Tangible reduction in learning sample complexity with large classical samples and small quantum system

Abstract

Quantum computation requires large classical datasets to be embedded into quantum states in order to exploit quantum parallelism. However, this embedding requires considerable resources in general. It would therefore be desirable to avoid it, if possible, for noisy intermediate-scale quantum (NISQ) implementation. Accordingly, we consider a classical-quantum hybrid architecture, which allows large classical input data, with a relatively small-scale quantum system. This hybrid architecture is used to implement a sampling oracle. It is shown that in the presence of noise in the hybrid oracle, the effects of internal noise can cancel each other out and thereby improve the query success rate. It is also shown that such an immunity of the hybrid oracle to noise directly and tangibly reduces the sample complexity in the framework of computational learning theory. This NISQ-compatible learning advantage is attributed to the oracle’s ability to handle large input features.

Appendices

Appendix A: Detailed calculations of $P_Q(\omega )$

Here, we present the procedure to calculate $P_Q(\omega )$ in Eq. (6) of the main manuscript. We start by analyzing the simple case, i.e., of $\omega =1$. In particular, we consider an input $\mathbf {x} = x_1 x_2 \cdots x_n$ satisfying $x_{l_1} = 1$ for arbitrary $l_1 \in [1,n]$ and $x_{j}=0$ for all $j \ne l_1$. Subsequently, only two gates ${\hat{a}}_0$ and ${\hat{a}}_{l_1}$ are activated with $\varOmega _\mathbf {x} = \{ 0, l_1 \}$. In a purely classical query, $P_C(\omega =1)$ is given as

$$\begin{aligned} P_C(\omega =1)=\left( 1- \eta _{l_1}\right) \left( 1- \eta _0\right) + \eta _{l_1}\eta _{0}, \end{aligned}$$

(13)

where $\eta _k$ is the probability that a bit-flip error will occur at ${\hat{a}}_k$ ($k \in \{0, l_1\}$). Meanwhile, $P_Q(\omega =1)$ is calculated as below:

$$\begin{aligned} P_Q(\omega =1)= & {} \left| \left<h^\star (\mathbf {x})\right| {\hat{\epsilon }_{l_1} {\hat{a}}_{l_1} \hat{\epsilon }_{0} {\hat{a}}_{0}} \left| \alpha \right>\right| ^2, \nonumber \\= & {} \left| \left<h^\star (\mathbf {x})\right| \left( \sqrt{1-\eta _{l_1}}\hat{1\!\!1} \pm i \sqrt{\eta _{l_1}}\hat{\sigma }_x \right) {\hat{a}}_{l_1} \left( \sqrt{1-\eta _0}\hat{1\!\!1} \pm i \sqrt{\eta _0}\hat{\sigma }_x \right) {\hat{a}}_{0} \left| \alpha \right> \right| ^2, \nonumber \\= & {} \left| \sqrt{1-\eta _{l_1}}\sqrt{1-\eta _0} \left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}{\hat{a}}_0\left| \alpha \right> \pm i\sqrt{1-\eta _{l_1}}\sqrt{\eta _0} \left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}\hat{\sigma }_x{\hat{a}}_0\left| \alpha \right> \right. \nonumber \\&\left. \pm i\sqrt{\eta _{l_1}}\sqrt{1-\eta _0} \left<h^\star (\mathbf {x})\right| \hat{\sigma }_x{\hat{a}}_{l_1}{\hat{a}}_0\left| \alpha \right> - \sqrt{\eta _{l_1}}\sqrt{\eta _0} \left<h^\star (\mathbf {x})\right| \hat{\sigma }_x{\hat{a}}_{l_1}\hat{\sigma }_x{\hat{a}}_0\left| \alpha \right> \right| ^2, \nonumber \\ \end{aligned}$$

(14)

where $\hat{\epsilon }_k=\sqrt{1-\eta _k}\hat{1\!\!1} \pm i \sqrt{\eta _k}\hat{\sigma }_x$ is the error operation, defined in the main manuscript. Using the properties in Eq. (7) of the main manuscript, i.e., $\hat{\sigma }_x {\hat{a}}_k = -{\hat{a}}_k \hat{\sigma }_x$ and $\left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}{\hat{a}}_{0}\left| \alpha \right>=1$, we can evaluate the following:

$$\begin{aligned} \left\{ \begin{array}{l} \left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}\hat{\sigma }_x{\hat{a}}_0\left| \alpha \right> = -\left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}{\hat{a}}_0\hat{\sigma }_x\left| \alpha \right>=0, \\ \left<h^\star (\mathbf {x})\right| \hat{\sigma }_x{\hat{a}}_{l_1}{\hat{a}}_0\left| \alpha \right> = \left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}{\hat{a}}_0\hat{\sigma }_x\left| \alpha \right>=0, \\ \left<h^\star (\mathbf {x})\right| \hat{\sigma }_x{\hat{a}}_{l_1}\hat{\sigma }_x{\hat{a}}_0\left| \alpha \right> = -\left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_1}{\hat{a}}_0\left| \alpha \right>=-1. \end{array} \right. \end{aligned}$$

(15)

Subsequently, using Eq. (15), we can obtain

$$\begin{aligned} P_Q(\omega =1) - P_C(\omega =1) = \varGamma _{0, l_1}, \end{aligned}$$

(16)

where

$$\begin{aligned} \varGamma _{0, l_1}=2\sqrt{1-\eta _{l_1}}\sqrt{1-\eta _0}\sqrt{\eta _{l_1}}\sqrt{\eta _0} \ge 0. \end{aligned}$$

(17)

Fig. 3

3D graphs of $P_{C}$ (left) and $P_{Q}$ (right) with respect to $\eta _0$ and $\eta _{l_1}$ for $\omega =1$. The advantage defined in Eq. (16) is observed; $P_Q$ is always larger than $P_C$. Here, the most remarkable feature is that our hybrid oracle always yields correct results when $P_Q=1$ provided that $\eta _0 = \eta _{l_1}$

This factor $\varGamma _{0, l_1}$ is from quantum superposition and clearly indicates the enhancement of the success probability with the condition $\varGamma _{0, l_1} \ge 0$. In Fig. 3, we depict the graphs of $P_{C,Q}$ with respect to $\eta _0$ and $\eta _{l_1}$. It is noteworthy that our hybrid oracle always yields correct results, i.e., $P_Q=1$, provided that $\eta _{l_1} = \eta _0$, even though $\eta _{l_1}$ and $\eta _{0}$ are large. This is the most remarkable feature in our classical–quantum hybrid query.

Subsequently, we consider the case of $\omega =2$, where a set of four gates, ${\hat{a}}_0$, ${\hat{a}}_{l_1}$, ${\hat{a}}_{l_2}$, and ${\hat{a}}_{l_3}$, are to be activated with $\varOmega _\mathbf {x}=\{ 0, l_1, l_2, l_3 \}$. We subsequently calculate $P_Q(\omega =2)$ as follows:

$$\begin{aligned} P_Q(\omega =2) = \left| \left<h^\star (\mathbf {x})\right| \hat{\epsilon }_{l_3} {\hat{a}}_{l_3} \hat{\epsilon }_{l_2} {\hat{a}}_{l_2} \hat{\epsilon }_{l_1} {\hat{a}}_{l_1} \hat{\epsilon }_{0} {\hat{a}}_{0} \left| \alpha \right>\right| ^2. \end{aligned}$$

(18)

To proceed with the calculation, we introduce an identity , where the state ($\beta \in \{0,1\}$) is defined with the following properties:

$$\begin{aligned} \left| \left<h^\star (\mathbf {x})\right| {\hat{a}}_{l_3}{\hat{a}}_{l_2} \left| \beta \right>\right| ^2 = 1~\text {and}~\left| \left<\beta \right| {\hat{a}}_{l_1} {\hat{a}}_{0} \left| \alpha \right>\right| ^2 =1. \end{aligned}$$

(19)

Using a mathematical method of substituting the identity $\hat{1\!\!1}_{\beta ,\beta ^\perp }$ between $\hat{\epsilon }_{l_3} {\hat{a}}_{l_3} \hat{\epsilon }_{l_2} {\hat{a}}_{l_2}$ and $\hat{\epsilon }_{l_1} {\hat{a}}_{l_1} \hat{\epsilon }_{0} {\hat{a}}_{0}$ in Eq. (18), we can obtain

$$\begin{aligned} P_Q(\omega =2)= & {} \left| \left( \sqrt{1-\eta _{l_3}}\sqrt{1-\eta _{l_2}} + \sqrt{\eta _{l_3}}\sqrt{\eta _{l_2}} \right) \left( \sqrt{1-\eta _{l_1}}\sqrt{1-\eta _{0}} + \sqrt{\eta _{l_1}}\sqrt{\eta _{0}} \right) \right. \nonumber \\&- \left. \left( \sqrt{1-\eta _{l_3}}\sqrt{\eta _{l_2}} - \sqrt{\eta _{l_3}}\sqrt{1-\eta _{l_2}} \right) \left( \sqrt{1-\eta _{l_1}}\sqrt{\eta _{0}} - \sqrt{\eta _{l_1}}\sqrt{1-\eta _{0}} \right) \right| ^2 \nonumber \\ \end{aligned}$$

(20)

Furthermore, after some algebraic simplifications, we can arrive at

$$\begin{aligned} P_Q(\omega =2) - P_C(\omega =2)= & {} \varGamma _{l_1, 0} \left( 1- 2\eta _{l_2}\right) \left( 1- 2\eta _{l_3}\right) + \varGamma _{l_2, l_1} \left( 1- 2\eta _{l_3}\right) \left( 1- 2\eta _{0}\right) \nonumber \\&+ \varGamma _{l_3, l_2} \left( 1- 2\eta _{l_1}\right) \left( 1- 2\eta _{0}\right) + \varGamma _{0, l_3} \left( 1- 2\eta _{l_2}\right) \left( 1- 2\eta _{l_1}\right) \nonumber \\&- \varGamma _{l_2, 0} \left( 1- 2\eta _{l_3}\right) \left( 1- 2\eta _{l_1}\right) - \varGamma _{l_3, l_1} \left( 1- 2\eta _{l_2}\right) \left( 1- 2\eta _{0}\right) \nonumber \\&+ 2 \varGamma _{l_3, l_2}\varGamma _{l_1, 0}, \end{aligned}$$

(21)

where

$$\begin{aligned} P_C(\omega = 2)= & {} \left( 1 - \eta _{l_3} \right) \left( 1 - \eta _{l_2} \right) \left( 1 - \eta _{l_1} \right) \left( 1 - \eta _{0} \right) \nonumber \\&+ \left( 1 - \eta _{l_3} \right) \left( 1 - \eta _{l_2} \right) \eta _{l_1} \eta _{0} + \left( 1 - \eta _{l_3} \right) \eta _{l_2} \left( 1 - \eta _{l_1} \right) \eta _{0} \nonumber \\&+ \eta _{l_3} \left( 1 - \eta _{l_2} \right) \left( 1 - \eta _{l_1} \right) \eta _{0} + \left( 1 - \eta _{l_3} \right) \eta _{l_2} \eta _{l_1} \left( 1 - \eta _{0} \right) \nonumber \\&+ \eta _{l_3} \left( 1 - \eta _{l_2} \right) \eta _{l_1} \left( 1 - \eta _{0} \right) + \eta _{l_3} \eta _{l_2} \left( 1 - \eta _{l_1} \right) \left( 1 - \eta _{0} \right) \nonumber \\&+ \eta _{l_3} \eta _{l_2} \eta _{l_1} \eta _{0}. \end{aligned}$$

(22)

Here, $\varGamma _{a, b}$ is defined as $\varGamma _{a,b}=2\sqrt{1-\eta _{a}}\sqrt{1-\eta _{b}}\sqrt{\eta _{a}}\sqrt{\eta _{b}}$ for $a \ne b \in \varOmega _\mathbf {x} =\{ 0, l_1, l_2, l_3 \}$, similarly to Eq. (17). Subsequently, using Eq. (20) and Eq. (21), we demonstrate that the quantum advantage can be achieved with the positive factors $\varGamma _{a, b}$. Note that Eq. (21) could be negative thus exhibiting the disadvantage, e.g., when $\eta _{l_3}=\eta _{l_1}=0$ or $\eta _{l_2}=\eta _{l_1}=0$ for all input $\mathbf {x}$. However, the aforementioned situation is not likely to occur in real physical systems. Consistent with the case of $\omega =1$, we observed that $P_Q(\omega =2)$ becomes unity when $\eta _{l_3}=\eta _{l_2}=\eta _{l_1}=\eta _{0}$.

By observing the two cases above, we can infer that the same method, i.e., of introducing the identities, can be used to calculate $P_Q(\omega )$ for arbitrary higher Hamming-weight inputs. The most remarkable construction, i.e., having unity query-success probability with equal error probabilities, can be generalized as well. Therefore, it can be sufficiently concluded that the enhancement in the query-success probability can be achieved for an arbitrary Hamming-weight in our hybrid query.

Appendix B: Numerical analyses with realistic conditions

As mentioned in the main manuscript, in a more realistic situation, the amplitudes related to the errors are not completely canceled out owing to a nonzero $\varDelta _\eta$, and $P_Q(\omega )$ exhibits an analogous form to $P_C(\omega )$ in Eq. (4) of the main manuscript, with an “effective” characteristic constant $c_\text {eff} \simeq (2\overline{\eta }_\text {eff})^{-1}$. Here, the effective average error $\overline{\eta }_\text {eff}$ is expected to be much smaller than c. This feature results in the quantum advantage that does not depend on the degree of $\overline{\eta }$ but only on $\varDelta _\eta$, i.e., how “varying” they are.

Fig. 4

We plot the graphs of $P_{C,Q}$ versus $\omega$. The simulation is performed for randomly chosen inputs $\mathbf {x}$ and $h^\star$. In each simulation, $P_{C,Q}$ is evaluated by counting success and failure events over $10^5$ queries. One single data point of $P_{C,Q}$ is obtained by averaging $\simeq 10^3$ trials of the simulation. a First, we present the simulation data of $P_{C,Q}$ evaluated for $\overline{\eta } = 10^{-4}$, $10^{-3}$, and $10^{-2}$ with $\varDelta _\eta =0$. The results show that $P_C$ rapidly approaches $\frac{1}{2}$ with increasing $\omega$, indicating good agreement with Eq. (6) of our main manuscript (see red, blue, and green solid lines). Meanwhile, $P_Q$ remains unity for all the cases of $\overline{\eta }$, as predicted. b Next, we consider the realistic situation, assuming a normal distribution ${{\mathcal {N}}}(\overline{\eta }, \varDelta _\eta )$. Here, we set $\overline{\eta }=10^{-3}$ with $\varDelta _\eta = 1\%$, $5\%$, and $10\%~\text {of}~\overline{\eta }$. The data of $P_{C,Q}$ are shown to decay, but $P_Q$ is much slower. In such cases, the data $P_{Q}$ are well fitted by Eq. (6) of the main manuscript with $c_\text {eff}$, indicating that the data agrees well with our theoretical predictions (Color figure online)

Table 2 Detailed values of $\overline{\eta }_\text {eff}$ and $c_\text {eff}$ for each $\varDelta _\eta$

To corroborate and extend our theoretical predictions, we perform a numerical analysis. It starts with an input $\mathbf {x}$ of $\omega (\mathbf {x})$. We subsequently evaluate $P_{C,Q}(\mathbf {x})$ by counting the number of “$h^\star (\mathbf {x})$” (e.g.,, “success”) and “$h^\star (\mathbf {x}) \oplus 1$” (e.g.,, “failure”), such that $P_{C,Q}(\mathbf {x}) = {N_S}/\left( {N_S + N_F}\right)$, where $N_S$ and $N_F$ denote the numbers of success and failure, respectively, and $N_S + N_F = 10^5$. Here, we use the Monte-Carlo approach to mimic quantum measurement statistics. This simulation is repeated for different values of $\eta _k$ (for $k \in \varOmega _{\mathbf {x}}$) satisfying $c=(2\overline{\eta })^{-1}$. This condition enables us to analyze the data statistically (i.e., by averaging over the trials) without losing generality, even though in each simulation $\eta _k$ is changed with different $h^\star$. First, as an extreme but illustrative example, we consider the case of $\varDelta _\eta =0$, i.e., by assuming $\eta _k = \overline{\eta }$ for all possible $k = 1, 2, \ldots , 2^n$. As results, we present the graphs of $P_{C,Q}$ versus $\omega$ as dots in Fig. 4a for $\overline{\eta }=10^{-4}$, $10^{-3}$, and $10^{-2}$, where each data point of $P_{C,Q}$ is obtained by averaging over $\simeq 10^3$ trials. Here, it is observed that $P_C$ decays fast to $\frac{1}{2}$, indicating good agreement with Eq. (6) of the main manuscript. The data of $P_Q$ are, meanwhile, shown to be unity without depending on the degree of $\overline{\eta }$, as predicted. Next, we consider a realistic situation, assuming that $\eta _k$ is drawn from a normal distribution ${{\mathcal {N}}}(\overline{\eta }, \varDelta _\eta )$ for all $k = 1,2,\ldots ,2^n$ (and hence for $k \in \varOmega _{\mathbf {x}}$). Here, we set $\overline{\eta }=10^{-3}$ with $\varDelta _\eta = 1\%$, $5\%$, and $10\%~\text {of}~\overline{\eta }$. The simulation results are shown in Fig. 4b. For all cases of $\varDelta _\eta$, both $P_{C}$ and $P_{Q}$ decay to $\frac{1}{2}$; however, $P_Q$ is much slower. It is also observed that the data of $P_Q$ matched well with Eq. (6) of the main manuscript, thus allowing us to identify the effective characteristic constant $c_\text {eff}$. The identified values of $c_\text {eff}$ and $\overline{\eta }_\text {eff}$ are listed in Table 2; they manifest the predicted condition in Eq. (8) of the main manuscript.

Fig. 5

Graphs of $P_Q$ with respect to $\omega$ for $\overline{\chi }=10^{-4}$, $10^{-3}$, and $10^{-2}$. Each data point is obtained by averaging over $10^3$ simulations. The data fitted well to Eq. (6) in the main manuscript, together with the parameter $c_\text {eff}$. The result shows that the quantum advantage becomes less pronounced as $\overline{\chi }$ is increased; however, it is still highly durable. It is noteworthy that the data obtained for $\varDelta _\chi = 1\%~\text {of}~\overline{\chi }$ (filled square, circle, and triangle points) and $\varDelta _\chi = 10\%~\text {of}~\overline{\chi }$ (empty square, circle, and triangle points) are almost identical (up to the order of $10^{-2}$); namely, $P_Q$ is not affected significantly by $\varDelta _\chi$. The identified $\overline{\eta }_\text {eff}$ and $c_\text {eff}$ are listed in Table 2 (Color figure online)

Table 3 Detailed values of $\overline{\eta }_\text {eff}$ and $c_\text {eff}$ for each $\overline{\chi }$

For a more realistic condition, we consider another type of error, i.e., phase-flip in the assistant qubit that would be crucial for maintaining a higher success rate of the query. In particular, we assume that the phase-flip errors primarily occur when the qubit travels between ${\hat{a}}_k$ and ${\hat{a}}_{k+1}$ with a certain probability $\chi _k \le \frac{1}{2}$. First, when $\varDelta _\eta =0$ (or equivalently, $\eta _k = \overline{\eta }$) for all k, the phase-flip errors do not affect the query process and $P_{Q}$ becomes unity. In the realistic case, namely of $\varDelta _\eta \ne 0$, however, it is predicted that the amplitudes of the bit-flip errors would interfere disorderly owing to the phase-flip, and eventually the quantum advantage becomes smaller, as described in our main manuscript. Thus, we perform the simulations and present the data of $P_Q$ in Fig. 5. Here, $\chi _k$ is assumed to be drawn from ${{\mathcal {N}}}(\overline{\chi }, \varDelta _\chi )$ for all $k=1,2,\ldots ,2^n$. The simulation data are generated for $\overline{\chi }=10^{-4}$, $10^{-3}$ and $10^{-2}$. Here, we set $\overline{\eta }=10^{-3}$ with $\varDelta _\eta = 5\%~\text {of}~\overline{\eta }$. The data are well fitted by Eq. (6) of our main manuscript, and $c_\text {eff}$ are well estimated from the data (see Table 3). As expected, the quantum advantage becomes less pronounced as $\overline{\chi }$ is increased; however, it can still exhibit a higher success rate of the query. It is noteworthy that the data obtained for both $\varDelta _\chi = 1\%~\text {and}~10\%~\text {of}~\overline{\chi }$ are almost identical (up to the second digit of a decimal).

Appendix C: Reduction in learning sample complexity in the framework of probably-approximately-correct (PAC) learning

In a probably-approximately-correct (PAC) learning model, a learner (or equivalently, a learning algorithm) samples a finite set of training data $\{ (\mathbf {x}_i, h^\star (\mathbf {x}_i)) \}$ ($i = 1,2,\ldots ,M$) by accessing an oracle, aiming at obtaining the best hypothesis h close to $h^\star$ for a given set, e.g., H, of the hypothesis h. Here, $\mathbf {x}_i$ is typically assumed to be drawn uniformly. Subsequently, a learning algorithm is a ($\epsilon$, $\delta$)-PAC learner (under uniform distribution), if the algorithm obtains an $\epsilon$-approximated correct h with probability $1-\delta$; more specifically, satisfying

$$\begin{aligned} P[E(h, h^\star ) \le \epsilon ] \ge 1-\delta , \end{aligned}$$

(23)

where $E(h, h^\star )$ denotes the error. Here, if h identified by the algorithm agrees with

$$\begin{aligned} M \ge \frac{1}{\epsilon }\ln {\frac{{\left| H\right| }}{\delta }} \end{aligned}$$

(24)

of samples constructed from the oracle, then Eq. (23) holds. Here, $\left| H\right| \le 2^{2^n}$ denotes the cardinality of H. Equation (24) is known as the bound of the sample complexity [23, 24], i.e., it yields the minimum number of training samples to successfully learn $h \in { H}$ satisfying Eq. (23). Such a sample complexity bound derived from the previous studies can directly be carried over to our scenario; in our classical–quantum hybrid query scheme, the same sample complexity bound exists, because $\mathbf {x}_i$ and $h^\star (\mathbf {x}_i)$ identified by the measurement on are classical.

However, in the case where the oracle is not perfect, the bound of sample complexity in Eq. (24) is modified as follows: First, we draw a sequence of the training data $\{ (\mathbf {x}_1, m_1), (\mathbf {x}_2, m_2), \ldots , (\mathbf {x}_M, m_M) \}$ sampled from our classical–quantum hybrid oracle, where $m_i \in \{ h^\star (\mathbf {x}_i), h^\star (\mathbf {x}_i) \oplus 1 \}$ denotes the outcome of the measurement performed on . Subsequently, if the sampling is performed with

$$\begin{aligned} M \ge 2 A_Q\ln {\left( \frac{2\left| H\right| }{\delta }\right) ^{\frac{1}{\epsilon ^2}}}, \end{aligned}$$

(25)

we can verify that Eq. (23) holds for the algorithm that obtains h maximizing ${\overline{P}}_Q$. In fact, it has been proven that the additional factor $A_Q$ is given as [30]

$$\begin{aligned} A_Q = \frac{1}{\left( 2 {\overline{P}}_Q(n) - 1\right) ^{2}}. \end{aligned}$$

(26)

It is noteworthy that in the purely classical case, the corresponding factor, e.g., $A_C$, is given with ${\overline{P}}_C$ instead of ${\overline{P}}_Q$. Thus, we can derive the reduction in the sample complexity with the condition $A_Q \le A_C$ from ${\overline{P}}_Q \ge {\overline{P}}_C$. To view this explicitly, we rewrite $A_Q$ in Eq. (26) to a more useful form:

$$\begin{aligned} A_Q = \left[ \frac{1}{2^n}\sum _{\omega =0}^n {{n}\atopwithdelims (){\omega }} e^{-\frac{2^\omega }{\gamma c}} \right] ^{-2} = \left[ \sum _{j=0}^\infty \frac{(-1)^j}{j !}\left( \frac{1}{\gamma c}\right) ^j \left( \frac{2^j+1}{2} \right) ^n \right] ^{-2} \end{aligned}$$

(27)

This implies that a small increment in the sample complexity bound when n is small increases abruptly from near $n \simeq 2\log _2{\gamma c}$. As $A_C$ is characterized by c without $\gamma$, we can interpret $\gamma$ as a quantum learning advantage in the PAC learning framework; i.e., for any large n, we can define a ($\epsilon$, $\delta$)-PAC learner with our hybrid oracle, unlike with a fully classical one. It is noteworthy that if n is excessively large, i.e., when $n \gg 2\log _2{\gamma c}$, it is impractical to define a legitimate PAC learner even with our hybrid oracle. This result is consistent with the recent theoretical study in Ref. [22]; however, in our case, such a quantum learning advantage is achieved with classical data.

To corroborate and extend our analysis, numerical simulations are performed: For a given n, we prepare a set of inputs $\{\mathbf {x}_1, \mathbf {x}_2, \ldots , \mathbf {x}_M \}$ that is sampled randomly. For the given $\omega (\mathbf {x}_i)$ ($i=1,2,\ldots ,M$), we evaluate $P_{C,Q}(\omega )$ by counting $10^5$ queries and identify their average value, i.e., $\frac{1}{10^5} \sum _{i=1}^{10^5} P_{C,Q}(\omega (\mathbf {x}_i))$. This process is repeated $\simeq 10^3$ times for different input sets to analyze ${\overline{P}}_{C,Q}(n)$ statistically. The data are generated from $n=8$ to $n=35$, assuming that $\eta _k$ and $\chi _k$ ($\forall k$) are drawn from ${{\mathcal {N}}}(\overline{\eta }, \varDelta _\eta )$ and ${{\mathcal {N}}}(\overline{\chi }, \varDelta _\chi )$, respectively. Here, we consider $\overline{\eta }=10^{-3}$ with $\varDelta _\eta = 0.05\overline{\eta }$ and $\overline{\chi }=10^{-2}$ with $\varDelta _\chi = 0.1\overline{\chi }$. In each simulation, M is fixed to 100. The obtained data agree well with our theoretical predictions (see Fig. 2a, b in the main manuscript).