Bipartite unitary gates and billiard dynamics in the Weyl chamber

Antonio Mandarino, Tomasz Linowski, and Karol Życzkowski

Phys. Rev. A 98, 012335 – Published 30 July 2018

Abstract

Long-time behavior of a unitary quantum gate $U$ , acting sequentially on two subsystems of dimension $N$ each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a three-dimensional (3D) billiard. Due to ergodicity of such a dynamics an average along a trajectory $V^{t}$ stemming from a generic two-qubit gate $V$ in the canonical form tends for a large $t$ to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber—the minimal 3D set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension $N$ the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on $U (N^{2})$ .

Received 6 November 2017
Revised 26 March 2018

DOI:https://doi.org/10.1103/PhysRevA.98.012335

Physics Subject Headings (PhySH)

Nonlocality Quantum entanglement Quantum gates

Ergodic theory Random matrix theory

Quantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

Antonio Mandarino^1,*, Tomasz Linowski², and Karol Życzkowski^1,3

¹Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
²Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland
³Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30–348 Kraków, Poland

^*mandarino@cft.edu.pl

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Vol. 98, Iss. 1 — July 2018

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Images

Figure 1
(a) Time evolution of $α_{1}$ of a generic gate $V_{α_{I}}^{t}$ after $t$ iterations according to Eq. (3); (b) exemplary 2D trajectory of a typical gate: $V_{α_{I I}}$ after six iterations; (c) 3D trajectory in the tetrahedron $Γ$ generated by a typical gate $V$ in $Γ_{I I I}$ as prescribed by Eq. (4).
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Figure 2
(a) Distribution of linear entropy $P (S_{L})$ for canonical gates $V_{I} \in Γ_{I}$ (left curve), $V_{I I} \in Γ_{I I}$ (middle curve), and $V_{I I I} \in Γ_{I I I}$ (right curve). Filled symbols in each plot refer to distribution of gates in the parameter space, while empty symbols to averaging in time over a single typical trajectory. Solid black curve represents the arcsine distribution (9) corresponding to the 1D case $Γ_{I}$ . Circles, squares, and triangles refer to $m = I, I I, I I I$ , respectively. For $m = I I, I I I$ a line joining the squares or the triangles is plotted to guide the eye. The red dashed line and black diamonds represent the average over the Haar measure on $U (4)$ . Inset shows similar data concerning distributions of entanglement entropy $P (S)$ . (b) Time dependence of information content $α$ for powers $V^{t}$ of a two-qubit gate $V$ in the canonical form (2) and for an ensemble of random locally equivalent gates $U$ . Black dots correspond to $α_{1} (V^{t})$ . Blue, green, and orange dots denote $α_{m} (U^{t})$ for $m = 1, 2, 3$ , respectively. The red dashed line corresponds to the time evolution of $α_{1} (V^{t})$ as in Fig. 1.
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Figure 3
Distribution of linear entropy $P (S_{L})$ averaged over a trajectory $V^{t}$ stemming from an initial gate $V \in Γ_{I I I}$ . Blue circles, green squares, and yellow triangles refer to an initial gate $V$ which approximates with fidelity $F ≃ 0.998$ (a) local gate, (b) SWAP gate, and (c) $\sqrt{SWAP}$ gate. Solid curve represents the inferred distribution. Inset shows similar data for initial gates $V \in Γ_{I}$ , which approximate with the same fidelity (a) local unitary, (b) $\sqrt{CNOT}$ , and (c) CNOT gate compared with the arcsine distribution represented by a solid line.
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Figure 4
(a) Average linear entropy $S_{L} (U)$ of random gates of size $N^{2}$ . Space averages $〈 y 〉$ denoted by filled symbols coincide with time averages $\bar{y}$ (empty symbols). Solid curves represent interpolation between mean values (10) over CUE, $(\circ)$ denote averaging over trajectories stemming from generic $U$ , while dashed lines and triangles correspond to mean values (11) over CPE and over iterated generic diagonal gates. Inset shows averages of Shannon entropy $S (U)$ . (b) Distribution $P (S_{L})$ of a random sample of $M = 10^{6}$ matrices: generated according to CUE (yellow), time trajectory initiated from a single CUE matrix (blue), and characterizing the spectra of Wishart random matrices (green) practically coincide already for matrices of size $N^{2} = 100$ . The differences between analogous distributions obtained for $N^{2} = 4$ are shown in the inset with the same notation.
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Figure 5
Spectral density $P (x)$ of three samples of random matrices of dimension $d = N^{2} = 100$ , where $x = d λ$ is a rescaled eigenvalue. Green triangles refer to Wishart random matrices, blue circles to an ensemble of matrices $ρ$ as in (A2), and yellow squares to an ensemble of nonindependent matrices stemming from a single trajectory $Z^{R} {(Z^{R})}^{†} / d$ , with $Z = U_{0}^{t}$ . The black line denotes Marchenko-Pastur distribution (A3). The differences between analogous distributions obtained for $d = N^{2} = 4$ with oscillations due to level repulsion are shown in the inset.
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