QUANTUM COMPUTING FOR MULTI-QUBIT SYSTEMS USING SCHWINGER REPRESENTATION OF ANGULAR MOMENTUM

An alternative approach to superdense coding of information in quantum computing is proposed on the basis of Schwinger’s two-boson representation of angular momentum. Since the effective spin S = 2n-1 - ½ corresponds to the n-qubit system, this representation can be used in the quantum computing. Operators of the logical elements of the quantum circuit were found, performing superdense coding of information in the paired bosons representation. It is shown that for superdense coding of information, the results obtained in the spinor representation and in the representation of paired bosons coincide. For one-qubit systems, one of the two representations cannot be favored. In the case of n-qubit systems for n >> 1, the representation of paired bosons is probably more convenient for applications, since in this representation the explicit form of the Pauli operators X, Y, and Z does not depend on n.

Boson theory of angular momentum: Heisenberg ferromagnetism at low temperatures

Exact wave functions and coherent states for a forced damped harmonic oscillator

We discuss how the observation of population localization effects in periodically driven systems can be used to quantify the presence of quantum coherence in interacting qubit arrays. Essential for our proposal is the fact that these localization effects persist beyond tight-binding Hamiltonian models. This result is of special practical relevance in those situations where direct system probing using tomographic schemes becomes infeasible beyond a very small number of qubits. As a proof of principle, we study analytically a Hamiltonian system consisting of a chain of superconducting flux qubits under the effect of a periodic driving. We provide extensive numerical support of our results in the simple case of a two-qubit chain. For this system we also study the robustness of the scheme against different types of noise and disorder. We show that localization effects underpinned by quantum coherent interactions should be observable within realistic parameter regimes in chains with a larger number of qubits.

本文根据n维耦合谐振子矩阵形式的薛定谔(Schrodinger)方程，在表象理论的基础上，得到了哈密顿(Hamiltonian)算子ˆH 的矩阵元Hnn。通过构建一类完备的赋范线性空间，由泛函理论，证明了Hnn在此空间中是有界算子，同时求得哈密顿算子的本征值E；进而利用矩阵理论得到E的谱条件数公式。从这个表达式出发，得到了能量E的算子范数与谐振子的状态之间的关系式；研究了E的谱条件数与算子范数之间的关系，并估算E的算子范数上、下界的值，给出了能量谱条件数值大小的原因。结果表明：求出谱条件数与算子范数的大致范围，就可以根据谱条件数的准确值，在表象理论框架内，估测谐振子两个状态之间的差异程度，分析谐振子的状态特征。 This paper according to the matrix form of Schrodinger equation of n-dimension coupled harmonic oscillator, on the basis of representation theory, the element of matrix Hnn of Hamiltonian operator ˆH is derived. Through con-structing one of complete normed linear space, the Hnn is proved to be boundedness in this space by using functional theory, and eigenvalue E of Hamiltonian operator is obtained; and then the author get the spectrum condition number formula of E that is made use of matrix theory. From this expressions, the formula of operator norm of energy E and harmonic oscillator’s state is acquired. Researching the relationship between spectrum condition number of E and operator norm, the supremum and infimum of E operator norm is estimated, the reason of numerical size of energy spectrum condition number is presented. It turned out that: when the approximately range of spectrum condition number and operator norm is achieved, under the representation theory frame, the difference degree between two states of har-monic oscillator are estimated, which in terms of the exactly value of spectrum condition number, and analysis the fea-ture states of harmonic oscillator.

Gaussian Klauder coherent states of general time-dependent harmonic oscillator

In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.

Spinors over a Cone, Dirac Operator, and Representations of Spin(4, 4)

The static fluctuation approximation (SFA) is applied to compute the thermodynamic properties of a trapped two-dimensional (2D) interacting hard-sphere (HS) Bose gas in the weakly and strongly interacting regime. A mean-field approach involving a variational wave function is used to compute the mean-field energy as a function of temperature for each harmonic oscillator (HO) state plugged into the SFA technique. In the variational approach, a parameter $\ensuremath{\alpha}$ is introduced into the harmonic oscillator wave function in order to take into account the changes in the width when the repulsive interactions between the bosons are increased. In the weakly interacting regime, below the critical temperature, the total energy of all HO states (evaluated by our model) matches the noninteracting result very well. However, beyond the critical temperature, we ``fit'' our energies to the classical limit for 2D bosons in a trap by using a suitably proposed weighting function. We compare our results to earlier results of mean-field theory. Further, we evaluate the density matrix arising from correlations between the HO orbitals.

When a harmonic oscillator is under the influence of a Gaussian process such as linear damping, parametric gain, and linear coupling to a thermal environment, its coherent states are transformed into states with Gaussian Wigner function. Qubit states can be encoded in the |0> and |1> Fock states of a quantum harmonic oscillator, and it is relevant to know the fidelity of the output qubit state after a Gaussian process on the oscillator. In this paper we present a general expression for the average qubit fidelity in terms of the first and second moments of the output from input coherent states subjected to Gaussian processes.

We analyze the problem of a quantum computer in a correlated environment protected from decoherence by quantum error correction using a perturbative renormalization group approach. The scaling equation obtained reflects the competition between the dimension of the computer and the scaling dimension of the correlations. For an irrelevant flow, the error probability is reduced to a stochastic form for a long time and/or a large number of qubits; thus, the traditional derivation of the threshold theorem holds for these error models. In this way, the "threshold theorem" of quantum computing is rephrased as a dimensional criterion.

Future universal quantum computers solving problems of practical relevance are expected to require at least 106 qubits, which is a massive scale-up from the present numbers of less than 50 qubits operated together. Out of the different types of qubits, solid state qubits are considered to be viable candidates for this scale-up, but interfacing to and controlling such a large number of qubits is a complex challenge that has not been solved yet. One possibility to address this challenge is to use qubit control circuits located close to the qubits at cryogenic temperatures. In this work we evaluate the feasibility of this idea, taking as a reference the physical requirements of a two-electron spin qubit and the specifications of a standard 65 nm complementary metal-oxide-semiconductor process. Using principles and flows from electrical systems engineering we provide realistic estimates of the footprint and of the power consumption of a complete control-circuit architecture. Our results show that with further research it is possible to provide scalable electrical control in the vicinity of the qubit, with our concept.

Quantum computing promises an exponentially higher computational power than classical computers; although all the building blocks have become available, certain constraints still prevent quantum advantage. The fundamental challenge in building a practical quantum computer is integrating thousands of highly coherent qubits with the control and readout electronics. The need for a high‐coherence qubit drives the effort for quantum error correction algorithms to create fault‐tolerant quantum systems. Error correction becomes tangible in a quantum processor only in large numbers of qubits. Thus, the other challenge is reducing the number of physical interconnects (coaxial lines) between the quantum–classical interface and bulky room‐temperature electronics. To interface thousands of qubits, interconnects can be reduced by bringing the control and readout electronics near the quantum processor. Cryogenic complementary metal–oxide–semiconductor (CMOS) technology has been an ideal candidate for this purpose. Integrated control and readout at cryogenic temperatures require low power dissipation circuit designs and techniques such as frequency‐division multiplexing (FDM) due to the finite cooling power of a dilution refrigerator. Herein, an overview of each building block in a superconducting quantum computer is provided, focusing on scalability. Furthermore, this article is concluded with an outlook discussing current challenges and future directions for the scalable superconducting control and readout.

The quantum adder is an essential attribute of a quantum computer, just as classical adder is needed for operation of a digital computer. We model the quantum full adder as a realistic complex algorithm on a large number of qubits in an Ising spin quantum computer. Our results are an important step toward effective modeling of the quantum modular adder which is needed for Shor's and other quantum algorithms. Our full adder has the following features. (i) The near-resonant transitions with small detunings are completely suppressed, which allows us to decrease errors by several orders of magnitude and to model a 1000-qubit full adder. (We add a 1000-bit number using 2001 spins.) (ii) We construct the full adder gates directly as sequences of radio-frequency pulses, rather than breaking them down into generalized logical gates, such as Control-Not and one qubit gates. This substantially reduces the number of pulses needed to implement the full adder. (The maximum number of pulses required to add one bit (F-gate) is 15.) (iii) Full adder is realized in a homogeneous spin chain. (iv) The phase error is minimized: the F-gates generate approximately the same phase for different states of the superposition. (v) Modeling of the full adder is performed using quantum maps instead of differential equations. This allows us to reduce the calculation time to a reasonable value.

The fragment molecular orbital (FMO) scheme is one of the popular fragmentation-based methods and has the potential advantage of making the circuit shallow for quantum chemical calculations on quantum computers. In this study, we used a GPU-accelerated quantum simulator (cuQuantum) to perform the electron correlation part of the FMO calculation as unitary coupled-cluster singles and doubles (UCCSD) with the variational quantum eigensolver (VQE) for hydrogen-bonded (FH) and (FH) -H O systems with the STO-3G basis set. VQE-UCCSD calculations were performed using both canonical and localized MO sets, and the results were examined from the point of view of size-consistency and orbital-invariance affected by the Trotter error. It was found that the use of localized MO leads to better results, especially for (FH) -H O. The GPU acceleration was substantial for the simulations with larger numbers of qubits, and was about a factor of 6.7-7.7 for 18 qubit systems.