Quantum Permutation Research Articles

Correction to: Tracing the orbitals of the quantum permutation group

Correction to: Tracing the orbitals of the quantum permutation group

Optimal control of quantum permutation algorithm with a molecular ququart

Optimal control of quantum permutation algorithm with a molecular ququart

Efficient implementation of quantum permutation algorithm using a polar SrO molecule in pendular states

Abstract Quantum algorithms offer enhanced computational efficiency compared to their classical counterparts in solving specific tasks. In this study, we implement the quantum permutation algorithm utilizing a polar molecule within an external electric field. The selection of the molecular qutrit involves the utilization of field-dressed states generated through the pendular modes of SrO. Through the application of multi-target optimal control theory, we strategically design microwave pulses to execute logical operations, including Fourier transform, oracle Uf operation, and inverse Fourier transform within a three-level molecular qutrit structure. The observed high fidelity of our outcomes is intricately linked to the concept of the quantum speed limit, which quantifies the maximum speed of quantum state manipulation. Subsequently, we design the optimized pulse sequence to successfully simulate the quantum permutation algorithm on a single SrO molecule, achieving remarkable fidelity. Consequently, a quantum circuit comprising a single qutrit suffices to determine permutation parity with just a single function evaluation. Therefore, our results indicate that the optimal control theory can be well applied to quantum computation of polar molecular systems.

Planar #CSP Equality Corresponds to Quantum Isomorphism – A Holant Viewpoint

Recently, Mančinska and Roberson proved that two graphs G and \(G^{\prime }\) are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of holographic transformation of Valiant, using the quantum permutation matrix \(\mathcal {U}\) defining the quantum isomorphism. Due to the noncommutativity of \(\mathcal {U}\) ’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group \(Qut(\mathcal {F})\) of a set \(\mathcal {F}\) of constraint functions/tensors and characterize the intertwiners of \(Qut(\mathcal {F})\) as the signature matrices of planar \(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\) quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.

Entanglement-invertible channels

In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].

A Concrete Model for the Quantum Permutation Group on 4 Points

In 2019, Jung–Weber gave an example of a concrete magic unitary M, which defines a C * -algebraic model of the quantum permutation group S 4 + . We show with the help of a computer that there exist no polynomials up to degree 50 separating the entries of M from the generators of C ( S 4 + ) . This indicates that the magic unitary M might already define a faithful model of S 4 + .

Discrete-Time Quantum Walks Community Detection in Multi-Domain Networks

Abstract The problem of detecting communities in real-world networks has been extensively studied in the past, but most of the existing approaches work on single-domain networks, i.e. they consider only one type of relationship between nodes. Single-domain networks may contain noisy edges and they may lack some important information. Thus, some authors have proposed to consider the multiple relationships that connect the nodes of a network, thus obtaining multi-domain networks. However, most community detection approaches are limited to multi-layer networks, i.e. networks generated from the superposition of several single-domain networks (called layers) that are regarded as independent of each other. In addition to being computationally expensive, multi-layer approaches might yield inaccurate results because they ignore potential dependencies between layers. This paper proposes a multi-domain discrete-time quantum walks (MDQW) model for multi-domain networks. First, the walking space of network nodes in multi-domain network is constructed. Second, the quantum permutation circuit of the coin state is designed based on the coded particle state. Then, using different coin states, the shift operator performs several quantum walks on the particles. Finally, the corresponding update rule is selected to move the node according to the measurement result of the quantum state. With continuous update iteration, the shift operator automatically optimizes the discovered community structure. We experimentally compared our MDQW method with four state-of-the-art competitors on five real datasets. We used the normalized mutual information (NMI) to compare clustering quality, and we report an increase in NMI of up to 3.51 of our MDQW method in comparison with the second-best performing competitor. The MDQW method is much faster than its competitors, allowing us to conclude that MDQW is a useful tool in the analysis of large real-life multi-domain networks. Finally, we illustrate the usefulness of our approach on two real-world case studies.

Weak Fault Feature Extraction of the Rotating Machinery Using Flexible Analytic Wavelet Transform and Nonlinear Quantum Permutation Entropy

Weak Fault Feature Extraction of the Rotating Machinery Using Flexible Analytic Wavelet Transform and Nonlinear Quantum Permutation Entropy

From Quantum Automorphism of (Directed) Graphs to the Associated Multiplier Hopf Algebras

This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that K [Mq(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele.

Quantum data communication protection with the quantum permutation pad block cipher in counter mode and Clifford operators

Background: This article integrates two cryptographic schemes for quantum data protection. The result achieves authentification, confidentiality, integrity, and replay protection. The authentication, integrity, and replay aspects leverage quantum Clifford operators. Confidentiality of quantum messages is achieved using the quantum permutation pad (QPP) cryptographic scheme. Methods: Clifford operators and the QPP are combined into a block cipher in counter mode. A shared secret is used to seed a random number generator for the arbitrary selection of Clifford operators and quantum permutations to produce a signature field and perform encryption. An encryption and signature algorithm and a decryption and authentication algorithm are specified to protect quantum messages. Results: A symmetric key block cipher with authentication is described. The plain text is signed with a sequence of randomly selected Clifford operators. The signed plaintext is encrypted with a sequence of randomly selected permutations. The algorithms are analyzed. As a function of the values selected for the security parameters, there is an unavoidable risk of collision. The probability of block collision is modelled versus the number of blocks encrypted, for block sizes two, three, four, and five qubits. Conclusions: The scheme is practical but does not achieve perfect indistinguishability because of the risk of message collision. This is normal and unavoidable when fixed-size fields are assumed to make a scheme practical. The model can be used to determine the values of the security parameters and the lifetime of session keys to mitigate the risk of information leakage according to the needs of the scheme’s users. The session key can be renewed when a tolerable maximum number of messages has been sent.

Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$^\ast $-algebra $B$, which includes the quantum permutation group $S_N^+$, where $N = \dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $\Gamma $ lead to isomorphic $\ast $-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups.

Quantum Permutation Matrices

Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.

Quantum encryption of superposition states with quantum permutation pad in IBM quantum computers

We present an implementation of Kuang and Bettenburg’s Quantum Permutation Pad (QPP) used to encrypt superposition states. The project was conducted on currently available IBM quantum systems using the Qiskit development kit. This work extends previously reported implementation of QPP used to encrypt basis states and demonstrates that application of the QPP scheme is not limited to the encryption of basis states. For this implementation, a pad of 56 2-qubit Permutation matrices was used, providing 256 bits of entropy for the QPP algorithm. An image of a cat was used as the plaintext for this experiment. The plaintext was randomized using a classical XOR function prior to the state preparation procedure. To create corresponding superposition states, we applied a novel operator defined in this paper. These superposition states were then encrypted using QPP, with 2-qubit Permutation Operators, producing superposition ciphertext states. Due to the lack of a quantum channel, we omitted the transmission and executed the decryption procedure on the same IBM quantum system. If a quantum channel existed, the superposition ciphertext states could be transmitted as qubits, and be directly decrypted on a different quantum system. We provide a brief discussion of the security, although the focus of the paper remains on the implementation. Previously we have demonstrated QPP operating in both classical and quantum computers, offering an interesting opportunity to bridge the security gap between classical and quantum systems. This work broadens the applicability of QPP for the encryption of basis states as well as superposition states. We believe that quantum encryption schemes that are not limited to basis states will be integral to a secure quantum internet, to reduce vulnerabilities introduced by using two separate algorithms for secure communication between a quantum and a classical computer.

Infinite quantum permutations

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups.We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.

Magic squares: Latin, semiclassical, and quantum

Quantum magic squares have recently been introduced as a “magical” combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e., dilated to a quantum permutation matrix)—only the so-called semiclassical ones can. Purifying establishes a relation to an ideal world of fundamental theoretical and practical importance; the opposite of purifying is described by the matrix convex hull. In this paper, we prove that semiclassical magic squares can be purified to quantum Latin squares, which are “magical” combinations of orthonormal bases. Conversely, we prove that the matrix convex hull of quantum Latin squares is larger compared to the semiclassical ones. This tension is resolved by our third result: we prove that the quantum Latin squares that are semiclassical are precisely those constructed from a classical Latin square. Our work sheds light on the internal structure of quantum magic squares, on how this is affected by the matrix convex hull, and, more generally, on the nature of the “magical” composition rule, both at the semiclassical and at the quantum level.

Tannaka–Krein Reconstruction and Ergodic Actions of Easy Quantum Groups

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

Quantum encryption with quantum permutation pad in IBMQ systems

Quantum permutation pad or QPP is a quantum-safe symmetric cryptographic algorithm proposed by Kuang and Bettenburg in 2020. The theoretical foundation of QPP leverages the linear algebraic representations of quantum gates which makes QPP realizable in both, quantum and classical systems. By applying the QPP with 64 of 8-bit permutation gates, holding respective entropy of over 100,000 bits, we accomplished quantum random number distributions digitally over today’s classical internet. The QPP has also been used to create pseudo quantum random numbers and served as a foundation for quantum-safe lightweight block and streaming ciphers. This paper continues to explore numerous applications of QPP, namely, we present an implementation of QPP as a quantum encryption circuit on today’s still noisy quantum computers. With the publicly available 5-qubit IBMQ devices, we demonstrate quantum secure encryption (256 bits of entropy) using 2-qubit QPP with 56 permutation gates, and 3-qubit QPP with 17 permutation gates respectively. Initial qubits of the encryption circuit correspond to the plaintext and after applying quantum encryption operations, cipher qubits are measured with probabilistic distributions, and the results with the highest probability are recorded as cipher bits. The cipher bits are then decrypted with an inverse QPP circuit. The output state plaintext qubits are measured and the most frequent count measurement results are recorded as plaintext bits. This quantum encryption and decryption process clearly demonstrates that QPP quantum implementations works exactly as symmetric encryption and decryption schemes should. The plaintext and ciphertext bits can also be encrypted and decrypted respectively by any classical computing device with the corresponding QPP algorithm as in quantum computers. This work reveals that it is possible to build quantum-secure communications between quantum-to-quantum and quantum-to-classical computers over today’s internet and the future quantum internet.

Quantum permutation pad for universal quantum-safe cryptography

Quantum permutation pad for universal quantum-safe cryptography

Cutoff profiles for quantum Lévy processes and quantum random transpositions

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time \(N\ln (N)\). Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.

Quantum Channels with Quantum Group Symmetry

In this paper we will demonstrate that any compact quantum group can be used as symmetries for quantum channels, which leads us to the concept of covariant channels. We then unearth the structure of the convex set of covariant channels by identifying all extreme points under the assumption of multiplicity-free condition for the associated fusion rule, which provides a wide generalization of the results of Mozrzymas et al. (J Math Phys 58(5):052204, 2017). The presence of quantum group symmetry in contrast to the group symmetry will be highlighted in the examples of quantum permutation groups and \(SU_q(2)\). In the latter example, we will see the necessity of the Heisenberg picture coming from the non-Kac type condition. This paper ends with the covariance with respect to projective representations, which leads us back to Weyl covariant channels and its fermionic analogue.