Quantum Implications of Huang's Sensitivity Theorem

04/28/2020 ∙ by Scott Aaronson, et al. ∙ University of Waterloo The University of Texas at Austin Microsoft berkeley college 0

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, the deterministic query complexity, D(f), is at most quartic in the quantum query complexity, Q(f): D(f) = O(Q(f)^4). This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We also use the result to resolve the quantum analogue of the Aanderaa-Karp-Rosenberg conjecture. We show that if f is a nontrivial monotone graph property of an n-vertex graph specified by its adjacency matrix, then Q(f) = Ω(n), which is also optimal.

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1 Introduction

Last year, Huang resolved a major open problem in the analysis of Boolean functions called the sensitivity conjecture [Hua19], which was open for nearly 30 years [NS94]. Surprisingly, Huang’s elegant proof takes less than 2 pages—truly a “proof from the book.” Specifically, Huang showed that for any total Boolean function, which is a function , we have

(1)

where is the real degree of and is the (maximum) sensitivity of . These measures and other measures appearing in this introduction are defined in Section 2.

In this note, we describe some implications of Huang’s resolution of the sensitivity conjecture to quantum query complexity. We observe that Huang actually proves a stronger claim, in which in Eq. 1 can be replaced by , a spectral relaxation of sensitivity that we define later. This observation has several implications for quantum query complexity.

We use this observation to settle the optimal relation between the deterministic query complexity, , and quantum query complexity, , for total functions. We know from the seminal results of Nisan [Nis91], Nisan and Szegedy [NS94] and Beals et al. [BBC01] that any total Boolean function satisfies555This means that for total functions, quantum query algorithms can only outperform classical query algorithms by a polynomial factor. On the other hand, for partial functions, which are defined on a subset of , exponential and even larger speedups are possible.

(2)

Grover’s algorithm [Gro96] shows that for the or function, a quadratic separation between and is possible. This was the best known quantum speedup for total functions until the work of Ambainis et al. [ABB17], who constructed a total function with

(3)

In this note, we show that the quartic separation (up to log factors) in Eq. 3 is actually the best possible:

Theorem 1.

For all Boolean functions , we have .

We deduce Theorem 1 as a corollary of a new tight quadratic relationship between and :

Theorem 2.

For all Boolean functions , we have .

Observe that Theorem 2 is tight for the or function on variables, whose degree is and whose quantum query complexity is [Gro96, BBBV97]. Prior to this work, the best relation between and was a sixth power relation, , which follows from Eq. 2.

As discussed earlier, our proof relies on the restatement of Huang’s result (Theorem 5), showing that , where is the spectral relaxation of sensitivity defined in Section 3. We then show that the measure lower bounds the original quantum adversary method of Ambainis [Amb02], which in turn lower bounds .

We now show how Theorem 1 straightforwardly follows from Theorem 2 using two previously known connections between complexity measures of Boolean functions.

Proof of Theorem 1 assuming Theorem 2.

In [Mid04], Midrijanis showed that for all total functions , we have

(4)

where is the block sensitivity of .

Theorem 2 shows that . Combining the relationship between block sensitivity and approximate degree from [NS94] with the results of [BBC01], we get that . (This can also be proved directly using the lower bound method in [BBBV97].)

Combining these three inequalities yields for all total Boolean functions . ∎

It remains to show the main result, Theorem 2, which we do in Section 3 using the proof of the sensitivity conjecture by Huang [Hua19] and the spectral adversary method in quantum query complexity [BSS03].

In Section 4, we also use Theorem 2 to prove the quantum analogue of the famous Aanderaa–Karp–Rosenberg conjecture. Briefly, this conjecture is about the minimum possible query complexity of a nontrivial monotone graph property, for graphs specified by their adjacency matrices.

There are variants of the conjecture for different models of computation. For example, the randomized variant of the Aanderaa–Karp–Rosenberg conjecture, attributed to Karp [SW86, Conjecture 1.2] and Yao [Yao77, Remark (2)], states that for all nontrivial monotone graph properties , we have . Following a long line of work, the current best lower bound is due to Chakrabarti and Khot [CK01].

The quantum version of the conjecture was raised by Buhrman, Cleve, de Wolf, and Zalka [BCdWZ99], who observed that the best one could hope for is , because the nontrivial monotone graph property “contains at least one edge” can be decided with queries using Grover’s algorithm [Gro96]. Buhrman et al. [BCdWZ99] also showed that all nontrivial monotone graph properties satisfy . The current best lower bound is , which was credited to Yao in [MSS07]. We resolve this conjecture by showing an optimal lower bound.

Theorem 3.

Let be a nontrivial monotone graph property. Then .

Theorem 3 follows by combining Theorem 2 with a known quadratic lower bound on the degree of monotone graph properties.

1.1 Known relations and separations

Figure 1: Relations between complexity measures. An upward line from a measure to denotes for all total functions .

Table 1 summarizes the known relations and separations between complexity measures studied in this paper (and more). This is an update to a similar table that appears in [ABK16] with the addition of and . Definitions and additional details about interpreting the table can be found in [ABK16].

For all the separations claimed in the table, we provide either an example of a separating function or a citation to a result that constructs such a function. All the relationships in the table follow by combining the relationships depicted in Figure 1 and the following inequalities that hold for all total Boolean functions:

  • [noitemsep,topsep=4pt]

  • [Nis91]

  • [BBC01]

  • [Mid04]

  • [KT16]

  • [KT16]

  • [Hua19]

  • (Lemma 15)


2, 2
[ABB17]
2, 3
[ABB17]
2, 2
2, 3
2, 3
3,6
[BHT17]
4,6
[ABB17]
2, 3
[ABB17]
2, 3
[GPW18]
4,4
[ABB17]
4, 6
[ABB17]

1, 1
2, 2
[ABB17]
2, 2
2, 3
2, 3
3,6
[BHT17]
3,6
[BHT17]
2, 3
[ABB17]
2, 3
[GJPW18]
3,4
[ABB17]
4, 6
[ABB17]

1, 1
1, 1
2, 2
2, 3
2, 3
3,6
[BHT17]
3,6
[BHT17]
, 3
[ABB17]
2, 3
[GJPW18]
,4
[Tal19]
4, 6
[ABB17]

1, 1
1, 1
1, 2
2, 2
[GSS13]
2, 2
[GSS13]
2.22,5
[BHT17]
2.22,6
[BHT17]
1.15, 3
[Amb13]
1.63, 3
[NW95]
2, 4
2, 4

1, 1
1, 1
1, 1
1, 1
, 2
[GSS13]
2,4
[Rub95]
2,4
1.15, 2
[Amb13]
1.63, 2
[NW95]
2, 2
2, 2

1, 1
1, 1
1, 1
1, 1
1, 1
2,4
[Rub95]
2,4
1.15, 2
[Amb13]
1.63, 2
[NW95]
2, 2
2, 2

1, 1
1, 1
1, 1
1, 1
1, 1
1, 1
2,2
1.15, 2
[Amb13]
1.63, 2
[NW95]
2, 2
2, 2

1, 1
1, 1
1, 1
1, 1
1, 1
1, 1
1, 1
1, 1
1, 2
1, 1
1, 2

1, 1
1.33, 2
-tree
1.33, 3
-tree
2, 2
2, 3
2, 3
3,6
[BHT17]
3,6
[BHT17]
2, 3
[ABK16]
2,4
4, 6
[ABK16]

1, 1
1.33, 2
-tree
1.33,2
-tree
2, 2
2,2
2,2
2,2
2,2
1, 1
2,2
2,4

1, 1
1, 1
1, 1
2, 2
[ABK16]
2, 3
[ABK16]
2, 3
[ABK16]
3,6
[BHT17]
3,6
[BHT17]
1, 1
2, 3
[ABK16]
4, 6
[ABK16]

1, 1
1, 1
1, 1
, 2
[BT17]
,2
[BT17]
,2
[BT17]
2,2
[BT17]
2,2
[BT17]
1, 1
1, 1
1, 1
  • [itemsep=3pt]

  • An entry in the row and column roughly means that for all total functions , and there exists a function with (see [ABK16] for a precise definition).

  • The second row of each cell contains an example of a function that achieves the separation (or a citation to an example), where , , , , and -tree is the balanced nand-tree function.

  • Cells have a white background if the relationship is optimal and a gray background otherwise.

  • Entries with a green background follow from Huang’s result. Entries with a red background follow from this work.

Table 1: Best known separations between complexity measures

1.2 Paper organization

Section 2 contains some preliminaries required to understand the proof of Theorem 2, which is proved in Section 3. Section 4 gives some background and motivation for the Aanderaa–Karp–Rosenberg conjecture and proves Theorem 3. We end with some open problems in Section 5.

Appendix A describes some properties of , its many equivalent formulations, and its relationship with other complexity measures.

2 Preliminaries

2.1 Query complexity

Let be a Boolean function. Let be a deterministic algorithm that computes on input by making queries to the bits of . The worst-case number of queries makes (over choices of ) is the query complexity of . The minimum query complexity of any deterministic algorithm computing is the deterministic query complexity of , denoted by .

We define the bounded-error randomized (respectively quantum) query complexity of , denoted by (respectively ), in an analogous way. We say an algorithm computes with bounded error if for all

, where the probability is over the internal randomness of

. Then (respectively ) is the minimum number of queries required by any randomized (respectively quantum) algorithm that computes with bounded error. It is clear that . For more details on these measures, see the survey by Buhrman and de Wolf [BDW02].

2.2 Sensitivity and block sensitivity

Let be a Boolean function, and let be a string. A block is a subset of . We say that a block is sensitive for (with respect to ) if , where is the -bit string that is on bits in and otherwise. We say a bit is sensitive for if the block is sensitive for . The maximum number of disjoint blocks that are all sensitive for is called the block sensitivity of (with respect to ), denoted by . The number of sensitive bits for is called the sensitivity of , denoted by . Clearly, , since is has the same definition as except that the size of the blocks is restricted to . We define and .

2.3 Degree measures

A polynomial is said to represent the function if for all . A polynomial is said to -approximate if for all and for all . The degree of , denoted by , is the minimum degree of a polynomial representing . The -approximate degree, denoted by , is the minimum degree of a polynomial -approximating . We will omit when . We know that , , and .

The degree of as a polynomial is also called the Fourier-degree of , which equals where . In particular, if and only if agrees with the Parity function, , on exactly half of the inputs.

3 Proof of main result (Theorem 2)

Before proving Theorem 2, which is based on Huang’s proof, we reinterpret his result in terms of a new complexity measure of Boolean functions that we call : the spectral norm of the sensitivity graph of .

Definition 4 (Sensitivity Graph , Spectral Sensitivity ).

Let be a Boolean function. The sensitivity graph of , is a subgraph of the Boolean hypercube, where , and . That is, is the set of edges between neighbors on the hypercube that have different -value. Let be the adjacency matrix of the graph . We define the spectral sensitivity of as .

Note that because is a real symmetric matrix,

is also the largest eigenvalue of

. Since is bipartite, the largest and smallest eigenvalues of are equal in magnitude.

Huang’s proof of the sensitivity conjecture can be divided into two steps:

The second step is the simple fact that the spectral norm of an adjacency matrix is at most the maximum degree of any vertex in the graph, which equals in this case.

We reprove the first claim, i.e., , for completeness.

Theorem 5 ([Hua19]).

For all Boolean functions , we have .

Proof.

Without loss of generality we can assume that since otherwise we can restrict our attention to a subcube of dimension in which the degree remains the same and the top eigenvalue is at most . Specifically, we can choose any monomial in the polynomial representing of degree and set all the variables not appearing in this monomial to .

For with , let and . By the fact that we know that as otherwise would have correlation with the -variate parity function, implying that ’s top Fourier coefficient is .

We also note that any edge in the hypercube that goes between and is an edge in since it changes the value of . This holds since for such an edge, , we have . Similarly, any edge in the hypercube that goes between and is an edge in .

Assume without loss of generality that . Thus,

. We will show that there exists a nonzero vector

supported only on the entries of , such that .

Let be the complete -dimensional Boolean Hypercube. That is, and . Take the following signing of the edges of the Boolean hypercube, defined recursively.

(5)

This gives a new matrix where if and only if is not a neighbor of in the hypercube.

Huang showed that has eigenvalues that equal and eigenvalues that equal . To show this, he showed that by induction on and thus all eigenvalues of must be either or . Then, observing that the trace of is , as all diagonal entries equal , we see that we must have an equal number of and eigenvalues.

Thus, the subspace of eigenvectors for

with eigenvalue is of dimension . Using , there must exists a nonzero eigenvector for with eigenvalue that vanishes on . Fix to be any such vector.

Let be the vector whose entries are the absolute values of the entries of . We claim that . To see so, note that for every we have

(6)

On the other hand, for we have . Thus the norm of is at least times the norm of , and hence . ∎

Finally, we prove that . We rely on a variant of the adversary method introduced by Barnum, Saks, and Szegedy [BSS03] (see also [SS06]).

Definition 6 (Spectral Adversary method).

Let and be matrices of size with entries in satisfying if and only if , and if and only if . Let denote a nonnegative symmetric matrix such that (i.e., the nonzero entries of are a subset of the the nonzero entries of ). Then .

Barnum, Saks, and Szegedy [BSS03] proved that .

Lemma 7.

For all Boolean functions .

Proof.

We prove that . Indeed, one can take to be simply the adjacency matrix of . That is, for any put if and only if in the hypercube and . We observe that . On the other hand, for any , is the restriction of the sensitive edges in direction . The maximum degree in the graph represented by is hence is at most . Thus we have

(7)

Combining this with  [BSS03], we get . ∎

From Theorem 5 and Lemma 7 we immediately get Theorem 2.

4 Monotone graph properties

The Aanderaa–Karp–Rosenberg conjectures are a collection of conjectures related to the query complexity of deciding whether an input graph specified by its adjacency matrix satisfies a given property in various models of computation.

Specifically, let the input be an -vertex undirected simple graph specified by its adjacency matrix. This means we can query any unordered pair , where , and learn whether there is an edge between vertex and . Note that the input size is .

A function on variables is a graph property if it treats the input as a graph and not merely a string of length . Specifically, the function must be invariant under permuting vertices of the graph. In other words, the function can only depend on the isomorphism class of the graph, not the specific labels of the vertices. A function is monotone (increasing) if for all , , where means for all . For a monotone function, negating a in the input cannot change the function value from to . In the context of graph properties, if the input graph has a certain monotone graph property, then adding more edges cannot destroy the property.

Examples of monotone graph properties include “ is connected,” “ contains a clique of size ,” “ contains a Hamiltonian cycle,” “ has chromatic number greater than ,” “ is not planar”, and “ has diameter at most .” Many commonly encountered graph properties (or their negation) are monotone graph properties. Finally, we say a function is nontrivial if there exist inputs and such that .

The deterministic Aanderaa–Karp–Rosenberg conjecture, also called the evasiveness conjecture,666A function is called evasive if its deterministic query complexity equals its input size. states that for all nontrivial monotone graph properties , . This conjecture remains open to this day, although the weaker claim that was proved over 40 years ago by Rivest and Vuillemin [RV76]. Several works have improved on the constant in their lower bound, and the best current result is due to Scheidweiler and Triesch [ST13], who prove a lower bound of . The evasiveness conjecture has been established in several special cases including when is prime [KSS84] and when restricted to bipartite graphs [Yao88].

The randomized Aanderaa–Karp–Rosenberg conjecture asserts that all nontrivial monotone graph properties satisfy . A sequence of increasingly stronger lower bounds, starting with a lower bound of due to Yao [Yao91], a lower bound of due to King [Kin88], and a lower bound of due to Hajnal [Haj91], has led to the current best lower bound of due to Chakrabarti and Khot [CK01]. There are also two lower bounds due to Friedgut, Kahn, and Wigderson [FKW02] and O’Donnell, Saks, Schramm, and Servedio [OSSS05] that are better than this bound for some graph properties.

The quantum Aanderaa–Karp–Rosenberg conjecture states that all nontrivial monotone graph properties satisfy . This is the best lower bound one could hope to prove since there exist properties with , such as the property of containing at least one edge. In fact, for any it is possible to construct a graph property with quantum query complexity using known lower bounds for the threshold function [BBC01].

As stated in the introduction, the question was first raised by Buhrman, Cleve, de Wolf, and Zalka [BCdWZ99], who showed a lower bound of . This was improved by Yao to using the technique in [CK01] and Ambainis’ adversary bound [Amb02]. Better lower bounds are known in some special cases, such as when the property is a subgraph isomorphism property, where we know a lower bound of due to Kulkarni and Podder [KP16].

As stated in Theorem 3, we resolve the quantum Aanderaa–Karp–Rosenberg conjecture and show an optimal lower bound. The proof combines Theorem 2 with a quadratic lower bound on the degree of nontrivial monotone graph properties. With some work, the original quadratic lower bound on the deterministic query complexity of nontrivial monotone graph properties by Rivest and Vuillemin [RV76] can be modified to prove a similar lower bound for degree. We were not able to find such a proof in the literature, and instead combine the following two claims to obtain the desired claim.

First, we use the result of Dodis and Khanna [DK99, Theorem 2]:

Theorem 8.

For all nontrivial monotone graph properties, .

Here is the minimum degree of a Boolean function when represented as a polynomial over the finite field with two elements, . We combine this with a standard lemma that shows that this measure lower bounds . A proof can be found in [O’D09, Proposition 6.23]:

Lemma 9.

For all Boolean functions , we have .

Combining these with Theorem 2, we get that all nontrivial monotone graph properties satisfy , which is the statement of Theorem 3.

5 Open questions

We saw that lower-bounds both , and thus , and also the sensitivity . One might conjecture that lower-bounds all the complexity measures in Figure 1, including .

Conjecture 1.

For all Boolean functions , we have .

If creftypecap 1 we true, Theorem 5 would imply that , settling a longstanding conjecture posed by Nisan and Szegedy [NS94]. The current best relation between the two measures is . The following conjecture is weaker, and might be easier to tackle first.

Conjecture 2.

For all Boolean functions , we have .

Another longstanding open problem is to show a quadratic relation between deterministic query complexity and block sensitivity:

Conjecture 3.

For all Boolean functions , we have .

If this conjecture were true, it would optimally resolve several relationships in Table 1, and would imply, for example, and .

After settling the best relation between and , the next pressing question is to settle the best relation between and . Recently, the fourth author [Tal19] showed a power separation between and , while the best known relationship is a power relationship (this work). We conjecture that both these bounds can be improved.

Conjecture 4.

For all Boolean functions , we have .

Conjecture 5.

There exists a Boolean function such that .

We note that there are candidate constructions based on the work of [AA18, ABK16, Tal19] that are conjectured to satisfy

. In particular, it suffices to prove a conjectured bound on the Fourier spectrum of deterministic decision trees

[Tal19] to prove creftype 5.

Finally, for the special case of monotone total Boolean functions , Beals et al. [BBC01] already showed in 1998 that . It would be interesting to know whether this can be improved, perhaps all the way to .

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Appendix A Properties of the measure

We show that the measure satisfies various elegant properties. First, it can be defined in multiple ways, one of which was introduced by Koutsoupias back in 1993 [Kou93]. It also has a formulation as a special case of the quantum adversary bound and hence can be expressed as as a semidefinite program closely related to that of the quantum adversary bound. Due to this characterization, can be viewed as both a maximization problem and a minimization problem. These equivalent formulations are described in Section A.1.

Second, we show that , which was already observed by Laplante, Lee, and Szegedy [LLS06] (though we give a slightly different proof). Finally, we show lower bounds on and an optimal quadratic separation between and .

a.1 Equivalent formulations

Theorem 10.

For all Boolean functions , we have

(8)

where the measures , , and are defined below. Furthermore, itself has several equivalent formulations: .

We now define all these measures before proving this theorem.

Koutsoupias complexity .

For a Boolean function , let , and let . Let be the matrix with rows and columns labeled by and respectively, with if the Hamming distance of and is , and otherwise. Koutsoupias [Kou93] observed that is a lower bound on formula size, for every such choice of and . We define to be the maximum value of over choices of and . Thus is a lower bound on the formula size of .

Single-bit positive adversary .

We define as a version of the adversary bound where we are only allowed to put nonzero weight on input pairs where and the Hamming distance between and is exactly . We will define in terms of the spectral adversary version, which we also denote by . is defined as the maximum of

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over matrices of a special form. We require satisfy the following: (1) its entries are nonnegative reals; (2) its rows and columns are indexed by ; (3) whenever ; (4) whenever the Hamming distance of and is not ; and (5) is not all . In the above expression, refers to the Hadamard (entrywise) product, is the domain of , and is the -valued matrix with if and only if .

Single-bit negative adversary .

We define using the same definition as above, except that the matrix is allowed to have negative entries. Note that since this is a relaxation of the conditions on , we clearly have .

Single-bit strong weighted adversary .

We define as a single-bit version of the strong weighted adversary method from [SS06]. For this definition, we say a weight function is feasible if it is symmetric (i.e., ) and if it satisfies the conditions on above (i.e., it places weight on a pair unless both and the Hamming distance between and is ). We view such a feasible weight scheme as the weights on a weighted bipartite graph, where the left vertex set is and the right vertex set is . We let denote the weighted degree of in this graph, i.e., the sum of the weights of its incident edges. Then is defined as the maximum, over such feasible weight schemes , of

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Here ranges over , ranges over , and denotes the string with bit flipped.777

Readers familiar with the adversary bound should note that this definition is analogous a weighted version of Ambainis’s original adversary method; in the original method, the denominator was the geometric mean of (a) the weight of the neighbors of

with disagree with at , and (b) the weight of the neighbors of which disagree with at ; but in our case, both (a) and (b) are simply , since is the only string that disagrees with on bit and is connected to in the bipartite graph.

Single-bit minimax adversary .

Unlike the other forms, we define as a minimization problem rather than a maximization problem. We say a weight function is feasible if for all with and Hamming distance , we have , where is the bit on which and disagree. is defined as the minimum, over such feasible weight schemes , of

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Semidefinite program version .

We define to be the optimal value of the following semidefinite program.