Quantum algorithm for multivariate polynomial interpolation

0.18 MB·

by  Jianxin Chen

QUANTUM ALGORITHM FOR MULTIVARIATE POLYNOMIAL INTERPOLATION
JIANXINCHEN‡,ANDREWM.CHILDS∗,†,‡,ANDSHIH-HANHUNG∗,‡
ABSTRACT. How many quantum queries are required to determine the coefficients of a degree-d
polynomialinnvariables? Wepresentandanalyzequantumalgorithmsforthismultivariatepoly-
7 nomialinterpolationproblemoverthefieldsFq,R,andC. WeshowthatkC and2kC queriessuffice
201 otinothgaec1rhsfieopvreecliaparlrgoceabsfiaebesli.dliFtyoorr1dFefoqr,rqwC.eTasnhhdeowRcla,tshrseaistcpa⌈elncq+dtiudve(ernlyy+d,dcw)o⌉hmqepureleerkxiCeitsy=sou⌈fffint+hc1ei1s(tonp+draodc)bh⌉lieeemvxeceipspr(tonfb+oardb)di,li=styo2aopaupnrrdroefasocuuhlr-t
d
n providesaspeedupbyafactorofn+1, n+21,and n+dd forC,R,andFq,respectively. Thuswefind
a a much largergap between classical and quantum algorithms than the univariate case, where the
J
speedupisbyafactorof2. ForthecaseofFq,weconjecturethat2kC queriesalsosufficetoachieve
5 probabilityapproaching1forlargefieldorderq,althoughweleavethisasanopenproblem.
1

]
h
p
- 1. INTRODUCTION
t
n
Let f(x ,...,x ) K[x ,...,x ] be a polynomial of degreed. Supposed is known and we are
a 1 n ∈ 1 n
u givenablackboxthatcomputes f onanydesiredinput. Thepolynomialinterpolationproblemis
q todetermineallthecoefficientsofthepolynomialbyqueryingtheblackbox.
[
Classically, a multivariate polynomial can be interpolated by constructing a system of linear

1 equations. Invertibility of the Vandermonde matrix implies that (n+d) queries are necessary and
v d
sufficienttodetermineallthecoefficients. Notethatonemustchoosetheinputvaluescarefullyto
0
>
9 constructafull-rankVandermondematrixforn 1[GS00].
9 Recent work has established tight bounds on the quantum query complexity of interpolating
3
univariate polynomials over finite fields. In particular, [CvDHS16] developed an optimal quan-
0
. tum algorithm that makes d+1 queries to succeed with bounded error and one more query to
1 2
achieve success probability 1 O(1/q). They also showed that the success probability of the al-
0
−
7 gorithm is optimal among all algorithms making the same number of queries. Previous work
1 [KK11, MP11] shows that no quantum algorithm can succeed with bounded error using fewer
:
v queries,sotheoptimal successprobability exhibits asharp transitionas thenumber ofqueriesis
i increased.
X
Formultivariatepolynomials,[CvDHS16]conjecturedthatastraightforwardanalogoftheuni-
r
a variatealgorithmsolvestheinterpolationproblemwithprobability1 o(1)using 1 (n+d) +1
− ⌊n+1 d ⌋
queries. However, the analysis of the algorithm appeared to require solving a difficult problem
inalgebraicgeometryandwasleftopen. (Inaddition,Montanaroconsideredthequantumquery
complexityofinterpolatingamultilinear polynomial[Mon12],butthisisquitedifferentfromthe
generalmultivariatecase.)
Tothe bestof ourknowledge,all previous workon quantumalgorithms for polynomial inter-
polationhasfocusedonfinitefields. Althoughcryptographicapplicationsofpolynomialinterpo-
lation typically use finite fields, the task of polynomial interpolation over infinite fields is also a
naturalproblem,especiallyconsideringtheubiquityofreal-andcomplex-valued polynomialsin
numericalanalysis.
∗DEPARTMENTOFCOMPUTERSCIENCE,UNIVERSITYOFMARYLAND
†INSTITUTEFORADVANCEDCOMPUTERSTUDIES,UNIVERSITYOFMARYLAND
‡JOINTCENTERFORQUANTUMINFORMATIONANDCOMPUTERSCIENCE,UNIVERSITYOFMARYLAND
1
2 JIANXINCHEN,ANDREWM.CHILDS,ANDSHIH-HANHUNG
Inthispaper,weproposeanapproachtodiscussingthequantumquerycomplexityofpolyno-
mial interpolation in the continuum limit. The algorithm prepares an initial state in a bounded
working region. The bounded region limits the precision that can be achieved due to the uncer-
tainty principle, although the algorithm can be made arbitrarily precise by taking an arbitrarily
large region. Using this strategy, we present a quantum algorithm for multivariate polynomial
interpolationovertherealandcomplexnumbers. Wealsoconsidermultivariatepolynomialinter-
polationoverfinitefields,whereouralgorithmcanbeviewedasageneralizationoftheunivariate
polynomialinterpolationalgorithmproposedin[CvDHS16].
Toanalyzethesuccessprobabilityofourapproach,werelateittothetensorrankproblem. The
rank ofa given tensor,which is thesmallest integer k such that the tensorcan be decomposedas
linearcombinationofksimpletensors(i.e.,thosethatcanbewrittenastensorproducts),wasfirst
introduced nearly a century ago. A half century later, with the advent of principal component
analysis on multidimensional arrays, the study of tensor rank attracted further attention. How-
ever, it has recently been shown that most tensor problems, including tensor rank, are NP-hard
[Ha˚s90, HL13], and restricting these problems to symmetric tensors does not seem to alleviate
theirNP-hardness[HL13]. Morespecifically,tensorrankisNP-hardoveranyfieldextensionofQ
andNP-completeoverafinitefieldF .
q
Fortunately, analyzing the success probability of multivariate polynomial interpolation does
not require exactly computing the rank of a symmetric tensor. The number of queries needed
to achieve success probability 1 can be translated to the smallest integer k such that almost every
symmetric tensor can be decomposed as a linear combination of no more than k simple tensors.
In turn, this quantity can be related to properties of certain secant varieties, which lets us take
advantageofrecentprogressinalgebraicgeometry[BBO15,BT15].
The success probability of our algorithm behaves differently as a function of the number of
queriesforthethreefieldsweconsider. Specifically,byintroducing
n+1 d = 2,n 2;
≥
kC(n,d) := ⌈n+11(n+dd)⌉+1 (n,d) = (4,3),(2,4),(3,4),(4,4); (1)
 1 (n+d) otherwise,
⌈n+1 d ⌉

wehavethefollowingupperboundsonthequerycomplexity:
Theorem1.1. Forpositiveintegersdandn,thereexistsaquantumalgorithmforinterpolatingann-variate
polynomialofdegree doverthefieldK usingatmost
(1) d (n+d)queriesforK = F ,succeedingwithprobability 1 O(1/q);
n+d d q −
(2) 2k queriesforK = R,succeedingwithprobability 1;
C
(3) k queriesforK = C,succeedingwithprobability 1.
C
The remainder of the paper is organized as follows. After introducing the notation and math-
ematical background in Section2, we describe the query model in Section3.1 and present our
algorithmsinSection3.2. Wethenanalyzethealgorithmtoestablishourquerycomplexityupper
bounds in Section3.3. In Section4, we show that our proposed algorithm is optimal for finite
fields. Finally,inSection5weconcludebymentioningsomeopenquestions.
2. PRELIMINARIES AND NOTATIONS
InSection2.1,weintroducesomenotationthatisusedinthepaper,especiallywhendescribing
the algorithm in Section3.2. Section2.2 reviews basic definitions and concepts from algebraic
geometrythatariseinourperformanceanalysisinSection3.3.
QUANTUMALGORITHMFORMULTIVARIATEPOLYNOMIALINTERPOLATION 3
2.1. Notationanddefinitions. Let f K[x ,...,x ]beapolynomialofdegreeatmostdoverthe
1 n
∈
fieldK. Weletxj := ∏n xji forj J,whereJ := j Nn : j + +j d isthesetofallowed
i=1 i ∈ { ∈ 1 ··· n ≤ }
exponentswithsize J := (n+d). Thusxj isamonomialin x ,...,x ofdegree j +j + +j .
d 1 n 1 2 ··· n
Access to the function f is given by a black box that performs x,y x,y+ f(x) for all
| i 7→ | i
x Kn and y K. We will compute the coefficients of f by performing phase queries, which are
∈ ∈
obtainedbyphasekickbackoverK,asdetailedinSection3.1.
For k-dimensional vectors x,y Kk, we consider the inner product : Kk Kk K defined
∈ · × →
by x y = ∑k x y , where x is the complex conjugate of x (where we let x = x for x F ). We
· i=1 i i ∈ q
denotetheindicator functionforaset A Rn byI (z) = 1if z A and0otherwise. Wedenote
A
⊆ ∈
aballofradiusr R+ centeredat0by B(r).
∈
Alattice Λ isadiscreteadditivesubgroupofRn forpositiveintegern generatedbye ,...,e
1 n
∈
Rn. For every element x Λ, we have x = ∑n c e for some c Z for i 1,...,n . A
∈ i=1 i i i ∈ ∈ { }
fundamentaldomain T ofΛcenteredatzeroisasubsetofRn suchthat
n 1 1
T = ∑a e : a , . (2)
i i i
∈ −2 2
(i=1 (cid:20) (cid:19))
The dual lattice of Λ, denoted by Λ, is an additive subgroup of Rn generated by f ,..., f Rn
1 n
∈
satsifyinge f = δ fori,j 1,...,n .
i j ij
· ∈ { }
The standard basis over the reael numbers is the set x : x Rn for positive integer n. The
{| i ∈ }
amplitude of a state ψ in the standard basis is denoted by ψ(x) or x ψ . The standard basis
| i h | i
vectors over real numbers are orthogonal in the sense of the Dirac delta function, i.e., x x =
′
h | i
δ(n)(x,x )for x,x Rn. Wealsodenoteδ(n)(x) := δ(n)(x,0).
′ ′
∈
2.2. Algebraic geometry concepts. A subset V of Kn is an algebraic set if it is the set of common
zerosofafinitecollectionofpolynomials g ,g ,...,g with g K[x ,x , ,x ]for1 i r.
1 2 r i 1 2 n
∈ ··· ≤ ≤
The union of a finite number of algebraic sets is an algebraic set, and the intersection of any
family of algebraic sets is again an algebraic set. Thus by taking the open subsetsto be the com-
plementsofalgebraicsets,wecandefineatopology,calledtheZariskitopologyonKn.
AnonemptysubsetVofatopologicalspaceXiscalledirreducibleifitcannotbeexpressedasthe
union V = V V oftwoproper(Zariski)closedsubsetsV ,V . Theemptysetis notconsidered
1 2 1 2
∪
tobeirreducible.
AnaffinealgebraicvarietyisanirreducibleclosedsubsetofsomeKn,withrespecttotheinduced
topology.
We define projective n-space, denoted by Pn, to be the set of equivalence classes of (n+1)-
tuples (a ,...,a ) of complex numbers, not all zero, under the equivalence relation given by
0 n
(a ,...,a ) (λa ,...,λa )forallλ K,λ = 0.
0 n 0 n
∼ ∈ 6
Anotionofalgebraicvarietymayalsobeintroducedinprojectivespaces,givingthenotionofa
projectivealgebraicvariety: asubsetV Pn isanalgebraic setifitisthesetofcommonzerosofa
⊆
finitecollectionofhomogeneouspolynomialsg ,g ,...,g withg K[x ,x ,...,x ]for1 i r.
1 2 r i 0 1 n
∈ ≤ ≤
Wecallopensubsetsofirreducibleprojectivevarietiesquasi-projectivevarieties.
Foranyintegersnandd,wedefinetheVeronesemapofdegreedasthefollowing:
V : [x : x : : x ] [ : xj : ] (3)
d 0 1 n
··· 7→ ··· ···
wherethenotation[ : x : ]denoteshomogeneouscoordinatesand xj rangesoverallmono-
i
··· ···
mials of degreed in x ,x ,...,x . Theimage oftheVeronesemap is an algebraic variety, usually
0 1 n
calledaVeronesevariety.
4 JIANXINCHEN,ANDREWM.CHILDS,ANDSHIH-HANHUNG
Now we define secant varieties. For an irreducible algebraic variety V, its kth secant variety
σ (V)istheZariskiclosureoftheunionofk-secantspacestoV,
k
σ (V) = P ,P , ,P . (4)
k 1 2 k
h ··· i
P1,P2,[...,Pk∈V
For more information about Veronese and secant varieties, refer to Example 2.4 and Example
11.30in[Har92].
3. QUANTUM ALGORITHM FOR POLYNOMIAL INTERPOLATION
3.1. The query model. Using the standard concept of phase kickback, we encode the results of
queriesinthephasebyperformingstandardqueriesintheFourierbasis. Webrieflyexplainthese
queriesforthethreetypesoffieldsweconsider.
3.1.1. Finite field F . The order of a finite field can always be written as a prime power q := pr.
q
Let e: F C be the exponential function e(z) := ei2πTr(z)/p where the trace function Tr: F
q q
→ →
Fp is defined by Tr(z) := z+zp +zp2 + +zpr−1. The Fourier transform over Fq is a unitary
···
transformation acting as x 1 ∑ e(xy) y for all x F . The k-dimensional quantum
| i 7→ √q y∈Fq | i ∈ q
Fouriertransformisgivenby x 1 ∑ e(x y) y forany x Fk.
| i 7→ qk/2 y∈Fkq · | i ∈ q
A phase query is simply the Fourier transform of a standard query. We can achieve this by
performinganinverseQFT,aquery,andthenaQFT,asfollows:
1
x,y ∑ e( yz) x,z (5)
| i 7→ √q − | i
z∈Fq
1
∑ e( yz) x,z+ f(x) (6)
7→ √q − | i
z∈Fq
1
∑ e( yz)e(w(z+ f(x))) x,w = e(yf(x)) x,y (7)
7→ √q − | i | i
z,w∈Fq
forany x,y F . Hereweusetheidentity∑ e(zv) = qδ .
∈ q z∈Fq v,0
Analgorithmmakingkqueriesinparallelgeneratesaphase∑k y f(x ) = ∑k ∑ y xjc . We
i=1 i i i=1 j J i i j
∈
defineZ: Fnk Fk FJ satisfyingZ(x,y) = ∑k y x j forj J,sothat∑k y f(x ) = Z(x,y) c.
q × q → q j i=1 i i ∈ i=1 i i ·
3.1.2. RealnumbersR. ThequantumFouriertransformoverR isthetransformationbetweenpo-
sition and momentum space. Let e: R C be the exponential function e(x) := ei2πx. For any
→
functionψwhoseFouriertransformexists,thequantumFouriertransformactsas
dxψ(x) x dyΨ(y) y , (8)
R | i 7→ R | i
Z Z
where Ψ(y) = dxe( xy)ψ(x). The unitarity of the quantum Fourier transform follows from
R −
Parseval’stheorem.
R
QUANTUMALGORITHMFORMULTIVARIATEPOLYNOMIALINTERPOLATION 5
Asinthefinitefieldcase,weconstructaphasequerybymakingastandardqueryintheFourier
basis:
dxdyψ(x,y) x,y dx dz dyψ(x,y)e( yz) x,z (9)
R2 | i 7→ R R R − | i
Z Z Z (cid:20)Z (cid:21)
dx dz dyψ(x,y)e( yz) x,z+ f(x) (10)
7→ R R R − | i
Z Z (cid:20)Z (cid:21)
dx dz dwe(w(z+ f(x))) dyψ(x,y)e( yz) x,w (11)
7→ R R R R − | i
Z Z Z (cid:20)Z (cid:21)
= dxdye(yf(x))ψ(x,y) x,y , (12)
R2 | i
Z
whereweusetherelationδ(x,x ) = dye(y(x x ))for x,x R.
′ R − ′ ′ ∈
Analgorithmmakingkqueriesinparallelgeneratesaphase∑k y f(x ) = ∑k ∑ y xjc . We
R i=1 i i i=1 j J i i j
∈
defineZ: Rnk Rk RJsatisfyingZ(x,y) = ∑k y x jforj J,sothat∑k y f(x ) = Z(x,y) c.
× → j i=1 i i ∈ i=1 i i ·
3.1.3. Complex numbers C. The complex numbers can be viewed as a field extension of the real
numbers of degree 2, namely C = R[√ 1]. For any positive integer n, let φ : Cn R2n be an
n
− →
isomorphismφ (x) := (Re(x ),Im(x ),Re(x ),Im(x ),...,Re(x ),Im(x )),whichwealsodenote
n 1 1 2 2 n n
inboldfacebyx. Acomplexnumberx Ccanbestoredinaquantumregisterasatensorproduct
∈
ofitsrealandimaginaryparts, x = Re(x) Im(x) .
| i | i| i
Acomplexfunctionψ: Cm Cn canbeseenasafunctionwith2mvariables. Letψ(x) = ψ˜(x).
→
By abuse of notation, we will neglect the tilde and write ψ(x) = ψ(x). Let e: C C be the
→
exponential function e(x) := ei2πRe(x). For any function ψ: C C whose Fourier transform
→
exists,wedefinethetransform
d2xψ(x) x d2yΨ(y) y , (13)
R2 | i 7→ R2 | i
Z Z
where Ψ(y) = d2xe( yx)ψ(x). Note that in general Ψ(y) cannot be written in the form of
R2 −
Ψ(y) withacomplexvariable y C. Toencodetheoutputinthephase,thequeriesactas
R ∈
d2x d2yψ(x,y) x,y
R2 R2 | i
Z Z
d2x d2y d2zψ(x,y)e( yz) x,z (14)
7→ R2 R2 R2 − | i
Z Z Z
d2x d2y d2zψ(x,y)e( yz) x,z+f(x) (15)
7→ R2 R2 R2 − | i
Z Z Z
d2x d2y d2z d2uψ(x,y)e( yz)e(u(z+ f(x))) x,u (16)
7→ R2 R2 R2 R2 − | i
Z Z Z Z
d2x d2yψ(x,y)e(yf(x)) x,y , (17)
7→ R2 R2 | i
Z Z
whereweusetheidentity d2ye(y(x x )) = δ(2)(x,x )for x,x C.
R2 − ′ ′ ′ ∈
Analgorithmmakingkqueriesinparallelgeneratesaphase∑k y f(x ) = ∑k ∑ y xjc . We
R i=1 i i i=1 j J i i j
∈
defineZ: Cnk Ck CJ satisfyingZ(x,y) = ∑k y x jforj J,sothat∑k y f(x ) = Z(x,y) c.
× → j i=1 i i ∈ i=1 i i ·
3.2. The algorithm. The algorithm follows the same idea as in [CvDHS16]. For a k-query quan-
tum algorithm, we consider the mapping Z: Knk Kk KJ defined in Section3.1 for K = F ,
q
× →
R, and C. The range of Z is denotedby R := Z(Knk,Kk). For each z R , we choose a unique
k k
∈
(x,y) Knk Kk such that Z(x,y) = z. To construct a reversible mapping, let T be some setof
k
∈ ×
uniquerepresentatives,sothat Z: T R isabijection.
k k
→
6 JIANXINCHEN,ANDREWM.CHILDS,ANDSHIH-HANHUNG
3.2.1. K = F . ThealgorithmgeneratesauniformsuperpositionoverT ,performskphasequeries,
q k
andcomputesZ inplace,giving
1 1
∑ x,y ∑ e(Z(x,y) c) x,y (18)
T | i 7→ T · | i
| k| (x,y)∈Tk | k| (x,y)∈Tk
p p1 ∑ e(z c) z . (19)
7→ R · | i
| k| z∈Rk
WethenmeasureinthebasisofFourierstaptes c := 1 ∑ e(z c) z . Asimplecalculation
| i √qJ z∈FqJ · | i
shows that the result of this measurement is the correct vector of coefficients with probability
R /qJ. e
k
| |
3.2.2. K = R. We consider a bounded subset S RJ and a set T of unique preimages of each
⊆ k′
element in R S such that Z(T ) = R S and Z: T R S is bijective. The algorithm on
k ∩ k′ k ∩ k′ → k ∩
input ψ withsupportsupp(ψ) R Sgives
k
| i ⊆ ∩
ψ = dJzψ(z) z dJzψ(z) z Z 1(z) (20)
−
| i | i 7→ | i| i
ZRk∩S ZRk∩S
dJzψ(z)e(z c) z Z 1(z) (21)
−
7→ · | i| i
ZRk∩S
dJzψ(z)e(z c) z =: ψ . (22)
c
7→ · | i | i
ZRk∩S
ThechoiceofSconstrainsthesetofinputsthatcanbeperfectlydistinguishedbythisprocedure,
ascapturedbythefollowinglemma.
Lemma3.1 (Orthogonality). For positive integer n, let m(A) := dnz be the measure of the set A
A ⊆
Rn. Let SbeaboundedsubsetofRn withnonzero measure. Let c = 1 dnze(c z) z andletU
| i R √m(S) S · | i
bethemaximalsubsetofRn suchthatforanyc,c′ U withc = c′, R
∈ 6 e
1
c′ c = dnze((c c′) z) = 0. (23)
h | i m(S) − ·
ZS
Thenthereisalattice ΛsuchthatU Rn/Λ.
e e∈
Proof. Bydefinition,c c mustbeazerooftheFouriertransform (I )oftheindicatorfunction
′ S
− F
I (z). We denote Λ := c : (I )(c) = 0 0 and let c U. Clearly U c +Λ as Λ
S S 0 0
{ F } ∪{ } ∈ ⊆
^
contains all zeros. Since c+c c = 0 for all c Λ 0 , we have c +Λ U and U =
0 0 0
h | i ∈ \{ } ⊆
^ ^
c +Λ. If c Λ 0 , then c +c c = c c c = 0 implies that c Λ. If c,c Λ 0 ,
0 0 0 0 0 ′
∈ \{ } h | i h | − i − ∈ ∈ \{ }
then c^+c ^c +c = c+^c +ec c = 0 implies c+c Λ 0 . Therefore Λ is an additive
0 ′ 0 ′ 0 0 ′
h |− i h | i ∈ \{ }
subgroupofRn. e e
NowweprovethatΛisalattice. Forǫ > 0,δ B(ǫ),andc Λ,
e
∈ ∈
2 2
c]+δ c 2 = dnze(δ z) dnz cos(2πδ z) > 0 (24)
|h | i| · ≥ ·
(cid:12)ZS (cid:12) (cid:12)ZS (cid:12)
ifS ⊆ B(r)forr < 41ǫ. ThuseB(ǫ)c(cid:12)(cid:12)(cid:12)ontainsexactl(cid:12)(cid:12)(cid:12)yone(cid:12)(cid:12)(cid:12) elementinΛandh(cid:12)(cid:12)(cid:12)enceΛisdiscrete. (cid:3)
Roughlyspeaking,Lemma3.1isaconsequenceoftheuncertaintyprinciple: restrictingthesup-
port to a finite window limits the precision with which we can determinethe Fourier transform.
Intheproof,notethatalargerwindowoffersbetterresolutionofthecoefficients.
We have shown that the set Λ of perfectly distinguishable coefficients forms a lattice. We also
require the set c : c Λ to be a complete basis. Since z c = 1 e(z c), completeness
{| i ∈ } h | i √m(S) ·
e e
QUANTUMALGORITHMFORMULTIVARIATEPOLYNOMIALINTERPOLATION 7
impliesthat z isoftheform∑ e( z c) c uptoanormalizationconstant. Moreformally,we
havethefollo|wiing: c∈Λ − · | i
e
Lemma 3.2 (Completeness). For positive integer n, let m(A) := dnz be the measure of the set A
A ⊆
Rn. Let Λ be a discrete additive subgroup of Rn. Let S be a bounded set with nonzero measure and
R
c = 1 dnze(z c) z . Then c : c Λ formsacompletebasis oversupport S ifandonlyifS
| i √m(S) S · | i {| i ∈ }
isafundamenRtaldomainoftheduallatticeofΛ.
e e
Proof. LetΛbetheduallatticeofΛ. Weobservethat(ignoringthenormalizationconstant)
e ∑ e( z c) c = dJz′ ∑ e((z′ z) c) z′ (25)
− · | i − ·
c Λ ZS c Λ
∈ ∈ (cid:12) (cid:11)
e = dJz′ ∑ δ(z′ z z0)(cid:12) z′ (26)
− −
ZS z0∈Λ (cid:12) (cid:11)
(cid:12)
= ∑ IS(z+e z0) z+z0 = (z+Λ) S . (27)
| i ∩
z0∈Λ (cid:12)(cid:12) E
e
(cid:12)
In Equation(26), ∑ e(z c) = ∑ δ(ez z ) up to a constant factor [Ho¨r83, Section 7.2]. The
c∈Λ · z0∈Λ − 0
set (z+Λ) S cannot be empty, so a fundamental domain of Λ is a subset of S. For z,z Rn,
′
∩ e ∈
(z+Λ) S (z +Λ) S = 0 if z / z+Λ, which implies that S is a subset of a fundamental
′ ′
h ∩ | ∩ i ∈
domainoefΛ. e (cid:3)
e e e
Lemma3e.2 further restricts the bounded set S has to be a fundamental region of Λ. Without
loss of generality, one may choose S to be a fundamental domain of a lattice centered at zero. In
thelaststep,thealgorithmappliestheunitaryoperator e
1
∑ dJze( z c′) c′ z (28)
m(S) c ΛZS − · h |
′∈ (cid:12) (cid:11)
tothestate ψ inEquation(22)p. Thealgorithmoutputsc (cid:12) Λwithprobability
c ′
| i ∈
1 2 m(R S)
dJzψ(z)e(z (c c′)) k∩ , (29)
m(R S)m(S) · − ≤ m(S)
k∩ (cid:12)ZRk∩S (cid:12)
(cid:12) (cid:12)
wheretheupperboundfollowsfro(cid:12)mtheCauchy-Schwarzineq(cid:12)uality. Themaximumisreachedif
(cid:12) (cid:12)
ψ(z) = 1 I (z) and c happens to be a lattice point. If c / Λ, the algorithm returns the
√m(Rk∩S) Rk∩S ∈
closestlatticepointwithhighprobability.
To achieve arbitrarily high precision, one may want to take S RJ. In this limit, the basis of
→
coefficientsisnormalizedtotheDiracdeltafunction,i.e., c c = δ(J)(c c ). Inthiscase,Λ RJ
′ ′
h | i − →
andtheunitaryoperatorinEquation(28)becomesthe J-dimensionalquantumFouriertransform
m(R S)
overtherealnumbers. However,fortheinterpolationproeblem,thesuccessprobability k∩ is
m(S)
not well-defined in the limit S RJ since different shapes for S can give different probabilities.
→
Thus it is necessary to choose a bounded region, and we leave the optimal choice as an open
question.
Though the size of the fundamental domain S affects the resolution of the coefficients, it does
m(R S)
notaffectthemaximalsuccessprobability k∩ . Thiscanbeseenbyscaleinvariance: forevery
m(S)
z R , there is a preimage (x,y) such that Z(x,y) = z. Then λz R since Z(x,λy) = λz
k k
for∈any λ R. In terms of the bijection ℓ: z λz for λ R , ∈we have ℓ(R ) = R and
× k k
∈ 7→ ∈
8 JIANXINCHEN,ANDREWM.CHILDS,ANDSHIH-HANHUNG
ℓ(R S) = R ℓ(S). Thenm(R ℓ(S)) = m(ℓ(R S)) = λJm(R S)andhence
k k k k k
∩ ∩ ∩ ∩ ∩
m(R ℓ(S)) m(R S)
k∩ = k∩ . (30)
m(ℓ(S)) m(S)
m(R S)
Thuswecan make theprecision arbitrarily high bytaking S arbitrarily large, andwecall k∩
m(S)
thesuccessprobabilityofthealgorithm.
3.2.3. K = C. WeconsideraboundedsetS CJ andasetT ofuniquepreimagesofeachelement
⊆ k′
in R S suchthat Z(T ) = R and Z: T R S is bijective. Thealgorithm oninput ψ with
k∩ k′ k k′ → k∩ | i
supportsupp(ψ) R Sgives
k
⊆ ∩
ψ = d2Jzψ(z) z d2Jzψ(z) z φ(Z−1(z)) (31)
| i Zφ(Rk∩S) | i 7→ Zφ(Rk∩S) | i| i
d2Jzψ(z)e(z c) z φ(Z−1(z)) (32)
7→ Zφ(Rk∩S) · | i| i
d2Jzψ(z)e(z c) z =: ψ . (33)
c
7→ Zφ(Rk∩S) · | i | i
ByLemma3.1andLemma3.2,thesetSmustbeafundamentaldomaininCJ. Let c : c Λ
{| i ∈ }
bethemeasurementbasis. Inthelaststepofthealgorithm,weapplytheunitaryoperator
1 e
∑ d2Jze( z c′) c′ z (34)
m(S) c φ(Λ)Zφ(S) − · h |
′∈ (cid:12) (cid:11)
tothestate ψ inEquation(p33). Thealgorithmoutputsc Λ(cid:12)withprobability
c ′
| i ∈
2
1
d2Jzψ(z)e(z (c c )) . (35)
′
m(Rk∩S)m(S) (cid:12)Zφ(Rk∩S) · − (cid:12)
(cid:12) (cid:12)
Again,since ψ isnormalized, Equation(cid:12) (35) cannotbearbitrarily large(cid:12). BytheCauchy-Schwarz
| i (cid:12) (cid:12)
m(R S)
inequality, Equation(35) is upper bounded by k∩ ; this maximal success probability is ob-
m(S)
tained if ψ(z) = 1 I (z) and c happens to be a lattice point. If c / Λ, the algorithm
√m(Rk∩S) φ(Rk∩S) ∈
returnstheclosestlatticepointwithhighprobability.
BythesameargumentasinSection3.2.2,wecanshowscaleinvarianceholdsforcomplexnum-
bers: for ℓ: z 7→ λz where z ∈ CJ and λ ∈ R×, m(mR(kS∩)S) = m(mR(kℓ∩(Sℓ()S))). Thus we can make the pre-
cisionofthealgorithmarbitrarilyhighbytakingSarbitrarilylargewithoutaffectingthemaximal
successprobability.
3.3. Performance. Wehave shownin Section3.2.1thattheoptimalsuccessprobability isatmost
R /qJ for K = F . For real and complex numbers, we consider a bounded support S in which
k q
| |
the algorithm is performed. The success probability of the algorithm with this choice is at most
m(R S)
k∩ , as shown in Equation(29) and Equation(35). To establish the querycomplexity, first we
m(S)
showthatifdimR = J,thealgorithmoutputsthecoefficientswithboundederror.
k
Lemma3.3. Forpositiveintegersn,k,d,let J := (n+d)andletm(A) := dJzbethevolumeof A RJ.
d A ⊆
Let Z: Knk Kk, Z(x,y) = ∑k y xj foraninfinitefieldK. Let R = Z(Knk,Kk) betherangeof Z. If
× i=1 i i k R
dimRk = J,then m(mR(kS∩)S) > 0ifSisafundamentaldomaincenteredat0.
Proof. R isaconstructiblesetforK = C anditisasemialgebraicsetforK = R. By[BBO15]and
k
[BT15], R hasnon-emptyinteriorifdim(R ) = J forbothcases.
k k
QUANTUMALGORITHMFORMULTIVARIATEPOLYNOMIALINTERPOLATION 9
S is a fundamental domain centered at 0 with finite measure, so we only need to prove that
m(R S) isofpositivemeasure,orequivalently,thattheinteriorof R andtheinteriorofShave
k k
∩
non-emptyintersection.
If this is not the case, then any interior point of S cannot be in the interior of R . By scale
k
invariance of R , any point in Kn except 0 cannot be in the interior of R , which contradicts the
k k
factthat R hasnon-emptyinteriorgivendim(R ) = J. (cid:3)
k k
Lemma3.3 shows that for infinite fields, although we perform the algorithm over a bounded
support, the query complexity can be understoodby considering the dimension of the entire set
R . Moreover, by invoking recent work on typical ranks, we can establish the minimum number
k
ofqueriestodeterminethecoefficientsalmostsurely.
Nowletv (x ,x ,...,x )bethe(n+d)-dimensionalvectorthatcontainsallmonomialswithvari-
d 1 2 n d
ables x ,...,x ofdegreenomorethandasitsentries. Let
1 n
X := v (x ,x ,...,x ) : x ,x ,...,x K (36)
n,d d 1 2 n 1 2 n
{ ∈ }
whereKisagivengroundfield. Forexample,wehave
X = (x2,x2,x2,x x ,x x ,x x ,x ,x ,x ,1)T : x ,x ,x K . (37)
3,2 { 1 2 3 1 2 1 3 2 3 1 2 3 1 2 3 ∈ }
Our question is to determine the smallest number k such that a generic vector in K(n+dd) can be
written as a linear combination of no more than k elements from X . More precisely, we have
n,d
R = ∑k c v : c K,v X , and we ask what is the smallest number k such that R has
k { i=1 i i i ∈ i ∈ n,d} k
fullmeasureinK(n+dd).
Our approach requires basic knowledge of algebraic geometry—specifically, the concepts of
Zariskitopology,Veronesevariety,andsecantvariety. FormaldefinitionscanbefoundinSection2.2.
Forthereader’sconvenience,wealsoexplaintheseconceptsbrieflywhenwefirstusethem.
Nowwemaketwosimpleobservations.
(1) Ingeneral,v (x ,x ,...,x )canbetreatedasan(n+d)-dimensionalvectorthatcontainsall
d 1 2 n d
monomials with variables x ,...,x ,x of degree d as its entries, by simply taking the
1 n n+1
map (x ,x ,...,x ) ( x1 ,..., xn ) and multiplying by xd . For example, applying
1 2 n 7→ xn+1 xn+1 n+1
thismappingtoX gives
3,2
X = (x2,x2,x2,x x ,x x ,x x ,x x ,x x ,x x ,x2)T : x ,x ,x ,x K .
3′,2 { 1 2 3 1 2 1 3 2 3 1 4 2 4 3 4 4 1 2 3 4 ∈ }
ThenewsetX isslightlybiggerthanX sinceitalsocontainsthosepointscorrespond-
n′,d n,d
ingtox = 0,butthiswillnotaffectourcalculationsincethedifferenceisjustameasure
n+1
zerosetin X .
n′,d
(2) The set X is the Veronese variety. One may also notice that this set is isomorphic to
n′,d
(x1,x2,...,xn+1)T ⊗d inthesymmetricsubspace.
TheseobservationsimplythatinsteadofstudyingR ,wecanstudythenewset
(cid:0) (cid:1) k
k
R = ∑c v : c K,v X . (38)
′k i ′i i ∈ ′i ∈ n′,d
(i=1 )
Ingeneral,wehaveasequenceofinclusions:
Xn′,d = R1′ ⊆ R2′ ⊆ ··· ⊆ R′k ⊆ ··· ⊆ K(n+dd). (39)
BytakingtheZariskiclosure,wealsohave
Xn′,d = R1′ ⊆ R2′ ⊆ ··· ⊆ R′k ⊆ ··· ⊆ K(n+dd) (40)
where R isthekthsecantvarietyoftheVeronesevariety X .
′k n′,d
10 JIANXINCHEN,ANDREWM.CHILDS,ANDSHIH-HANHUNG
Palatinishowedthefollowing[Pal09,Adl87]:
Lemma3.4. IfdimR dimR +1,then R islinear.
′k+1 ≤ ′k ′k+1
Inparticular,thisshowsthatifdimR′k = (n+dd),thenR′k = K(n+dd).
m(R S)
ForaninfinitefieldK,definekK tobethesmallestintegersuchthat mk(KS∩) = 1. Forthefinite
m(R S)
fieldcaseK = F ,weonlyrequirethat kFq∩ goesto1whenqtendstoinfinity.
q m(S)
3.3.1. K = C. AtheoremduetoAlexanderandHirschowitz[AH95]impliesan upperboundon
thequerycomplexityofpolynomialinterpolationoverC.
Theorem3.5(Alexander-HirschowitzTheorem,[AH95]). Thedimensionof R satisfies
′k
k(n+1) k(k−1) d = 2,2 k n;
− 2 ≤ ≤
dimR′k = (n+dd)−1 (d,n,k) = (3,4,7),(4,2,5),(4,3,9),(4,4,14); (41)
min k(n+1),(n+d) otherwise.
{ d }
Thus,theminimumktomake R′k = C(n+dd) is
n+1 d = 2,n 2;
≥
kC(n,d) := ⌈n+11(n+dd)⌉+1 (n,d) = (4,3),(2,4),(3,4),(4,4); (42)
 1 (n+d) otherwise.
⌈n+1 d ⌉
Byparametercounting,weseethatRk isoffullmeasurein R′k. ItremainstoshowthatR′k isoffull
measureinitsZariskiclosure R :
′k
Theorem3.6. R isoffullmeasurein R .
′k ′k
Proof. R is just theimage ofthe map (Q ,Q ,...,Q ) (Q +Q + +Q ). By Exercise3.19
′k 1 2 k 7→ 1 2 ··· k
in Chapter II of[Har77], R is a constructible set,soit contains an opensubsetofeach connected
′k
(cid:3)
componentof R . Hence,itscomplementisofmeasure0,whichcompletestheproof.
′k
Thefollowingcorollaryfollowsimmediately.
Corollary3.7. Rk hasmeasure0inC(n+dd) fork < kC(n,d)andmeasure1inC(n+dd) fork kC(n,d).
≥
Thus, as the integer k increases, m(mR(′kS∩)S) suddenly jumps from 0 to 1 at the point kC(n,d), and
m(R S)
sodoes k∩ . Thisimpliespart(3)ofTheorem1.1.
m(S)
3.3.2. K = R. Now consider the case K = R. For d = 2, (n+1)-variate symmetric tensors are
simply (n+1) (n+1) symmetricmatrices, so a random (n+1)-variate symmetric tensorwill
×
be ofrank n+1with probability 1. However,if theorderofthesymmetrictensorsis larger than
2, thesituationis much morecomplicated. For example, arandombivariate symmetrictensorof
order3willbeoftwodifferentranks,2and3,bothwithpositiveprobabilities.
Fromtheperspectiveofalgebraic geometry,itstill holdsthat R′k = R(n+dd) for k ≥ kC(n,d), and
for k < kC(n,d), R′k is of measure zero in R(n+dd). It also holds that Rk is of full measure in R′k.
However,theclaimthatR hasfullmeasurein R nolongerholdsoverR. Asweexplainedinthe
′k ′k
proofofTheorem3.6,R istheimageofthemap(Q ,Q ,...,Q ) (Q +Q + +Q ). Foran
′k 1 2 k 7→ 1 2 ··· k
algebraically closedfieldK,itisknownthattheimageofanymapisalwaysaconstructiblesetin
itsZariskiclosure. ThusR isoffullmeasurein R . OverR,itiseasytoverifythattheimagemay
′k ′k
not be of full measure in its Zariski closure (a simple counterexample is x x2). Consequently,
7→