Fast computation of spherical phase-space functions of quantum many-body states

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Fast computation of spherical phase-space functions of quantum many-body states

Bálint Koczor, Robert Zeier, and Steffen J. Glaser

Phys. Rev. A 102, 062421 – Published 22 December 2020

Abstract

Quantum devices are preparing increasingly more complex entangled quantum states. How can one effectively study these states in light of their increasing dimensions? Phase spaces such as Wigner functions provide a suitable framework. We focus on spherical phase spaces for finite-dimensional quantum states of single qudits or permutationally symmetric states of multiple qubits. We present methods to efficiently compute the corresponding spherical phase-space functions which are at least an order of magnitude faster than traditional methods. Quantum many-body states in much larger dimensions can now be effectively studied by experimentalists and theorists using spherical phase-space techniques.

Received 22 September 2020
Accepted 17 November 2020

DOI:https://doi.org/10.1103/PhysRevA.102.062421

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum correlations, foundations & formalism Quantum metrology

Phase space methods

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Bálint Koczor ^1,2,3,*, Robert Zeier ^4,†, and Steffen J. Glaser ^2,3,‡

¹Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
²Department of Chemistry, Technical University of Munich, Lichtenbergstrasse 4, 85747 Garching, Germany
³Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany
⁴Forschungszentrum Jülich GmbH, Peter Grünberg Institute, Quantum Control (PGI-8), 54245 Jülich, Germany

^*balint.koczor@materials.ox.ac.uk
^†r.zeier@fz-juelich.de
^‡glaser@tum.de

Article Text

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Issue

Vol. 102, Iss. 6 — December 2020

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Images

Figure 1
(a) Runtimes of earlier methods A and B relative to our method C for computing phase-space functions of quantum states with an increasing dimension $d = 2 J + 1$ are at least an order of magnitude slower. Methods A and B both employ tensor-operator decompositions (Sec. 3). Method A relies on mathematica's built-in method to compute Clebsch-Gordan coefficients and method B uses an efficient recursive algorithm [51, 52, 53, 54]. Our method C (Sec. 5) combines spherical sampling techniques [55, 56], explicit descriptions of rotation operators [57, 58], Fourier series expansions, and fast Fourier transforms. The runtimes depend only on $d$ and not the quantum state. (b) Root mean square (rms) errors for certain quantum states relative to their analytically known formula. (We computed Wigner functions of tensor operators of high rank $j > 1$ , whose functional form we also know analytically as spherical harmonics; these decompose into a large number of nontrivial Fourier components.) Method C shows a high numerical precision comparable to machine precision.
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Figure 2
Applications highlighted by numerically computed Wigner functions of single-qudit states with $d = 2 J + 1$ which are equivalent to permutation-symmetric states of $N = d - 1$ qubits: (a) GHZ states for $N \in {8, 16, 32, 64}$ and (b) Dicke state $| J m 〉$ for $N = 128$ and $m = 0$ . The runtime (without three-dimensional graphics and rasterization) using method D (Sec. 5) on a laptop in mathematica is dominated by the FFT on a $1024 \times 1024$ grid. (c) Similarly, Wigner functions of squeezed states $| ξ 〉 : = exp [- i ξ I_{x}^{2}] {| 0 〉}^{\otimes N}$ with varying squeezing angle $ξ$ and fixed $d = N + 1 = 500$ in a plane for a small spherical subset (runtime approximately equal to 1 min): Gaussian for $ξ = 0$ (leftmost panel) and squeezed Gaussians for $ξ < 0.05$ . Larger $ξ \geq 0.05$ lead to nontrivial and rapidly oscillating shapes which are nicely recovered, while analytical approximations fail in this regime. Red (dark gray) and green (light gray) represent positive and negative values, respectively. The brightness indicates the absolute value of the function relative to its global maximum $η$ .
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Figure 3
Runtimes for computing phase-space functions with methods A–D (Secs. 3 and 5) for dimensions $d = 2 J + 1 \leq 500$ , while ignoring asymptotically negligible contributions from spherical-harmonic transformations (for methods A and B) or FFTs (for methods C and D). Methods A–C show a similar asymptotic behavior $O (d^{4})$ , but our method C is at least an order of magnitude faster (using c code), which allows for much larger dimensions. Building on method C, method D is even faster and empirically shows a lower asymptotic time complexity $O (d^{3})$ but relies on precomputations and additional disk storage (Table 1). The runtimes depend only on $d$ and not the quantum state. (All data points were obtained on a desktop computer with an Intel^® Xeon^® W-2133 processor at 3.60 GHz using a single thread.)
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