Protocol for secure quantum machine learning at a distant place

Abstract

The application of machine learning to quantum information processing has recently attracted keen interest, particularly for the optimization of control parameters in quantum tasks without any pre-programmed knowledge. By adapting the machine learning technique, we present a novel protocol in which an arbitrarily initialized device at a learner’s location is taught by a provider located at a distant place. The protocol is designed such that any external learner who attempts to participate in or disrupt the learning process can be prohibited or noticed. We numerically demonstrate that our protocol works faithfully for single-qubit operation devices. A trade-off between the inaccuracy and the learning time is also analyzed.

Appendices

Appendix 1: Construction of SU(d) group generators

For any given d, we can generally define $\mathbf {G}$ in Eq. (1), systematically constructing ($d^2-1$) Hermitian operators as follows [14, 15]:

$$\begin{aligned} \left\{ \begin{array}{ll} \hat{u}_{jk} &{}= \hat{P}_{jk} + \hat{P}_{jk}, \\ \hat{v}_{jk} &{}= i\left( \hat{P}_{jk} - \hat{P}_{jk}\right) {,} \\ \hat{w}_{l} &{}= -\sqrt{\frac{2}{l(l+1)}} \left( \sum \nolimits _{i=1}^{l} \hat{P}_{ii} - l \hat{P}_{l+1 l+1} \right) {,} \end{array}\right. \end{aligned}$$

where $1 \le l \le d-1$ and $1 \le j \le k \le d$. Here, $\hat{P}_{jk} = \left| j\right\rangle \left\langle k\right|$ is a general projector. Then, the elements $\hat{G}_j$ of $\mathbf {G}$ can be given from the set $\{\hat{u}_{12}, \hat{u}_{13}, \cdots , \hat{v}_{12}, \hat{v}_{13}, \cdots , \hat{w}_{1}, \cdots , \hat{w}_{d-1} \}$, satisfying (i) hermiticity $\hat{G}_j=\hat{G}_j^\dagger$, (ii) traceless $\mathrm {tr}(\hat{G}_j)=0$ and (iii) orthogonality $\mathrm {tr}(\hat{G}_j^\dagger \hat{G}_k)=2\delta _{jk}$. The elements $\hat{G}_j, \hat{G}_k \in \mathbf {G}$ hold the relation,

$$\begin{aligned} \left[ \hat{G}_j, \hat{G}_k \right] = 2i \sum _l f_jkl \hat{G}_l, \end{aligned}$$

(12)

where $f_{jkl}$ is the (antisymmetric) structural constant of SU(d) algebra. Here, if $d=2$ (single qubit), we have Pauli spin operators as $\mathbf {G} = \{ \hat{\sigma }_x, \hat{\sigma }_y, \hat{\sigma }_z \}$.

Appendix 2: Detailed data in Figs. 4 and 5

Here we provide the detailed data in Figs. 4 and 5. By performing numerical simulations while increasing $N_L$ from 50 to 500 at intervals of 50, we characterize the learning probabilities $P_L(n)$ and survival probabilities $P_S(n)$. The simulations are performed 1000 times for each $N_L$. For all the cases of $N_L$, the survival probabilities $P_S(n)$ are well fitted to the fitting function $e^{-(n+1-N_L)/n_c}$ [as in Eq. (10)] with the characteristic constant $n_c$. The parameters $n_c$ and the (estimated) average number of iterations $\overline{n}=N_L + n_c$ are listed in Table 1. Here, $\overline{n}_\text {sim}$ denotes the average number of iterations actually counted in the simulations. We also find the learning error $\epsilon _L$ (averaged over 1000 simulations) for each $N_L$. The identified values of $\epsilon _L$ are also given in Table 1. We note again that the fitting parameters $n_c$ have finite values for all cases. We thus expect that learning can be completed faithfully for the given $N_L$.

Appendix 3: Approximation of $n_c$ in a random learning strategy

Here we approximately estimate $n_c$ in a random learning strategy. To this end, we first consider the probability $Pr(0|\mathbf {a})^{N_L}$ that the learning is completed for any fixed $\mathbf {a}$. $Pr(0|\mathbf {a})$ is the probability of the success event (namely, of measuring $\left| 0\right\rangle$ in $M_{0/1}$) [see Eq. (8)]. To proceed, we introduce a continuous function,

$$\begin{aligned} \frac{1}{2} \le \varXi (\mathbf {a}) = \xi _1(a_1)\xi _2(a_2)\cdots \xi _{d^2-1}(a_{d^2-1}) \le 1, \end{aligned}$$

(13)

satisfying $\varXi (\mathbf {a}\ne \mathbf {a}_\text {opt}) < \varXi (\mathbf {a}_\text {opt}) = 1$. We note that this function $\varXi (\mathbf {a})$ is made by minimizing $\left| \varXi (\mathbf {a}) - P(0|\mathbf {a})\right|$ for all $\mathbf {a}$. Thus, we infer that $P(0|\mathbf {a})^{N_L} \rightarrow 1$ when $\mathbf {a} \rightarrow \mathbf {a}_\text {opt}$, whereas $P(0|\mathbf {a})^{N_L} \rightarrow 0$ when $\mathbf {a}$ is far from $\mathbf {a}_\text {opt}$, and consequently, we can assume that $P(0|\mathbf {a})^{N_L} \simeq \varXi (\mathbf {a})^{N_L}$ ($\forall \mathbf {a}$) when $N_L$ is very large.

We then use a trick by approximating $\xi _j(a_j)^K$ with a delta function as

$$\begin{aligned} \xi _j(a_j)^{K} \approx \exp \left[ -\frac{(a_j - a_{j,\text {opt}})^2}{2\varDelta ^2}\right] {,} \end{aligned}$$

(14)

where $a_{j,\text {opt}}$ is a component of $\mathbf {a}_\text {opt}$ and K is assumed to be sufficiently large but $K \ll N_L$. Thus, we can also assume that $\varXi (\mathbf {a})^K \simeq P(0|\mathbf {a})^K$. In the circumstance, we estimate the average probability $P(0|\mathbf {a})_\text {avg}^{N_L}$, such that (for $\varDelta \ll 1$ ¹⁰)

$$\begin{aligned} P(0|\mathbf {a})_\text {avg}^{N_L}\simeq & {} \int da_1\xi _1(a_1)^{N_L} \int da_2\xi _1(a_2)^{N_L} \cdots \int da_{d^2-1}\xi _{d^2-1}(a_{d^2-1})^{N_L} \nonumber \\\approx & {} \prod _{j=1}^{d^2-1} \int _{-\infty }^{\infty } da_j \exp \left[ -\frac{(a_j - a_{j,\text {opt}})^2}{2\varDelta ^2} \frac{N_L}{K}\right] \approx \left( \frac{K 2\pi \varDelta ^2}{N_L}\right) ^\frac{d^2-1}{2}. \end{aligned}$$

(15)

Then, let us consider a probability that the learning is terminated at n iteration step:

$$\begin{aligned} \left( 1-P(0|\mathbf {a}^{(1)})^{N_L}\right) \left( 1{-}P(0|\mathbf {a}^{(2)})^{N_L}\right) \cdots \left( 1{-}P(0|\mathbf {a}^{(n-1)})^{N_L}\right) P(0|\mathbf {a}^{(n)})^{N_L}{,} \end{aligned}$$

(16)

for any sequence $\mathbf {a}^{(0)} \rightarrow \mathbf {a}^{(1)} \rightarrow \mathbf {a}^{(2)} \rightarrow \ldots \rightarrow \mathbf {a}^{(n)}$ of updating the parameter vector in the learning. Thus, in a random learning strategy, we can approximate the learning probability $P_L(n)$, introduced in Sect. 4, such that

$$\begin{aligned} P_L(n)\approx & {} P(0|\mathbf {a}^{(1)})_\text {avg}^{N_L} \nonumber \\&+ \left( 1-P(0|\mathbf {a}^{(1)})_\text {avg}^{N_L} \right) P(0|\mathbf {a}^{(2)})_\text {avg}^{N_L} \nonumber \\&+ \left( 1-P(0|\mathbf {a}^{(1)})_\text {avg}^{N_L} \right) \left( 1-P(0|\mathbf {a}^{(2)})_\text {avg}^{N_L} \right) P(0|\mathbf {a}^{(3)})_\text {avg}^{N_L} \nonumber \\&\vdots \nonumber \\&+ \left( 1-P(0|\mathbf {a}^{(1)})_\text {avg}^{N_L}\right) \left( 1-P(0|\mathbf {a}^{(2)})_\text {avg}^{N_L}\right) \nonumber \\&\cdots \left( 1-P(0|\mathbf {a}_\text {avg}^{(n-1)})^{N_L}\right) P(0|\mathbf {a}_\text {avg}^{(n)})^{N_L} \nonumber \\\approx & {} \sum _{i=0}^{n-1} \left( 1 - P(0|\mathbf {a})_\text {avg}^{N_L} \right) ^{i} P(0|\mathbf {a})_\text {avg}^{N_L} = 1 - \left( 1-P(0|\mathbf {a})_\text {avg}^{N_L}\right) ^n{,} \end{aligned}$$

(17)

where $P(0|\mathbf {a}^{(1)})_\text {avg}^{N_L}=P(0|\mathbf {a}^{(2)})_\text {avg}^{N_L} =\ldots =P(0|\mathbf {a}^{(n)})_\text {avg}^{N_L}=P(0|\mathbf {a})_\text {avg}^{N_L}$. Here, using Eq. (15), we finally arrive at (for $N_L \gg 1$ and $\varDelta \ll 1$)

$$\begin{aligned} P_L(n) \approx 1-e^{-\frac{n}{n_c}},~\text {or equivalently},~P_S(n) \approx e^{-\frac{n}{n_c}}, \end{aligned}$$

(18)

where $n_c \simeq O\left( \sqrt{N_L^{(d^2-1)/2}}\right)$.

Appendix 4: Effect on learning of any alterations in ${\mathcal {C}}_o^{AB}$

Here we consider a situation in which particles in the state $\left| \widetilde{\tau }_A(\mathbf {a})\right\rangle$ moving through ${\mathcal {C}}_o^{AB}$ are altered with a certain probability $p_\text {int}$ by some malicious Eve. Here, we assume a super-Eve who can sort out Alice’s estimation state $\left| \widetilde{\tau }_A(\mathbf {a})\right\rangle$, discarding the blinded state $\left| \widetilde{\chi }_A(\mathbf {r}_h)\right\rangle$, in ${\mathcal {C}}_o^{AB}$ for his/her own effective learning. Eve’s aim is to learn Alice’s vector $\mathbf {a}$ and thus to obtain the optimal vector as close to $\mathbf {a}_\text {opt}$ as possible when Alice’s learning is complete. Eve can thus adopt the strategy of learning Alice’s vector $\mathbf {a}$ using a stolen particle for each trial and resend the newly generated particle of his/her estimated state $\left| \widetilde{\tau }_E(\mathbf {e})\right\rangle$ to Bob, where $\mathbf {e}$ is a vector of Eve’s own device.

However, in this case, it takes much longer to complete the learning process because some particles of $\left| \widetilde{\tau }_A(\mathbf {a})\right\rangle$ are altered as $\left| \widetilde{\tau }_A(\mathbf {a})\right\rangle \rightarrow \left| \widetilde{\tau }_E(\mathbf {e})\right\rangle$. To corroborate this, we perform numerical simulations of single-qubit target states ($d=2$). Here, we set $N_L=100$ and consider three cases: $p_\text {int}=0.1$, 0.2, and 0.3. We assume further that Eve can use the best strategy for each stolen particle, i.e., $\left| \left\langle {\widetilde{\tau }_E(\mathbf {a}')}|{\widetilde{\tau }_A (\mathbf {a})}\right\rangle \right| =\frac{2}{3}$ [23]. In Fig. 7, we present the learning and survival probabilities for $p_\text {int}=0.1$ (red), 0.2 (green), and 0.3 (blue) on a log scale. The survival probabilities are also well matched to Eq. (10). The data are listed in Table 2. Note here that $\overline{n}$ increases exponentially with increasing alteration probability $p_\text {int}$. In this sense, the learning efficiency is very sensitive to the alterations. Thus, by monitoring the learning time, Alice can sense even any super-Eve; if learning is too late or cannot be completed, Alice stops the learning so that Eve cannot complete the process $\mathbf {e} \rightarrow \mathbf {a}_\text {opt}$.

Fig. 7

(Color online) a Learning probability $P_L(n)$ and b survival probability $P_S(n)$ on a log scale, assuming some Eve who can steal particles moving in ${\mathcal {C}}_o^{AB}$ with a certain probability $p_\text {int}$. We assume that Eve can adopt the best learning strategy for her learning (see the main text). Here, we set $N_L=100$ and consider the qubit target states, i.e., $d=2$. We consider three cases: $p_\text {int}=0.1$ (red), 0.2 (green), and 0.3 (blue). We perform 1000 simulations to draw the graphs. In each simulation, the target state $\left| \tau \right\rangle$ is randomly chosen. The survival probabilities $P_S(n)$ are also well fitted to Eq. (10) (black solid lines)

Table 2 Values of $n_c$, $\overline{n}$ ($\overline{n}_\text {sim}$), and $\epsilon _L$ in Fig. 7

Here we briefly note that, in a realistic application, Alice should evaluate and analyze the learning time, i.e., $n_c$, by performing the learning with her own devices, before starting the protocol with Bob. Such task is carried out taking into account the errors due to the imprecise control or contaminated devices. The maximum tolerable noise in the channels should also be estimated in this stage.