9.3 Matrix arithmetics using block-encodings
9.3.2 Sparse access to matrices
We now motivate the above singular value transformation techniques by showing how they can be used to compute block-encodings of smooth functions of Hermitian matrices to which we only have sparse access. In particular, in certain regimes, we show that this can be done more efficiently than by computing the singular value decomposition of the Hermitian matrix and then applying the smooth function. Let us first define the sparse access model.
Let A be an s-sparse (i.e., having at most s non-zero entries per row/column) Hermitian matrix acting on q qubits and let n = 2q. We assume sparse black-box access to the elements of A in the following way: for input (k, `) ∈ [n] × [s] we can query the location and value of the `th non-zero entry in the kth row of the matrix A.
In the quantum setting this means we assume access to two oracles, as described in [BCK15]. We have an oracle OI that calculates the function indexA: [n] × [s] → [n] that for input (k, `) gives the column index of the `th non-zero element in the kth row of A. We assume this oracle computes the index “in place”:
OI|k, `i = |k, indexA(k, `)i for k ∈ [n], ` ∈ [s]. (9.2) (In the degenerate case where the kth row has fewer than ` non-zero entries, indexA(k, `) is defined to be ` together with some special symbol.) We also as-sume we can apply the inverse of OI. Throughout we assume that the entries of A can each be represented using b bits and we furthermore assume access to an oracle OAthat returns a b-bit binary description of the entries of A:
OA|k, i, zi = |k, i, z ⊕ Akii for k, i ∈ [n], z ∈ {0, 1}b. (9.3) When we count queries we make no distinction between OI and OA. We say that we make M queries to A if the number of applications of OI plus the number of applications of OAis upper bounded by M .
166 Chapter 9. Quantum algorithms Lemma 9.9 ([GSLW18, Lem. 48 and Thm. 30]). Let A ∈ C2q×2q be a Hermitian operator that is s-sparse, satisfies kAk ≤ 1, and to which we have access through the oracles described in (9.2) and (9.3), and let ε > 0. Then we can implement a (2, q + 4, ε)-block-encoding of A using O(s log(s/ε)) queries to A and O(sq log(s/ε)) 2-qubit gates.
Combining the above block-encoding of A with Theorem 9.7 allows us to effi-ciently compute block-encodings of polynomials of A. Naturally we can combine this with polynomial approximations of more complicated functions f to provide block-encodings of f (A). We finish this chapter with two examples that will be useful in Chapter 11.
Approximating the square-root function. We show how to efficiently con-struct a block-encoding of the matrix p1 + A/4/4, given sparse access to A. We do so by combining Lemma 9.9 and Theorem 9.7. For the latter it is necessary to have a good polynomial approximation of the functionp1 + x/2/4 on the interval [−1, 1]. Notice that we consider the function p1 + x/2/4 instead of p1 + x/4/4, we do so since Lemma 9.9 only provides a block-encoding of A with α = 2. We show below that a good polynomial approximation ofp1 + x/2/4 can be obtained from its Taylor expansion around 0.
it follows that for d = log(1/δ) we have that the degree-d Taylor expansion is a δ-approximation on the interval [−1, 1]. One can verify that each Taylor expansion of the function p1 + x/2/4 around 0 is bounded in absolute value by 1/2 on the interval [−1, 1]. Therefore, for any 0 < δ ≤ 1/2, there exists a univariate polynomial p of degree d = O(log(1/δ)) such that |p(x) −
√
1+x/2
4 | ≤ δ and |p(x)| ≤ 12 for all x ∈ [−1, 1]. We may therefore apply Theorem 9.7.
Lemma 9.10. Let A ∈ C2q×2q be a Hermitian operator that is s-row-sparse, sat-isfies kAk ≤ 1, and to which we have access through the oracles described in (9.2) and (9.3). Let 0 < δ < 1/2 − 2-qubit gates. Moreover, we can compute a description of such a circuit with a classical computer in time O(poly(log(1/δ))).
Proof. Let ε, δ0 > 0 be constants to be determined later. We want the polynomial p to be a δ/2-approximation of the function p1 + x/2/4 on the interval [−1, 1].
9.3. Matrix arithmetics using block-encodings 167 Therefore we let p be the Taylor expansion of degree d = O(log(1/δ)) of the function p1 + x/2/4 around 0. We now construct a block-encoding of pI + A/4/4. First construct a (2, q+4, ε)-block-encoding of A using Lemma 9.9. Then use Theorem 9.7 with this block-encoding and the polynomial p to construct a (1, q + 6, 4dpε/2 + δ0+δ/2)-block-encoding ofp1 + A/4/4 (here the linear δ/2-term in the error comes from the polynomial approximation ofp1 + x/2/4 up to error δ/2). Finally pick δ0 = δ/10 and ε > 0 such that 4dpε/2 + δ0 ≤ δ/2, note that ε = Θ(δ2/ log(1/δ)2) suffices. The complexity statement follows from Lemma 9.9 and Theorem 9.7.
Approximating the exponential function. Assume we are given sparse access to a Hermitian matrix A that satisfies 0 A KI for some known K ∈ R. We show how to efficiently construct a block-encoding of e−A/4. We again want to use Theorem 9.7. For that we need a polynomial on the interval [−1, 1] that allows us to approximate the function e−x on the interval [0, K]. First, we use the identity
exp(−A) = exp
−K 2
A − KI/2 K/2 + I
,
to see that it suffices to obtain a good approximation of the function exp(−K2(x+1)) on the interval [−1, 1]. Next, as we did before for the function p1 + x/2/4, one can show that, for β > 0, the function e−β(x+1)/4 can be δ-approximated on the interval [−1, 1] by its Taylor expansion of degree O(β + log(1/δ)) around 0. We thus use this with β = K/2.
Lemma 9.11. Let A ∈ C2q×2q be a Hermitian operator that is s-sparse, satisfies 0 A KI for some K ∈ R, and to which we have access through the oracles described in (9.2) and (9.3). Let 0 < δ < 1/4. Then we can implement a (1, q+6, δ)-block-encoding of e−A/4 with eO(sK log(1/δ)) queries to A and eO(sKq log(1/δ)) other 2-qubit gates. Moreover, we can compute a description of such a circuit with a classical computer in time O(poly(log(K), log(1/δ))).
Proof. Let ε, δ0 > 0 be constants to be determined later. We want the polynomial p to be a δ/2-approximation of the function e−K2(x+1)/4 on the interval [−1, 1].
Therefore we let p be the Taylor expansion of degree d = O(K + log(1/δ)) of the function e−K2(x+1)/4 around 0. We now construct a block-encoding of e−A/4.
First construct a (2, q + 4, ε)-block-encoding of A−K/2·IK/2 using Lemma 9.9. Then use Theorem 9.7 with this block-encoding and the polynomial p to construct a (1, q + 6, 4dpε/2 + δ0 + δ/2)-block-encoding of e−A/4 (here the linear δ/2-term in the error comes from the polynomial approximation of e−K2(x+1)/4 up to error δ/2). Finally pick δ0= δ/10 and ε > 0 such that 4dpε/2 + δ0≤ δ/2, note that ε = Θ(δ2/(K + log(1/δ))2) suffices. The complexity statement follows from Lemma 9.9 and Theorem 9.7.
Chapter 10
Quantum query complexity and semidefinite
programming
This chapter is based on the paper “Semidefinite programming formulations for the completely bounded norm of a tensor”, by S. Gribling and M. Laurent [GL19].
We can try to understand the power and limitations of quantum computers by determining how efficiently they can compute Boolean functions. Let us first give an informal introduction to this topic, we refer to Section 10.4 for formal definitions. Given a Boolean function f : {±1}n→ {±1}, how many queries to an input x ∈ {±1}n do we need in order to compute f (x)? Here, a classical query would be of the form “what is the ith bit of x?”. We allow a quantum computer to make a superposition (over i ∈ [n]) of such queries. The minimum number of queries required to succeed with error probability ≤ 1/3 is respectively the classical and quantum query complexity of the function. A first natural question is whether there is a difference between the notions of classical and quantum query complexity.
Interestingly, the answer is yes for some class of functions. In Chapter 9 we have seen that there is a quantum algorithm (Grover’s search) that computes the OR function, the function that is the logical OR of n bits, using O(√
n) quantum queries to the input string. It is not too hard to see that Θ(n) classical queries are needed.
The study of the classical and quantum query complexity of Boolean func-tions has a long history, we refer to for instance the survey [BW02] and the pa-per [ABDK16] for more information. In that long history, several general lower bound techniques and characterizations have been developed, see Section 10.4 for an overview. In this chapter we consider a recent characterization of quantum query complexity due to Arunachalam, Bri¨et and Palazuelos [ABP19]. Let us first give a brief, high-level, summary of our results before explaining them in more detail in Section 10.1.
In this chapter we provide a new semidefinite programming characterization of 169
170 Chapter 10. Quantum query complexity and semidefinite programming the quantum query complexity of Boolean functions. Our new SDP characterization is based on a recent result of Arunachalam, Bri¨et and Palazuelos [ABP19]. They showed that the quantum query complexity of a Boolean function can be character-ized using tensors that have a completely bounded norm of at most one. Our main result is that the completely bounded norm of a t-tensor can be computed using an SDP involving matrices of size O(ndt/2e) and O(n2dt/2e) linear constraints. As an application of our result, the quantum query complexity of a Boolean function f can be obtained by checking feasibility of some SDPs. Using the duality theory of semidefinite programming we obtain a new type of certificates for large query complexity. We show that our class of certificates encompasses the linear program-ming certificates corresponding to the approximate degree of f and we propose an intermediate class of certificates based on second-order cone programming.
Organization. This chapter is organized as follows. We first explain our results in Section 10.1. We introduce some notation in Section 10.2. We then prove our main result in Section 10.3. In Section 10.4 we use our main result to derive a new SDP characterization of the quantum query complexity of Boolean functions. We compare our SDP to existing SDP characterizations of quantum query complexity in Section 10.4.3. Finally, using the duality theory of semidefinite programming, we obtain a new type of certificates for large query complexity in Section 10.5.
10.1 Our results
Throughout, we let T = (Ti1,...,it) ∈ Rn×···×n be a t-tensor acting on Rn. The completely bounded norm of T , denoted kT kcb, is defined as
supn matrices. Note that in (10.1) one could equivalently optimize over complex unitary matrices Uj(i).
We show that kT kcb can be expressed as the optimal value of a semidefinite program (SDP). This SDP involves matrices of size O(ndt/2e) and O(n2dt/2e) linear constraints, so that an additive ε-approximation of its optimal value can be obtained in time poly(nt, log(1/ε)) (see Theorem 10.5 in Section 10.3).
To put this result in perspective, if we replace the product U1(i1) · · · Ut(it) by the Kronecker (or tensor) product U1(i1) ⊗ · · · ⊗ Ut(it) then we obtain the jointly completely bounded norm of T . It is known that there is a one-to-one correspondence between the jointly completely bounded norm of a t-tensor and the entangled bias of an associated t-partite XOR game (see, e.g., [PV16]). The latter can be computed in polynomial time when t = 2 [Tsi87] (as we have seen in Section 3.3.1), but it is an NP-hard problem to give any constant-factor multiplicative approximation of the entangled bias of a 3-partite XOR game [Vid16]. Hence the jointly completely bounded norm of a 3-tensor is hard to approximate up to any constant factor.
10.2. Preliminaries 171