We present two experimental schemes that can be used to implement the Factorized Quantum Lattice-Gas Algorithm for the 1D Diffusion Equation with Persistent-Current Qubits. One scheme involves biasing the PC Qubit at multiple flux bias points throughout the course of the algorithm. An implementation analogous to that done in Nuclear Magnetic Resonance Quantum Computing is also developed. Errors due to a few key approximations utilized are discussed and differences between the PC Qubit and NMR systems are highlighted.
Solving differential equations is one of the most promising applications of quantum computing. Recently we proposed an efficient quantum algorithm for solving one-dimensional Poisson equation avoiding the need to perform quantum arithmetic or Hamiltonian simulation. In this letter, we further develop this algorithm to make it closer to the real application on the noisy intermediate-scale quantum (NISQ) devices. To this end, we first develop a new way of performing the sine transformation, and based on it the algorithm is optimized by reducing the depth of the circuit from n2 to n. Then, we analyze the effect of common noise existing in the real quantum devices on our algorithm using the IBM Qiskit toolkit. We find that the phase damping noise has little effect on our algorithm, while the bit flip noise has the greatest impact. In addition, threshold errors of the quantum gates are obtained to make the fidelity of the circuit output being greater than 90%. The results of noise analysis will provide a good guidance for the subsequent work of error mitigation and error correction for our algorithm. The noise-analysis method developed in this work can be used for other algorithms to be executed on the NISQ devices.
The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $mathcal{O}(text{polylog}(N_v) t/delta)$ where $delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.
The Bethe-Ansatz local density approximation (LDA) to lattice density functional theory (LDFT) for the one-dimensional repulsive Hubbard model is extended to current-LDFT (CLDFT). The transport properties of mesoscopic Hubbard rings threaded by a magnetic flux are then systematically investigated by this scheme. In particular we present calculations of ground state energies, persistent currents and Drude weights for both a repulsive homogeneous and a single impurity Hubbard model. Our results for the ground state energies in the metallic phase compares favorably well with those obtained with numerically accurate many-body techniques. Also the dependence of the persistent currents on the Coulomb and the impurity interaction strength, and on the ring size are all well captured by LDA-CLDFT. Our study demonstrates that CLDFT is a powerful tool for studying one-dimensional correlated electron systems with high accuracy and low computational costs.
We report on an experiment on Grovers quantum search algorithm showing that {em classical waves} can search a $N$-item database as efficiently as quantum mechanics can. The transverse beam profile of a short laser pulse is processed iteratively as the pulse bounces back and forth between two mirrors. We directly observe the sought item being found in $simsqrt{N}$ iterations, in the form of a growing intensity peak on this profile. Although the lack of quantum entanglement limits the {em size} of our database, our results show that entanglement is neither necessary for the algorithm itself, nor for its efficiency.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
David M. Berns
,T. P. Orlando
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(2005)
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"Implementation Schemes for the Factorized Quantum Lattice-Gas Algorithm for the One Dimensional Diffusion Equation using Persistent-Current Qubits"
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David Berns
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