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We introduce SA-DQAS in this paper, a novel framework that enhances the gradient-based Differentiable Quantum Architecture Search (DQAS) with a self-attention mechanism, aimed at optimizing circuit design for Quantum Machine Learning (QML) challenges. Analogous to a sequence of words in a sentence, a quantum circuit can be viewed as a sequence of placeholders containing quantum gates. Unlike DQAS, each placeholder is independent, while the self-attention mechanism in SA-DQAS helps to capture relation and dependency information among each operation candidate placed on placeholders in a circuit. To evaluate and verify, we conduct experiments on job-shop scheduling problems (JSSP), Max-cut problems, and quantum fidelity. Incorporating self-attention improves the stability and performance of the resulting quantum circuits and refines their structural design with higher noise resilience and fidelity. Our research demonstrates the first successful integration of self-attention with DQAS., Comment: 4 pages

Utilizing unsupervised representation learning for quantum architecture search (QAS) represents a cutting-edge approach poised to realize potential quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices. Most QAS algorithms combine their search space and search algorithms together and thus generally require evaluating a large number of quantum circuits during the search process. Predictor-based QAS algorithms can alleviate this problem by directly estimating the performance of circuits according to their structures. However, a high-performance predictor generally requires very time-consuming labeling to obtain a large number of labeled quantum circuits. Recently, a classical neural architecture search algorithm Arch2vec inspires us by showing that architecture search can benefit from decoupling unsupervised representation learning from the search process. Whether unsupervised representation learning can help QAS without any predictor is still an open topic. In this work, we propose a framework QAS with unsupervised representation learning and visualize how unsupervised architecture representation learning encourages quantum circuit architectures with similar connections and operators to cluster together. Specifically, our framework enables the process of QAS to be decoupled from unsupervised architecture representation learning so that the learned representation can be directly applied to different downstream applications. Furthermore, our framework is predictor-free eliminating the need for a large number of labeled quantum circuits. During the search process, we use two algorithms REINFORCE and Bayesian Optimization to directly search on the latent representation, and compare them with the method Random Search. The results show our framework can more efficiently get well-performing candidate circuits within a limited number of searches., Comment: 9 Pages, quantum architecture search, unsupervised representation learning

The Job shop scheduling problem (JSSP) plays a pivotal role in industrial applications, such as signal processing (SP) and steel manufacturing, involving sequencing machines and jobs to maximize scheduling efficiency. Before, JSSP was solved using manually defined circuits by variational quantum algorithm (VQA). Finding a good circuit architecture is task-specific and time-consuming. Differentiable quantum architecture search (DQAS) is a gradient-based framework that can automatically design circuits. However, DQAS is only tested on quantum approximate optimization algorithm (QAOA) and error mitigation tasks. Whether DQAS applies to JSSP based on a more flexible algorithm, such as variational quantum eigensolver (VQE), is still open for optimization problems. In this work, we redefine the operation pool and extend DQAS to a framework JSSP-DQAS by evaluating circuits to generate circuits for JSSP automatically. The experiments conclude that JSSP-DQAS can automatically find noise-resilient circuit architectures that perform much better than manually designed circuits. It helps to improve the efficiency of solving JSSP., Comment: 4+1 Pages, ICASSP24

Differentiable quantum architecture search (DQAS) is a gradient-based framework to design quantum circuits automatically in the NISQ era. It was motivated by such as low fidelity of quantum hardware, low flexibility of circuit architecture, high circuit design cost, barren plateau (BP) problem, and periodicity of weights. People used it to address error mitigation, unitary decomposition, and quantum approximation optimization problems based on fixed datasets. Quantum reinforcement learning (QRL) is a part of quantum machine learning and often has various data. QRL usually uses a manually designed circuit. However, the pre-defined circuit needs more flexibility for different tasks, and the circuit design based on various datasets could become intractable in the case of a large circuit. The problem of whether DQAS can be applied to quantum deep Q-learning with various datasets is still open. The main target of this work is to discover the capability of DQAS to solve quantum deep Q-learning problems. We apply a gradient-based framework DQAS on reinforcement learning tasks and evaluate it in two different environments - cart pole and frozen lake. It contains input- and output weights, progressive search, and other new features. The experiments conclude that DQAS can design quantum circuits automatically and efficiently. The evaluation results show significant outperformance compared to the manually designed circuit. Furthermore, the performance of the automatically created circuit depends on whether the super-circuit learned well during the training process. This work is the first to show that gradient-based quantum architecture search is applicable to QRL tasks., Comment: 4+1 pages, 3 figures, QCE23 workshop QML

In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4$ by merging two different systems of an existing $7$-qubit UPB of size $11$. Moreover, a new family of $7$-qubit positive-partial-transpose (PPT) entangled states of rank $2^7-11$ is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound are given by two methods., Comment: 15 pages, 4 figures

Multipartite quantum system is complex. Characterizing the relations among the three bipartite reduced density operators $\rho_{AB}$, $\rho_{AC}$ and $\rho_{BC}$ of a tripartite state $\rho_{ABC}$ has been an open problem in quantum information. One of such relations has been reduced by [Cadney et al, LAA. 452, 153, 2014] to a conjectured inequality in terms of matrix rank, namely $r(\rho_{AB}) \cdot r(\rho_{AC})\ge r(\rho_{BC})$ for any $\rho_{ABC}$. It is denoted as open problem $41$ in the website "Open quantum problems-IQOQI Vienna". We prove the inequality, and thus establish a complete picture of the four-party linear inequalities in terms of the $0$-entropy. Our proof is based on the construction of a novel canonical form of bipartite matrices under local equivalence. We apply our result to the marginal problem and the extension of inequalities in the multipartite systems, as well as the condition when the inequality is saturated., Comment: 20 pages, 1 figure

An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in [Quantum Information Processing 19:185 (2020)]. Next, we graph-theoretically show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they are in three different orbits up to local unitary equivalence. Moreover, we also present the number of sorts of nonisomorphic graphs of complete graphs of some known UPBs and their orbits., Comment: 23 pages, 11 figures

The determination of genuine entanglement is a central problem in quantum information processing. We investigate the tripartite state as the tensor product of two bipartite entangled states by merging two systems. We show that the tripartite state is a genuinely entangled state when the range of both bipartite states are entanglement-breaking subspaces. We further investigate the tripartite state when one of the two bipartite states has rank two. Our results provide the latest progress on a conjecture proposed in the paper [Yi Shen $\textit{et al}$, J. Phys. A 53, 125302 (2020)]. We apply our results to construct multipartite states whose bipartite reduced density operators have additive EOF. Further, such states are distillable across every bipartition under local operations and classical communications.

It is an interesting problem to construct genuine tripartite entangled states based on the collective use of two bipartite entangled states. We consider the case that the states are two-qubit Werner states, we construct the interval of parameter of Werner states such that the tripartite state is genuine entangled. Further, we present the way of detecting the tripartite genuine entanglement using current techniques in experiments. We also investigate the lower bound of genuine multipartite entanglement concurrence.

Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation $I_2\otimes V$ and $I_2\otimes W$. We show that $V$ and $W$ have no zero entry, and apply it to exclude examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is $\log_2 3$ ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.

Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)$ holds for any bipartite matrix $M=\sum_i R_i \otimes S_i$ and Schmidt rank $K$. We prove that the conjecture holds for $M$ with $K=3$ and some special $M$ with arbitrary $K$.

We investigate the number of real entries of an $n\times n$ complex Hadamard matrix (CHM). We analytically derive the numbers when $n=2,3,4,6$. In particular, the number can be any one of $0-22,24,25,26,30$ for $n=6$. We apply our result to the existence of four mutually unbiased bases (MUBs) in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exists then the real entries in any one of the remaining three matrices does not exceed $22$., Comment: 28 pages

The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size $6$ and six UPBs of size $9$ in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size $9$ in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size $7$ in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a special $\r$.