Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory encompassing quantum systems and gravity is expected to allow causally nonseparable processes featuring operations in indefinite causal order, defying that events be causally ordered at all. The first challenge has been addressed through the recent development of intrinsically quantum causal models, allowing causal explanations of quantum processes -- provided they admit a definite causal order, i.e. have an acyclic causal structure. This work addresses causally nonseparable processes and offers a causal perspective on them through extending quantum causal models to cyclic causal structures. Among other applications of the approach, it is shown that all unitarily extendible bipartite processes are causally separable and that for unitary processes, causal nonseparability and cyclicity of their causal structure are equivalent.
Charging efficiency bursts in a quantum battery with cyclic indefinite causal order
National Cheng Kung University（国立成功大学）；Center for Quantum Frontiers of Research and Technology, NCKU（国立成功大学量子前沿科技研究中心）；University of Rochester（罗切斯特大学）；National Center for Theoretical Sciences（国家理论科学中心）
Completely depolarising channels are often regarded as the prototype of physical processes that are useless for communication: any message that passes through them along a well-defined trajectory is completely erased. When two such channels are used in a quantum superposition of two alternative orders, they become able to transmit some amount of classical information, but still no quantum information can pass through them. Here we show that the ability to place N completely depolarising channels in a superposition of N alternative causal orders enables a high-fidelity, heralded transmission of…
In this article initial steps in an analysis of cyclic networks of quantum logic gates is given. Cyclic networks are those in which the qubit lines are loops. Here we have studied one and two qubit systems plus two qubit cyclic systems connected to another qubit on an acyclic line. The analysis includes the group classification of networks and studies of the dynamics of the qubits in the cyclic network and of the perturbation effects of an acyclic qubit acting on a cyclic network. This is followed by a discussion of quantum algorithms and quantum information processing with cyclic networks of …
We transfer the concept of linear feed-back shift registers to quantum circuits. It is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error-correcting codes.
We provide a construction for quantum codes (hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4^m). We also provide examples of these codes some of which meet the known bounds for quantum codes.
The thesis is mainly about the construction and implementation of cyclic mutually unbiased bases, dealing with different entanglement structures by discussing the related group structures. A recursive construction for Fermat number dimensions is given and related to Wiedemann's conjecture for systems with more than 2048 qubits. The second part of the thesis analyses a quantum public-key encryption scheme.
We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-\varepsilon$ with (polylog) complexity $O(\log(q)^{R}\log(\varepsilon)^{2})$ where $R<\infty.$
The quantum diffusion of a particle in an initially localized state on a cyclic lattice with N sites is studied. Diffusion and reconstruction time are calculated. Strong differences are found for even or odd number of sites and the limit N->infinit is studied. The predictions of the model could be tested with micro - and nanotechnology devices.
Fast and robust quantum gates is the cornerstone of fault-tolerance quantum computation. In this paper, we propose to achieve quantum gates based on non-cyclic geometric evolution. Dynamical phase during the evolution is cancelled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabatic manner. Different from geometric gates based on cyclic evolution, the superiority of non-cyclic scheme is that the operation time is proportional to the rotation angle (but not the geometric phase) of quantum state. Therefore, the non-cyclic scheme becomes fairly fast in the c…
We present a formula for an infinite number of universal quantum logic gates, which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We obtain this family from a certain representation of the cyclic group of order $n$. We then show that this {\it discrete} family, parametrized by integers $n$, is in fact, a small sub-class of a larger {\it continuous} family, parametrized by real numbers $θ$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.
The hardness to solve an unstructured quantum search problem by a standard quantum search algorithm mainly originates from the low efficiency to amplify the amplitude of the marked state by the oracle unitary operation associated with other known quantum operations. In order to bypass the square speedup limitation of a standard quantum search algorithm it is necessary to develop other type of quantum search algorithms. It is described in detail in the paper for a quantum dynamical method to solve the quantum search problems in the cyclic group state space. The binary dynamical representation f…
We propose an experimentally feasible scheme to achieve quantum computation based on a pair of orthogonal cyclic states. In this scheme, quantum gates can be implemented based on the total phase accumulated in cyclic evolutions. In particular, geometric quantum computation may be achieved by eliminating the dynamic phase accumulated in the whole evolution. Therefore, both dynamic and geometric operations for quantum computation are workable in the present theory. Physical implementation of this set of gates is designed for NMR systems. Also interestingly, we show that a set of universal geomet…
We quantitatively differentiate between the spreads of discrete-time quantum and classical random walks on a cyclic graph. Due to the closed nature of any cyclic graph, there is additional "collision"- like interference in the quantum random walk along with the usual interference in any such walk on any graph, closed or otherwise. We find that the quantum walker remains localized in comparison to the classical one, even in the absence of disorder, a phenomenon that is potentially attributable to the additional interference in the quantum case. This is to be contrasted with the situation on ope…
Quantum causality extends the conventional notion of fixed causal structure by allowing channels and operations to act in an indefinite causal order. The importance of such an indefinite causal order ranges from the foundational---e.g. towards a theory of quantum gravity---to the applied---e.g. for advantages in communication and computation. In this review, we will walk through the basic theory of indefinite causal order and focus on experiments that rely on a physically realisable indefinite causal ordered process---the quantum switch.
Causal modelling provides a powerful set of tools for identifying causal structure from observed correlations. It is well known that such techniques fail for quantum systems, unless one introduces `spooky' hidden mechanisms. Whether one can produce a genuinely quantum framework in order to discover causal structure remains an open question. Here we introduce a new framework for quantum causal modelling that allows for the discovery of causal structure. We define quantum analogues for many of the core features of classical causal modelling techniques, including the Causal Markov Condition and F…
Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to-date. In this work, we prove some critical properties of CSS codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla…
Finding a causal model for a set of classical variables is now a well-established task---but what about the quantum equivalent? Even the notion of a quantum causal model is controversial. Here, we present a causal discovery algorithm for quantum systems. The input to the algorithm is a process matrix describing correlations between quantum events. Its output consists of different levels of information about the underlying causal model. Our algorithm determines whether the process is causally ordered by grouping the events into causally-ordered non-signaling sets. It detects if all relevant com…
In studies of entanglement, finding out if a state is entangled and quantifying the amount of entanglement contained in a state are related but different questions. Similarly in studies of causality, finding out the causal structures compatible with a model and quantifying the causal strengths are related but different questions. Recently much research have been directed towards the first question but considerably less attention is paid to the second one. In this paper we propose axioms for all reasonable quantitative measures of causality to obey. The axioms apply to a broad family of operati…
Higher-order quantum processes respecting closed labs in a spacetime have quantum controlled causal order
International Centre for Theory of Quantum Technologies, University of Gdańsk（格但斯克大学量子理论国际中心）；Institute for Theoretical Physics, ETH Zurich（苏黎世联邦理工学院理论物理研究所）；Université Grenoble Alpes, Inria（格勒诺布尔阿尔卑斯大学法国国家信息与自动化研究所）
We investigate the ground-state Riemannian metric and the cyclic quantum distance of an inhomogeneous quantum Ising spin-1/2 chain in a transverse field. This model can be diagonalized by using a general canonical transformation to the fermionic Hamiltonian mapped from the spin system. The ground-state Riemannian metric is derived exactly on a parameter manifold ring $S^1$, which is introduced by performing a gauge transformation to the spin Hamiltonian through a twist operator. The ground-state cyclic quantum distance and the second derivative of the ground-state energy are studied in differe…
We characterize the algebraic structure of semi-direct product of cyclic groups, $\Z_{N}\rtimes\Z_{p}$, where $p$ is an odd prime number which does not divide $q-1$ for any prime factor $q$ of $N$, and provide a polynomial-time quantum computational algorithm solving hidden symmetry subgroup problem of the groups.
In this paper we investigate the relationship between the efficiency of a cyclic quantum heat engine with the Hilbert space dimension of the thermal baths. By means of a general inequality, we show that the Carnot efficiency can be obtained only when both the hot and cold baths are infinitely large. By further introducing a specific model where the baths are constituted of ensembles of finite-dimensional particles, we further demonstrate the relationship between the engine's power and efficiency, with the dimension of the working substance and the bath particles.
In this note, we present a construction of new nonbinary quantum codes with good parameters. These codes are obtained by applying the Calderbank-Shor-Steane (CSS) construction. In order to do this, we show the existence of (classical) cyclic codes whose defining set consists of only one cyclotomic coset containing at least two consecutive integers.
The spatial modes of photons are one realization of a QuDit, a quantum system that is described in a D-dimensional Hilbert space. In order to perform quantum information tasks with QuDits, a general class of D-dimensional unitary transformations is needed. Among these, cyclic transformations are an important special case required in many high-dimensional quantum communication protocols. In this paper, we experimentally demonstrate a cyclic transformation in the high-dimensional space of photonic orbital angular momentum (OAM). Using simple linear optical components, we show a successful four-f…
We give sufficient conditions for self-orthogonality with respect to symplectic, Euclidean and Hermitian inner products of a wide family of quasi-cyclic codes of index two. We provide lower bounds for the symplectic weight and the minimum distance of the involved codes. Supported in the previous results, we show algebraic constructions of good quantum codes and determine their parameters.
Entanglement-assisted quantum (QUENTA) codes are a subclass of quantum error-correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. In this paper, it is shown a general method to construct QUENTA codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of QUENTA codes. Two families of QUENTA codes are maximal distance separable (MDS), and one is almost MDS or alm…
A new Gray map which is both an isometry and a weight preserving map from R=F_2+vF_2+v^2F_2 to (F_2)^3 is defined. A construction for quantum error correcting codes from cyclic codes over finite ring R=F_2+vF_2+v^2F_2, v^3=v is given. The parameters of quantum codes which are obtained from cyclic codes over R are determined.
In fundamental theories that accounts for quantum gravitational effects, the spacetime causal structure is expected to be quantum uncertain. Previous studies of quantum causal structure focused on finite-dimensional systems. Here we present an algebraic framework that incorporates both finite- and infinite-dimensional systems including quantum fields. Thanks to the absence of a definite spacetime causal structure, Lagrangian quantum field theories can be studied on a quantum superposition of spacetimes with a point identification structure.
Quantum teleportation is a very helpful information-theoretic protocol that allows to transfer an unknown arbitrary quantum state from one location to another without having to transmit the quantum system through the intermediate region. Quantum states, quantum channels, and indefinite causal structures are all examples of quantum causal structures that not only enable advanced quantum information processing functions, but can also model causal structures in nonclassical spacetimes. In this letter, we develop quantum teleportation of arbitrary quantum causal structures, as formalized by the pr…
It is recently conjectured in quantum information processing that phase-shift errors occur with high probability than qubit-flip errors, hence the former is more disturbing to quantum information than the later one. This leads us to construct asymmetric quantum error controlling codes to protect quantum information over asymmetric channels, $\Pr Z \geq \Pr X$. In this paper we present two generic methods to derive asymmetric quantum cyclic codes using the generator polynomials and defining sets of classical cyclic codes. Consequently, the methods allow us to construct several families of asymm…
In this paper, we construct a tensor network representation of quantum causal histories, as a step towards directly representing states in quantum gravity via bulk tensor networks. Quantum causal histories are quantum extensions of causal sets in the sense that on each event in a causal set is assigned a Hilbert space of quantum states, and the local causal evolutions between events are modeled by completely positive and trace-preserving maps. Here we utilize the channel-state duality of completely positive and trace-preserving maps to transform the causal evolutions to bipartite entangled sta…
We develop a generic model for a cyclic quantum heat engine that makes it possible to coherently amplify a periodically modulated input signal without the need to couple the working medium to multiple reservoirs at the same time. Instead, we suggest an operation principle that is based on the spontaneous creation of population inversion in incomplete relaxation processes induced by periodic temperature variations. Focusing on Lindblad dynamics and systems with equally spaced energy levels, e.g. qubits or quantum harmonic oscillators, we derive a general working criterion for such cyclic quantu…
It is well known that quantum codes can be constructed by means of classical symplectic dual-containing codes. This paper considers a family of two-generator quasi-cyclic codes and derives sufficient conditions for these codes to be symplectic dual-containing. Then, a new method for constructing binary quantum codes using symplectic dual-containing codes is proposed. As an application, we construct 8 binary quantum codes that exceed the best-known results. Further, another 36 new binary quantum codes are obtained by propagation rules, all of which improve the lower bound on the minimum distanc…
The Hamiltonian of an evolving Universe is shown to be formally equivalent to that of a driven quantum system, whose driving follows from the temporal dependence of the spacetime metric. This analogy allows insights from the field of driven quantum systems to be applied to cosmological settings. In particular, it is shown that periodic cyclic cosmologies are generically prohibited (except under certain limiting constraints) due to their correspondence with periodically-driven quantum systems (which are typically expected to experience unbounded heating in the infinite future). This result high…
Uncovering whether strong system-bath coupling can be an advantageous operation resource for energy conversion can facilitate the development of efficient quantum heat engines (QHEs). Yet, a consensus on this ongoing debate is still lacking owing to challenges arising from treating strong couplings. Here we conclude the debate for optimal linear cyclic QHEs operated under a small temperature difference by revealing the detrimental role of strong system-bath coupling in their optimal operations. We analytically demonstrate that both the efficiency at maximum power and maximum efficiency of stro…
It has been shown classically that combining two chaotic random walks can yield an ordered(periodic) walk. Our aim in this paper is to find a quantum analog for this rather counter-intuitive result. We study chaotic and periodic nature of cyclic quantum walks and focus on a unique situation wherein a periodic quantum walk on a 3-cycle graph is generated via a deterministic combination of two chaotic quantum walks on the same graph. We extend our results to even-numbered cyclic graphs, specifically a 4-cycle graph too. Our results will be relevant in quantum cryptography and quantum chaos contr…
In this paper, a construction of a pair of "regular" quasi-cyclic LDPC codes as ingredient codes for a quantum error-correcting code is proposed. That is, we find quantum regular LDPC codes with various weight distributions. Furthermore our proposed codes have lots of variations for length, code rate. These codes are obtained by a descrete mathematical characterization for model matrices of quasi-cyclic LDPC codes. Our proposed codes achieve a bounded distance decoding (BDD) bound, or known as VG bound, and achieve a lower bound of the code length.
