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arXiv:1506.06894 · Physical Review A

Compressed simulation of thermal and excited states of the one-dimensional XY model

W. L. Boyajian, Barbara Kraus

Published Sep 24, 2015• 8 citations
arXivHTMLPDF

Abstract

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) $XY$ model using very few qubits. We extend here this notion to simulate the behavior of thermal as well as excited states of the 1D $XY$ model. In particular, we consider the 1D $XY$ model of a spin chain of $n$ qubits and derive a quantum circuit processing only $log(n)$ 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 $log(n)$ 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.

DOI
10.1103/physreva.92.032323

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We extend the notion of compressed quantum simulation to the XYXYXY model. We derive a quantum circuit processing log(n)\mathrm{log}(n)log(n) qubits which simulates the one-dimensional XYXYXY model describing nnn 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.
Jozsa et al. [Proc. R. Soc. A 466, 809 2009)] have shown that a match gate circuit running on n qubits can be compressed to a universal quantum computation on log(n)+3 qubits. Here, we show how this compression can be employed to simulate the Ising interaction of a 1D chain consisting of n qubits using a universal quantum computer running on log(n) qubits. We demonstrate how the adiabatic evolution can be realized on this exponentially smaller system and how the magnetization, which displays a quantum phase transition, can be measured. This shows that the quantum phase transition of very large systems can be observed experimentally with current technology.
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.
The field of quantum computing has grown fast in recent years, both in theoretical advancements and the practical construction of quantum computers. These computers were initially proposed, among other reasons, to efficiently simulate and comprehend the complexities of quantum physics. In this paper, we present a comprehensive scheme for the exact simulation of the 1-D XY model on a quantum computer. We successfully diagonalize the proposed Hamiltonian, enabling access to the complete energy spectrum. Furthermore, we propose a novel approach to design a quantum circuit to perform exact time evolution. Among all the possibilities this opens, we compute the ground and excited state energies for the symmetric XY model with spin chains of n=4 n = 4 and n=8 n = 8 spins. Further, we calculate the expected value of transverse magnetization for the ground state in the transverse Ising model. Both studies allow the observation of a quantum phase transition from an antiferromagnetic to a paramagnetic state. Additionally, we have simulated the time evolution of the state all spins up in the transverse Ising model. The scalability and high performance of our algorithm make it an ideal candidate for benchmarking purposes, while also laying the foundation for simulating other integrable models on quantum computers.
Certain n-qubit quantum systems can be faithfully simulated by quantum circuits with only O(log(n)) qubits [B. Kraus, Phys. Rev. Lett. 107, 250503 (2011)]. Here we report an experimental realization of this compressed quantum simulation on a one-dimensional Ising chain. By utilizing an nuclear magnetic resonance quantum simulator with only five qubits, the property of ground-state magnetization of an open-boundary 32-spin Ising model is experimentally simulated, prefacing the expected quantum phase transition in the thermodynamic limit. This experimental protocol can be straightforwardly extended to systems with hundreds of spins by compressing them into up to merely 10-qubit systems. Our experiment paves the way for exploring physical phenomena in large-scale quantum systems with quantum simulators under current technology.
Quantum time dynamics (QTD) is considered a promising problem for quantum supremacy on near-term quantum computers. However, QTD quantum circuits grow with increasing time simulations. This study focuses on simulating the time dynamics of one-dimensional (1D) integrable spin chains with nearest-neighbor interactions. We have proved the existence of a reflection symmetry in the quantum circuit employed for simulating the time evolution of certain classes of 1D Heisenberg model Hamiltonians by virtue of the quantum Yang-Baxter equation, and how this symmetry can be exploited to compress and produce a shallow quantum circuit. With this compression scheme, the depth of the quantum circuit becomes independent of step size and only depends on the number of spins. We show that the depth of the compressed circuit is rigorously a linear function of the system size for the studied Heisenberg model Hamiltonians in the present work. As a consequence, the number of CNOT gates in the compressed circuit only scales quadratically with the system size, which allows for the simulations of time dynamics of very large 1D spin chains. We derive the compressed circuit representations for different special cases of the Heisenberg Hamiltonian. We compare and demonstrate the effectiveness of this approach by performing simulations on quantum computers.