Quantum Random Access Codes for Boolean Functions

An $n\overset{p}{\mapsto}m$ random access code (RAC) is an encoding of $n$ bits into $m$ bits such that any initial bit can be recovered with probability at least $p$, while in a quantum RAC (QRAC), the $n$ bits are encoded into $m$ qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function $f$ on any subset of fixed size of the initial bits, which we call $f$-random access codes. We study and give protocols for $f$-random access codes with classical ($f$-RAC) and quantum ($f$-QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement ($f$-EARAC) and Popescu-Rohrlich boxes ($f$-PRRAC). The success probability of our protocols is characterized by the \emph{noise stability} of the Boolean function $f$. Moreover, we give an \emph{upper bound} on the success probability of any $f$-QRAC with shared randomness that matches its success probability up to a multiplicative constant (and $f$-RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.


Introduction
One of the possible origins of quantum computers' power is the exponential size of the Hilbert space: a n-qubit quantum state is a unit vector in a 2 n dimensional complex vector space. On the other hand, one of the fundamental results in quantum information theory -Holevo's theorem [29] -states that no more than n bits of classical information can be transmitted by n qubits without entanglement. Nonetheless, interesting scenarios arise when allowing a small chance of transmitting the wrong message or/and obtaining partial information at the expense of losing information about the rest of the system. One of these scenarios is the concept of quantum random access codes (QRACs), where a number of bits are encoded into a smaller number of qubits such that any one of the initial bits can be recovered with some probability of success. A QRAC is normally denoted by n randomness expansion [38], studies of no-signaling resources [23], and characterization of quantum correlations from information theory [49]. The 2 → 1 and 3 → 1 QRACs were first experimentally demonstrated in [56]. See [21,25,42,59,61] for subsequent demonstrations.
In this paper we further generalize the idea of (quantum) random access codes to recovering not just an initial bit, but the value of a fixed Boolean function on any subset of the initial bits with fixed size. We call them f -random access codes. The case of the Parity function was already considered in [8], and here we generalize to arbitrary Boolean functions f : {−1, 1} k → {−1, 1}.

Related Work
An n p → m (Q)RAC is an encoding of n bits into m (qu)bits such that any initial bit can be recovered with probability at least p. This probability is the worst case success probability over all possible pairs (x, i) of input string x ∈ {−1, 1} n and recoverable bit i ∈ {1, . . . , n}. Many different resources can be used during the encoding and decoding, e.g. private randomness (PR), shared randomness (SR), shared entanglement, and even super-quantum correlations like Popescu-Rohrlich boxes [50].
Regarding the classical RAC, Ambainis et al. [4] proved that there is no 2 p → 1 RAC (and 2 m p → m RAC by extension) with PR and worst case success probability p > 1/2. On the other hand, Ambainis et al. [5] showed that RACs with SR can achieve success probability p > 1/2. (25)]). The optimal n p → 1 RAC with SR has success probability

Theorem 1 ([5, Equation
For a general number of encoded bits, Ambainis et al. [4] developed a RAC with PR using a specific code from [14] which matches their classical lower bound m ≥ (1−H(p))n up to an additive logarithmic term, where H(p) = −p log 2 p − (1 − p) log 2 (1 − p) is the binary entropy function.
The idea of decoding a function of the initial bits instead of a single bit was already considered by Ben-Aroya, Regev and de Wolf [8] (who also considered recovering multiple bits rather than 1 Usually private randomness is already assumed in QRACs under the encoding onto density matrices. 2 Ambainis et al. [5] do not give the high order terms, but these can be calculated by following their procedure together with [30,Equation 2.198]. just one). More specifically, they defined an n p → m XOR k -QRAC, where n bits are encoded into m qubits such that the parity of any k initial bits can be recovered with success probability at least p. 3 Using their hypercontractive inequality for matrix-valued functions, they proved the following upper bound on the success probability.
Theorem 4 ([8,Theorem 7]). For any η > 2 ln 2 there is a constant C η such that, for any n p → m XOR k -QRAC with SR and k = o(n), They conjectured that the factor η > 2 ln 2 can be dropped from the above bound, and thus extended to m/n > 1/(2 ln 2) ≈ 0.72, although it might require a strengthening of their hypercontractive inequality.
The use of shared entanglement in random access codes was first considered by Klauck [35,36].
Here the encoding and decoding parties are allowed to use an arbitrary amount of shared entangled states (note that shared entanglement can be used to obtain both private and shared randomness). The figure of merit in this generalization is the relation between n, m and p, while the amount of shared entanglement is not taken into account. Klauck [35,36] considered an n p → m QRAC with shared entanglement and, by its equivalence to the quantum one-way communication complexity for the index function, proved the lower bound m ≥ (1 − H(p))n/2, similar to Nayak's bound. Later Pawłowski and Żukowski [48] coined the term entanglement-assisted random access code (EARAC), which is a RAC with shared entanglement, and studied the case when m = 1, giving protocols with better decoding probabilities compared to the usual n p → 1 QRAC with SR. Recently Tănăsescu et al. [58] expanded the idea of n p → 1 EARACs to recovering an initial bit under a specific request distribution.
The idea of (Q)RAC was generalized in other ways, e.g. parity-oblivious [7,13,56] and multiparty [53] versions, encoding on d-valued qubits (qudits) [6,12,19,39,59], a wider range of information retrieval tasks [18] and a connection to Popescu-Rohrlich boxes. It was shown [66] that a Popescu-Rohrlich box can simulate a RAC by means of just one bit of communication, while in [23] the converse was proven. An object called racbox [23] was defined, which is a box that implements a RAC when supported with one bit of communication, and it was shown that a non-signaling racbox is equivalent to a Popescu-Rohrlich box. A quantum version of a racbox was later proposed in [24]. Finally, we mention that RACs were also studied within "theories" that violate the uncertainty relation for anti-commuting observables and present stronger-than-quantum correlations [57].

Our Results
This paper focuses on generalizing the classical, quantum and entanglement-assisted random access codes. Instead of recovering a single bit from the initial string x ∈ {−1, 1} n , we are interested in evaluating a Boolean function f : {−1, 1} k → {−1, 1} on any sequence of k bits from x. We generically call them f -random access codes.
. . , n} k | S i = S j ∀i, j} be the set of sequences of different elements from {1, . . . , n} with length k and let x S ∈ {−1, 1} k denote the substring of x ∈ {−1, 1} n specified by S ∈ S k n . Alice gets x ∈ {−1, 1} n and she needs to encode her data and send it to Bob, so that he can decode f (x S ) for any S ∈ S k n with probability p > 1/2. Such problem was already considered by Sherstov in a two-way communication complexity setting [54] and later used in his pattern matrix method [55] in order to prove other communication complexity lower bounds. Even though our results are expressed in a random access code language, they can also be seen as in a one-way communication complexity setting. If two-way communication is allowed, Bob can send the identity of his sequence to Alice with O(k log n) bits of communication, whereas (as we will see) significantly more communication may be required in the one-way scenario.
In the following, Π will refer to a sample space with some probability distribution. As before, PR and SR stand for private and shared randomness, respectively. Moreover, since we require the success probability to always be greater than 1/2, given that one can always guess the correct result with probability 1/2, from now on it will be convenient to use the bias ε of the prediction, defined as ε = 2p − 1, instead of its success probability p.
We start with n ε → m f -RAC, the f -classical random access code on m bits with bias ε.
We define n ε → m f -QRAC, the f -quantum random access code on m qubits with bias ε.

Definition 8. An n
ε → m f -QRAC with PR is an encoding map E : {−1, 1} n → C 2 m ×2 m that assigns an m-qubit density matrix to every x ∈ {−1, 1} n and satisfies the following: for every Due to shared entanglement being a source of SR, we already include SR in f -EARACs. We note that [48] focused on EARACs without SR. In Section 3 we devise encoding-decoding strategies for all the f -random access codes just defined, thus deriving lower bounds on their biases given the encoding/decoding parameters n, m and k. These f -random access codes are built based on previous ideas from [4,5,48]. The Boolean function f : {−1, 1} k → {−1, 1} that needs to be evaluated directly influences the final bias and such influence in our results is captured by the single quantity called noise stability [9,45]. Informally it is a measure of how resilient to noise a Boolean function is. Given a uniformly random input x ∈ {−1, 1} n , one might imagine a process that flips each bit of x independently with some 1], which leads to some final string y ∈ {−1, 1} n . The noise stability Stab q [f ] of f with parameter q is the correlation between f (x) and f (y) (see Section 2 for a formal definition).
Our positive results can be summarized by the following theorem.
(e) For any n ∈ N, there is an n Results (a), (b), (c) and (d) use an encoding scheme reminiscent of the concatenation idea from [48,49,58] (and suggested to us by Ronald de Wolf). The underlying idea is to randomly break the initial string x ∈ {−1, 1} n into different 'blocks' and encode them via a standard RAC/QRAC/EARAC. Result (a) breaks x into blocks and employs the n/ → m/ RAC from Theorem 2 on every block, each with n/ elements, while in results (b)/(c)/(d) we employ the n/m → 1 RAC/QRAC/EARAC from Theorems 1/3/5 in order to encode m blocks, each with n/m elements, into a single (qu)bit each, resulting in m encoded (qu)bits. With high probability all the bits from the needed string x S ∈ {−1, 1} k will be encoded into different blocks and therefore can be decoded and f evaluated. The decoded string y ∈ {−1, 1} k can be viewed as a 'noisy' x S , to which the noise stability framework can be applied. The bias of the base RAC/QRAC/EARAC thus becomes the parameter q in the noise stability of the corresponding f -random access code. As a quick remark, since we opted to lower-bound the parameters q in Theorem 12, in result (b) q does not exactly equal the bias from Theorem 1. One could write, though, q ≈ 2m πn . Result (a) is our strongest bound, since it also applies to all other f -random access codes. Moreover, there is some freedom in setting the number of blocks , since the number of encoded bits in Theorem 2 is not fixed to a single number (as opposed to Theorems 1, 3 and 5). The result is a trade-off between the number of bits k of the Boolean function and the number of encoded bits m. However, the number of encoded bits in result (a) is limited to m = Ω(log n), a characteristic inherited from the RAC in Theorem 2. It is possible to go below this limit by using SR, as demonstrated by results (b), (c) and (d).
The above results show that quantum resources offer a modest advantage over the classical f -random access code. On the other hand, result (e) demonstrates that stronger-than-quantum resources like Popescu-Rohrlich boxes can lead to extremely powerful f -random access codes. This is a consequence of violating Information Causality [49], since one bit transfer allows the access to any bit in a database via Popescu-Rohrlich boxes. From x ∈ {−1, 1} n a long bit-string All bits from x f are readable with the aid of Popescu-Rohrlich boxes, with non-signaling constraining the readout to just one bit. The protocol for f -PRRACs is taken from [49] and uses a pyramid of Popescu-Rohrlich boxes and nests a van Dam's protocol [16].
In Section 4 we prove an upper bound on the bias of any f -QRAC with SR (and f -RAC) using the same method of the hypercontractive inequality for matrix-valued functions from [8].
1} be a Boolean function. For any n ε → m f -QRAC with SR and k = o(n) the following holds: for any η > 2 ln 2 there is a constant C η such that where One can see that the above result is a generalization of Theorem 4. Indeed, for Parity on k bits, L 1, (XOR k ) = 1 iff = k, and so Eq. (1) is recovered. The following corollary from Theorem 13 helps to compare the bias upper bound to the bias lower bounds from Theorem 12.
where deg(f ) = max{|S| : f (S) = 0} is the degree of f and q = ηm n . Taking deg(f ) to be upper-bounded by a constant (for example, if k = O(1)), our bias upper bound matches our bias lower bounds for f -RAC/QRAC with SR up to a global multiplicative constant and a multiplicative constant √ η in the parameter q. We conjecture that the parameter q can be improved to m n , which might require a stronger version of the hypercontractive inequality. Other corollaries from Theorem 13 are derived in Section 4 and compared to our bias lower bounds.
Upper bound (2) does not apply to f -EARACs. Previously, it was known that for the special case of standard EARACs (m = 1), the bias ε is upper-bounded by 1/ √ n (Theorem 5). This upper bound can generalised to EARACs with m > 1 assuming an independence condition (Section 3.4). The resulting bound is ε ≤ m/n. We view this as evidence that the bias lower bound for the general case of f -EARACs given in Theorem 12 should actually be tight.
Regarding the quantity Stab q [f ] itself, it can be nicely related to the Fourier coefficients of f (see Theorem 17 in the next section). We briefly mention the noise stability for a few functions. For Parity (XOR k ), Stab q [XOR k ] = q k , and, more generally, for any function χ S (x) = i∈S x i , Stab q [χ S ] = q |S| . As for the Majority function (MAJ k ), one can show that [46,Theorem 5 Other examples can be found in [41].
Moreover, a randomized algorithm for approximating the noise stability of monotone Boolean functions up to relative error was proposed in [52].
In Section 3 we present protocols for all our f -random access codes, and in Section 4 we derive an upper bound on the bias of f -QRACs with SR.

Preliminaries
We shall briefly revise some results from Boolean analysis that are going to be useful. For an introduction to the analysis of Boolean functions, see O'Donnell's book [46] or de Wolf's paper [65]. In the following, we write [n] = {1, . . . , n} and S n is the set of all permutations of [n]. As before, j} be the set of sequences of different elements from [n] with length k and let x S ∈ {−1, 1} k denote the substring of x ∈ {−1, 1} n specified by S ∈ S k n . The inner product ·, · on the vector space of all functions f : .
Every function f : {−1, 1} n → R can be uniquely expressed as a multilinear polynomial, its Fourier expansion, as The real number f (S) is called the Fourier coefficient of f on S and is given by An important and useful concept for Boolean functions is noise stability. As previously mentioned, it is a measure of how resilient to noise a Boolean function is, and is defined from the concept of q-correlated pairs of random strings given below.
We say that y is q-correlated to x. If x ∼ {−1, 1} n is drawn uniformly at random and then y ∼ N q (x), we say that (x, y) is a q-correlated pair of random strings.
Given these definitions, we can formally define the concept of noise stability, which measures the correlation between f (x) and f (y) when (x, y) is a q-correlated pair.
The noise stability of f is nicely related to f 's Fourier coefficients as stated in the following theorem.
It is not hard to prove from the above results that Stab q [f ] = f, T q f . Some of the above concepts can be generalized to matrix-valued functions. The Fourier transform f of a matrix-valued function f : {−1, 1} n → C m×m is defined similarly as for scalar functions: it is the function f : {−1, 1} n → C m×m defined by Here the Fourier coefficients f (S) are also m × m complex matrices. Moreover, given A ∈ C m×m with singular values σ 1 , . . . , σ m , its trace norm is defined as A tr = Tr |A| = m i=1 σ i . We shall make use of the following result from Ben-Aroya, Regev and de Wolf [8], which stems from their hypercontractive inequality for matrix-valued functions. 3 Bias Lower Bounds

f -RAC with PR
We start by studying the f -RAC with PR. The following result is based on Ambainis et al. [4] and uses a procedure reminiscent of the concatenation idea from [48,49,58]: the initial string is broken in blocks, which in turn are encoded using the code from [14]. First we state a slightly modified version of Newman's Theorem [44] (see also [37,Theorem 3.14] and [51, Theorem 3.5]) which is going to be useful to us.  Proof. Consider a code C ⊆ {−1, 1} n such that, for every x ∈ {−1, 1} n , there is a y ∈ C within Hamming distance (1 − p − 1 n )n, with p > 1/2 (the extra 1/n term will be used to counterbalance the decrease in probability from Newman's theorem). It is known [14] that there is such a code C of size log 2 |C| = (1 − H(p + 1/n)) n + 2 log 2 n ≤ (1 − H(p)) n + 4 log 2 n.
Let C(x) denote the closest codeword to x. Hence at least (p + 1/n)n out of n bits of C(x) are the same as x, and the probability over a uniformly random i that x i = C(x) i is at least p + 1/n.
Let ∈ N such that divides n. Our protocol involves breaking up x ∈ {−1, 1} n into parts and encoding each part with the above code C ⊆ {−1, 1} n/ . Define the map that applies C to the first bits of x, and to next bits of x and so on. Hence the probability that x i = C ( ) (x) i over a uniformly random i is at least p + /n. In order to consider this probability for every bit instead of just an average over all bits, we employ the following randomization process. Let r ∈ {−1, 1} n and π ∈ S n , both taken uniformly at random. Given x ∈ {−1, 1} n , denote π(x) = x π(1) x π (2) . . . x π(n) . We define the encoding C ( ) π,r (x) = π −1 (C ( ) (π(x · r))) · r, where x · r denotes the bit-wise product of x and r. Let E S be the event that all indices in S ∈ S k n are encoded in different codes C, i.e., in different blocks from C ( ) . There are blocks, each with n/ elements. The probability that k specific elements fall into k different blocks is where inequality (a) can easily be proven by induction or the union bound.
We shall first present a protocol using shared randomness, and at the end we shall transform it into a protocol with private randomness by using Newman's theorem. The protocol is the following. Select r ∈ {−1, 1} n and π ∈ S n uniformly at random. Encode x as C where q := 2p − 1, and the inequality follows from monotonicity of the noise stability of f . With these considerations, the success probability of the protocol is where we used that k = o( √ ).
We now transform the shared randomness into private randomness. By Newman's theorem (Theorem 22) there is a small set of permutation-string pairs (note that Alice's input is size n bits and Bob's input is at most n bits) with size In the protocol from Theorem 23 we broke the initial string into different blocks and used different copies of C. This was done in order to guarantee the independence of the C(x) Si 's and hence analyse the influence of the code C on the function f . Interestingly enough, for the special case of the Parity function this is not required and a single copy of C can be used. There is an n ε → m XOR k -RAC with PR and bias where Proof. Consider the encoding C π,r (x) = π −1 (C(π(x · r))) · r, where C ⊆ {−1, 1} n is the code described in Theorem 23. Let δn be the Hamming distance between x and C(x), with δ ≤ 1−p−1/n by the properties of C. Then where we used k =0 (1−δ)n k− δn = n k on the second equality and K k,n (δn) − K k,n (δn + 1) = 2 on the final inequality, which can be obtained via the recurrence relation K k,n (x) − K k,n (x − 1) = K k−1,n (x) − K k−1,n (x − 1) and K 1,n (x) = n − 2x (see e.g. [15]). By Newman's theorem (Theorem 22) there is a small set of permutation-string pairs with size t = n n k 2 such that Si continues to hold with bias at least K k,n ((1−p)n)/ n k for any x and S if π, r are chosen uniformly at random from this set.
Our protocol is the following. Select j ∈ [t] uniformly at random. Encode x as C πj ,rj (x). To decode The result follows by using the first inequality from Theorem 21 to observe that Remark. Since K 1,n (x) = n − 2x, we note that, for k = 1, Eq. (4) reduces to ε ≥ m n − 7 log 2 n n , which is the result from Ambainis et al. [4] (see Theorem 2). Remark. If k = O(1), then the Krawtchouk polynomial has the asymptotic limit as n → ∞ of [17,Eq. (29)], thus the bias from Theorem 24 has the asymptotic limit Note that this result is very similar to the one that would follow from Theorem 23, but slightly tighter (without the parameter and the multiplicative constant 1 − o n (1)).

f -RAC with SR
There is a lower limit of m = Ω(log n) on the number of encoded bits in Theorem 23. It is possible to go below this limit by using SR: the blocks are now encoded via the n ε → 1 RAC with SR from Theorem 1 instead of the code C.
Let E S be the event that all indices in S ∈ S k n are encoded in different sets. Similarly to Eq. (3), The bias of correctly recovering any of the n/m encoded bits is q = 2 2 n/m n/m−1 n/m−1 2 by Theorem 1. Conditioning on E S happening, we see that x S and y T are q-correlated according to Definition 15. Therefore Pr T1,...,Tm where we used an input randomization via SR. With these considerations, the success probability of the protocol is Pr T1,...,Tm Remark. It is possible to use Newman's theorem in the above theorem in order to transform SR into PR, but then Ω(log n) encoding bits would need to be used to encode the randomization procedure, thus leading to m = Ω(log n). Moreover, the final f -RAC would have worse bias compared to the one from Theorem 23. Remark. If m = n/2 or m = n/3, the usual (and optimal) 2

Remark. The requirement m|n can be dropped by adding extra bits into
−→ 1 QRAC with PR can be used, respectively. The resulting biases have q m=n/2 = 1/ √ 2 and q m=n/3 = 1/ √ 3.

f -EARAC
The same protocol can also be used for f -EARACs, now with the n/m → 1 EARAC from Theorem 5. Proof. Replace the n/m → 1 RAC in the proof of Theorem 25 with the n/m → 1 EARAC from Theorem 5 with bias = m/n.
Remark. We could also define an entanglement-assisted f -QRAC (f -EAQRAC) similarly to Definition 10, i.e., as an f -QRAC with SR where both parties share an unlimited amount of entanglement. Due to super-dense coding and teleportation, an n ε → m f -EAQRAC is equivalent to an n ε → 2m f -EARAC, meaning that there is an n i.e., when considering the usual n → m EARAC, the above result tells us that the success probability is just Moreover, we note that the n → m EARAC is formed by a grouping of n/m → which is maximized by taking r i = n/m for all i ∈ [m]. Since the n/m → 1 EARACs are optimal by Theorem 5, so is Eq. (5) (under the assumption that the n → m EARAC is formed by m independent EARACs on 1 encoding bit).
We can use Theorem 17 to obtain the following Corollary from Theorems 23, 25, 26 and 27.

f -PRRAC
We now present a protocol for the f -PRRAC, based on reducing the problem to the standard random access code setting, and then using a protocol defined in [49]. This protocol was used to show the violation of information causality by means of a pyramid of Popescu-Rohrlich boxes and nesting van Dam's protocol [16], which allows us to decode the value of f (x S ) for any S ∈ S k n with just one encoded bit. This procedure of pyramiding and nesting was also used in the context of EARACs in [48,58] under the name of concatenation. Proof. To ease the notation, we shall use {0, 1} instead of {−1, 1} during the proof. We shall also name the encoding and decoding parties Alice and Bob, respectively, and refer to a Popescu-Rohrlich box as PR-box. Let 5 t := |S k n | = k! n k and define the string a ∈ {0, 1} t as a S := f (x S ), where S k n is arranged in lexicographic order. 6 Bob is interested in bit a S , whose index position can be described by a t-bit The remainder of the argument is the same as the protocol of [49], but we include the details for completeness. For t = 1 we have that Alice inputs a 0 ⊕ a 1 into a PR-box, while Bob inputs b 0 . Alice obtains the output A and sends the message y = a 0 ⊕ A to Bob, who can obtain y ⊕ B = a b using his output B, since, by the PR-box property, A ⊕ B = b 0 (a 0 ⊕ a 1 ).
For t > 1, write a = a a , where a = a 0 . . . a t/2−1 ∈ {0, 1} t/2 and a = a t/2 . . . a t−1 ∈ {0, 1} t/2 . Then one can show that 1 (a , b )) , 5 If f is symmetric, the size t can be decreased to n k by ignoring all the redundant permutations of the k-sets. 6 Here we use a S to denote a single bit of a, whereas x S denotes a subsequence of x.
Therefore we can construct a recursive protocol in t, which will encompass all values of n. The protocol uses a pyramid of 2 t − 1 Popescu-Rohrlich boxes placed on t levels. The case t = 1 was explained above. For t > 1, Alice and Bob use the protocol on inputs (a , b ) and (a , b ), which involves 2 t/2 − 1 PR-boxes in each one. Alice's outputs of each protocol are y and y , which she inputs into the last PR-box, similarly to the case t = 1, as y ⊕ y , while Bob inputs b t−1 . Given Alice's final output A, she sends y = y ⊕ A to Bob, who uses his output B t−1 to obtain y ⊕ b t−1 (y ⊕ y ). If b t−1 = 0, he gets y , otherwise, if b t−1 = 1, he gets y . With these, he can recursively go up the pyramid based on the protocol for t − 1 bits, which tells him which boxes to read. Looking at the binary decomposition of b, Bob goes (t − r) times to the left bit, and r times to the right bit, where r = t−1 i=0 b i . His final output will be y ⊕ B 0 ⊕ · · · ⊕ B t−1 , where B j is the output for the PR-box that Bob uses at level j. Bob will only need the outputs of t PR-boxes, while Alice uses 2 t − 1 PR-boxes in total.

Bias Upper Bounds
In order to prove an upper bound on the bias of any n ε → m f -QRAC with SR, we shall use the following equivalent version of Definition 9, which comes from input randomization, i.e., from considering the average success probability over the inputs, and from the following fact. ([28]). Let ρ be an unknown state picked from the set {ρ 0 , ρ 1 } with probability p and 1 − p, respectively. The optimal success probability of predicting which state it is by a POVM is

Fact 30
where |T |= f (T ) is the 1-norm of the -th level of the Fourier transform of f . Proof. We start by writing the following.
but, for a given T ⊆ [k] with |T | = , ρ(S) tr , and thus, using Jensen's inequality, We now use Theorem 20 with δ = (2 ln 2)m , taking only the sum on S with |S| = , to finally obtain . From here we can use Stirling's approximation n! = Θ( √ n(n/e) n ) to obtain We use the fact that for large enough n/ we have (1 + /(n − )) (n− )/ > (2e ln 2)/η, where η > 2 ln 2, and that the factor n/ (n − ) ≥ 1/k can be absorbed by this approximation. Then there is a constant C η such that Eq. (6) holds.
A few different bounds can be obtained from the above theorem, some with a clearer meaning.  Proof. From Theorem 32 we know that for any η > 2 ln 2 there is a constant C η such that There are a couple of ways to bound the above quantity. We start by proving Eq. (7a). Define g : {−1, 1} k → R, g = S∈supp( f ) sgn( f (S))χ S . Let T q be the noise operator with parameter q = ηm n . Let r, s ∈ [0, 1] be such that r + s = 1. By Cauchy-Schwarz,   S⊆ [k] q |S| | f (S)|   2 = | T q r f, T q s g | 2 ≤ T q r f, T q r f T q s h, T q s g = Stab q 2r Another comparison is between Eq. (7b) and Corollary 28 in terms of the pure high degree of f . Again both bounds match up to a global multiplicative constant and up to the constant η. We conjecture that the constant η can be dropped from all these bounds with a more careful analysis.

Conclusions
In this paper we proposed a simple generalization of the concept of random access to recovering the value of a given Boolean function on any subset of fixed size of the initial bits. This generalization was made assuming different resources as encoding maps, i.e., encoding the initial string into bits or qubits, and different auxiliary resources, e.g. private and shared randomness, shared entanglement and Popescu-Rohrlich boxes. Given the lower bounds from our protocols, it seems reasonable to assume that the bias Stab q [f ] with q = m n is, if not optimal, at least close to optimal. The case with the weakest resources, the n → m f -RAC with PR, already achieves such bias up to an additive term O((log n/ )/(n/ )) in the parameter q. For more general values of m = O(log n), the use of quantum resources progressively improves q: from q ≈ 2m πn using encoding bits and SR to q ≥ 8m 3πn using encoding qubits and SR and finally to q = m n using encoding bits and shared entanglement. Such an improvement offered by quantum resources is relatively modest, specially when compared to stronger-than-quantum resources like Popescu-Rohrlich boxes, which allows the recovery of f (x S ) with certainty for any S.
On the other hand, the techniques from Fourier analysis lead to bias upper bounds that match our bias lower bounds up to a global multiplicative constant and a factor √ η ≈ √ 2 ln 2 in the parameter q. We conjecture that such upper bounds can be improved and the factor η dropped. Moreover, the upper bounds apply only to f -QRACs with SR, therefore not including f -EARACs. The understanding of EARACs is still limited, and even though we obtained an upper bound by making an independence assumption, a general upper bound for the case m > 1 is yet unknown.
No. 817581). JFD was supported by the Bristol Quantum Engineering Centre for Doctoral Training, EPSRC Grant No. EP/L015730/1.