3 Characterization of quantum query algorithms

S. Arunachalam and J. Briët and C. Palazuelos 3:9

.. . W

A

Q U

0 Ox

U

1

. . .

^Ox

U

t

Figure 1 At-query quantum algorithm that starts with the all-zero state and concludes by measuring the registerA.

A quantum query algorithm consists of a fixed sequence of unitary operations acting on (Q,A) in addition to aworkspace registerW. At-query quantum algorithm begins by initializing the joint register (Q,A,W) in the all-zero state and continues by interleaving a sequence of unitariesU0, . . . , Uton (Q,A,W) with oraclesOxon (Q,A). Finally, the algorithm performs a 2-outcome measurement onAand returns the measurement outcome.

For a Boolean function f :{0,1}ⁿ → {0,1}, the algorithm is said to compute f with errorε >0 if for everyx, the measurement outcome of registerAequalsf(x) with probability at least 1−ε. The bounded-error query complexity of f, denotedQ_ε(f), is the smallest t for which such an algorithm exists. Note that in this model, we are not concerned with the amount of time (i.e., the number of gates) it takes to implement the interlacing unitaries, which could be much bigger than the query complexity itself.

Here we will work with a slightly less standard oracle sometime referred to as a phase oracle, in which the standard oracle is preceded and followed by a Hadamard onA. Since the Hadamards can be undone by the unitaries surrounding the queries in a quantum query algorithm, using the phase oracle does not reduce generality. A query to this oracle, sometimes denotedO_x,±, applies the (controlled) unitary Diag((1,(−1)^x)) to joint register (A,Q). To avoid having to write (−1)^xlater on, we shall work in the equivalent setting where Boolean functions send{−1,1}ⁿ to{−1,1}.

3:10 Quantum Query Algorithms are Completely Bounded Forms

ICorollary 7. Let m, n, tbe positive integers such thatt≥2and m=n^t. LetT ∈C^n×···×n be a t-tensor. Then, kTkcb≤1 if and only if there exist a positive integerd, unit vectors u, v∈C^mand contractions U_i, V_i∈L(C^m,C^dn)such that for any x₁, . . . , x_t∈Cⁿ, we have

T(x₁, . . . , x_t) =u^∗U₁^∗ Diag(x₁)⊗Idd

V₁· · ·U_t^∗ Diag(x_t)⊗Idd

V_tv. (7) The proof of the above corollary uses the following fact about the completely bounded norm of∗-representations ofC^∗-algebras [42, Theorem 1.6].

ILemma 8. LetX be a finite-dimensionalC^∗-algebra,H,H⁰be Hilbert spaces,π:X →L(H) be a ∗-representation andU ∈L(H,H⁰)andV ∈L(H⁰,H) be linear maps. Then, the map

σ:X →L(H⁰), defined asσ(x) =U π(x)V, satisfies thatkσkcb≤ kUkkVk.

In addition, we use the famous Fundamental Factorization Theorem [39, Theorem 8.4].

Below we state the theorem when restricted to finite-dimensional spaces (see also remark after [28, Theorem 16]).

ITheorem 9(Fundamental factorization theorem). Let σ:L(Cⁿ)→L(C^m)be a linear map and let d=nm. Then, there existU, V ∈L(C^m,C^dn)such that kUkkVk ≤ kσk_cb and for any M ∈L(Cⁿ), we have σ(M) =U^∗(M⊗Idd)V.

Proof of Corollary 7. The setX = Diag(Cⁿ) of diagonal matrices is a (finite-dimensional) C^∗-algebra (endowed with the standard matrix product and conjugate-transpose involution).

Define thet-linear formR:X × · · · × X →CbyR(X₁, . . . , X_t) =T(diag(X₁), . . . ,diag(X_t)).

We claim thatkRkcb=kTkcb. Observe that for every positive integerd, the set{B∈Md(X) : kBk ≤1}can be identified with the set of block-diagonal matrices B=Pn

i=1E_i,i⊗B(i) of sizend×ndand blocksB(1), . . . , B(n) of sized×dsatisfyingkB(i)k ≤1 for alli∈[n]. It follows that

Rd(B1, . . . , Bt) =

n

X

i1,...,it=1

R(Ei₁,i₁, . . . , Ei_t,i_t)B1(i1)· · ·Bt(it)

=

n

X

i₁,...,i_t=1

Ti₁,...,i_tB1(i1)· · ·Bt(it), which shows the claim.

Next, we show that (6) is equivalent to (7). The fact that (7) implies (6) follows immediately from the fact that the map Diag(x) 7→ Diag(x)⊗Idd is a ∗-representation.

Now assume (6). Without loss of generality, we may assume that each of the Hilbert spacesH1, . . . ,Hthas dimension at leastm. If not, we can expand the dimensions of the ranges and domains of the representationsπ_iand contractionsV_i by dilating with appropriate isometries into larger Hilbert spaces (“padding with zeros”). For eachi∈[t], letSi⊆ Hi be the subspace

S_i= Span

π_i(x_i)V_i+1· · ·V_tπ_t(x_t)V_t+1: x_i, . . . , x_t∈ X .

Since dim(X) = n, we have that dim(S_i) ≤m. For each i ∈ [t], let Q_i ∈ L(C^m,Hi) be an isometry such that Si ⊆ Im(Qi). Note that Vi+1 is a vector in the unit ball of Ht. Let Q_t+1 ∈ L(C^m,H_t) be an isometry such that V_t+1 ∈ Im(Q_t+1). Note that for each i∈[t+ 1], the mapQiQ^∗_i acts as the identity on Im(Qi). For eachi∈ {2, . . . , t} define the mapσi : X →L(C^m) byσi(x) =Q^∗_iViπi(x)Qi+1 andσ1(x) =Q^∗₁π1(x)Q2. Finally define u=Q^∗₁V₁^∗ andv=Q^∗_t+1V_t+1. Then, the right-hand side of (6) can be written as

u^∗σ1(x1)· · ·σt(xt)v.

S. Arunachalam and J. Briët and C. Palazuelos 3:11

It follows from Lemma 8 that kσikcb ≤ 1. Let σ_i⁰ : L(Cⁿ) :→ L(C^m) be the linear map given by σ_i⁰(M) = σi(Diag(M11, . . . , Mnn)) for anyM ∈ L(C^m). Then, for any diagonal matrixx∈ X, we haveσ_i(x) =σ_i⁰(x) andkσ_i⁰kcb=kσikcb. It follows from Theorem 9 that there exists a positive integer di and contractionsUi, Vi : L(C^m,C^dn) such that σi(x) = U_i^∗(x⊗Id_di)V_i for anyx∈ X. We can take alld_iequal tod= max_i{d_i} by suitably dilating the contractions Ui, Vi. Setting u⁰ = u/kuk`<sub>2</sub> and U<sub>1</sub><sup>0</sup> = kuk`₂U1, and similarly defining

v⁰, V_i+1⁰ gives the remaining implication. J

Corollary 7 implies the following lemma, from which Theorem 3 easily follows.

ILemma 10. Letβ:{−1,1}ⁿ→[−1,1]be some map and let tbe a positive integer. Then, the following are equivalent.

1. There exists a(2t)-tensorT ∈R^{2n×···×2n} such thatkTk_cb≤1and for everyx∈ {−1,1}ⁿ andy= (x,1), we have

2n

X

i1,...,i2t=1

Ti₁,...,i_2tyi₁· · ·yi_2t =β(x).

2. There exists at-query quantum algorithm that, on inputx∈ {−1,1}ⁿ, returns a random sign with expected valueβ(x).

Proof. We first prove that (2) implies (1). As discussed in Section 2, a t-query quantum algorithm with phase oracles initializes the joint register (A,Q,W) in the all-zero state on which it then performs a sequence of unitariesU₁, . . . , U_t interlaced with queriesD(x) = Diag((1, x))⊗IdW. Let{P0, P1}be the the two-outcome measurement done at the end of the algorithm and assume that it returns +1 on measurement outcome zero and−1 otherwise.

LetQ=P0−P1 and note thatQis a contraction sinceP0, P1are positive semi-definite and satisfyP0+P1= Id. The expected value of the measurement outcome is then given by

e^∗₀U₁^∗D(x)U₂^∗· · ·D(x)U_t^∗QUtD(x)· · ·U2D(x)U1e0. (8) By assumption, this expected value equals β(x) for everyx∈ {−1,1}ⁿ. Forz∈C²ⁿ, denote D⁰(z) = Diag((zn+1, . . . , z2n, z1, . . . , zn))⊗IdW and Uet = U_t^∗QUt. Define the (2t)-linear formT by

T(y₁, . . . , y_2t) =u^∗U₁^∗D⁰(y₁)U₂^∗· · ·D⁰(y_t)Ue_tD⁰(y_t+1)· · ·U₂D⁰(y_2t)U₁u.

ClearlyT((x,1). . . ,(x,1)) =β(x) for everyx∈ {−1,1}ⁿ. Moreover, by definitionT admits a factorization as in (7). It thus follows from Corollary 7 that kTkcb ≤ 1. We turn T into a real tensor by taking its real part T⁰ = (T+T)/2, where T is the coordinate-wise complex conjugate ofT. Since for anyx∈ {−1,1}ⁿ andy= (x,1), the valueT(y, . . . , y) is real, we haveT⁰(y, . . . , y) =β(x). We need to show thatkT⁰kcb≤1. To this end, consider an arbitrary positive integer d, unit vectors v, w ∈C^d and sequences of unitary matrices V1(i), . . . , V2t(i) fori∈[n] such that

2n

X

i1,...,i2t=1

Ti₁,...,i_2tV1(i1)· · ·V2t(i2t) =

2n

X

i1,...,i2t=1

Ti₁,...,i_2tv^∗V1(i1)· · ·V2t(i2t)w .

Note thatkTk_cbis given by the supremum overdandV_j(i). Taking the complex conjugate of the above summands on the right-hand side allows us to express the above absolute value as

2n

X

i₁,...,i_2t=1

T_i1,...,i2tv¯^∗V₁(i₁)· · ·V_2t(i_2t) ¯w

, (9)

I T C S 2 0 1 8

3:12 Quantum Query Algorithms are Completely Bounded Forms

...

H X X

V U _Diag(

y) W2

X

W∗ 2t

X

. . .

Diag(y) Wt

X

W∗ t+2 Diag(y) W∗ t+1

C H

W Q

Figure 2The registersC,Q,Wdenote the control, query and workspace registers. LetU, V be unitaries withW1e^uandW2t+1e^vas their first rows, respectively and forx∈ {−1,1}ⁿ andy= (x,1), let Diag(y) be the query operator. The algorithm begins by initializing the joint register (C,Q,W) in the all-zero state and proceeds by performing the displayed operations. The algorithm returns +1 if the outcome of the measurement onCequals zero and−1 otherwise.

where ¯v,w, V¯ j(i) denote the coordinate-wise complex conjugates. Since each Vj(i) is still unitary, it follows that (9) is at most kTk_cb and so kTk_cb ≤ kTk_cb ≤ 1. Hence, by the triangle inequality,kT⁰kcb≤(kTkcb+kTkcb)/2≤1 as desired.

Next, we show that (1) implies (2). Let T be a (2t)-tensor as in item 1. Since any matrix with operator norm at most 1 is a convex combination of unitary matrices (by the Russo-Dye Theorem), it follows from Corollary 7 that T admits a factorization as in (7). LetV₀, U_2t+1 ∈ L(C^m,C^2dn) be isometries. For each i∈ [2t+ 1], define the map Wi ∈ L(C^2dn) by Wi = V_i−1U_i^∗. Observe that each Wi is a contraction and recall that unitaries are contractions. For the moment, assume for simplicity that eachW_i is in fact unitary. Define two vectorseu=V0uandev=U2t+1v and observe that these are unit vectors inC^2dn. The right-hand side of (7) then gives us

T(y1, . . . , y2t) =ue^∗W1D(ye 1)W2D(ye 2)W3· · ·W2tD(ye 2t)W2t+1ev, (10) where D(ye i) = Diag(yi)⊗Idd for i∈ [2t]. Based on this, we obtain the quantum query algorithm described in Figure 2.

To see why this algorithm satisfies the requirements, first note that the algorithm makes tqueries to the inputx. For the correctness of the algorithm, we begin by observing that before the application of the first query, the state of the joint register (C,Q,W) is

√1

2(e0⊗W1eu+e1⊗Wt+1ev).

Before the final Hadamard gate, the state of the joint register is given by

√1

2e₀⊗ (Diag(y)⊗Id_d)W_t· · ·W₂(Diag(y)⊗Id_d)W₁eu + 1

√2e1⊗ W_t+1^∗ (Diag(y)⊗Idd)W_t+2^∗ · · ·W_2t^∗(Diag(y)⊗Idd)W_2t+1^∗ ev .

A standard calculation and (10) then show that after the final Hadamard gate, the expected output of the algorithm is preciselyT((x,1), . . . ,(x,1)) =β(x). In the general case where the W_is are not necessarily unitary, we can use the fact that, by the Russo–

Dye Theorem and Carathéodory’s Theorem, eachWi is a convex combination of at most (dn)²+ 1 unitaries. The algorithm can thus use randomness to effect eachW_i on expectation.

Alternatively, by linear algebra there exists a unitary matrix W_i⁰∈C^2dn×2dnthat has Wi as its upper-left corner (see [2, Lemma 7]), through which the algorithm could implementWi

by working on a larger quantum register. J

S. Arunachalam and J. Briët and C. Palazuelos 3:13

Using Lemma 10, we now prove our main Theorem 3.

Proof of Theorem 3. We first show that (2) implies (1). Using the equivalence in Lemma 10, it follows that there exists a (2t)-tensorT ∈R^{2n×···×2n} such that kTkcb≤1 and for every x∈ {−1,1}ⁿ andy= (x,1), we have

2n

X

i₁,...,i_2t=1

Ti₁,...,i_2tyi₁· · ·yi_2t =β(x).

Define the symmetric 2t-tensorT_p=_(2t)!¹ P

σ∈S_2tT◦σ. Letp∈R[x₁, . . . , x_2n] be the form of degree 2tassociated withTpby (2) (note that there is a unique polynomial associated with the symmetric tensor T_p). Then,p((x,1)) =β(x) for everyx∈ {−1,1}ⁿ. Moreover, if we setT^σ=T for eachσ∈St, it follows from the above decomposition of Tpand Definition 2 thatkpk_cb≤ kTk_cb≤1.

Next, we show that (1) implies (2). Let pbe a degree-(2t) form satisfying kpkcb ≤1.

SupposeTpas defined in Eq. (2) can be written asTp=P

σ∈S2tT^σ◦σandP

σ∈S2tkT^σkcb= kpk_cb ≤ 1. Define T = P

σ∈S2tT^σ. Then, using the triangle inequality, it follows that kTkcb≤P

σ∈S2tkT^σkcb≤1. Also note that for anyy∈R²ⁿ, T(y, . . . , y) = X

σ∈S2t

T^σ(y, . . . , y) = X

σ∈S2t

(T^σ◦σ)(y, . . . , y) =T_p(y, . . . , y) =p(y).

Using Lemma 10 (in particular (1) =⇒ (2)) for the tensor T, the theorem follows. J We now prove Corollary 5, which is an immediate consequence of our main theorem.

Proof of Corollary 5. We first show cb-deg_ε(f) ≥Qε(f): Suppose cb-deg_ε(f) = d, then there exists a degree-(2d) formpsatisfying: |p(x)−f(x)| ≤2εfor everyx∈Dandkpk_cb≤1.

Using our characterization in Theorem 3, it follows that there exists a d-query quantum algorithmA, that on input x∈D, returns a random sign with expected valuep(x). So, our ε-error quantum algorithm forf simply runsAand outputs the random sign.

We next show cb-deg_ε(f)≤Qε(f). SupposeQε(f) =t. There exists at-query quantum algorithm that, on input x∈D, outputs a random sign with expected valueβ(x) satisfying

|β(x)−f(x)| ≤2ε. Note that we could also run the quantum algorithm forx /∈Dand letβ(x) be the expected value of the quantum algorithm for suchxs. Using Theorem 3, we know that there exists a degree-(2t) formpsatisfyingβ(x) =p(x) for everyx∈ {−1,1}ⁿ andkpk_cb≤1.

Clearlypsatisfies satisfies the conditions of Definition 4, hence cb-deg_ε(f)≤t. J

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