Optimizing quantum circuits with Riemannian gradient flow

Variational quantum algorithms are a promising class of algorithms that can be performed on currently available quantum computers. In most settings, the free parameters of a variational circuit are optimized using a classical optimizer that updates parameters in Euclidean geometry. Since quantum circuits are elements of the special unitary group, we can consider an alternative optimization perspective that depends on the structure of this group. In this work, we investigate a Riemannian optimization scheme over the special unitary group and we discuss its implementation on a quantum computer. We illustrate that the resulting Riemannian gradient-flow algorithm has favorable optimization properties for deep circuits and that an approximate version of this algorithm can be performed on near-term hardware. We highlight the connections of our work with previously proposed heuristics like ADAPT-VQE and show that they can be understood as variants of our algorithm.

geometry: Euclidean
unitarity
hardware
flow
structure
quantum circuit
optimization
quantum algorithm: variational

References(60)

Figures(9)

[1]

A variational eigenvalue solver on a photonic quantum processor

Alberto Peruzzo
(
- Bristol U.
)
,
Jarrod McClean
(
- Harvard U. (main)
)
,
Peter Shadbolt
(
- Bristol U.
)
,
Man-Hong Yung
(
- Harvard U. (main) and
- Tsinghua U., Beijing
)
,
Xiao-Qi Zhou
(
- Bristol U.
)

et al.

- Nature Commun. 5 (2014) 1, 4213
•
e-Print:
- 1304.3061
•
DOI:
- 10.1038/ncomms5213

[2]

A Quantum Approximate Optimization Algorithm

Edward Farhi
(
- MIT, Cambridge, CTP
)
,
Jeffrey Goldstone
(
- MIT, Cambridge, CTP
)
,
Sam Gutmann
(
- MIT, Cambridge, CTP
)

e-Print:
- 1411.4028

[3]

Using models to improve optimizers for variational quantum algorithms

Kevin J. Sung
(
- Unlisted, US, CA and
- Michigan U.
)
,
Jiahao Yao
(
- UC, Berkeley, Math. Dept.
)
,
Matthew P. Harrigan
(
- Unlisted, US, CA
)
,
Nicholas C. Rubin
(
- Unlisted, US, CA
)
,
Zhang Jiang
(
- Unlisted, US, CA
)

et al.

- Quantum Sci.Technol. 5 (2020) 4, 044008
•
DOI:
- 10.1088/2058-9565/abb6d9

[4]

Quantum circuit learning

Kosuke Mitarai
(
- Osaka U.
)
,
Makoto Negoro
(
- Osaka U. and
- ERATO, JST, Tokyo
)
,
Masahiro Kitagawa
(
- Osaka U.
)
,
Keisuke Fujii
(
- ERATO, JST, Tokyo and
- Kyoto U., FIHS
)

- Phys.Rev.A 98 (2018) 3, 032309,
- Phys.Rev.A 98 (2018) 032309
•
e-Print:
- 1803.00745
•
DOI:
- 10.1103/PhysRevA.98.032309

[5]

Evaluating analytic gradients on quantum hardware

Maria Schuld
(
- U. Toronto (main)
)
,
Ville Bergholm
(
- U. Toronto (main)
)
,
Christian Gogolin
(
- U. Toronto (main)
)
,
Josh Izaac
(
- U. Toronto (main)
)
,
Nathan Killoran
(
- U. Toronto (main)
)

- Phys.Rev.A 99 (2019) 3, 032331
•
e-Print:
- 1811.11184
•
DOI:
- 10.1103/PhysRevA.99.032331

[6]

On the Mathematical Foundations of Theoretical Statistics. Philosophical Transactions of the Royal Society of London. Series A

R.A. Fisher

[7]

Shun-ichi Amari. Natural Gradient Works Efficiently in Learning

- Neural Comput. 10 (1998) 251-276

[8]

Razvan Pascanu and Yoshua Bengio. Revisiting Natural Gradient for Deep Networks. In International Conference on Learning Representations, 4

[9]

Guodong Zhang, Shengyang Sun, David Duvenaud, and Roger Grosse. Noisy Natural Gradient as Variational Inference. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 5852-5861. PMLR, 7

[10]

Sandro Sorella. Green Function Monte Carlo with Stochastic Reconfiguration

- Phys.Rev.Lett. 80 (1998) 4558-4561

[11]

Quantum Natural Gradient

James Stokes
(
- Flatiron Inst., New York
)
,
Josh Izaac
(
- Xanadu Quant. Tech.
)
,
Nathan Killoran
(
- Xanadu Quant. Tech.
)
,
Giuseppe Carleo
(
- Flatiron Inst., New York
)

- Quantum 4 (2020) 269
•
e-Print:
- 1909.02108
•
DOI:
- 10.22331/q-2020-05-25-269

[12]

Connecting geometry and performance of two-qubit parameterized quantum circuits

Amara Katabarwa
(
- Unlisted, US
)
,
Sukin Sim
(
- Unlisted, US and
- Harvard U.
)
,
Dax Enshan Koh
(
- A-STAR, Singapore
)
,
Pierre-Luc Dallaire-Demers
(
- Unlisted, US
)

- Quantum 6 (2022) 782,
- Quantum 6 (2022) 782
•
e-Print:
- 2106.02593
•
DOI:
- 10.22331/q-2022-08-23-782

[13]

Gary Becigneul and Octavian-Eugen Ganea. Riemannian Adaptive Optimization Methods. In International Conference on Learning Representations

[14]

Constantin Udrişte. Convex Functions and Optimization Methods on Riemannian Manifolds, volume 297 of Mathematics and Its Applications

[15]

Thomas Schulte-Herbrüggen, Steffen j. Glaser, Gunther Dirr, and Uwe Helmke. Gradient Flows for Optimization in Quantum Information and Quantum Dynamics: Foundations and Applications

- Rev.Math.Phys. 22 (2010) 597-667

[16]

Uwe Helmke and John B. Moore. Optimization and Dynamical Systems. Communications and Control Engineering. SpringerVerlag London

[17]

Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies

Ilia A. Luchnikov
(
- Moscow, MIPT and
- Skoltech and
- Russian Quantum Ctr., Moscow
)
,
Mikhail E. Krechetov
(
- Skoltech
)
,
Sergey N. Filippov
(
- Moscow, MIPT and
- Steklov Math. Inst., Moscow and
- Moscow, INR
)

- New J.Phys. 23 (2021) 073006
•
e-Print:
- 2007.01287
•
DOI:
- 10.1088/1367-2630/ac0b02

[18]

and Th. Schulte-Herbrüggen. Gradient Flows Computing the C-numerical Range with Applications in NMR Spectroscopy. Journal of Global Optimization, 23(3-4):283-308, 8

U. Helmke
,
K. Hüper
,
J.B. Moore

[19]

Unitary Control in Quantum Ensembles: Maximizing Signal Intensity in Coherent Spectroscopy. Science, 280(5362):421-424

S.J. Glaser
,
T. Schulte-Herbrüggen
,
M. Sieveking
,
O. Schedletzky
,
N.C. Nielsen

et al.

[20]

Zhiwu Huang, Jiqing Wu, and Luc Van Gool. Building Deep Networks on Grassmann Manifolds. In Proceedings of the ThirtySecond AAAI Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence, AAAI’18/IAAI’18/EAAI’18. AAAI Press

[21]

Simone Fiori. Quasi-Geodesic Neural Learning Algorithms Over the Orthogonal Group:

A. Tutorial

- J.Machine Learning Res. 6 (2005) 743-781

[22]

Simone Fiori. Learning by Natural Gradient on Noncompact Matrix-Type PseudoRiemannian Manifolds

- IEEE Trans.Neural Networks 21 (2010) 841-852

[23]

Scott Wisdom, Thomas Powers

John R. Hershey

[24]

Mile Gu, and Andrew C. Doherty. Quantum Computation as Geometry. Science, 311(5764):1133-1135

Michael A. Nielsen
,
Mark R. Dowling

[25]

The geometry of quantum computation

Mark R. Dowling
(
- Queensland U.
)
,
Michael A. Nielsen
(
- Queensland U.
)

- Quant.Inf.Comput. 8 (2008) 10, 0861-0899
•
e-Print:
- quant-ph/0701004
•
DOI:
- 10.26421/QIC8.10-1

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