Quantum Coupon Collector

We study how efficiently a k-element set S?[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=?_{i?S}|i>/?|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of |S> suffice, in contrast to the ?(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=?(k), then ?(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case

Quantum coupon collector

We study how efficiently a k-element set S ? [n] can be learned from a uniform superposition |Si of its elements. One can think of |Si = Pi?S |ii/p|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the “coupon collector problem.” We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n - k = O(1) missing elements then O(k) copies of |Si suffice, in contrast to the T(k log k

Experimental quantum advantage with quantum coupon collector

An increasing number of communication and computational schemes with quantum advantages have recently been proposed, which implies that quantum technology has fertile application prospects. However, demonstrating these schemes experimentally continues to be a central challenge because of the difficulty in preparing high-dimensional states or highly entangled states. In this study, we introduce and analyse a quantum coupon collector protocol by employing coherent states and simple linear optical elements, which was successfully demonstrated using realistic experimental equipment. We showed that our protocol can significantly reduce the number of samples needed to learn a specific set compared with the classical limit of the coupon collector problem. We also discuss the potential values and expansions of the quantum coupon collector by constructing a quantum blind box game. The information transmitted by the proposed game also broke the classical limit. These results strongly prove the advantages of quantum mechanics in machine learning and communication complexity.Comment: 10 pages, 3 figures, 3 tables, Accepted by Researc

Optimal lower bounds for Quantum Learning via Information Theory

Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, Arunachalam and de Wolf (JMLR, 2018) proved that quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. They established lower bounds on sample complexity via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both the PAC and agnostic models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC, 2020) characterized the quantum sample complexity of this problem up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, while the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharper lower bound for the Quantum Coupon Collector problem, via the generalized Holevo-Curlander bounds on the distinguishability of an ensemble. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.Comment: v3: 40 pages; Added references; edited extensively; simplified the proof of Theorem 3.2; results unchanged. A preliminary version of the results in Section 3 was included in the S.B.H.'s PhD thesis at University of Waterloo (Dec. 2020). An extended abstract of the results in Section 4 was included in the P.S.' bachelor's project report at Indian Institute of Science (Apr. 2022

Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems

Recently Chen and Gao~\cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over \CC, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over \CC by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables. Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over \FF_q improving on subsequent work by Chen, Gao and Yuan \cite{ChenGao2018}. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.Comment: 27 page

All-Photonic Quantum Repeater for Multipartite Entanglement Generation

Quantum network applications like distributed quantum computing and quantum secret sharing present a promising future network equipped with quantum resources. Entanglement generation and distribution over long distances is critical and unavoidable to utilize quantum technology in a fully-connected network. The distribution of bipartite entanglement over long distances has seen some progresses, while the distribution of multipartite entanglement over long distances remains unsolved. Here we report a two-dimensional quantum repeater protocol for the generation of multipartite entanglement over long distances with all-photonic framework to fill this gap. The yield of the proposed protocol shows long transmission distance under various numbers of network users. With the improved efficiency and flexibility of extending the number of users, we anticipate that our protocol can work as a significant building block for quantum networks in the future

Breaking universal limitations on quantum conference key agreement without quantum memory

Quantum conference key agreement is an important cryptographic primitive for future quantum network. Realizing this primitive requires high-brightness and robust multiphoton entanglement sources, which is challenging in experiment and unpractical in application because of limited transmission distance caused by channel loss. Here we report a measurement-device-independent quantum conference key agreement protocol with enhanced transmission efficiency over lossy channel. With spatial multiplexing nature and adaptive operation, our protocol can break key rate bounds on quantum communication over quantum network without quantum memory. Compared with previous work, our protocol shows superiority in key rate and transmission distance within the state-of-the-art technology. Furthermore, we analyse the security of our protocol in the composable framework and evaluate its performance in the finite-size regime to show practicality. Based on our results, we anticipate that our protocol will play an indispensable role in constructing multipartite quantum network

The Quantum Supremacy Tsirelson Inequality

A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit $C$ on $n$ qubits and a sample $z \in \{0,1\}^n$, the benchmark involves computing $|\langle z|C|0^n \rangle|^2$, i.e. the probability of measuring $z$ from the output distribution of $C$ on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given $C$ can output a string $z$ such that $|\langle z|C|0^n\rangle|^2$ is substantially larger than $\frac{1}{2^n}$ (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit $C$, sampling $z$ from the output distribution of $C$ achieves $|\langle z|C|0^n\rangle|^2 \approx \frac{2}{2^n}$ on average (Arute et al., 2019). In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than $\frac{2}{2^n}$? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to $C$. We show that, for any $\varepsilon \ge \frac{1}{\mathrm{poly}(n)}$, outputting a sample $z$ such that $|\langle z|C|0^n\rangle|^2 \ge \frac{2 + \varepsilon}{2^n}$ on average requires at least $\Omega\left(\frac{2^{n/4}}{\mathrm{poly}(n)}\right)$ queries to $C$, but not more than $O\left(2^{n/3}\right)$ queries to $C$, if $C$ is either a Haar-random $n$-qubit unitary, or a canonical state preparation oracle for a Haar-random $n$-qubit state. We also show that when $C$ samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from $C$ is the optimal 1-query algorithm for maximizing $|\langle z|C|0^n\rangle|^2$ on average.Comment: 26 pages. V2: corrected typos, added additional discussion, added journal reference. V3: additional minor corrections. V4: final journal version, various writing improvement

Phase-Matching Quantum Key Distribution without Intensity Modulation

Quantum key distribution provides a promising solution for sharing secure keys between two distant parties with unconditional security. Nevertheless, quantum key distribution is still severely threatened by the imperfections of devices. In particular, the classical pulse correlation threatens security when sending decoy states. To address this problem and simplify experimental requirements, we propose a phase-matching quantum key distribution protocol without intensity modulation. Instead of using decoy states, we propose a novel method to estimate the theoretical upper bound on the phase error rate contributed by even-photon-number components. Simulation results show that the transmission distance of our protocol could reach 305 km in telecommunication fiber. Furthermore, we perform a proof-of-principle experiment to demonstrate the feasibility of our protocol, and the key rate reaches 22.5 bps under a 45 dB channel loss. Addressing the security loophole of pulse intensity correlation and replacing continuous random phase with 6 or 8 slices random phase, our protocol provides a promising solution for constructing quantum networks.Comment: Comments are welcome! 12 pages, 6 figure

Breaking Rate-Distance Limitation of Measurement-Device-Independent Quantum Secret Sharing

Quantum secret sharing is an important cryptographic primitive for network applications ranging from secure money transfer to multiparty quantum computation. Currently most progresses on quantum secret sharing suffer from rate-distance bound, and thus the key rates are limited and unpractical for large-scale deployment. Furthermore, the performance of most existing protocols is analyzed in the asymptotic regime without considering participant attacks. Here we report a measurement-device-independent quantum secret sharing protocol with improved key rate and transmission distance. Based on spatial multiplexing, our protocol shows it can break rate-distance bounds over network under at least ten communication parties. Compared with other protocols, our work improves the secret key rate by more than two orders of magnitude and has a longer transmission distance. We analyze the security of our protocol in the composable framework considering participant attacks. Based on the security analysis, we also evaluate their performance in the finite-size regime. In addition, we investigate applying our protocol to digital signatures where the signature rate is improved more than $10^7$ times compared with existing protocols. Based on our results, we anticipate that our quantum secret sharing protocol will provide a solid future for multiparty applications on quantum network.Comment: arXiv admin note: text overlap with arXiv:2212.0522