SlowGuess/NeurIPS_2023_OCR huggingface dataset:
A fundamental issue in proving our QSQ result is, what techniques could one use to prove these lower bounds? Prior to our work, in [27] they introduced two new techniques based on differential privacy and communication complexity that give lower bounds on QSQ complexity. However, both these lower bounds are exponentially weak! In particular, the lower bounds that they could prove were linear in  $n$  for learning an  $n$ -bit concept class. Classically, there have been a sequence of works [42-44] with the goal of proving SQ lower bounds and finally the notion of statistical dimension was used to obtain close-to-optimal bounds for SQ learning certain concept classes and the breakthrough works of [44] used it to settle the complexity of learning the planted  $k$ -biclique distribution.
In this work, our technical contribution is a combinatorial parameter to lower bound QSQ complexity akin to the classical parameter. To this end, we follow a three-step approach.
1. Reduction to Decision Problems. We show that an algorithm  $\mathcal{A}$  that learns a concept class below error  $\varepsilon$  in trace distance using Qstat queries of tolerance  $\tau$  can also be used to solve the following decision problem: for a fixed  $\sigma$  such that  $\min_{\rho \in \mathcal{C}} d_{\mathrm{Tr}}(\rho, \sigma) > 2(\tau + \varepsilon)$ , decide if an unknown state is either some  $\rho \in \mathcal{C}$  or equals  $\sigma$ . Calling QQC the complexity of such decision problem, we show that
$$
\operatorname {Q S Q} (\mathcal {C}) \geq \max  _ {\sigma} \{\operatorname {Q Q C} (\mathcal {C}, \sigma) - 1: \min  _ {\rho \in \mathcal {C}} d _ {\mathrm {T r}} (\rho , \sigma) > 2 (\tau + \varepsilon) \}.
$$
2. Quantum Statistical Dimension. Next, we define the notion of quantum statistical dimension QSD: for  $\tau > 0$ , a class of states  $\mathcal{C}$  and a  $\sigma \notin \mathcal{C}$ , the  $\mathrm{QSD}_{\tau}(\mathcal{C},\sigma)$  is the smallest integer such that there exists a distribution  $\nu$  over Qstat queries  $M$  satisfying  $\operatorname*{Pr}_{M\sim \nu}[\left|\mathrm{Tr}(M(\rho -\sigma))\right| > \tau] \geq 1 / d$  for all  $\rho \in \mathcal{C}$ . From an operational perspective QSD is natural, as it can be viewed as the smallest expected number of observables that can distinguish all states in  $\mathcal{C}$  from  $\sigma$ . We then show that if the decision algorithm succeeds with probability at least  $1 - \delta$ , we have that:
$$
\operatorname {Q Q C} (\mathcal {C}, \sigma) \geq (1 - 2 \delta) \mathrm {Q S D} _ {\tau} (\mathcal {C}, \sigma).
$$
3. Lower Bounds on QSD. Even with this lower bound, proving bounds on  $\mathrm{QSD}(\mathcal{C},\sigma)$  is non-trivial. To this end, we further give two lower bounding techniques for  $\mathrm{QSD}(\mathcal{C},\sigma)$ , one based on the variance of Qstat queries across  $\mathcal{C}$  (inspired by the work of Kearns [28]) and one based on average correlation (inspired by the work of Feldman [43]). In particular, we define two combinatorial quantities  $\mathrm{Var}(\mathcal{C})$  and  $\mathrm{QAC}(\mathcal{C},\sigma)$  which can be associated with every class and use it to lower bound QSD.
Let  $\mu$  be a distribution over  $\mathcal{C}$ , such that  $\sigma_{1} := \mathbb{E}_{\rho \sim \mu}[\rho] \notin \mathcal{C}$ . We define  $\operatorname{Var}(\mathcal{C})$  as follows:
$$
\operatorname {V a r} (\mathcal {C}) = \tau^ {2} \cdot \min  _ {M, \| M \| \leq 1} \left(\operatorname {V a r} [ \operatorname {T r} [ \rho M ] ]\right) ^ {- 1}, \tag {3}
$$
$$
\mathop{\rm Var}_{\rho \sim \mu}[\operatorname {Tr}(\rho M)] = \underset {\rho \sim \mu}{\mathbb{E}}\left[\operatorname {Tr}(\rho M)^{2}\right] - \Bigl(\underset {\rho \sim \mu}{\mathbb{E}}\left[\operatorname {Tr}(\rho M)\right]\Bigr)^{2}.
$$
We show that  $\mathrm{QSD}_{\tau}(\mathcal{C},\sigma_1)\geq \mathrm{Var}(\mathcal{C})$  , which we eventually use to lower bound  $\mathrm{QSQ}(\mathcal{C})$  Next, for  $\sigma_{2}\notin \mathcal{C}$  , we define the average correlation  $\mathrm{QAC}(\mathcal{C},\sigma_2)$  as
$$
\operatorname {Q A C} \left(\mathcal {C}, \sigma_ {2}\right) = \sup  _ {\mathcal {C} _ {0} \subset \mathcal {C}} \left(\kappa_ {\tau} ^ {\gamma} - \operatorname {f r a c} \left(\mathcal {C} _ {0}, \sigma_ {2}\right)\right) ^ {- 1}, \tag {4}
$$
where  $\kappa_{\tau}^{\gamma}$ -frac  $(\mathcal{C}_0, \sigma_2)$  is a combinatorial parameter capturing correlations between states in  $\mathcal{C}_0$  and  $\sigma_2$ . We then show that  $\mathrm{QSD}_{\tau}(\mathcal{C}, \sigma_2) \geq \mathrm{QAC}(\mathcal{C}, \sigma_2)$  and use this in turn to lower bound QSQ( $\mathcal{C}$ ).<sup>5</sup>
Putting together the three bullets above, the QSQ complexity of learning can be lower bounded by the variance bound and the average correlation bound that we define in this work. We remark that although, our quantum combinatorial parameters are inspired by the classical works of Feldman et al. [42-44], proving that they lower bound QSQ complexity and also giving lower bounds for the corresponding concept class using these parameters is non-trivial and is a key technical contribution of our work. Below, we apply these lower bounds to obtain our learning results. For further details we defer the formal definitions and proofs to Section 3 in the Supplementary material.
3.2.2 QSQ versus noisy QPAC
We now sketch the proof of Result 2. As mentioned earlier, previous to our work we did not have a candidate class to separate QPAC from QSQ (let alone with noise). There have been a few works that have shown exponential lower bounds for learning using separable measurements [26, 37, 45], but all these lower bounds correspond to learning classes of mixed quantum states. Hence it was open if there is very simple structured function class such that quantum examples corresponding to this function class is hard for QSQ (in fact given our polynomial relation between entangled and separable learning, it is conceivable that for the small class of function states, QSQ are QPAC are polynomially related as well). In this work, we look at the degree-2 concept class  $\mathcal{C}$  defined in Eq. (2).
Recently, it was observed that [24] this class is learnable using  $O(n)$  quantum examples with entangled measurements and  $O(n^2)$  quantum examples with separable measurements. Our main contribution is in showing that the QSQ complexity of learning  $\mathcal{C}$  with tolerance  $\tau$  is  $\Omega(2^{n/2} \cdot \tau^2)$ . When  $\tau = 1/poly(n)$  this is an exponential,  $2^{\Omega(n)}$ , lower bound. We prove the hardness for algorithms using Qstat queries using the variance lower bounding technique. In particular, we show that for every  $n + 1$  qubit operator  $M$  such that  $\|M\| \leq 1$ , we have that
$$
\operatorname {V a r} _ {f} \left(\operatorname {T r} \left[ M \psi_ {f} \right]\right) = O \left(2 ^ {- n / 2}\right), \tag {5}
$$
where we let  $\psi_f = |\psi_f\rangle \langle \psi_f|$  for notational simplicity. Combined with our variance lower bound introduced in Section 3.2.1, we obtain our lower bound on the QSQ complexity of learning  $\mathcal{C}$ . It remains to establish Eq. (5). To this end, we need to understand
$$
\operatorname {V a r} _ {f} \left(\operatorname {T r} [ M \psi_ {f} ]\right) = \mathbb {E} _ {f} \left[ \operatorname {T r} [ M \psi_ {f} ] ^ {2} \right] - \left(\mathbb {E} _ {f} \left[ \operatorname {T r} [ M \psi_ {f} ] \right]\right) ^ {2} \tag {6}
$$
To do so, we decompose  $\psi_f$  as follows. For every  $f: \{0,1\}^n \to \{0,1\}$  let  $|\psi_f\rangle = \frac{1}{\sqrt{2^n}}\sum_x |x,f(x)\rangle$  and  $|\phi_f\rangle = \sum_x (-1)^{f(x)}|x\rangle$ . For convenience we let  $|u\rangle = \frac{1}{\sqrt{2^n}}\sum_x |x\rangle$ . Then we see that
$$
|\psi_{f}\rangle \langle \psi_{f}| = \frac{1}{2}\Bigl(\underbrace{|\phi_{f}\rangle\langle\phi_{f}|\otimes| - \rangle\langle - |}_{\rho_{1}^{f}} - \underbrace{|\phi_{f}\rangle\langle u|\otimes| - \rangle\langle + |}_{\rho_{2}^{f}} - \underbrace{|u\rangle\langle\phi_{f}|\otimes| + \rangle\langle - |}_{\rho_{3}^{f}} + \underbrace{|u\rangle\langle u|\otimes| + \rangle\langle + |}_{\rho_{4}^{f}}\Bigr).
$$
We now note that any  $n + 1$  qubit observable  $M$  can be decomposed as  $M = \sum_{a,b} M_{a,b} \otimes |a\rangle \langle b|$  where now  $a, b \in \{+, -\}$ . Since  $\|M\| \leq 1$  we also have that  $\|M_{a,b}\| \leq 1$ , however the off-diagonal blocks now no longer need be Hermitian. In an abuse of notation we now discard the last qubit of  $\rho_i^f$  and denote the resulting state also as  $\rho_i^f$ . For ease of notation we further introduce the notation  $M_1 = M_{-, -}$ ,  $M_2 = M_{-, +}$ ,  $M_3 = M_{+, -}$ , and  $M_4 = M_{+, +}$ . Thus, we see that  $\mathrm{Tr}[M\psi_f] = \frac{1}{2}\sum_i\mathrm{Tr}[M_i\rho_i^f]$  and further the variance can be written as
<sup>5</sup>We point out two subtleties, discussed in detail in Section 3 of the Supplementary Material. Firstly, our definition of QAC is defined only for  $\sigma$ s that are full rank. Secondly, as stated, the  $\mathsf{Var}(\mathcal{C})$  lower bound only yields a QSQ lower bound if  $\min_{\rho \in \mathcal{C}} d_{\mathrm{Tr}}(\rho, \sigma_1) > 2(\tau + \varepsilon)$ . This does not impact the lower bound proofs by this technique that we present here. In Section 3 of the supplementary material, we also briefly discuss an alternative approach to hiding decision problems in  $\mathcal{C}$  that relaxes this condition.
$$
\operatorname {V a r} _ {f} \left(\operatorname {T r} \left[ M \psi_ {f} \right]\right) = \frac {1}{4} \sum_ {i, j} \left[ \mathbb {E} _ {f} \operatorname {T r} \left(M _ {i} \rho_ {i} ^ {f}\right) \cdot \operatorname {T r} \left(M _ {j} \rho_ {j} ^ {f}\right) - \operatorname {T r} \left(M _ {i} \mathbb {E} _ {f} \left[ \rho_ {i} ^ {f} \right]\right) \cdot \operatorname {T r} \left(M _ {j} \mathbb {E} _ {f} \left[ \rho_ {j} ^ {f} \right]\right) \right]. \tag {7}
$$
At this point, we upper bound all these terms by  $\exp(-n/2)$ . Proving this upper bound is fairly combinatorial but crucially it involves understanding the properties of the ensemble  $\{|\psi_f\rangle\}_f$  and its moments for a uniformly random degree-2 functions  $f$ . Finally, we observe that the concept class can be learned given noisy quantum examples like in Eq. (1) using  $\mathrm{poly}(n, 1/(1 - 2\eta))$  examples (proving this uses the standard procedure to take derivatives of quantum states and the observation that this procedure is noise-resilient). This gives us the claimed separation between QSQ and noisy-QPAC, the "quantum analogue" of the seminal result of Blum et al. [1] for a natural class and with non-constant error rate close to  $1/2$ . We refer the interested reader to Section 4.1 in the Supplementary material.
3.3 Smallest class separation
Above we saw that the concept class of quadratic functions separated QPAC from QSQ. Observe that states in this concept class can be prepared by circuits of size  $O(n^{2})$  and depth  $O(n)$  consisting of  $\{\mathsf{Had},\mathsf{X},\mathsf{CX}\}$  gates. A natural question is, can states prepared by smaller circuits also witness such a separation between QPAC and QSQ? In Section 2 we saw that trivial states are learnable in QSQ, so it is necessary to have super-constant depth in order to show super-polynomial lower bounds? In the theorem below we answer this in the positive, by using a simple padding argument inspired by a prior work of Hinshe et al. [3]. In particular, we show that the class of states produced by  $\omega (\log n)$ -depth is already hard to learn for QSQ algorithms using a polynomial number of queries. For further details we defer the proof to Section 4.2 in the Supplementary material.
Theorem 3. For any  $\alpha \in (0,1)$  there exists a family of  $n$  qubit Clifford circuits of depth  $d = (\log n)^{1 / \alpha}$  and size  $d^2$  that requires  $2^{\Omega (d)}$  Qstat queries to learn the state to error  $\leq 1 / 2$  in trace distance.
4 Further applications
In this section we first use our lower bound techniques to give QSQ lower bounds for interesting classes of states and then use our QSQ lower bounds for further applications outside learning. For further details we defer the proof to Sections 4 and 5 in the Supplementary material.
4.1 QSQ lower bounds for learning states
Extending our lower bounds from above, we consider fundamental problems in quantum computing and prove QSQ lower bounds for these tasks. We summarize our results in the table below, for  $\tau = O\left(\frac{1}{\mathrm{poly}(n)}\right)$ , before expanding upon these in the following subsections.
Problem	QSQ Bound	General Complexity
Shadow tomography with Pauli observables	Ω(4n)	O(2n) with separable measurements [26]
Learning coset states from Abelian hidden subgroup problem	2Ω(n)	O(n) with separable measurements
Purity testing	22Ω(n)	O(1) with entangled measurements Θ(2n) with separable measurements [26]
Learning states from O(2n) approximate 2-designs	2Ω(n)	O(n) for stabilizer states with entangled measurements
Hidden subgroup problem. Coset states appear often in the hidden subgroup problem (HSP) [46, 47], a fundamental problem in quantum computing. It is well-known that coset states of the Abelian HSP can be learned exactly from separable measurements in polynomial sample complexity and for non-Abelian groups, it was well-known that separable measurements [48, 49] require exponential many copies to learn a coset state. A natural question is, what is the QSQ complexity of learning coset states? Given that the standard approach for HSP is based on Fourier sampling and [27] showed that a version of Fourier sampling is easy in QSQ, it is natural to expect that HSP is implementable in QSQ. Surprisingly, in this work, we show that, even for Abelian groups, the QSQ sample complexity of learning the unknown coset state is exponentially large. In particular, we show a lower bound of
$\Omega (\tau^4\cdot 2^n)$  on the QSQ complexity of learning using Qstat  $(\tau)$  queries and the proof of this is done via the average correlation method we introduced in Section 3.2.1.
**Shadow tomography.** In recent times, there have been a lot of works surrounding the framework of shadow tomography [21, 37]. The goal here is, given copies of an unknown quantum state  $\rho$ , the learner has to predict the expectation value  $\mathrm{Tr}[O_i\rho]$  of a collection of known observables  $\{O_i\}_{i\in [k]}$  up to error  $\varepsilon$ . It is well-known to be solvable using  $\mathrm{poly}(n,\log k)$  copies of  $\rho$ . In [26] the authors show  $\Theta (2^n)$  copies of  $\rho$  are necessary and sufficient for shadow tomography using separable measurements. To prove the lower bounds the authors construct a many-vs-one decision task where  $\sigma = \mathbb{I} / 2^n$  and  $\mathcal{C} = \{\rho_i = \frac{\mathbb{I} + 3\varepsilon O_i}{2^n}\}$ . Assuming that  $\mathrm{Tr}[O_i] = 0$  and  $\mathrm{Tr}[O_i^2 ] = 2^n$  for all  $O_{i}$ , then an algorithm which solves the shadow tomography problem with high probability also solves the decision problem. Thus, a lower bound on the latter is also a lower bound on the sample complexity of shadow tomography. Here we give a quadratically stronger lower bound of  $\Omega (4^n)$  when given access to only Qstat measurements, which we prove using the average correlation method. Our result shows that even separable measurements and not just statistics play a non-trivial role in shadow tomography.
Does tolerance matter? A natural question when discussing QSQ learning is, is there a natural distribution learning task that can be solved with tolerance  $\tau_{Q} \geq \tau_{C}$  such that classical  $\operatorname{Stat}(\tau_{C})$  queries cannot solve the task but  $\mathrm{Qstat}(\tau_{Q})$  can solve the task? Here we consider the class of bi-clique states introduced in the seminal work of Feldman et al. [44]. In their work they showed that for detecting a planted bipartite  $k$ -clique distributions when the planted  $k$ -clique has size  $n^{1/2 - \varepsilon}$  (for constant  $\varepsilon > 0$ ), it is necessary and sufficient to make superpolynomial in  $n$  many  $\mathrm{Stat}(k/n)$  queries. Here we show that one can achieve the same query complexity quantumly but with  $\mathrm{Qstat}(\sqrt{k/n})$ , i.e., with quadratically larger tolerance we can detect a  $k$ -biclique. A classical SQ algorithm cannot solve this task with  $\tau_{C} = \sqrt{k/n}$  queries.
A doubly exponential lower bound? So far all our lower bounds for learning  $n$ -qubit quantum states are exponential in  $n$ . A natural question is, can one prove a doubly exponential lower bound for some task? In this work, we show that the natural problem of testing purity, i.e., given a quantum state  $\rho$  return an estimate of  $\mathrm{Tr}[\rho^2]$ , requires  $\exp(2^n\tau^2)$  many QSQ queries to solve. Previous work of [26] showed that it is necessary and sufficient to use  $\Theta(2^n)$  many copies of  $\rho$  to test purity if we were allowed separable measurements, but our work considers the weaker QSQ model and proves a doubly-exponential lower bound. The proof of this uses Levy's lemma and the ensemble of Haar random states to lower bound the quantum statistical dimension in a manner similar to that of the variance based technique.
4.2 Applications outside learning
Using our QSQ results we present two applications. First we give an exponential separation between weak and strong error mitigation, resolving an open question of Quek et al. [2]. Second, we show super-polynomial lower bounds for learning output distributions (in the computational basis) of  $n$ -qubit Clifford circuits of depth  $\omega (\log n)$  and Haar random circuit of depth-  $O(n)$ , extending the works of [3, 4]. All these results [2-4] proved these lower bounds for QSQ algorithms assuming diagonal observables. For further details we defer the proof to Section 6 in the Supplementary material.
Error mitigation. Error mitigation (EM) was introduced as an algorithmic technique to reduce the noise-induced in near-term quantum devices, hopefully with a small overhead, in comparison to building a full-scale fault-tolerant quantum computer to harness general quantum advantages [50]. In recent times, EM has obtained a lot of attention with several works understanding how to obtain near-term quantum speedups as a surrogate to performing error correction. EM has been an important component in recent QML demonstrations [37, 51, 10].
More formally, an EM algorithm  $\mathcal{A}$  takes as input a quantum circuit  $C$ , noise channel  $\mathcal{N}$  and copies of  $|\psi^{\prime}\rangle = \mathcal{N}(C)|0^{n}\rangle$ . In a strong EM protocol,  $\mathcal{A}$  needs to produce samples from a distribution  $D$  that satisfies  $d_{\mathrm{TV}}(D,\{\langle x|C|0^n\rangle^2\}_x)\leq \varepsilon$  and in the weak EM setting, given observables  $M_1,\ldots ,M_k$  the goal is to approximate  $\langle \psi |M_i|\psi \rangle$  upto  $\varepsilon$ -error. In [2], they asked the question: how large should  $k$  be in order to simulate weak EM by strong EM? They show that when  $M_{i}s$  are diagonal, then  $k = \Omega (2^{n})$ , i.e., they gave an exponential separation between weak and strong EM. In this work, our main contribution is to use Result 2 to remove the assumption that  $M_{i}s$  are diagonal and show an exponential separation unconditionally between weak and strong EM.
Learning distributions. Recently, the works of Hinsche et al. [3] and Nietner et al. [4] initiated the study of learning output distributions of quantum circuits. In particular, they considered the following general question: Let  $|\psi_U\rangle = U|0^n\rangle$  where  $U\in \mathcal{U}$  is a family of interesting unitaries and let  $P_U(x) = \langle x|U|0^n\rangle^2$  be a distribution. How many QSQ queries does one need to learn the  $P_U$  to  $d_{\mathrm{TV}}$  at most  $\varepsilon$ ? To this end, the works of [3, 4] looked at diagonal  $M$ s, i.e.,  $M = \sum_{X}\phi (x)|x\rangle \langle x|$  for  $\phi :\{0,1\} ^n\to [-1,1]$  and showed the hardness of approximately learning  $P_U$  for  $\mathcal{U}$  being  $\omega (\log n)$ -depth Clifford circuits and depth-  $d\in \{\omega (\log n),O(n)\}$  and  $d\rightarrow \infty$ -depth Haar random circuits.
In this work, we improve upon their lower bounds by removing the assumption that  $M$  is diagonal and prove a general QSQ lower bounds for these circuit families that is considered in their work. In order to prove this bound, we follow the following three step approach (i) We first observe the following simple fact: for distributions  $p,q:\mathcal{Z}\to [0,1]$ , define  $|\psi_p\rangle = \sum_{z\in \mathcal{Z}}\sqrt{p(z)} |z\rangle$  and  $|\psi_q\rangle$  similarly. Then  $\| |\psi_p\rangle -|\psi_q\rangle \|_{\mathrm{Tr}}^2\leq 2\| p - q\|_{tvd}$ . So it suffices to prove the hardness of learning the output state, in order to prove the hardness of learning the distribution. (ii) Next we consider the class  $\mathcal{C}$  of states forming a  $\gamma$ -approximate 2-design where  $\gamma = O(2^{-n})$  and show that learning states from  $\mathcal{C}$  with error  $\leq 2 / 3$  in trace distance requires  $\Omega (\tau^2\cdot 2^n)$  Qstat  $(\tau)$  queries. (iii) Finally, using that depth- $O(n)$  circuits form a design, we invoke our QSQ lower bound in order to prove the hardness of learning the output states of these circuits. These three steps prove the hardness of learning output distributions (in the computational basis) of quantum circuits.
Open questions. There are a few natural questions that our work opens up: (i) Can we show that for every concept class  $\mathcal{C}$ , we have that  $\mathsf{SepExact} \leq O(n \cdot \mathsf{EntExact})$ ?, (ii) Following [3, 4] what is QSQ complexity of learning the output distribution of constant-depth circuits assuming we only use diagonal operators? (iii) Theoretically our work separates weak and strong error mitigation, but in practice there are often assumptions in the mitigation protocols, can we show theoretical separations even after making these assumptions? (iv) Classically it is well-known that several algorithms can be cast into the QSQ framework, is the same true quantumly? If so, that would suggest that QSQ as a unifying framework for designing new learning algorithms. (v) What is the QSQ complexity of the Hidden subgroup problem when given access to function states, instead of coset states (which is the case only in the standard approach).
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They prove that learning even a  $(1 - \exp (-n))$ -fraction of the circuits in these families is hard in SQ model.