The -approximate degree of a function , denoted , is the least degree of a multivariate real-valued polynomial such that for all inputs . Lower bounds on approximate degree have many applications in theoretical computer science, ranging from quantum query and communication lower bounds, to oracle separations and cryptographic secret sharing schemes. Upper bounds on approximate degree have important algorithmic implications in learning theory and differential privacy, and underlie state-of-the-art circuit and formula size lower bounds. The interested reader can find a bibliographic overview of these applications in [BKT18, She18].
This paper focuses on three well-studied functions whose approximation by polynomials has applications to quantum computing and beyond. The first function is element distinctness where the input is a list of numbers from and the objective is to determine if the numbers are pairwise distinct. The second function is surjectivity where the input is a list of numbers from the range and the goal is to check whether every range element appears on the list. The canonical setting is for some constant The third problem that we study is permutation testing parameterized by a constant Here, the input is a list of numbers from and the objective is to distinguish the case when the list contains every range element from the case when the list contains at most range elements. In the context of polynomial approximation, it is customary to represent the input to these functions as a Boolean matrix , where if and only if the th element on the list equals .
Vanishing-error approximate degree
Much work in the area has focused on bounded-error approximate degree, defined for a Boolean function as the quantity The choice of constant here is arbitrary, as for all constants . In particular, the bounded-error approximate degrees of element distinctness, surjectivity, and permutation testing are known to be and respectively [AS04, Amb05, Kut05, Aar12, She18, BKT18]. Our understanding of approximate degree with vanishing error, is far less complete. Among the very few functions whose vanishing-error approximate degree has been determined is the -bit AND function, with the asymptotic bound due to Buhrman et al. [BCWZ99]. We give a new and entirely different proof of their result. Our technique further allows us to settle the vanishing-error approximate degrees of the much more complicated functions of element distinctness, surjectivity, and permutation testing:
Let and be arbitrary constants. Then
This theorem is optimal with respect to all parameters. The lower bounds for element distinctness and surjectivity match the vanishing-error constructions in [She18], whereas the lower bound for permutation testing is tight by a quantum query argument which we include as Theorem 3.8. A comment is in order on -approximate degree in the complementary range,
Routine interpolation gives an exact representation for each of the functions in Theorem1.1 as a polynomial of degree at most Theorem 1.1 shows that this upper bound is asymptotically tight, settling the -approximate degree for as well.
We prove a result analogous to Theorem 1.1 for -element distinctness , a well-studied generalization of . Specifically, we prove that if has bounded-error approximate degree , then it has -approximate degree . The best known lower bound on the bounded-error approximate degree of is [BKT18], so this yields
For large , this comes close to the best known upper bound [She18]:
Our techniques are quite general, and we are confident that they will find other applications in the area. The technical core of our results establishes that for any function that contains as a subfunction for each , any bounded-error approximate degree lower bound for automatically implies a strong lower bound for the -approximate degree of . This allows us to prove tight lower bounds on the vanishing-error approximate degrees of and . To handle
we generalize our technique to other outer functions. Our analysis is based on the so-called method of dual polynomials, whereby one proves approximate degree lower bounds by constructing explicit dual solutions to a certain linear program capturing the approximate degree of the given function.
In the remainder of the introduction, we focus on an application of Theorem 1.1 to quantum Merlin–Arthur complexity.
The Merlin–Arthur model
The Merlin–Arthur (MA) model of query complexity features a function and two asymmetric players, Merlin and Arthur. Arthur’s goal is to compute on some unknown input while querying as few bits of as possible. Merlin, who knows , can help Arthur compute by sending him a single witness, i.e., an arbitrary message of some bit length . However, Merlin is untrusted. The model requires that, for any
, there is some Merlin message causing Arthur to output 1 with probability at least, and for any , no Merlin message can cause Arthur to output 1 with probability more than . The cost of the protocol is the sum of the witness length and the number of bits of queried by Arthur. In quantum Merlin-Arthur (QMA) query complexity, the witness sent by Merlin is allowed to be an arbitrary
-qubit quantum message, and Arthur is permitted to query bits of the input
in superposition. The MA and QMA query models have important analogues in communication complexity and Turing machine complexity. In the former setting, Arthur is replaced by two parties Alice and Bob, and the inputis split between them.
The complexity class is a quantum analog of and accordingly has received considerable attention. It is well known that any protocol can be simulated by an protocol with at most a quadratic blowup in cost, i.e., [Vya03].111An SBQP protocol is a quantum protocol for which there is some such that accepts every input in with probability at least , and every input in with probability at most [Kup15]. In turn, the existence of an SBQP query protocol that makes at most queries implies that the one-sided approximate degree of is at most . Here, the one-sided -approximate degree of is the least degree of a real polynomial such that for all , and for all [BT15]. As a consequence, one can prove QMA query lower bounds for by lower bounding the one-sided approximate degree of .
Only a handful of additional results are known about QMA query and communication complexity. Raz and Shpilka [RS04] showed that has QMA query complexity . Klauck [Kla11] showed that the QMA communication complexity of the disjointness problem is . Neither of these results follows from a naïve application of the bound .
QMA complexity of permutation testing
The permutation testing problem has played an important role in the study of interactive proof systems because it possesses a simple non-interactive perfect zero knowledge () protocol of logarithmic cost, yet is a hard problem in many other models. Hence, it has been used to prove a variety of complexity class separations. In particular, Aaronson [Aar12] showed that the QMA query complexity of is , and thereby gave an oracle separating from . Bouland et al. [BCH17] built on Aaronson’s result to give an oracle separating non-interactive statistical zero knowledge () from the complexity class , answering a question of Watrous from 2002. Gur, Liu, and Rothblum [GLR18] showed that the MA query complexity of is . Despite this progress, the precise QMA complexity of has remained open, with the best upper bound being [BHT16, Aar12] and the best lower bound being Aaronson’s . We obtain a polynomially stronger lower bound.
Let be an arbitrary constant. Then any QMA query protocol for with witness length has query cost . In particular, has QMA complexity
This result quantitatively matches the MA lower bound of Gur et al. [GLR18] but holds in the more powerful quantum setting. Theorem 1.2 comes reasonably close to matching the known QMA query upper bound of which holds even if Merlin does not send any message to Arthur; see Theorem 3.8.
To prove Theorem 1.2, we derive a connection between QMA query complexity and vanishing-error approximate degree for a class of functions that includes , , and . This connection amounts to the observation that, for these particular functions, the one-sided -approximate degree is equal to the -approximate degree. Prior work on QMA complexity (e.g., [Kla11]) has implicitly exploited a similar observation in the special case of . Our analysis substantially generalizes the insights of prior work, and makes explicit the key phenomenon at play, namely the equivalence of one-sided vs. standard approximate degree for these functions. Combining this connection with our new vanishing-error approximate degree lower bounds in Theorem 1.1 establishes Theorem 1.2.
For a function we let and stand for the domain and image of respectively. We view Boolean functions as mappings for a finite set For functions and , we let denote the block-composition of and . In more detail, is the function that maps to . We generalize block-composition to the case when the domain of is properly contained in by defining the domain of as the set of such that
2.1. Polynomial approximation
For a multivariate real polynomial , we let denote the total degree of , i.e., the largest degree of any monomial of It will be convenient to define the degree of the zero polynomial by For a real-valued function supported on a finite subset of , we define the orthogonal content of denoted , to be the minimum degree of a real polynomial for which We adopt the convention that if no such polynomial exists. For two functions , let denote the correlation of and , and let . For any real-valued function , its
-th tensor poweris given by .
The -approximate degree of a function , denoted , is the least degree of a polynomial such that for all . We emphasize that no restriction is placed on the behavior of at inputs outside ’s domain of definition, For most functions of interest to us, the domain is a proper subset of and thus their approximating polynomials may take on arbitrary values on The following dual characterization of approximate degree is well known and can be verified using linear programming duality.
Fix and a function . Then if and only if there exists a function such that
The simplest function of interest to us is given as usual by Its bounded-error approximate degree was determined by Nisan and Szegedy [NS94], as follows.
Let stand for the set of Boolean matrices of size in which every row has exactly one Every matrix has a natural interpretation as specifying a mapping , where if and only if Our next three functions are defined on and can thus be regarded as “function properties.” To start with, the surjectivity problem with elements and range size is defined as where
Thus, takes as input an Boolean matrix in which every row contains exactly one , and evaluates to 1 if and only if every column of the input contains at least one 1. Interpreting the input matrix as a mapping, evaluates to if and only if that mapping is surjective. This surjectivity property is trivially false for and the standard setting of parameters is for some constant The choice of constant is unimportant because it affects by at most a multiplicative constant. It was shown in [She18] that the surjectivity function has bounded-error approximate degree Bun et al. [BKT18] gave an alternate proof of this upper bound and additionally proved that it is tight up to a polylogarithmic factor. We thus have:
Let be an arbitrary constant. Then
2.3. Element distinctness
Another well-studied function is element distinctness defined by if and only if every column of the input matrix has at most one Switching to the interpretation of as a mapping, evaluates to true if and only if the mapping is one-to-one. This property is trivially false for In the complementary case, Ambainis [Amb05] proved that for any given the -approximate degree of is the same for all This means that one may without loss of generality focus on the special case with the shorthand notation . Aaronson and Shi [AS04], Ambainis [Amb05], and Kutin [Kut05] showed that has bounded-error approximate degree
Element distinctness generalizes in a natural way to a function called -element distinctness, denoted . This new function evaluates to true if and only if the input matrix has no column with or more 1s. Viewing the input as a mapping, evaluates to true if and only if no range element occurs or more times. With these definitions, we have
2.4. Permutation testing
The final problem of interest to us is a restriction of element distinctness . In more detail, fix an integer and a real number . The domain of the permutation testing problem is the set of all matrices in which the number of columns containing a is either exactly or at most The function evaluates to true in the former case and to false in the latter. Equivalently, if and only if is a permutation matrix. In the regime of interest to us, is a constant independent of
The permutation testing problem was introduced by Aaronson [Aar12], who defined it somewhat differently. In his variant of permutation testing, which we denote by one is given a matrix that is either (i) a permutation matrix, or (ii) disagrees from every permutation matrix in at least rows. The function evaluates to true in case (i) and to false in case (ii). As the following proposition shows, Aaronson’s is precisely the same function as our
Let and be given. Then as functions,
Specifically, the l.h.s. and r.h.s. have the same domain and agree at every point thereof.
This claim is easiest to verify by interpreting an input as a mapping
A moment’s reflection shows thatdisagrees from every permutation in at least points, and there is a permutation that achieves this lower bound. Restating this in matrix terminology, a matrix disagrees from every permutation matrix in at least rows if and only if the number of columns of containing a is at most ∎
By adapting earlier analyses of element distinctness, Aaronson [Aar12] obtained the following result.
Let be an arbitrary constant. Then
Let be an arbitrary constant. Then
We close this section with a remark on input encoding. In this work, functions like take as input a Boolean matrix in which every row has exactly one 1. Some other works [BM12, BKT18] represent the input as a list where encodes the location of the unique 1 in the -th row of the matrix representation Switching to this alternate representation affects the approximate degree by at most a logarithmic factor. See [She18] for a detailed treatment of the relationship between these representations.
3. Approximate Degree Lower Bounds
In this section, we study the vanishing-error approximate degree of element distinctness, surjectivity, and permutation testing, and in particular settle Theorem 1.1 from the introduction. The core of our technique is the following auxiliary result.
For any and any function on a finite subset of Euclidean space,
In particular, every function satisfies
The proof of Proposition 3.1 applies more generally to the conjunction of distinct functions, but we will not need this generalization.
To illustrate our technique in the simplest possible setting, we consider the well-studied function. Buhrman et al. [BCWZ99] proved that its -error approximate degree is We give a new and simple proof of their lower bound.
3.2. Element distinctness
Our next result is a tight lower bound on the vanishing error approximate degree of element distinctness, matching the upper bound from [She18].
For any , we claim that is a subproblem of . To see why, recall that the input to is an Boolean matrix in which every row contains exactly one , corresponding to the value of the th element. Now, fix and consider the restriction of to input matrices that are block-diagonal, with blocks of size each and an additional block of ones on the diagonal. Each of the first blocks corresponds to an instance of and the overall problem amounts to computing the AND of these instances. Therefore, is a subproblem of , and
for all and all
The previous proof shows more generally that is a subfunction of for any As a result, our analysis of element distinctness proves the following statement.
Fix constants and such that
An instance of the surjectivity problem can be embedded inside a larger instance of surjectivity in many ways, e.g., by duplicating a row of or by forming a block-diagonal matrix with blocks and These two transformations yield
respectively. We will now prove an essentially tight lower bound on the vanishing-error approximate degree of surjectivity, matching the upper bound from [She18] up to a logarithmic factor.
Let be an arbitrary constant. Then
The proof is a cosmetic adaptation of the analysis of element distinctness. To start with, we claim that for any positive integers such that and the composition is a subproblem of . Indeed, the input to is an Boolean matrix in which every row contains exactly one . Consider the restriction of to input matrices that are block-diagonal, with blocks of size each. Each of these blocks corresponds to an instance of and the overall problem amounts to computing the AND of these instances. This settles the claim.
Now let be arbitrary. Then for all positive integers
where the first step uses (3.5); the second and third steps use (3.4); the fourth step applies the claim from the opening paragraph of the proof; the fifth step is valid by Proposition 3.1; and the sixth step invokes Theorem 2.3. This settles the theorem. ∎
3.4. Permutation testing
We now turn to the permutation testing problem, which requires a more subtle analysis than the functions that we have examined so far. The difficulty is that permutation testing does not admit a self-reduction with AND as an outer function. To address this, we will need to generalize Proposition 3.1 appropriately. For a real and an integer we define to be the restriction of to inputs whose Hamming weight is either or at most The following result subsumes Proposition 3.1 as the special case
Fix a real number and an integer Then for any and any function on a finite subset of Euclidean space,
We may assume that since the proposition is trivial otherwise. Let be an -error dual polynomial for , as guaranteed by Fact 2.1:
Abbreviate and define by
Observe that is supported on the domain of Moreover, we have the pointwise inequality
For a permutation testing instance can be extended in a natural way to a larger instance by letting for This gives
We are now in a position to prove our lower bound on the -approximate degree of permutation testing.
Let be a given constant. Then
Let be arbitrary. We claim that for any positive integers and with the permutation testing function contains
as a subfunction. The proof is similar to that for element distinctness. Specifically, view instances of (3.9) as block-diagonal matrices with blocks of size each. Then a positive instance of (3.9) is a permutation matrix and therefore a positive instance of . A negative instance of (3.9), on the other hand, features at least blocks from and therefore corresponds to a mapping with a range of size at most
In particular, any negative instance of (3.9) is also a negative instance of . This completes the proof of the claim.
Now for any and any we have
where the first inequality uses (3.7), and the second inequality follows from the claim established in the previous paragraph. The rest of the proof is analogous to those for and . For
We will now show that Theorem 3.7 is optimal with respect to all parameters. In fact, we will prove the stronger result that permutation testing has an -error quantum query algorithm with cost . Our quantum algorithm is inspired by the well-known algorithm for the collision problem due to Brassard et al. [BHT16].
Let be a given constant. Then for all and the permutation testing problem has an -error quantum query algorithm with cost In particular,
We give an algorithm whose only quantum component is Grover search. Specifically, we will only use the fact that, given query access to items of which are marked, Grover search finds a marked item with probability using queries (see, e.g., [BHT16, BHT98]). We will follow the convention in the quantum query literature and view the input to as a function where the algorithm has query access to
Let be an integer parameter to be determined later. Our algorithm starts by choosing a uniformly random subset of cardinality Next, we query at every point of If is not one-to-one on we output “false.” In the complementary case, we execute Grover search times independently, each time looking for a point with the property that We output “false” if such a point is found, and “true” otherwise.
If is a permutation, the described algorithm is always correct. In the complementary case when there are at least points such that Call such points special. We will henceforth assume that contains at least special points, which happens with probability at least If is not one-to-one on the algorithm correctly outputs “false.” If is one-to-one on and contains at least special points, then each of the Grover executions has eligible points to output from among a total of possibilities; this means that each Grover execution finds an eligible point with probability at least using queries, thereby forcing the correct output. In summary, the described algorithm has error probability at most and query cost In particular, error can be achieved with query cost This query bound in turn implies (3.11) using the standard transformation of a quantum query algorithm to a polynomial; see, e.g., Ambainis [Amb05]. ∎
4. QMA Lower Bounds
The objective of this section is to “lift” the approximate degree lower bound of Theorem 3.7 to QMA query complexity. As our first step, we generalize our lower bound to one-sided approximation. The one-sided -approximate degree of a function , denoted , is the least degree of a polynomial such that for all and for all Thus, approximates uniformly on but may take on arbitrarily large values on It is clear from the definition that The gap between these quantities can be large in general, such as versus for the bounded-error approximation of . However, we will show that these two notions of approximation are equivalent for the permutation testing function.
For all and
This equality of approximate degree and one-sided approximate degree for permutation testing has the important consequence that the lower bound of Theorem 3.7 applies to the one-sided setting as well. The proof of Proposition 4.1 is based on the observation that any one-sided approximant for permutation testing can be symmetrized to be constant on effectively making it a two-sided approximant. This technique was used previously in [BT15, Theorem 2] to argue that
Proof of Proposition 4.1..
Let be a one-sided approximant for with error so that on and on Define
where are uniformly random permutations on and denotes the matrix obtained by permuting the rows of according to and the columns according to Then is also a one-sided approximant for because and are closed under permutations of rows and columns. Moreover, takes on the same value, call it , at all because in (4.2) is a uniformly random permutation matrix in that case. As a result, the normalized polynomial approximates pointwise within Finally, because is an average of polynomials, each obtained from by permuting the input variables. ∎
Suppose that has a QMA query protocol with witness length and query cost . Then there is a polynomial such that
Marriott and Watrous [MW05] showed that the soundness and completeness errors of the QMA query protocol for can be driven down to without an increase in witness length, and with only a factor of