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arXiv:1305.5895 · Physical Review A

Compressed simulation of evolutions of the XY model

W. L. Boyajian, V. Murg, Barbara Kraus

Published Nov 25, 2013• 16 citations
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Abstract

We extend the notion of compressed quantum simulation to the $XY$ model. We derive a quantum circuit processing $\mathrm{log}(n)$ qubits which simulates the one-dimensional $XY$ model describing $n$ qubits. In particular, we demonstrate how the adiabatic evolution can be realized on this exponentially smaller system and how the magnetization, which witnesses a quantum phase transition, can be observed. Furthermore, we analyze several dynamical processes, like quantum quenching and finite time evolution, and derive the corresponding compressed quantum circuit.

DOI
10.1103/physreva.88.052329

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Since several years, the preparation and manipulation of a small number of quantum systems in a controlled and coherent way is feasible in many experiments. In fact, these experiments are nowadays commonly used for quantum simulation and quantum computation. As recently shown, such a system can, however, also be utilized to simulate specific behaviors of exponentially larger systems. That is, certain quantum computations can be performed by an exponentially smaller quantum computer. This compressed quantum computation can be employed to observe, for instance, the quantum phase transition of the one-dimensional (1D) XYXYXY model using very few qubits. We extend here this notion to simulate the behavior of thermal as well as excited states of the 1D XYXYXY model. In particular, we consider the 1D XYXYXY model of a spin chain of nnn qubits and derive a quantum circuit processing only log(n)log(n)log(n) qubits which simulates the original system. We demonstrate how the behavior of thermal as well as any eigenstate of the system can be efficiently simulated in this compressed fashion and present a quantum circuit on log(n)log(n)log(n) qubits to measure the magnetization, the number of kinks, and correlations occurring in the thermal as well as any excited state of the original systems. Moreover, we derive compressed circuits to study time evolutions.
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Here, I summarize the results presented in B. Kraus, Phys. Rev. Lett. 107, 250503 (2011). Recently, it has been shown that certain circuits, the so-called match gate circuits, can be compressed to an exponentially smaller universal quantum computation. We use this result to demonstrate that the simulation of a 1-D Ising chain consisting of n qubits can be performed on a universal quantum computer running on only log(n) qubits. We show how the adiabatic evolution can be simulated on this exponentially smaller system and how the magnetization can be measured. Since the Ising model displays a quantum phase transition, this result implies that a quantum phase transition of a very large system can be observed with current technology.
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The notion of compressed quantum computation is employed to simulate the Ising interaction of a one-dimensional chain consisting of nnn qubits using the universal IBM cloud quantum computer running on log2(n){log}_{2}(n)log2​(n) qubits. The external field parameter that controls the quantum phase transition of this model translates into particular settings of the quantum gates that generate the circuit. We measure the magnetization, which displays the quantum phase transition, on a two-qubit system, which simulates a four-qubit Ising chain, and show its agreement with the theoretical prediction within a certain error. We also discuss the relevant point of how to assess errors when using a cloud quantum computer with a limited amount of runs. As a solution, we propose to use validating circuits, that is, to run independent controlled quantum circuits of similar complexity to the circuit of interest.
We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase-transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on NNN spins, such a metrology protocol can be efficiently simulated on an exponentially smaller quantum computer while maintaining the same Heisenberg scaling for the squared error, i.e., $O({N}^{\ensuremath{-}2})$ precision, and derive the explicit circuit that accomplishes the simulation.