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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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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