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arXiv:quant-ph/0306203 · Physical Review A

Quantum computation of the Anderson transition in the presence of imperfections

A. A. Pomeransky, Dima L. Shepelyansky

Published Jan 12, 2004• 25 citations
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Abstract

We propose a quantum algorithm for simulation of the Anderson transition in disordered lattices and study numerically its sensitivity to static imperfections in a quantum computer. In the vicinity of the critical point the algorithm gives a quadratic speedup in computation of diffusion rate and localization length, comparing to the known classical algorithms. We show that the Anderson transition can be detected on quantum computers with 7--10 qubits.

DOI
10.1103/physreva.69.014302

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We propose a quantum algorithm for simulation of the Anderson transition in disordered lattices and study numerically its sensitivity to static imperfections in a quantum computer. In the vicinity of the critical point the algorithm gives a quadratic speedup in computation of diffusion rate and localization length, comparing to the known classical algorithms. We show that the Anderson transition can be detected on quantum computers with 7--10 qubits.
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We study the effects of static interqubit interactions on the accuracy of various quantum algorithms. Extensive numerical simulations show that their effect is stronger compared to external decoherence. Analytical approach based on Random Matrix Theory is developed. It gives universal law for fidelity decay induced by interqubit static interactions. This determines the time scale for reliable quantum computation in presence of realistic static imperfections and external decoherence. New polynomial algorithms are developed for simulation of complex dynamics in the regime of classical and quantum chaos, and Anderson metal- insulator transition. A generic quantum error correction method is developed; it allows to eliminate coherent effect of static errors. The theoretical results are confirmed by numerical computations with up to 28 qubits.
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We show how a quantum computer may efficiently simulate a disordered Hamiltonian, by incorporating a pseudo-random-number generator directly into the time evolution circuit. This technique is applied to quantum simulation of few-body disordered systems in the large volume limit, in particular, Anderson localization. The method requires a number of (error corrected) qubits proportional to the logarithm of the volume of the system, and each time evolution step requires a number of gates polylogarithmic in the volume. We simulate the method to observe the metal-insulator transition on a three-dimensional lattice. Additionally, we demonstrate the algorithm on a one-dimensional lattice, using physical quantum processors.
We describe how to use quantum linear algebra to simulate a physically realistic model of disordered non-interacting electrons. The physics of disordered electrons outside of one dimension challenges classical computation due to the critical nature of the Anderson localization transition or the presence of large localization lengths, while the atypical distribution of the local density of states limits the power of disorder averaged approaches. Starting from the block-encoding of a disordered non-interacting Hamiltonian, we describe how to simulate key physical quantities, including the reduced density matrix, Green's function, and local density of states, as well as bulk-averaged observables such as the linear conductivity, using the quantum singular value transformation, quantum amplitude estimation, and trace estimation. We further discuss a quantum advantage that scales polynomially with system size and exponentially with lattice dimension.