A quantum algorithm to estimate the Gowers U 2 norm and linearity testing of Boolean functions
C. A. Jothishwaran, Anton Tkachenko, Sugata Gangopadhyay, Constanza Riera, Pantelimon Stanica
aa r X i v : . [ c s . D M ] J un A quantum algorithm to estimate the Gowers U normand linearity testing of Boolean functions C. A. Jothishwaran , Anton Tkachenko , Sugata Gangopadhyay Constanza Riera , Pantelimon St˘anic˘a Department of Computer Science and Engineering,Indian Institute of Technology Roorkee, Roorkee 247667, INDIA [email protected], [email protected] Department of Computer Science, Electrical Engineering and Mathematical Sciences,Western Norway University of Applied Sciences, 5020 Bergen, NORWAY
[email protected], [email protected] Department of Applied MathematicsNaval Postgraduate School, Monterey, CA 93943–5216, USA [email protected]
Abstract.
We propose a quantum algorithm to estimate the Gowers U norm of aBoolean function, and extend it into a second algorithm to distinguish between linearBoolean functions and Boolean functions that are ǫ -far from the set of linear Booleanfunctions, which seems to perform better than the classical BLR algorithm. Finally, weoutline an algorithm to estimate Gowers U norms of Boolean functions. Keywords:
Boolean functions, Fourier spectrum, Gowers uniformity norms, quantumalgorithms
Gowers uniformity norms were introduced by Gowers [6] to prove Szmer´edi’s theorem. Intheir full generality, Gowers uniformity norms operate over functions from finite sets tothe field of complex numbers. The Gowers uniformity norm of dimension d of a function f tells us the extent of correlation of f to the polynomial phase functions of degree upto d −
1. In this paper, we consider the Gowers uniformity norm U of dimension 2 forBoolean functions, and find a quantum estimate of its upper bound. We also propose alinearity test of Boolean functions based on the same quantum algorithm, which seemsto perform better than the classical BLR algorithm. We denote the ring of integers, the set of positive integers, and the fields of real num-bers and complex numbers by Z , Z + , R , and C , respectively. For any n ∈ Z + , the set[ n ] = { i ∈ Z + : 1 ≤ i ≤ n } , and F n = { x = ( x , . . . , x n ) : x i ∈ F , for all i ∈ [ n ] } where F is the prime field of characteristic 2. Addition in each of the above algebraicsystems is denoted by ‘+’. An n -variable Boolean function F is a function from F n to F . The set of all such functions is denoted by B n . Each function F ∈ B n has its char-acter form f : F n → R defined by f ( x ) = ( − F ( x ) , for all x ∈ F n . In this article,abusing notation, we refer to the character form f as Boolean functions and go to theextent of writing f ∈ B n , whenever F ∈ B n , if there is no danger of confusion. For any x, y ∈ F n , the inner product x · y = P i ∈ [ n ] x i y i where the sum is over F . The (Hamming)weight of a vector u = ( u , . . . , u n ) ∈ F n is wt( u ) = P i ∈ [ n ] u i , where the sum is over Z . The weight of a Boolean function F ∈ B n , or equivalently f ∈ B n is the cardinalitywt( F ) = (cid:12)(cid:12) { x ∈ F n : F ( x ) = 0 } (cid:12)(cid:12) , or equivalently wt( f ) = (cid:12)(cid:12) { x ∈ F n : f ( x ) = 1 } (cid:12)(cid:12) . The Ham-ming distance between F, G ∈ B n , or equivalently, between f, g ∈ B n is d H ( F, G ) = (cid:12) { x ∈ F n : F ( x ) = G ( x ) } (cid:12)(cid:12) , or, d H ( f, g ) = (cid:12)(cid:12) { x ∈ F n : f ( x ) = g ( x ) } (cid:12)(cid:12) . Any Boolean function F ∈ B n can be expressed as a polynomial, called the algebraic normal form (ANF), F ( x , . . . , x n ) = X u ∈ F n λ u x u where λ u ∈ F , and x u = Y i ∈ [ n ] x u i i . (1)The algebraic degree of a Boolean function deg( F ) = max { wt( u ) : λ u = 0 } . A Booleanfunction with algebraic degree at most 1 is said to be an affine function. An affine functionin B n is of the form ϕ ( x ) = u · x + ε for some u ∈ F n and ε ∈ F . An affine function with ε = 0 is said to be a linear function. We denote the set of all n -variable affine functionsby A n , and the set of all n -variable linear functions by L n .The Fourier series expansion of f ∈ B n is f ( x ) = X u ∈ F n b f ( u )( − u · x . (2)The coefficients b f ( u ) are said to be the Fourier coefficients of f . The transformation f b f is the Fourier transformation of f . It is known that X x ∈ F n ( − v · x = ( v = 02 n if v = 0 . (3)Equations (2) and (3) yield X x ∈ F n f ( x )( − u · x = X x ∈ F n X v ∈ F n b f ( v )( − ( u + v ) · x = X v ∈ F n b f ( v ) X x ∈ F n ( − ( u + v ) · x = 2 n b f ( u ) , (4)that is, b f ( u ) = 2 − n P x ∈ F n f ( x )( − u · x . The sum X x ∈ F n f ( x ) = X x ∈ F n X u ∈ F n b f ( u )( − u · x X v ∈ F n b f ( v )( − v · x = X u ∈ F n X v ∈ F n b f ( u ) b f ( v ) X x ∈ F n ( − ( u + v ) · x = 2 n X u ∈ F n b f ( u ) . The identity P x ∈ F n b f ( x ) = 2 − n P x ∈ F n f ( x ) , is known as the Plancherel’s identity .This is true for f : F n → R . If f ∈ B n , we have the Parseval’s identity P u ∈ F n b f ( u ) = 1.For f, g ∈ B n the convolution product, f ∗ g is defined as( f ∗ g )( x ) = 2 − n X y ∈ F n f ( y ) g ( x + y ) = 2 − n X y ∈ F n f ( x + y ) g ( y ) . (5)Using (4) on (5) [ f ∗ g ( u ) = 2 − n X x ∈ F n ( f ∗ g )( x )( − u · x = 2 − n X x ∈ F n X y ∈ F n f ( y ) g ( x + y )( − u · x = 2 − n X x ∈ F n X y ∈ F n f ( y )( − u · y g ( x + y )( − u · ( x + y ) = − n X y ∈ F n f ( y )( − u · y − n X x ∈ F n g ( y )( − u · x = b f ( u ) b g ( u ) . (6)For each x ∈ F n , ( f ∗ f )( x ) = 2 − n P y ∈ F n f ( y ) f ( x + y ) is said to be the autocorrelation of f at x , and [ f ∗ f ( x ) = b f ( x ) .he derivative of f ∈ B n at c ∈ F n is the function ∆ c f ( x ) = f ( x ) f ( x + c ) , for all x ∈ F n . (7)We write ∆ x (1) ,...,x ( k ) f ( x ) = Y S ⊆ [ k ] f x + X i ∈ S x ( i ) , (8)where x ( i ) ∈ F n , for all i ∈ [ k ], and some k ∈ Z + . In Equation (7) we have definedderivatives of a Boolean function when the codomain of the function is { , − } . In thatcase, the resulting derivative turns out to be function from F n to { , − } . The derivativeof a Boolean function F : F n → F , at a point a ∈ F n is ∆ a F ( x ) = F ( x ) + F ( x + a ) , for all x ∈ F n . (9)For any a, b ∈ F n ∆ a,b F ( x ) = F ( x ) + F ( x + b ) + F ( x + a ) + F ( x + a + b ) . (10)For a, b, c ∈ F n , ∆ a,b,c F ( x ) = F ( x ) + F ( x + c ) + F ( x + b ) + F ( x + b + c ) + F ( x + a )+ F ( x + a + c ) + F ( x + a + b ) + F ( x + a + b + c ) . (11)In general for x (1) , . . . , x ( k ) ∈ F n , ∆ x (1) ,...,x ( k ) F ( x ) = X S ⊆ [ k ] F x + X i ∈ S x ( i ) . (12) Gowers [6] introduced (now, called Gowers) uniformity norms in his work on Szmer´edi’stheorem. For an introductory reading on the topic, we refer to the Ph.D. thesis of Chen [4].The Gowers U k norm of f ∈ B n , denoted by k f k U k , is defined as k f k U k = − ( k +1) n X x,x (1) ,...,x ( k ) ∈ F n Y S ⊆ [ k ] f x + X i ∈ S x ( i ) − k . (13)The Gowers U norm is k f k U = − n X x ∈ F n X a ∈ F n X b ∈ F n f ( x ) f ( x + a ) f ( x + b ) f ( x + a + b ) − = − n X a ∈ F n X x ∈ F n f ( x ) f ( x + a ) X b ∈ F n f ( x + b ) f ( x + a + b ) − = − n X a ∈ F n X x ∈ F n f ( x ) f ( x + a ) X y ∈ F n f ( y ) f ( y + a ) − = − n X a ∈ F n X x ∈ F n b f ( x ) ( − x · a X y ∈ F n b f ( y ) ( − y · a − − n X x ∈ F n X y ∈ F n b f ( x ) b f ( y ) X a ∈ F n ( − ( x + y ) · a − = X x ∈ F n b f ( x ) − . The Gowers U norm is k f k U = (cid:16) − n X x ∈ F n X a ∈ F n X b ∈ F n X c ∈ F n f ( x ) f ( x + a ) f ( x + b ) f ( x + a + b ) f ( x + c ) f ( x + a + c ) f ( x + b + c ) f ( x + a + b + c ) (cid:17) − . (14)Substituting the derivative in (14) k f k U = (cid:16) − n X c ∈ F n X x,a,b ∈ F n ∆ c f ( x ) ∆ c f ( x + a ) ∆ c f ( x + b ) ∆ c f ( x + a + b ) (cid:17) − = (cid:16) − n X c ∈ F n (cid:13)(cid:13) ∆ c f ( x ) (cid:13)(cid:13) U (cid:17) − . (15)In general, the Gowers U k norm of f ∈ B n is k f k U k = − ( k +1) n X x,x (1) ,...,x ( k ) ∈ F n Y S ⊆ [ k ] \ [2] Y T ⊆ [2] f x + X i ∈ S x ( i ) + X j ∈ T x ( j ) − k = − ( k +1) n X x,x (1) ,...,x ( k ) ∈ F n Y T ⊆ [2] Y S ⊆ [ k ] \ [2] f x + X i ∈ S x ( i ) + X j ∈ T x ( j ) − k = − ( k +1) n X x (3) ,...,x ( k ) ∈ F n X x (1) ,x (2) ∈ F n Y T ⊆ [2] ∆ x (3) ,...,x ( k ) f x + X j ∈ T x ( j ) − k = − ( k − n X x (3) ,...,x ( k ) ∈ F n (cid:13)(cid:13)(cid:13) ∆ x (3) ,...,x ( k ) f ( x ) (cid:13)(cid:13)(cid:13) U − k . (16)Equation (16) shows the relation between the Gowers U k norm and the U norms of the( k − f . The time complexity of computing the Gowers U norm of aBoolean function f ∈ B n is O ( n n ). Arguing in the same way, the time complexity ofcomputing Gowers U k norm is O ( n kn ).In this paper, we propose a quantum algorithm to estimate an upper bound of Gowers U norm and based upon that, we find a quantum counterpart of the BLR linearity testing [2]that tends to perform better than the classical version, assuming the availability of aquantum computer with sufficient number of qubits. The complexities of the quantumalgorithms are independent of the number of variables n , of course, again with the strongassumption of the availability of a fairly large quantum computer. In this section, we discuss the connection between the Gowers uniformity norms and theapproximation of Boolean functions by low degree Boolean functions. The nonlinearity,enoted by nl ( f ), of a Boolean function f ∈ B n is the minimum Hamming distance from f to all affine functions in A n . That is nl ( f ) = min { d H ( f, ϕ ) : ϕ ∈ A n } . (17)The r th-order nonlinearity of a Boolean function f , denoted by nl r ( f ), is the minimumHamming distance from f to the functions having algebraic degree less than or equalto r . The first-order nonlinearity nl ( f ) = nl ( f ). It is well known that (cf. [5]) nl ( f ) = 2 n − (cid:16) − max x ∈ F n | b f ( x ) | (cid:17) . (18)Carlet [3] obtained lower bounds of r th-order nonlinearity of Boolean functions by usingnonlinearities of their higher-order derivatives. This establishes a relationship between the r th-order nonlinearities of Boolean functions and Fourier coefficients of their derivatives.Gowers uniformity norms involve Fourier coefficients of higher-order derivatives (16), andserve the same purpose as evident from the following theorem. Theorem 1 ([4], Fact 2.2.1).
Let k ∈ Z + , ǫ > . Let P : F n → F be a polynomialof degree at most k , and f : F n → R . Suppose (cid:12)(cid:12)(cid:12) − n P x ∈ F n f ( x )( − P ( x ) (cid:12)(cid:12)(cid:12) ≥ ǫ . Then k f k U k +1 ≥ ǫ . For k = 1, informally, this means that if, for some f , the norm k f k U is small then itsFourier coefficients are small, and therefore f has high nonlinearity. On the other hand, k f k U = X x ∈ F n b f ( x ) ≤ max x ∈ F n | b f ( x ) | X x ∈ F n b f ( x ) = max x ∈ F n | b f ( x ) | (applying Parseval identity)= (1 − − n nl ( f )) (using (18)) . (19)Equation (19) tells us that if a Boolean function has high nonlinearity then its U normis small, and if U norm is large, then the nonlinearity is small.The second-order nonlinearity of a Boolean function is the minimum of the distancesof that function from the quadratic Boolean function (i.e., the Boolean functions withalgebraic degree at most 2). By Theorem 1, for all polynomials P : F n → F of degreeat most 2, if k f k U < ǫ , then (cid:12)(cid:12)(cid:12) − n P x ∈ F n f ( x )( − P ( x ) (cid:12)(cid:12)(cid:12) < ǫ . Therefore, the second-ordernonlinearity of such functions ought to be high. Green and Tao [7] proved that just asfor U , if a Boolean function has high second-order nonlinearity, then its U norm is low.They also proved that such an implication is not valid for U k norms for k ≥ U and U norms havethe promise of being good indicators for the first and second-order nonlinearities ofa Boolean function. Determination of these nonlinearities have complexities that scaleexponentially with the number of input variables of Boolean functions. In the followingsection, we propose a quantum algorithm to estimate an upper bound of the Gowers U norm that is probabilistic in nature, the probability converges as e − m t where m is thenumber of trials and t is a positive error margin. In this section, we will introduce some notation that we use throughout the paper. Foran introduction to quantum computing, we refer to Rieffel and Polak [12], or Nielsen andChuang [11]. qubit or qu-bit can be described by a vector | ψ i = ( a, b ) T ∈ C , where ‘ T ’ indicates thetranspose, | a | is the probability of observing the value 0 when we measure the qubit, and | b | is the probability of observing 1. If both a and b are nonzero, the qubit has both thevalue 0 and 1 at the same time, and we call this a superposition . Once we have measuredthe qubit, however, the superposition collapses, and we are left with a classical state thatis either 0 or 1 with certainty. A state of n qubits is represented by a normalized complexvector with 2 n elements. We define h ψ | as the conjugate transpose of | ψ i . This notationis known as the bra-ket notation. We denote the standard basis (column) vectors as | i and | i , and then | ψ i = ( a, b ) T = a | i + b | i .In the following, we will use the conventional notation | a i | b i := | a i ⊗ | b i , or | ab i := | a i ⊗ | b i . A state on n qubits can be represented as a C -linear combination of the vectorsof the standard basis | ψ i = P x ∈ F n a x | x i , where a x ∈ C , ∀ x ∈ F n , and P x ∈ F n | a x | = 1.Let | n i be the quantum state associated with the zero vector in F n . Let | + i = | i + | i√ and |−i = | i−| i√ . For any x ∈ F n and ǫ ∈ F , the bit oracle implementation U F of F is | ε i | x i U F −−→ | ε + F ( x ) i | x i . (20)Here, n -qubits in | x i specify the input state that changes the target qubit | ǫ i accordingto the value of the Boolean function F ( x ).If the first qubit is |−i , then |−i | x i U F −−→ ( − F ( x ) |−i | x i . We write | x i U F −−→ ( − F ( x ) | x i with the understanding that there is an additional target qubit in the |−i state thatremains unchanged and refer to this as the phase oracle implementation of the function F .Suppose that a computational basis state is of the form | x (1) k x (2) k . . . k x ( m ) i where forany two vectors x ∈ F r and y ∈ F s , the concatenation x k y = ( x , . . . , x r , y , . . . , y s ) isa vector in F r + s . It is reasonable to write | x (1) k x (2) k . . . k x ( m ) i = | x (1) i | x (2) i . . . | x ( m ) i .The vector x ( i ) ∈ F r i , for some r i ∈ Z + is said to be the content of the i th register. If x ( i ) , x ( j ) ∈ F r , for some r ∈ Z + , we define MCNOT ji as | x (1) i . . . | x ( i ) i . . . | x ( j ) i . . . | x ( m ) i MCNOT ji −−−−−−−→ | x (1) i . . . | x ( i ) + x ( j ) i . . . | x ( j ) i . . . | x ( m ) i . We can realize the transformation induced by
MCNOT ji by using an appropriate numberof conventional CNOT gates.Let I = ! be the 2 × H = √ − ! be the 2 × ⊗ . The matrix H n is recursivelydefined as: H = H ⊗ H,H n = H n − ⊗ H n − , for all n ≥ . (21)Note that, for x ∈ F n , H n | x i = 2 − n P x ′ ∈ F n ( − x · x ′ | x ′ i .In the next section we propose an algorithm to compute Gowers U norm of Boolean func-tions. Our approach resembles that employed by Bera, Maitra, and Tharrmashashtha [1]to estimate the autocorrelation spectra of Boolean functions. We prepare the quantum state 2 − n P b ∈ F n P a ∈ F n P x ∈ F n | x i | a i | b i , and apply the fol-lowing transformations:2 − n X b ∈ F n X a ∈ F n X x ∈ F n | x i | a i | b i F ⊗ I ⊗ I −−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x ) | x i | a i | b i MCNOT −−−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x ) | x + a i | a i | b i U F ⊗ I ⊗ I −−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x )+ F ( x + a ) | x + a i | a i | b i MCNOT −−−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x )+ F ( x + a ) | x i | a i | b i MCNOT −−−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x )+ F ( x + a ) | x + b i | a i | b i (22) U F ⊗ I ⊗ I −−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x )+ F ( x + a )+ F ( x + b ) | x + b i | a i | b i MCNOT −−−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − F ( x )+ F ( x + a )+ F ( x + b ) | x + a + b i | a i | b i U F ⊗ I ⊗ I −−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − ∆ a,b F ( x ) | x + a + b i | a i | b i MCNOT −−−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − ∆ a,b F ( x ) | x + b i | a i | b i MCNOT −−−−−−−→ − n X b ∈ F n X a ∈ F n X x ∈ F n ( − ∆ a,b F ( x ) | x i | a i | b i H ⊗ n −−−→ X a ′ ,b ′ ,x ′ ∈ F n (2 − n X x,a,b ∈ F n ( − ∆ a,b F ( x )+ a · a ′ + b · b ′ + x · x ′ ) | x ′ i | a ′ i | b ′ i . It should be remembered that in addition to the three n -qubit registers used, there is anadditional target qubit that is in the |−i state and remains unchanged throughout, owingto this fact the qubit has been dropped from the sequence of operations, for brevity.The above sequence of operations excluding the last H ⊗ n is summarized as2 − n X a,b,x ∈ F n | x, a, b i D F −−→ − n X a,b,x ∈ F n ( − ∆ a,b F ( x ) | x, a, b i . (23)The probability that a measurement of the resultant state yields the result | n i | n i | n i is given by Pr[ x ′ = a ′ = b ′ = 0 n ] = (cid:16) − n X x,a,b ∈ F n ( − ∆ a,b F ( x ) (cid:17) and since f ( x ) = ( − F ( x ) , using (10)Pr[ x ′ = a ′ = b ′ = 0 n ] = ( k f k U ) = k f k U . (24) U norm Let the final output state at the end of the transformation described in (22) be | Ψ i = X a ′ ,b ′ ,x ′ ∈ F n C ( x ′ , a ′ , b ′ ) | x ′ a ′ b ′ i . (25)The probability amplitude of the state | x ′ a ′ b ′ i is C ( x ′ , a ′ , b ′ ) = 2 − n X x,a,b ∈ F n ( − ∆ a,b F ( x )+ a · a ′ + b · b ′ + x · x ′ . (26)he outcome of a measurement, with respect to the computational basis, performed onthe output state is a 3 n bit string ( x ′ k a ′ k b ′ ), where x ′ , a ′ , b ′ ∈ F n , and the probabilityof measuring said string is (cid:12)(cid:12) C ( x ′ , a ′ , b ′ ) (cid:12)(cid:12) . Therefore, the eighth power of the Gowers U norm is given by (cid:12)(cid:12) C (0 n , n , n ) (cid:12)(cid:12) . The next theorem outlines a strategy to determine aprobabilistic upper bound of the Gowers U norm. Theorem 2.
We assume that the measurements are done with respect to the compu-tational basis. Suppose that Y is a random variable defined on the set of all possiblemeasurement outcomes on the quantum state H ⊗ n ◦ D F (cid:16) − n P a,b,x ∈ F n | x i | a i | b i (cid:17) as Y ( x ′ , a ′ , b ′ ) = 2 − n ( x ′ k a ′ k b ′ ) , where ( x ′ k a ′ k b ′ ) is the decimal value of the concatenated n bit string. Following theusual convention, we write Y instead of Y ( x ′ , a ′ , b ′ ) . Suppose that ( Y , . . . , Y m ) be arandom sample such that each Y i is independent and identically distributed as Y . Let Y = m P i ∈ [ m ] Y i . Then Pr h k f k U ≤ (1 + t − Y ) / i ≥ − exp( − m t ) , for any positive real number t .Proof. Let the expectation of Y , E [ Y ] = µ . Let Pr[ Y = 0] = k f k U = p , so Pr[ Y = 0] =1 − p . The range of the random variable Y has 2 n distinct values in the interval [0 , y , y , . . . , y n − , where y j = 2 − n j . The expectationof Y is µ = E [ Y ] = y Pr[ Y = 0] + y Pr[ Y = y ] + · · · + y n − Pr[ Y = y n − ]= y Pr[ Y = y ] + y Pr[ Y = y ] + · · · + y n − Pr[ Y = y n − ] < Pr[ Y = y ] + Pr[ Y = y ] + · · · + Pr[ Y = y n − ]= Pr[ Y = 0] = 1 − p. (27)Suppose that ( Y , . . . , Y m ) be a random sample of size m . The sample mean is Y = m P i ∈ [ m ] Y i . By the Hoeffding’s inequality [9]Pr h Y ≥ µ + t i ≤ exp( − m t ) . (28)where t is any positive real number. Using equations (27) and (28),Pr h − p > µ ≥ Y − t i ≥ − exp( − m t ) , which implies Pr h p < t − Y i ≥ − exp( − m t ) , that is , Pr h k f k U < (1 + t − Y ) / i ≥ − exp( − m t ) . (29)The theorem is shown. ⊓⊔ The last line of (29) tells us that if we measure m times and compute Y , then theprobability that k f k U is bounded above by (1 + t − Y ) / is 1 − exp( − m t ). Thereforewith an appropriate choice of m and t we can estimate an upper bound of the Gowers U norm of f with a very high probability. U norm We start by defining distance between Boolean functions in terms of probabilities. efinition 3.
For any two functions f, g ∈ B n , dist( f, g ) = Pr x ∼ F n [ f ( x ) = g ( x )] = d H ( f, g )2 n where x is a random variable uniformly distributed over F n . The function f is said to be ǫ -close to g if dist( f, g ) ≤ ǫ , and ǫ -far from g if dist( f, g ) > ǫ .We will now design an algorithm to determine whether a function is linear or ǫ -far fromlinear; we refer to Hillery and Anderson [8, Section III] for a discussion on such tests. Algorithm 1
Linearity checking with the Gowers U norm. Input: Quantum implementation of f ∈ B n .1: Initial state: 2 − n P a,b,x ∈ F n | x, a, b i .2: Perform the following sequence of transformations:2 − n X a,b,x ∈ F n | x, a, b i D F −−→ − n X a,b,x ∈ F n ( − ∆ a,b F ( x ) | x, a, b i H ⊗ n −−−→ X a ′ ,b ′ ,x ′ ∈ F n (2 − n X x,a,b ∈ F n ( − ∆ a,b F ( x )+ a · a ′ + b · b ′ + x · x ′ ) | x ′ , a ′ , b ′ i .
3: Measure the output state with respect to the computational basis.4: If the measurement result is | n , n , n i then “ACCEPT” (the function is linear).5: Else “REJECT”. Theorem 4. If f is a linear function then the output is “ACCEPT” with probability .If f is ǫ -far from linear functions, then probability of “REJECT” is greater than − exp( − ǫ ) .Proof. If f is a linear functions, then the output is “ACCEPT” with certainty. Thisdirectly follows from the definition of Gowers U norm. If f is ǫ -far from linear functions,then k f k U ≤ (cid:18) − nl ( f )2 n (cid:19) ≤ (1 − ǫ ) . This means that the probability that the output is “ACCEPT” is less than or equalto (1 − ǫ ) ; therefore the probability of “REJECT” is greater than 1 − (1 − ǫ ) ≈ − exp( − ǫ ). ⊓⊔ The result concerning the BLR test is:
Theorem 5. [10, Theorem 1.30]
Suppose the BLR Test accepts F : F n → F with prob-ability − ǫ . Then f is ǫ -close to being linear. By the BLR test, if a function is ǫ -far from the linear functions, and it is promised thatwe have such functions and linear functions only, then given a function from the latterclass, the probability that the algorithm will REJECT is greater than ǫ . Remark 6.
Appendix: generalization to higher Gowers norms
The same technique can be used for other Gower’s norms. For instance, we can apply theunitary transformation H ⊗ n ◦ D F to the state 2 − n P x ∈ F n P a ∈ F n P b ∈ F n P c ∈ F n | x i | a i | b i | c i ,where, with notation M ji = MCNOT ji and U F = U F ⊗ I ⊗ I ⊗ I , D F = M ◦ U F ◦ M ◦ M ◦ U F ◦ M ◦ U F ◦ M ◦ U F ◦ M ◦ U F ◦ M ◦ U F ◦ M ◦ U F . We obtain thus the state P a ′ ,b ′ ,b ′ ,x ′ ∈ F n − n P a,b,c,x ∈ F n ( − ∆ a,b,c F ( x ) | x, a, b, c i . Then, P r [ x ′ = a ′ = b ′ = c ′ = 0 n ] = (cid:16) − n P a,b,c,x ∈ F n ( − ∆ a,b,c F ( x ) (cid:17) , and, using (11), P r [ x ′ = a ′ = b ′ = c ′ = 0 n ] = (cid:16) k f k U (cid:17) = k f k U . Acknowledgment:
Research of C. A. Jothishwaran and Sugata Gangopadhyay is a partof the project “Design and Development of Quantum Computing Toolkit and CapacityBuilding” sponsored by the Ministry of Electronics and Information Technology (MeitY)of the Government of India.
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