Quantum Computation as Geometry

By Morten Nielsen, Mark R. Dowling, Mile Gu, Andrew C. Doherty. Science, 2006

559 citations recorded by the source corpus.

Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/dd6eb70a67c55e1e7c91b2e459af904bd441f6f8

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Citations in the corpus

Qudits and High-Dimensional Quantum Computing (2020, methodology): “A method exists to do a general systematic evaluation of the circuit efficiency with the mathematical techniques of Riemannian geometry [126].”
Linear growth of quantum circuit complexity (2021, methodology): “Another opportunity involves Nielsen’s geometric notion of complexity [18, 38, 40, 41], which underlies Brown and Susskind’s conjecture.”
Measurement-Induced Phase Transitions in the Dynamics of Entanglement (2018, background): “[83] Michael A Nielsen, Mark R Dowling, Mile Gu, and Andrew C Doherty.”
Comments on holographic complexity (2016, background): “Perhaps, the techniques developed in [46, 47, 48] or in [49] can provide better insight towards developing such a covariant approach.”
Complexity and entanglement for thermofield double states (2018, methodology): “To overcome this challenge, Nielsen and collaborators [70–72] introduced a geometric approach to identify the optimal circuit, which ref.”
Quantum complexity and negative curvature (2016, background): “Although we do not fully understand the reasons for this correspondence it most likely has its roots in Nielsen’s geometrized approach to complexity [4, 5, 6].”
Complexity, action, and black holes (2015, methodology): “However, [71] uses an L2-norm so the complexity growth is proportional to ffiffiffi n p instead of n in that formulation.”