Probing nonlinear adiabatic paths with a universal integrator

Michael Hofmann and Gernot Schaller

Phys. Rev. A 89, 032308 – Published 10 March 2014

Abstract

We apply a flexible numerical integrator to the simulation of adiabatic quantum computation with nonlinear paths. We find that a nonlinear path may significantly improve the performance of adiabatic algorithms versus the conventional straight-line interpolations. The employed integrator is suitable for solving the time-dependent Schrödinger equation for any qubit Hamiltonian. Its flexible storage format significantly reduces cost for storage and matrix-vector multiplication in comparison to common sparse matrix schemes.

Received 4 November 2013
Revised 13 February 2014

DOI:https://doi.org/10.1103/PhysRevA.89.032308

Authors & Affiliations

Michael Hofmann^1,2,* and Gernot Schaller²

¹Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany
²Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany

^*michael.hofmann@wias-berlin.de

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Issue

Vol. 89, Iss. 3 — March 2014

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Images

Figure 1
Instance of EC3, where the labeled vertices represent the 13 bits $z_{i}$ , edges of same color and line style represent a clause $c_{i}$ . Bold-bordered vertices indicate the upbits in the solution $z = 1098 = (0010001001010)$ .
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Figure 2
Scheme of the used algorithm for randomly generating hard instances of EC3. Rectangles and circles indicate instructions and if clauses, respectively. The algorithm starts and stops at diamonds.
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Figure 3
Scaling of the dimensionless runtime $Ω T_{s} (n)$ (symbols, fit lines only serve to guide the eye) for algorithms $X$ (solid line), $XYZ$ (long-dashed line), $XY$ (dotted line) compared with the quadratically scaling Ising model (dashed-dotted line). Symbols represent the median and error bars the first and third quartiles out of 100 randomly chosen hard instances. Although the scaling behavior is inconclusive, it is evident that different initial states may drastically improve the performance of the AQA.
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Figure 4
Sketch of alternative paths for adiabatic algorithms all starting from $H^{XY}$ . As an example, the number of clauses is set to $m = 6$ . The red solid line is the straight-line algorithm. The green dashed curve denotes a path with an additional, nonlinear term $s (1 - s) H_{5}$ , which is defined in the text. A clause-by-clause algorithm is shown by the blue dotted lines.
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Figure 5
Top: Final energies versus dimensionless runtimes for different paths between $H^{XY}$ and $H_{f}$ for a 13-qubit example. Symbol shapes and colors correspond to the paths in Fig. 4. The $X$ algorithm is also shown as an example of a slow adiabatic behavior. For the nonlinear path, the coupling is chosen as $α = 8$ . Bottom: Energy gap between ground and first excited states of $H (s)$ for the same algorithms. The symbols for the curves correspond to those in the upper panel. For the clause-by-clause algorithm, the symbols indicate the start and end of the secondary interpolations in (22). Comparing both panels, a correspondence between a large minimal gap and a fast decrease in the final energy is clearly seen.
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Figure 6
A very hard instance. Top: Final energy of the nonlinear path for different choices of $H_{m - 1}$ with coupling $α = 8$ compared to the $X$ and $XY$ algorithms. Clauses removed from $H_{f}$ for the nonlinear term are $A = (1, 10, 11), B = (1, 4, 13)$ , and $C = (9, 11, 13)$ . Symbol shapes and colors correspond to the clauses in Fig. 1. Whereas the short-time performance was much better for all nonlinear paths, it turned out that these differ strongly for large evolution times $T$ , and choice $A$ performs worse than the straight-line interpolations. Bottom: Energy gap for the same algorithms. Again, the value of the minimum gap clearly corresponds to the large-time performance in the above panel. It depends on the choice of $H_{m - 1}$ whether this gap is increased or decreased compared to straight-line algorithms.
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