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

Compressed simulation of thermal and excited states of the one-dimensional $X Y$ model

W. L. Boyajian and B. Kraus

Phys. Rev. A 92, 032323 – Published 24 September 2015

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) $X Y$ model using very few qubits. We extend here this notion to simulate the behavior of thermal as well as excited states of the 1D $X Y$ model. In particular, we consider the 1D $X Y$ 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.

Received 7 July 2015

DOI:https://doi.org/10.1103/PhysRevA.92.032323

Authors & Affiliations

W. L. Boyajian and B. Kraus

Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Austria

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
The magnetization [see Eq. (80)] of the thermal states of the $X Y$ Hamiltonian as a function of $g$ is shown. We have set $δ = 0.2$ and considered system sizes of $n = 8$ qubits (a) and $n = 128$ qubits (b). Each curve corresponds to a different value of the temperature, which is chosen equally spaced between $T = 0$ (uppermost curve on the left side of the figures) and $T = 0.9$ (lowermost curve on the left side of the figures). At $T = 0$ the magnetization shows a discontinuity at the critical point (dashed line) as a consequence of the level crossing between the ground state and the first excited state. The discontinuity diminishes as the system size increases [compare figures (a) and (b)] and vanishes if $T > 0$ .
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Figure 2
The number of kinks $ν$ as a function of the parameter $g$ , occurring during the quantum quenching [see Eq. (86)] for a system size of $n = 128$ spins is depicted. The different curves correspond to various total quenching times ranging from $t = 1$ (uppermost curve) to $t = 300$ (continuous bottommost curve). The number of Trotter steps was set to $L = 100 t$ . The number of kinks for $T = 0$ and adiabatic evolution is depicted in (a) as a dashed line. Figures (c) and (d) show a standard linear fitting function of the logarithm of the correlations $ν$ at the end of the quenching, i.e., at $g = 0$ , vs the logarithm of the quenching speed $(1 / t)$ . To fit the data points, we have only considered the points depicted by filled circles, which correspond to long quenching times, although not long enough to correspond to an adiabatic evolution. The slope of the fitted function (represented by the dashed line) is the parameter $p$ , corresponding to the exponent in the relation $ν \propto t^{- p}$ . We have obtained a value $p = 0.51$ for $T = 0$ , and $p = 0.03$ for $T = 2$ .
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Figure 3
The parameter $p$ that corresponds to the exponent in the relation $ν \propto t^{- p}$ is depicted as a function of the temperature $T$ . Each point has been obtained by simulating a quantum quenching for a system of $n = 64$ qubits. For each value of the temperature we have extracted $p$ by fitting the correlations $ν$ measured in the system after the quantum quenching as a function of the quenching time $t$ (see Fig. 2). Note that for low temperatures, the parameter $p$ is close to the one for $T = 0$ . This is expected as the correlations $ν$ in each of the first four excited states scale similarly as the ones of the ground state.
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Figure 4
The linear fit of the logarithm of the correlations $ν$ at the end of the quenching, i.e., at $g = 0$ , vs the logarithm of the quenching speed $(1 / t)$ is shown. The simulated system size is of $n = 128$ qubits. The fitting was done in the same way as explained in Fig. 2. To obtain the correlations depicted in (a) and (b) we have considered a system initially prepared in the first and fourth excited states of the Ising Hamiltonian, respectively. The corresponding slopes are $p = 0.61$ (a) and $p = 0.48$ (b).
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Figure 5
The correlations of the form $C_{j, k}$ [see Eq. (87)] between the $j th$ qubit and the $(\frac{n}{2}) th$ qubit as a function of $j$ are depicted for $n = 64$ qubits (not all depicted). In (a) the dependency of the correlations on temperature is shown. Each curve corresponds to a different temperature whose values have been chosen to be equally spaced between $T = 0$ (corresponding to the uppermost curve) and $T = 0.9$ (corresponding to the bottommost curve); the value of $g$ has been fixed at $g = 0.8$ . In (b) we show the dependency of the correlations on the value of $g$ , at $T = 0$ .
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Figure 6
The magnetization for different eigenstates of the $X Y$ Hamiltonian as a function of $1 / g$ and for $δ = 0.2$ and $n = 8$ is shown. Due to the boundary conditions, several magnetization curves exhibit a discontinuity at the critical point (see Sec. 2), as indicated by a vertical line in the figure at the value $g_{c}^{- 1} = {(1 + δ)}^{- 1} = 0.83$ .
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