Two-Unitary Decomposition Algorithm and Open Quantum System Simulation

Nishchay Suri1,2,3, Joseph Barreto1,2,4, Stuart Hadfield1,2, Nathan Wiebe5,6, Filip Wudarski1,2, and Jeffrey Marshall1,2

1QuAIL, NASA Ames Research Center, Moffett Field, California 94035, USA
2USRA Research Institute for Advanced Computer Science, Mountain View, California 94043, USA
3Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
4QuTech, Delft University of Technology, Delft, The Netherlands
5Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3E1, Canada
6Pacific Northwest National Laboratory, Richland, Washington 99352, USA

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Abstract

Simulating general quantum processes that describe realistic interactions of quantum systems following a non-unitary evolution is challenging for conventional quantum computers that directly implement unitary gates. We analyze complexities for promising methods such as the Sz.-Nagy dilation and linear combination of unitaries that can simulate open systems by the probabilistic realization of non-unitary operators, requiring multiple calls to both the encoding and state preparation oracles. We propose a quantum two-unitary decomposition (TUD) algorithm to decompose a $d$-dimensional operator $A$ with non-zero singular values as $A=(U_1+U_2)/2$ using the quantum singular value transformation algorithm, avoiding classically expensive singular value decomposition (SVD) with an $O(d^3)$ overhead in time. The two unitaries can be deterministically implemented, thus requiring only a single call to the state preparation oracle for each. The calls to the encoding oracle can also be reduced significantly at the expense of an acceptable error in measurements. Since the TUD method can be used to implement non-unitary operators as only two unitaries, it also has potential applications in linear algebra and quantum machine learning.

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[2] Dhrumil Patel and Mark M. Wilde, "Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials", Open Systems & Information Dynamics 30 03, 2350014 (2023).

[3] Dhrumil Patel and Mark M. Wilde, "Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics", Open Systems & Information Dynamics 30 02, 2350010 (2023).

[4] Rahul Bandyopadhyay, Alex H. Rubin, Marina Radulaski, and Mark M. Wilde, "Efficient Quantum Algorithms for Testing Symmetries of Open Quantum Systems", Open Systems & Information Dynamics 30 03, 2350017 (2023).

[5] Huo Chen, Niladri Gomes, Siyuan Niu, and Wibe Albert de Jong, "Adaptive variational simulation for open quantum systems", Quantum 8, 1252 (2024).

[6] Juha Leppäkangas, Nicolas Vogt, Keith R. Fratus, Kirsten Bark, Jesse A. Vaitkus, Pascal Stadler, Jan-Michael Reiner, Sebastian Zanker, and Michael Marthaler, "Quantum algorithm for solving open-system dynamics on quantum computers using noise", Physical Review A 108 6, 062424 (2023).

[7] Hans Hon Sang Chan, David Muñoz Ramo, and Nathan Fitzpatrick, "Simulating non-unitary dynamics using quantum signal processing with unitary block encoding", arXiv:2303.06161, (2023).

[8] Chiara Leadbeater, Nathan Fitzpatrick, David Muñoz Ramo, and Alex J. W. Thom, "Non-unitary Trotter circuits for imaginary time evolution", arXiv:2304.07917, (2023).

[9] Joseph Peetz, Scott E. Smart, Spyros Tserkis, and Prineha Narang, "Simulation of Open Quantum Systems via Low-Depth Convex Unitary Evolutions", arXiv:2307.14325, (2023).

[10] I J David, I Sinayskiy, and F Petruccione, "Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial", arXiv:2302.02953, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 07:54:15) and SAO/NASA ADS (last updated successfully 2024-03-28 07:54:16). The list may be incomplete as not all publishers provide suitable and complete citation data.