Concrete Quantum Research Articles

Storage properties of a quantum perceptron.

Driven by growing computational power and algorithmic developments, machine learning methods have become valuable tools for analyzing vast amounts of data. Simultaneously, the fast technological progress of quantum information processing suggests employing quantum hardware for machine learning purposes. Recent works discuss different architectures of quantum perceptrons, but the abilities of such quantum devices remain debated. Here, we investigate the storage capacity of a particular quantum perceptron architecture by using statistical mechanics techniques and connect our analysis to the theory of classical spin glasses. Specifically, we focus on one concrete quantum perceptron model and explore its storage properties in the limit of a large number of inputs.

Concrete quantum cryptanalysis of binary elliptic curves via addition chain

Thus far, several papers reported concrete resource estimates of Shor’s quantum algorithm for solving the elliptic curve discrete logarithm problem. In this paper, we study quantum FLT-based inversion algorithms over binary elliptic curves. There are two major algorithms proposed by Banegas et al. and Putranto et al., where the former and latter algorithms achieve fewer numbers of qubits and smaller depths of circuits, respectively. We propose two quantum FLT-based inversion algorithms that essentially outperform previous FLT-based algorithms and compare the performance for NIST curves of the degree n. Specifically, for all n, our first algorithm achieves fewer qubits than Putranto et al.’s one without sacrificing the number of Toffoli gates and the depth of circuits, while our second algorithm achieves smaller depths of circuits without sacrificing the number of qubits and Toffoli gates. For example, when n=571\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 571$$\\end{document}, the number of qubits of our first algorithm is 74 % of that of Putranto et al.’s one, while the depth of our second algorithm is 83 % of that of Banegas et al.’s one. The improvements stem from the fact that FLT-based inversions can be performed with arbitrary sequences of addition chains for n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n - 1$$\\end{document} although both Banegas et al. and Putranto et al. follow fixed sequences that were introduced by Itoh and Tsujii’s classical FLT-based inversion. In particular, we analyze how several properties of addition chains, which do not affect the computational resources of classical FLT-based inversions, affect the computational resources of quantum FLT-based inversions and find appropriate sequences.

Quantum entanglement dynamics of spacetime and matter

It was known long ago that quantum theory and general relativity are in sharp conflict in their foundations. Their fundamental inconsistencies render a consistent theory of quantum gravity the most challenging problem in physics. Here we propose an information-complete quantum field theory (ICQFT), which describes elementary fermions, their gauge fields, and gravity (together, called the trinary fields) as an elementary trinity without any conceptual inconsistency of existing theories. The ICQFT unifies matter and spacetime (gravity) as information via spacetime-matter entanglement, which encodes complete physical predictions of the theory and leads to a compelling solution to the problem of time. We consider two particular forms of spacetime-matter entangled states and their physical consequences. One of them results in a universal relation between entanglement entropy and geometry (area and volume), allowing us to determine the cosmological constant term in the classical Einstein equation. Based on a quantum-information definition of dark energy, our Universe is not strictly holographic. We predict the interior quantum state of a Schwarzschild black hole to be maximally information-complete. As a concrete quantum formulation of gravity coupled with matter, the ICQFT is quantum entanglement dynamics for spacetime and matter and eliminates the conceptual obstacles of existing quantum gravity theory.

Solitonic Symmetry beyond Homotopy: Invertibility from Bordism and Noninvertibility from Topological Quantum Field Theory.

Solitonic symmetry has been believed to follow the homotopy-group classification of topological solitons. Here, we point out a more sophisticated algebraic structure when solitons of different dimensions coexist in the spectrum. We uncover this phenomenon in a concrete quantum field theory, the 4D CP^{1} model. This model has two kinds of solitonic excitations-vortices and hopfions-which would follow two U(1) solitonic symmetries according to homotopy groups. Nevertheless, we demonstrate the nonexistence of the hopfion U(1) symmetry by evaluating the hopfion charge of vortex operators. We clarify that what conserves hopfion numbers is a noninvertible symmetry generated by 3D spin topological quantum field theories (TQFTs). Its invertible part is just Z_{2}, which we recognize as a spin bordism invariant. Compared with the 3D CP^{1} model, our work suggests a unified description of solitonic symmetries and couplings to topological phases.

Are inertial vacua equivalent in Lorentz-violating theories? Does it matter?

Several approaches to quantum gravity suggest violations of Lorentz symmetry as low-energy signatures. This article uses a concrete Lorentz-violating quantum field theory to study different inertial vacua. We show that they are unitarily inequivalent and that the vacuum in one inertial frame appears, in a different inertial frame, to be populated with particles of arbitrarily high momenta. At first sight, this poses a critical challenge to the physical validity of Lorentz-violating theories, since we do not witness vacuum excitations by changing inertial frames. Nevertheless, we demonstrate that inertial Unruh-De Witt detectors are insensitive to these effects. We also discuss the Hadamard condition for this Lorentz-violating theory.

Generation of large-scale continuous-variable cluster states multiplexed both in time and frequency domains.

A large-scale continuous variable (CV) cluster state is necessary in quantum information processing based on measurement-based quantum computing (MBQC). Specially, generating a large-scale CV cluster state multiplexed in a time domain is easier to implement and has strong scalability in experiment. Here one-dimensional (1D) large-scale dual-rail CV cluster states multiplexed both in time and frequency domains are parallelly generated, which can be further extended to a three-dimensional (3D) CV cluster state by combining two time-delay nondegenerate optical parametric amplification systems with beam-splitters. It is shown that the number of parallel arrays depends on the corresponding frequency comb lines, the partite number of each array can be very large (million), and the scale of the 3D cluster state can be ultra-large. In addition, the concrete quantum computing schemes of applying the generated 1D and 3D cluster states are also demonstrated. Our schemes may pave the way for fault-tolerant and topologically protected MBQC in hybrid domains, by further combining with efficient coding and quantum error correction.

Towards practical Quantum Credit Risk Analysis

In recent years a CRA (Credit Risk Analysis) quantum algorithm with a quadratic speedup over classical analogous methods has been introduced [1]. We propose a new variant of this quantum algorithm with the intent of overcoming some of the most significant limitations (according to business domain experts) of this approach. In particular, we describe a method to implement a more realistic and complex risk model for the default probability of each portfolio’s asset, capable of taking into account multiple systemic risk factors. In addition, we present a solution to increase the flexibility of one of the model’s inputs, the Loss Given Default, removing the constraint to use integer values. This specific improvement addresses the need to use real data coming from the financial sector in order to establish fair benchmarking protocols.Although these enhancements come at a cost in terms of circuit depth and width, they nevertheless show a path towards a more realistic software solution. Recent progress in quantum technology shows that eventually, the increase in the number and reliability of qubits will allow for useful results and meaningful scales for the financial sector, also on real quantum hardware, paving the way for a concrete quantum advantage in the field.The paper also describes experiments conducted on simulators to test the circuit proposed and contains an assessment of the scalability of the approach presented.

Verification of Information Thermodynamics in a Trapped Ion System.

Information thermodynamics has developed rapidly over past years, and the trapped ions, as a controllable quantum system, have demonstrated feasibility to experimentally verify the theoretical predictions in the information thermodynamics. Here, we address some representative theories of information thermodynamics, such as the quantum Landauer principle, information equality based on the two-point measurement, information-theoretical bound of irreversibility, and speed limit restrained by the entropy production of system, and review their experimental demonstration in the trapped ion system. In these schemes, the typical physical processes, such as the entropy flow, energy transfer, and information flow, build the connection between thermodynamic processes and information variation. We then elucidate the concrete quantum control strategies to simulate these processes by using quantum operators and the decay paths in the trapped-ion system. Based on them, some significantly dynamical processes in the trapped ion system to realize the newly proposed information-thermodynamic models is reviewed. Although only some latest experimental results of information thermodynamics with a single trapped-ion quantum system are reviewed here, we expect to find more exploration in the future with more ions involved in the experimental systems.

Direct measurement of density-matrix elements using a phase-shifting technique

A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. However, the direct measurement protocols to date are primarily based on weak or strong measurements with an ancillary pointer, which interacts with the investigated system to extract information about the specified elements. Here, we present a direct measurement scheme based on phase-shifting operations which do not need ancillary pointers. In this method, estimates of at most six expectation values of projective observables suffice to determine any specific element of an unknown quantum density matrix. A concrete quantum circuit to implement this direct measurement protocol for multiqubit states is provided, which is composed of just single-qubit gates and two multiqubit controlled-phase gates. This scheme is also extended for the direct measurement of the density matrix of continuous-variable quantum states. Our method can be used in quantum information applications where only partial information about the quantum state needs to be extracted, for example, problems such as entanglement witnessing, fidelity estimation of quantum systems, and quantum coherence estimation.

Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as the barren plateau phenomenon, in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown that, for the hardware efficient ansatz and the MPS-inspired ansatz, there exist barren plateaus, while for the QCNN ansatz and the tree tensor network ansatz, there exists no barren plateau.

Computational Adequacy for Substructural Lambda Calculi

Substructural type systems, such as affine (and linear) type systems, are type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum programming. We describe one linear and one affine type systems and we formulate abstract categorical models for both of them which are sound and computationally adequate. We also show, under basic assumptions, that interpreting lambda abstractions via a monoidal closed structure (a popular method for linear type systems) necessarily leads to degenerate and inadequate models for call-by-value affine type systems with recursion. In our categorical treatment, a solution to this problem is clearly presented. Our categorical models are more general than linear/non-linear models used to study linear logic and we present a homogeneous categorical account of both linear and affine type systems in a call-by-value setting. We also give examples with many concrete models, including classical and quantum ones.

Tools for quantum network design

Quantum networks will enable the implementation of communication tasks with qualitative advantages with respect to the communication networks known today. While it is expected that the first demonstrations of small scale quantum networks will take place in the near term, many challenges remain to scale them. To compare different solutions, optimize over parameter space, and inform experiments, it is necessary to evaluate the performance of concrete quantum network scenarios. Here, the authors review the state-of-the-art of tools for evaluating the performance of quantum networks. The authors present them from three different angles: information-theoretic benchmarks, analytical tools, and simulation.

Concrete quantum cryptanalysis of binary elliptic curves

This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2(n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2(n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography.

Generalized Riesz systems and orthonormal sequences in Krein spaces

We analyze special classes of bi-orthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with generalized Riesz systems. In this way, the notion of the first/second type sequences is introduced and studied. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.

Improving fidelity of quantum secret sharing by weak measurement

Quantum secret sharing is an important branch of quantum cryptography. In this paper, we firstly propose a simple four-party quantum secret sharing scheme with entangled states. Since the noisy channel has a great influence on the shared quantum secret, we analyze the impact of the noisy channel on quantum secret sharing and obtain the expression among the fidelity, noisy coefficient and shared quantum state coefficients. To enhance the fidelity of the shared secret, we use the weak measurement method to protect the scheme in noisy environment. Furthermore, we analyze the fidelity between the obtained secret and the initial one after the amplitude damped channel. Finally, we take the concrete quantum state as an example to compare with the original way and find that the weak measurement method has a certain significance to improve the fidelity of the secret state in the scheme.

Quantum Circuit Realization of Morphological Gradient for Quantum Grayscale Image

In this paper, based on the principle of classical morphology operations, the flat grayscale dilation and erosion operations are proposed for NEQR quantum image model. Furthermore, through combining these two morphology operations, we further realize the morphological gradient operation. As the basis of designing of grayscale morphology operations, a series of quantum circuit designs arepresented, which includes special add one operation UA1(n) and special subtract one operation US1(n) both for an n-length qubits sequence, quantum unitary operation UC, parallel subtractor (PS) module, quantum comparator output the large QCOL and quantum comparator output the small QCOS modules. When designsthe concrete quantum circuit, a sequence of UA1(n) and US1(n) modules are used to obtain the quantum image sets based on the shape of specific structuring element. Then, the searching for maximaor minima in a certain space is involved, which can be solved by cascading a series of QCOL and QCOS modules in certain order. Finally, the PS module can be used to calculate the difference of the maxima and minima for producing the morphological gradient. The circuit’s complexity analysis illustrate that our scheme is very lower to the classical morphology operations.

Dynamical and observational constraints on the warm little inflaton scenario

We explore the dynamics and observational predictions of the Warm Little Inflaton scenario, presently the simplest realization of warm inflation within a concrete quantum field theory construction. We consider three distinct types of scalar potentials for the inflaton, namely chaotic inflation with a quartic monomial potential, a Higgs-like symmetry breaking potential and a non-renormalizable plateau-like potential. In each case, we determine the parametric regimes in which the dynamical evolution is consistent for 50-60 e-folds of inflation, taking into account thermal corrections to the scalar potential and requiring, in particular, that the two fermions coupled directly to the inflaton remain relativistic and close to thermal equilibrium throughout the slow-roll regime and that the temperature is always below the underlying gauge symmetry breaking scale. We then compute the properties of the primordial spectrum of scalar curvature perturbations and the tensor-to-scalar ratio in the allowed parametric regions and compare them with Planck data, showing that this scenario is theoretically and observationally successful for a broad range of parameter values.

Geometric structure of quantum resources for Bell-diagonal states

Two-qubit Bell-diagonal states can be depicted as a tetrahedron in three dimensions. We investigate the structure of quantum resources, including coherence and quantum discord, in the tetrahedron. The ordering of different resources measures is a common problem in resource theories, and which measure should be chosen to investigate the structure of resources is still an open question. We consider the structure of quantum resources which is not affected by the choice of measure. Our work provides a complete structure of coherence and quantum discord for Bell-diagonal states. The pictorial approach also indicates how to explore the structure of resources even when we don't have consistent measure of a concrete quantum resource.

Quantum machine learning.

Fuelled by increasing computer power and algorithmic advances, machine learning techniques have become powerful tools for finding patterns in data. Quantum systems produce atypical patterns that classical systems are thought not to produce efficiently, so it is reasonable to postulate that quantum computers may outperform classical computers on machine learning tasks. The field of quantum machine learning explores how to devise and implement quantum software that could enable machine learning that is faster than that of classical computers. Recent work has produced quantum algorithms that could act as the building blocks of machine learning programs, but the hardware and software challenges are still considerable.

Quantum realization of the bilinear interpolation method for NEQR

In recent years, quantum image processing is one of the most active fields in quantum computation and quantum information. Image scaling as a kind of image geometric transformation has been widely studied and applied in the classical image processing, however, the quantum version of which does not exist. This paper is concerned with the feasibility of the classical bilinear interpolation based on novel enhanced quantum image representation (NEQR). Firstly, the feasibility of the bilinear interpolation for NEQR is proven. Then the concrete quantum circuits of the bilinear interpolation including scaling up and scaling down for NEQR are given by using the multiply Control-Not operation, special adding one operation, the reverse parallel adder, parallel subtractor, multiplier and division operations. Finally, the complexity analysis of the quantum network circuit based on the basic quantum gates is deduced. Simulation result shows that the scaled-up image using bilinear interpolation is clearer and less distorted than nearest interpolation.