Model Of Quantum Computation Research Articles

Realistic photon-number resolution in Gaussian boson sampling

Gaussian boson sampling (GBS) is a model of nonuniversal quantum computation that claims to demonstrate quantum supremacy with current technologies. This model entails sampling photocounting events from a multimode Gaussian state at the outputs of a linear interferometer. In this scheme, collision events—those with more than one photon for each mode—are infrequent. However, they are still used for validation purposes. Therefore, the limitation of realistic detectors to perfectly resolve adjacent photon numbers becomes pivotal. We derive a photocounting probability distribution in GBS schemes which is applicable for use with general detectors and photocounting techniques. This probability distribution is expressed in terms of functionals of the field-quadrature covariance matrix, e.g., Hafnian and Torontonian in the well-known special cases of photon-number-resolving and on-off detectors, respectively. Based on our results, we consider a GBS validation technique involving detectors with realistic photon-number resolution. Published by the American Physical Society 2024

S-слівна арифметика та високоточні обчислення

The intricacies of using S-word arithmetic, the influence of the value of the parameter S on the estimation of the rounding error are analyzed; what are high-precision calculations and where they are used. The problems of two-key cryptography, computer steganography and the problem of transcomputational complexity are considered as areas of application of S-word arithmetic. For the development of S-word arithmetic algorithms, sequential, parallel, quantum computing models are used, and systems of residual classes are used. The architectural features of the computer system for the implementation of an effective algorithm in various models of calculations are considered. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Peculiarities of transferring algorithms to another computing model are considered. An analysis of the complexity of implementing S-word arithmetic operations in sequential, parallel, and quantum computing models is carried out. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Information is provided about the ongoing scientific forum «Calculation optimization issues», the subject of which is closely related to the topic (1969-2023)

Quantum computation from dynamic automorphism codes

We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of N triangular patches, the DA color code encodes N logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.

Photonic cellular automaton simulation of relativistic quantum fields: Observation of Zitterbewegung

Quantum cellular automaton (QCA) is a model for universal quantum computation and a natural candidate for digital quantum simulation of relativistic quantum fields. Here we introduce the first photonic platform for implementing QCA simulation of a free relativistic Dirac quantum field in 1+1 dimension, through a Dirac quantum cellular automaton (DQCA). Encoding the field position degree of freedom in the orbital angular momentum (OAM) of single photons, our state-of-the-art setup experimentally realizes eight steps of a DQCA, with the possibility of having complete control over the input OAM state preparation and the output measurement making use of two spatial light modulators. Therefore, studying the distribution in the OAM space at each step, we were able to reproduce the time evolution of the free Dirac field observing the , an oscillatory movement extremely difficult to see in a real-case experimental scenario that is a signature of the interference of particle and antiparticle states. The accordance between the expected and measured oscillations certifies the simulator performances, paving the way towards the application of photonic platforms to the simulation of more complex relativistic effects. Published by the American Physical Society 2024

Classical computation over quantum architectures

Abstract The lack of purely Quantum Programming Languages constitutes a hurdle in the general description of quantum computational processes; the implementation is heavily dependent on the considered quantum computational model. To bypass the obstacle, this paper pursues a new direction, investigating the compilation of classical programming paradigms over different quantum computational models: Gate-Based, Measurement-Based and Adiabatic Quantum Computation. Since graphs can be exploited to describe both classical and quantum computations, the problem of graph encoding on quantum hardware is tightly connected to our purposes. As such, it holds a major relevance in our quest for quantum compilation. While studying these topics through the lenses of Graph Theory, declarative programming emerges as the ideal candidate for such endeavour. In this paper we consider some existing quantum computational models and for each of them we identify the main subtleties in the compilation of classical languages. In turn, we break these complexities down into easier problems to stimulate further developments in this area of research. As it turns out, the observations for each model differ widely. Nevertheless, as for the tasks here considered, no model seems to claim supremacy over the others. In contrast, declarative programming maintains the spot as the ideal candidate for quantum compilation, independently of the model.

Bell Sampling from Quantum Circuits.

A central challenge in the verification of quantum computers is benchmarking their performance as a whole and demonstrating their computational capabilities. In this Letter, we find a universal model of quantum computation, Bell sampling, that can be used for both of those tasks and thus provides an ideal stepping stone toward fault tolerance. In Bell sampling, we measure two copies of a state prepared by a quantum circuit in the transversal Bell basis. We show that the Bell samples are classically intractable to produce and at the same time constitute what we call a "circuit shadow": from the Bell samples we can efficiently extract information about the quantum circuit preparing the state, as well as diagnose circuit errors. In addition to known properties that can be efficiently extracted from Bell samples, we give several new and efficient protocols: an estimator of state fidelity, an error-mitigated estimator of Pauli expectation values, a test for the depth of a circuit, and an algorithm to estimate a lower bound on the number of T gates in the circuit. With some additional measurements, the latter algorithm can be used to learn a full description of states prepared by circuits with low T count.

Unifying flavors of fault tolerance with the ZX calculus

There are several models of quantum computation which exhibit shared fundamental fault-tolerance properties. This article makes commonalities explicit by presenting these different models in a unifying framework based on the ZX calculus. We focus on models of topological fault tolerance – specifically surface codes – including circuit-based, measurement-based and fusion-based quantum computation, as well as the recently introduced model of Floquet codes. We find that all of these models can be viewed as different flavors of the same underlying stabilizer fault-tolerance structure, and sustain this through a set of local equivalence transformations which allow mapping between flavors. We anticipate that this unifying perspective will pave the way to transferring progress among the different views of stabilizer fault-tolerance and help researchers familiar with one model easily understand others.

An atomic boson sampler.

A boson sampler implements a restricted model of quantum computing. It is defined by the ability to sample from the distribution resulting from the interference of identical bosons propagating according to programmable, non-interacting dynamics1. An efficient exact classical simulation of boson sampling is not believed to exist, which has motivated ground-breaking boson sampling experiments in photonics with increasingly many photons2-12. However, it is difficult to generate and reliably evolve specific numbers of photons with low loss, and thus probabilistic techniques for postselection7 or marked changes to standard boson sampling10-12 are generally used. Here, we address the above challenges by implementing boson sampling using ultracold atoms13,14 in a two-dimensional, tunnel-coupled optical lattice. This demonstration is enabled by a previously unrealized combination of tools involving high-fidelity optical cooling and imaging of atoms in a lattice, as well as programmable control of those atoms using optical tweezers. When extended to interacting systems, our work demonstrates the core abilities required to directly assemble ground and excited states in simulations of various Hubbard models15,16.

Hidden variable model for quantum computation with magic states on qudits of any dimension

It was recently shown that a hidden variable model can be constructed for universal quantum computation with magic states on qubits. Here we show that this result can be extended, and a hidden variable model can be defined for quantum computation with magic states on qudits with any Hilbert space dimension. This model leads to a classical simulation algorithm for universal quantum computation.

Efficient Implementation of Discrete-Time Quantum Walks on Quantum Computers.

Quantum walks have proven to be a universal model for quantum computation and to provide speed-up in certain quantum algorithms. The discrete-time quantum walk (DTQW) model, among others, is one of the most suitable candidates for circuit implementation due to its discrete nature. Current implementations, however, are usually characterized by quantum circuits of large size and depth, which leads to a higher computational cost and severely limits the number of time steps that can be reliably implemented on current quantum computers. In this work, we propose an efficient and scalable quantum circuit implementing the DTQW on the 2n-cycle based on the diagonalization of the conditional shift operator. For t time steps of the DTQW, the proposed circuit requires only O(n2+nt) two-qubit gates compared to the O(n2t) of the current most efficient implementation based on quantum Fourier transforms. We test the proposed circuit on an IBM quantum device for a Hadamard DTQW on the 4-cycle and 8-cycle characterized by periodic dynamics and by recurrent generation of maximally entangled single-particle states. Experimental results are meaningful well beyond the regime of few time steps, paving the way for reliable implementation and use on quantum computers.

Development of quantum computational model for analysis of drug delivery: Loading of human beta-defensin 3 and mitoxantrone onto composite polymers

A molecular dynamics simulation study evaluated the co-loading of mitoxantrone (MTX) and human beta-defensin 3 (HBD3) on PLGA and chitosan chains. Simulations were performed on four different polymeric systems. Electrostatic and vdW interactions between MTX and HBD3 are examined. According to the obtained results, MTX strongly interacts with PLGA. In addition, HBD3 interacts strongly with CHI. Moreover, the radial distribution function (RDF) was estimated for the compounds. The structural changes of HBD3 and the interaction potentials were considered when selecting a suitable polymeric carrier for HBD3. For this purpose, HBD3 in a water medium was considered a reference system. RMSD, RMSF, radius of gyration, and the secondary structure of HBD3 were analyzed. According to the results, HBD3 has a secondary structure similar to water when adsorbed on 10C and 5PC. CHI could stabilize the structure of HBD3. 5PC (5 PLGA and 5 CHI chains) is suitable for co-loading MTX and HBD3. Polymeric carriers can be designed using these molecular results.

Quantum Implementation of the SAND Algorithm and Its Quantum Resource Estimation for Brute-Force Attack.

The SAND algorithm is a family of lightweight AND-RX block ciphers released by DCC in 2022. Our research focuses on assessing the security of SAND with a quantum computation model. This paper presents the first quantum implementation of SAND (including two versions of SAND, SAND-64 and SAND-128). Considering the depth-times-width metric, the quantum circuit implementation of the SAND algorithm demonstrates a relatively lower consumption of quantum resources than that of the quantum implementations of existing lightweight algorithms. A generalized Grover-based brute-force attack framework was implemented and employed to perform attacks on two versions of the SAND algorithm. This framework utilized the g-database algorithm, which considered different plaintext-ciphertext pairs in a unified manner, reducing quantum resource consumption. Our findings indicate that the SAND-128 algorithm achieved the NIST security level I, while the SAND-64 algorithm fell short of meeting the requirements of security level I.

Construction of two-qubit gates using B Gate

Abstract In the circuit model of quantum computation, an entangling two-qubit gate and a set of single-qubit gates are used as universal gate set or basis gates for doing quantum computation. CNOT and iSWAP gates, the perfect entanglers that can create maximally entangled two-qubit states in one application, are broadly used entangling two-qubit basis gates in quantum computers. In this paper, we analyze the potentiality of B gate, an unexplored non-perfect entangler of the form, exp i π ( σ ˆ x ⨂ σ ˆ x ) 8 + i π ( σ ˆ y ⨂ σ ˆ y ) 16 , as an entangling two-qubit basis gate in quantum computers by studying its ability to generate other two-qubit gates. We derive a necessary condition for a two-qubit gate to be generated by n applications of B gate. Using this condition, we show that the gates that can be generated by two and three applications of B gate are contained in the 50% and 92.97% of the volume of Weyl chamber of two-qubit gates, respectively. We prove that two applications of B gate can generate both SWAP and SWAP † which is not possible for CNOT and iSWAP gates; further, we conjecture that three applications of B gate can generate all perfect entanglers. Finally, we discuss about the construction of a three independent parameter universal two-qubit quantum circuit using four B gates that can generate all two-qubit gates. In the end, we mention about the schemes to implement B gate in ion-trap quantum computers.

Deep Q-learning with hybrid quantum neural network on solving maze problems

Quantum computing holds great potential for advancing the limitations of machine learning algorithms to handle higher dimensions of data and reduce overall training parameters in deep learning (DL) models. This study uses a trainable variational quantum circuit (VQC) on a gate-based quantum computing model to investigate the potential for quantum benefit in a model-free reinforcement learning problem. Through a comprehensive investigation and evaluation of the current model and capabilities of quantum computers, we designed and trained a novel hybrid quantum neural network based on the latest Qiskit and PyTorch framework. We compared its performance with a full-classical CNN with and without an incorporated VQC. Our research provides insights into the potential of deep quantum learning to solve a maze problem and, potentially, other reinforcement learning problems. We conclude that reinforcement learning problems can be practical with reasonable training epochs. Moreover, a comparative study of full-classical and hybrid quantum neural networks is discussed to understand these two approaches’ performance, advantages, and disadvantages to deep Q-learning problems, especially on larger-scale maze problems larger than 4times 4.

Quantum simulation for partial differential equations with physical boundary or interface conditions

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions.

An Unbiased Quantum Random Number Generator Based on Boson Sampling

AbstractIt is proven that Boson sampling is a much promising model of optical quantum computation, which is applied to designing quantum computer successfully, such as “Jiuzhang”. However, the meaningful randomness of Boson sampling results has not been utilized or exploited. In this research, Boson sampling is applied to design a quantum random number generator (QRNG) by fully exploiting the randomness of Boson sampling results, and its prototype system is constructed with the programmable silicon photonic processor, which can generate unbiased random sequences and overcome the shortcomings of the existing discrete QRNGs such as source‐restricted, high demand for the photon number resolution capability of detector and slow self‐detection generator speed. Boson sampling is implemented as a random entropy source, and random bit strings with satisfactory randomness and uniformity can be obtained after post‐processing the sampling results. It is the first approach for applying the randomness of Boson sampling results to develop a practical prototype system for actual tasks, and the experiment results demonstrate that the designed Boson sampling‐based QRNG prototype system passes 15 tests of the NIST SP 800‐22 statistical test component, which proves that Boson sampling has great potential for practical applications with desirable performance besides quantum advantage.

Optimal, hardware native decomposition of parameterized multi-qubit Pauli gates

We show how to efficiently decompose a parameterized multi-qubit Pauli (PMQP) gate into native parameterized two-qubit Pauli (P2QP) gates minimizing both the circuit depth and the number of P2QP gates. Given a realistic quantum computational model, we argue that the technique is optimal in terms of the number of hardware native gates and the overall depth of the decomposition. Starting from PMQP gate decompositions for the path and star hardware graph, we generalize the procedure to any generic hardware graph and provide exact expressions for the depth and number of P2QP gates of the decomposition. Furthermore, we show how to efficiently combine the decomposition of multiple PMQP gates to further reduce the depth as well as the number of P2QP gates for a combinatorial optimization problem using the Lechner–Hauke–Zoller mapping.

High-Threshold Quantum Computing by Fusing One-Dimensional Cluster States.

We propose a measurement-based model for fault-tolerant quantum computation that can be realized with one-dimensional cluster states and fusion measurements only; basic resources that are readily available with scalable photonic hardware. Our simulations demonstrate high thresholds compared with other measurement-based models realized with basic entangled resources and 2-qubit fusion measurements. Its high tolerance to noise indicates that our practical construction offers a promising route to scalable quantum computing with quantum emitters and linear-optical elements.

Shor's Factoring Algorithm and Modular Exponentiation Operators

We provide a pedagogical presentation of Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor such large numbers. Shor's algorithm therefore has profound implication for public-key encryption such as RSA and Diffie–Hellman key exchange. We assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. Shor's algorithm contains a number of moving parts, and can be rather daunting at first. The literature is replete with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. They require a thicket of appendices and assume a knowledge of quantum algorithms and classical mathematics with which the reader might not be familiar. We therefore start with first principle derivations of the quantum Fourier transform (QFT) and quantum phase estimation (QPE), which are the essential building blocks of Shor's algorithm. We then go on to develop the theory of modular exponentiation (ME) operators, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also delve into the number theory that establishes the link between factorization and the period of the modular exponential function. We then apply the QPE algorithm to obtain Shor's factoring algorithm. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger integers, developing a systematic procedure that will find the ME operators for any semi-prime N=p×q (where q and p are prime). Finally, we factor the composite numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected.Quanta 2023; 12: 41–130.

Pauli-based model of quantum computation with higher-dimensional systems

Pauli-based model of quantum computation with higher-dimensional systems