Boolean functions of low polynomial degree for quantum query complexitytheory

Rusins FreivaldsInstitute of Mathematics and Computer Science

University of LatviaRaina bulvaris. 29, Riga, Latvia

Rusins.Freivalds@mii.lu.lv

Lıva GarkajeAgenskalna Valsts gimnazija

Lavızes iela 2a, Riga, LV-1002, LatviaLiva.Garkaje@gmail.com

Abstract

The degree of a polynomial representing (or approximat-ing) a function f is a lower bound for the quantum querycomplexity of f . This observation has been a source ofmany lower bounds on quantum algorithms. It has beenan open problem whether this lower bound is tight.

This is why Boolean functions are needed with a highnumber of essential variables and a low polynomial degree.Unfortunately, it is a well-known problem to construct suchfunctions. The best separation between these two complex-ity measures of a Boolean function was exhibited by Ambai-nis [5]. He constructed functions with polynomial degreeM and number of variables Ω(M2). We improve such aseparation to become exponential. On the other hand, weuse a computerized exhaustive search to prove tightness ofthis bound.

1. Introduction

Quantum computing provides speedups for factoring[15], search [10] and many related problems. Thesespeedups can be quite surprising. For example, Grover’ssearch algorithm [10] solves an arbitrary exhaustive searchproblem with N possibilities in time O(

√N). Classically,

it is obvious that time Ω(N) would be needed.This makes lower bounds particularly important in the

quantum world. If we can search in time O(√

N), why canwe not search in time O(logc N)? (Among other things,that would have meant NP ⊆ BQP .) Lower boundof Bennett et al. [7] shows that this is not possible andGrover’s algorithm is exactly optimal.

Let [N ] denote 1, . . . , N.We consider computing a Boolean function

f(x1, . . . , xN ) : 0, 1N → 0, 1 in the quantumquery model (for a survey on query model, see [9]). In

this model, the input bits can be accessed by queries to anoracle X and the complexity of f is the number of queriesneeded to compute f . A quantum computation with Tqueries is just a sequence of unitary transformations

U0 → O → U1 → O → . . . → UT−1 → O → UT .

The Uj’s can be arbitrary unitary transformations that donot depend on the input bits x1, . . . , xN . The O’s are query(oracle) transformations which depend on x1, . . . , xN . Todefine O, we represent basis states as |i, z〉 where i con-sists of dlog(N + 1)e bits and z consists of all other bits.Then, Ox maps 0, z to itself and i, z to (−1)xii, z fori ∈ 1, ..., N (i.e., we change phase depending on xi, un-less i = 0 in which case we do nothing).

The computation starts with a state |0〉. Then, we applyU0, Ox, . . ., Ox, UT and measure the final state. The resultof the computation is the rightmost bit of the state obtainedby the measurement.

The quantum computation computes f exactly if,for every x = (x1, . . . , xN ), the rightmost bit ofUT Ox . . . OxU00 equals f(x1, . . . , xN ) with certainty.

The quantum computation computes f with boundederror if, for every x = (x1, . . . , xN ), the probabilitythat the rightmost bit of UT OxUT−1 . . . OxU00 equalsf(x1, . . . , xN ) is at least 1− ε for some fixed ε < 1/2.

QE(f) (Q2(f)) denotes the minimum number T ofqueries in a quantum algorithm that computes f exactly(with bounded error). D(f) denotes the minimum numberof queries in a deterministic query algorithm computing f .

2. Main results

Currently, we have good lower bounds on the quantumcomplexity of many problems. They mainly follow by twomethods: the hybrid/adversary method[7, 3] and the poly-nomials method [6].

Proceedings of the 37th International Symposium on Multiple-Valued Logic (ISMVL'07)0-7695-2831-7/07 $20.00 © 2007

For any Boolean function f , there is a unique multilinearpolynomial g such that f(x1, . . . , xN ) = g(x1, . . . , xN ) forall x1, . . . , xN ∈ 0, 1. We say that g represents f . Letdeg(f) denote the degree of polynomial representing f .

A polynomial g(x1, . . . , xN ) approximates f if 1− ε ≤g(x1, . . . , xN ) ≤ 1 whenever f(x1, . . . , xN ) = 1 and0 ≤ g(x1, . . . , xN ) ≤ ε whenever f(x1, . . . , xN ) = 0.Let ˜deg(f) denote the minimum degree of a polynomial ap-proximating f . It is known that

Theorem 1 [6]

1. QE(f) = Ω(deg(f));

2. Q2(f) = Ω( ˜deg(f));

This theorem has been a source of many lower boundson quantum algorithms.

The polynomials method is useful for proving lowerbounds both in classical [12] and quantum complexity [6].It is known that

1. the number of queries QE(f) needed to compute aBoolean function f by an exact quantum algorithm ex-actly is at least deg(f)

2 , where deg(f) is the degree ofthe multilinear polynomial representing f ,

2. the number of queries Q2(f) needed to compute f bya quantum algorithm with two-sided error is at least˜deg(f)

2 , where ˜deg(f) is the smallest degree of a mul-tilinear polynomial approximating f .

This reduces proving lower bounds on quantum algo-rithms to proving lower bounds on degree of polynomials.This is a well-studied mathematical problem with meth-ods from approximation theory available. Quantum lowerbounds shown by polynomials method include a Q2(f) =Ω( 6

√D(f)) relation for any total Boolean function f [6].

Polynomials method is also a key part of recent Ω(√

N)lower bound on set disjointness which resolved a longstand-ing open problem in quantum communication complexity[14].

Given the usefulness of polynomials method, it is animportant question how tight is the polynomials lowerbound. Buhrman et al. [9] have proved that, for all totalBoolean functions, Q2(f) = O(deg6(f)) and QE(f) =O(deg4(f)). The second result was recently improved toQE(f) = O(deg3(f)) [11]. Thus, the bound is tight up topolynomial factor.

Even stronger result would be QE(f) = O(deg(f)) orQ2(f) = O( ˜deg(f)). Then, determining the quantum com-plexity would be equivalent to determining the degree of afunction as a polynomial. It has been an open problem toprove or disprove either of these two equalities [6, 9].

Unfortunately, it is not easy to construct a Boolean func-tion such that its number of variables is larger than the de-gree of the representing polynomial. We have only severalsuch examples.

Ambainis’ function[5]: f(x) is equal to 1 iff x =x1x2x3x4 is one of the following values: 0011, 0100, 0101,0111, 1000, 1010, 1011, 1100. This function has the de-gree of 2, as witnessed by polynomial f(x1, x2, x3, x4) =x1 +x2 +x3x4−x1x4−x2x3−x1x2 and the deterministiccomplexity D(f) = 3.This function can be iterated.Define a sequence f1 = f , f2,. . . with fd being a function of 4d variables by

fd+1 = f(fd(x1, . . . , x4d), fd(x4d+1, . . . , x2·4d),

fd(x2·4d+1, . . . , x3·4d), fd(x3·4d+1, . . . , x4d+1)). (1)

Then, deg(fd) = 2d, D(fd) = 3d and, on every input x,sx(fd) = 2d and bsx(fd) = 3d.

Kushilevitz’s function (cited in[12]):

fK(x1, x2, x3, x4, x5, x6) = 0

iff either x1 = x2 = x3 = x4 = x5 = x6 = 0 or

x1 = x2 = x3 = 1

x2 = x3 = x4 = 1

x3 = x4 = x5 = 1

x4 = x5 = x1 = 1

x5 = x1 = x2 = 1

x1 = x3 = x6 = 1

x3 = x5 = x6 = 1

x5 = x2 = x6 = 1

x2 = x4 = x6 = 1

x4 = x1 = x6 = 1

The function fK is represented by the polynomial

x1 + x2 + x3 + x4 + x5 + x6−

−x1x2 − x1x3 − x1x4 − x1x5 − x1x6−

−x2x3 − x2x4 − x2x5 − x2x6 − x3x4−

−x3x5 − x3x6 − x4x5 − x4x6 − x5x6+

x1x2x4 + x2x3x5 + x3x4x1 + x4x5x2 + x5x1x3+

x1x5x6 + x3x2x6 + x5x4x6 + x2x1x6 + x4x3x6.

It is relatively easy to understand that a small degree ofthe representing polynomial implies a small number of theessential variables of the function. It is much more hardto make precise numerical estimates. The subsequent two

Proceedings of the 37th International Symposium on Multiple-Valued Logic (ISMVL'07)0-7695-2831-7/07 $20.00 © 2007

theorems are very much alike. However the methods of theproof are very much different. Theorem 2 below was provedexplicitly and its proof is not difficult. On the other hand,Theorem 3 was proved by (rationalized) exhaustive searchon a computer.

Theorem 2If for a Boolean function f(x1, x2, · · · , xn) the degree

of the representing polynomial deg(f) equals 2 and all thevariables are essential, then n ≤ 4.

Theorem 3If for a Boolean function f(x1, x2, · · · , xn) the degree

of the representing polynomial deg(f) equals 3 and all thevariables are essential, then n ≤ 9.

We have also proved an exponential gap between thedeg(f) and the number of essential variables in f .

Theorem 4For arbitrary k ≥ 2 there is a (2k + 2k−2 − 1)-variable

Boolean function fk with degfk = k.Proof:By induction. For k = 2 the assertion follows from the

existence of the Ambainis’ function.Let P (x1, x2, · · · , x(2k+2k−2−1)) be the representing

polynomial for the function fk. The polynomial

x(2k+1+2k−1−1)P (x1, x2, · · · , x(2k+2k−2−1))+

+(1− x(2k+1+2k−1−1))(x(2k+2k−2), x(2k+2k−2+1), · · · ,

x(2k+1+2k−1))

represents the Boolean function fk+1.CorollaryThere is a Boolean function f of 9 variables with

degf = 3.This function is represented by the polynomial

x9(x1 + x2 + x3x4 − x1x4 − x2x3 − x1x2)+

+(1− x9)(x5 + x6 + x7x8 − x5x8 − x6x7 − x5x6).

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Proceedings of the 37th International Symposium on Multiple-Valued Logic (ISMVL'07)0-7695-2831-7/07 $20.00 © 2007