Quantum Computation Theory Research Articles

Contraction of ZX diagrams with triangles via stabiliser decompositions

Recent advances in classical simulation of Clifford+T circuits make use of the ZX calculus to iteratively decompose and simplify magic states into stabiliser terms We improve on this method by studying stabiliser decompositions of ZX diagrams involving the triangle operation. We show that this technique greatly speeds up the simulation of quantum circuits involving multi-controlled gates which can be naturally represented using triangles. We implement our approach in the QuiZX library (2022 A. Kissinger amd J. van de Wetering Quantum Science and Technology 7, 044001), (2022 A. Kissinger et al F. Le Gall and T. Morimae, ed. 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), Leibniz International Proceedings in Informatics (LIPIcs) 232, Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, pp 5:1–5:13) and demonstrate a significant simulation speed-up (up to multiple orders of magnitude) for random circuits and a variation of previously used benchmarking circuits. Furthermore, we use our software to contract diagrams representing the gradient variance of parametrised quantum circuits, which yields a tool for the automatic numerical detection of the barren plateau phenomenon in ansätze used for quantum machine learning. Compared to traditional statistical approaches, our method yields exact values for gradient variances and only requires contracting a single diagram. The performance of this tool is competitive with tensor network approaches, as demonstrated with benchmarks against the quimb library (2018 J. Gray Journal of Open Source Software 3, 819).

Bases for optimising stabiliser decompositions of quantum states

Stabiliser states play a central role in the theory of quantum computation. For example, they are used to encode computational basis states in the most common quantum error correction schemes. Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as a superposition of stabiliser states. Understanding the structure of stabiliser decompositions has significant applications in verifying and simulating near-term quantum computers. We introduce and study the vector space of linear dependencies of n-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in n. We construct elegant bases of linear dependencies of constant size three. Critically, our sparse bases can be computed without first compiling a dictionary of all n-qubit stabiliser states. We utilise them to explicitly compute the stabiliser extent of states of more qubits than is feasible with existing techniques. Finally, we delineate future applications to improving theoretical bounds on the stabiliser rank of magic states.

Limitations of Classically Simulable Measurements for Quantum State Discrimination.

In the realm of fault-tolerant quantum computing, stabilizer operations play a pivotal role, characterized by their remarkable efficiency in classical simulation. This efficiency sets them apart from nonstabilizer operations within the quantum computational theory. In this Letter, we investigate the limitations of classically simulable measurements in distinguishing quantum states. We demonstrate that any pure magic state and its orthogonal complement of odd prime dimensions cannot be unambiguously distinguished by stabilizer operations, regardless of how many copies of the states are supplied. We also reveal intrinsic similarities and distinctions between the quantum resource theories of magic states and entanglement in quantum state discrimination. The results emphasize the inherent limitations of classically simulable measurements and contribute to a deeper understanding of the quantum-classical boundary.

Schrödinger as a Quantum Programmer: Estimating Entanglement via Steering

Quantifying entanglement is an important task by which the resourcefulness of a quantum state can be measured. Here, we develop a quantum algorithm that tests for and quantifies the separability of a general bipartite state by using the quantum steering effect, the latter initially discovered by Schrödinger. Our separability test consists of a distributed quantum computation involving two parties: a computationally limited client, who prepares a purification of the state of interest, and a computationally unbounded server, who tries to steer the reduced systems to a probabilistic ensemble of pure product states. To design a practical algorithm, we replace the role of the server with a combination of parameterized unitary circuits and classical optimization techniques to perform the necessary computation. The result is a variational quantum steering algorithm (VQSA), a modified separability test that is implementable on quantum computers that are available today. We then simulate our VQSA on noisy quantum simulators and find favorable convergence properties on the examples tested. We also develop semidefinite programs, executable on classical computers, that benchmark the results obtained from our VQSA. Thus, our findings provide a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory. They also demonstrate the value of a parameterized mid-circuit measurement in a VQSA.

Magic behaviors in non-Markovian environment

Magic as an essential resource in fault-tolerant quantum computation may shed insight into quantum information processing of open quantum system. In this work, we study the dynamics of magic for atom system immersed in non-Markovian reservoir. It is found that magic decay process can be delayed and adjusted by the strength of classical driving field and non-Markovian memory effect. In addition, it is shown that the magic state is less susceptible to decay the process of system-environment coupling than stabilizer state, and for two atoms coupling to two independent reservoirs, magic can be generated. The results may be beneficial for theory and experiment of quantum computation and information in open quantum system.

The fundamental theory of quantum computing, applications in practical fields, and future challenges

Building on the fundamental principles of quantum computing, this paper sequentially presents the development of quantum computing and its associated application domains. We also analyze and compare the differences between quantum and classical computing, highlighting the benefits of quantum computing for specific problems. To comprehend the advantages of quantum computing in particular fields, extensive research has been conducted in three areas: cryptography, chemistry, and artificial intelligence. We also discuss the wider applications of this emerging technology for future investigation, considering the current state of quantum computing development and anticipated trends.

The ZX-calculus as a language for topological quantum computation

Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.

Quantum double structure in cold atom superfluids

The theory of topological quantum computation is underpinned by two important classes of models. One is based on non-abelian Chern–Simons theory, which yields the so-called SU(2)k anyon models that often appear in the context of electrically charged quantum fluids. The physics of the other is captured by symmetry broken Yang–Mills theory in the absence of a Chern–Simons term and results in the so-called quantum double models. Extensive resources have been invested into the search for SU(2)k anyon quasi-particles, in particular, the so-called Ising anyons (k = 2) of which Majorana zero modes are believed to be an incarnation. In contrast to the SU(2)k models, quantum doubles have attracted little attention in experiments despite their pivotal role in the theory of error correction. Beyond topological error correcting codes, the appearance of quantum doubles has been limited to contexts primarily within mathematical physics, and as such, they are of seemingly little relevance for the study of experimentally tangible systems. However, recent works suggest that quantum double anyons may be found in spinor Bose–Einstein condensates. In light of this, the core purpose of this article is to provide a self-contained exposition of the quantum double structure, framed in the context of spinor condensates, by constructing explicitly the quantum doubles for various ground state symmetry groups and discuss their experimental realisability. We also derive analytically an equation for the quantum double Clebsch–Gordan coefficients from which the relevant braid matrices can be worked out. Finally, the existence of a particle-vortex duality is exposed and illuminated upon in this context.

Target Tracking Control of UAV Through Deep Reinforcement Learning

This study presents an innovative reinforcement-learning-based control algorithm for a vertical take-off and landing (VTOL) aircraft under wind disturbances. In our approach, the tracking control problem of the VTOL aircraft is formulated as a Markov decision process, and the appropriate system state, reward function, and soft update method are presented. To improve the control accuracy under wind disturbances, three kinds of wind fields were added in the learning environment to expand the exploration space and simulate the effect of wind disturbances on the flight control. Moreover, to ensure the tracking accuracy and the practical implementation, a quantum-inspired experience replay strategy was developed based on quantum computation theory. In this strategy, the preparation operation scheme was designed to encourage the exploration and speed up the convergence. The depreciation operation method was developed to enrich the sample diversity, which increased the robustness of the controller and allowed the control strategy learned in the numerical simulations to be directly transferred into real-world environments. Numerical simulations, hardware-in-the-loop experiments, and real-world flight experiments were conducted to evaluate the performance and merits of the proposed method. The results demonstrated high accuracy and effectiveness and good robustness of the proposed control algorithm in terms of standoff target tracking control and flight stability.

КВАНТОВОЕ ПРЕВОСХОДСТВО В ПАРАДИГМЕ МАТЕМАТИЧЕСКОЙ ТЕОРИИ СЛОЖНОСТИ И ТЕОРИИ РЕКУРСИВНЫХ ФУНКЦИЙ

The article is devoted to the issues of quantum computational number theory. In particular, the main approaches used in quantum computing and their differences from the classical ones, according to the basic mathematical canons laid down in the mathematical theory of complexity and the theory of recursive functions.

Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes (‘quasi codes’). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.

General Conditions for Universality of Quantum Hamiltonians

Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the result concerns the theory of analogue Hamiltonian simulation - a promising application of near-term quantum technology - the proof relies on abstract complexity theoretic concepts and the theory of quantum computation. As well as providing simplified proofs of previous Hamiltonian universality results, and offering a route to new universal constructions, the results in this paper give insight into the origins of universality. For example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.

КВАНТОВАЯ ВЫЧИСЛИТЕЛЬНАЯ ТЕОРИЯ ЧИСЕЛ

The article is devoted to the issues of quantum computational number theory. The prerequisites that gave impetus to its formation, as well as the main provisions regulating its further development, are revealed.

Development and simulation of the quantum microarchitecture operation using an optimization algorithm for a quantum calculation process

The theory of quantum computing is being actively developed. Despite the fact that quantum computing device has some peculiarities, any task designed for a classical computer can be reproduced on a quantum system. The aim of the research is to develop a methodology for constructing models of quantum systems simulators using hardware simulators. The subject of this research is methods for constructing quantum computing devices and systems. In the course of the research, the system analysis methods and computer modeling, methods of object-oriented design were used. Modern high-level languages were used to solve the problems of constructing a modular quantum system with an open architecture. The work of a quantum microarchitecture using an algorithm for optimizing a quantum computational process has been developed and modeled. A universal methodology for modeling algorithms of a quantum nature using the hardware core and requirements for the mutual operation of software and hardware components for the efficient operation of a quantum system is derived.

Quantum computing methods for supervised learning

The last two decades have seen an explosive growth in the theory and practice of both quantum computing and machine learning. Modern machine learning systems process huge volumes of data and demand massive computational power. As silicon semiconductor miniaturization approaches its physics limits, quantum computing is increasingly being considered to cater to these computational needs in the future. Small-scale quantum computers and quantum annealers have been built and are already being sold commercially. Quantum computers can benefit machine learning research and application across all science and engineering domains. However, owing to its roots in quantum mechanics, research in this field has so far been confined within the purview of the physics community, and most work is not easily accessible to researchers from other disciplines. In this paper, we provide a background and summarize key results of quantum computing before exploring its application to supervised machine learning problems. By eschewing results from physics that have little bearing on quantum computation, we hope to make this introduction accessible to data scientists, machine learning practitioners, and researchers from across disciplines.

Intelligent Trajectory Planning in UAV-Mounted Wireless Networks: A Quantum-Inspired Reinforcement Learning Perspective

In this letter, we consider a wireless uplink transmission scenario in which an unmanned aerial vehicle (UAV) serves as an aerial base station collecting data from ground users. To optimize the expected sum uplink transmit rate without any prior knowledge of ground users (e.g., locations, channel state information and transmit power), the trajectory planning problem is optimized via the quantum-inspired reinforcement learning (QiRL) approach. Specifically, the QiRL method adopts novel probabilistic action selection policy and new reinforcement strategy, which are inspired by the collapse phenomenon and amplitude amplification in quantum computation theory, respectively. Numerical results demonstrate that the proposed QiRL solution can offer natural balancing between exploration and exploitation via ranking collapse probabilities of possible actions, compared to the traditional reinforcement learning approaches that are highly dependent on tuned exploration parameters.

Blind Source Separation Based on Quantum Slime Mould Algorithm in Impulse Noise

In order to resolve engineering problems that the performance of the traditional blind source separation (BSS) methods deteriorates or even becomes invalid when the unknown source signals are interfered by impulse noise with a low signal-to-noise ratio (SNR), a more effective and robust BSS method is proposed. Based on dual-parameter variable tailing (DPVT) transformation function, moving average filtering (MAF), and median filtering (MF), a filtering system that can achieve noise suppression in an impulse noise environment is proposed, noted as MAF-DPVT-MF. A hybrid optimization objective function is designed based on the two independence criteria to achieve more effective and robust BSS. Meanwhile, combining quantum computation theory with slime mould algorithm (SMA), quantum slime mould algorithm (QSMA) is proposed and QSMA is used to solve the hybrid optimization objective function. The proposed method is called BSS based on QSMA (QSMA-BSS). The simulation results show that QSMA-BSS is superior to the traditional methods. Compared with previous BSS methods, QSMA-BSS has a wider applications range, more stable performance, and higher precision.

Optimized geometric quantum computation with a mesoscopic ensemble of Rydberg atoms

We propose a nonadiabatic non-Abelian geometric quantum operation scheme to realize universal quantum computation with mesoscopic Rydberg atoms. A single control atom entangles a mesoscopic ensemble of target atoms through long-range interactions between Rydberg states. We demonstrate theoretically that both the single qubit and two-qubit quantum gates can achieve high fidelities around or above 99.9% in ideal situations. Besides, to address the experimental issue of Rabi frequency fluctuation (Rabi error) in Rydberg atom and ensemble, we apply the dynamical-invariant-based zero systematic-error sensitivity (ZSS) optimal control theory to the proposed scheme. Our numerical simulations show that the average fidelity could be 99.98% for single ensemble qubit gate and 99.94% for two-qubit gate even when the Rabi frequency of the gate laser acquires 10% fluctuations. We also find that the optimized scheme can also reduce errors caused by higher-order perturbation terms in deriving the Hamiltonian of the ensemble atoms. To address the experimental issue of decoherence error between the ground state and Rydberg levels in Rydberg ensemble, we introduce a dispersive coupling regime between Rydberg and ground levels, based on which the Rydberg state is adiabatically discarded. The numerical simulation demonstrate that the quantum gate is enhanced. By combining strong Rydberg atom interactions, nonadiabatic geometric quantum computation, dynamical invariant and optimal control theory together, our scheme shows a new route to construct fast and robust quantum gates with mesoscopic atomic ensembles. Our study contributes to the ongoing effort in developing quantum information processing with Rydberg atoms trapped in optical lattices or tweezer arrays.

Symmetry-Protected Self-Correcting Quantum Memories

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D "cluster-state" model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally.

Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits.

A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order of the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the lengths of error strings do not grow by more than a constant factor. When applied to the Fibonacci code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. These results also apply directly to braiding of topological defects in surface codes. Our results reformulate the notion of braiding in general as an effectively instantaneous process, rather than as an adiabatic, slow process.