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Universal programmable quantum circuit schemes to emulate an operator

### Abstract

Unlike fixed designs, programmable circuit designs support an infinite number of operators. The functionality of a programmable circuit can be altered by simply changing the angle values of the rotation gates in the circuit. Here, we present a new quantum circuit design technique resulting in two general programmable circuit schemes. The circuit schemes can be used to simulate any given operator by setting the angle values in the circuit. This provides a fixed circuit design whose angles are determined from the elements of the given matrix–which can be non-unitary–in an efficient way. We also give both the classical and quantum complexity analysis for these circuits and show that the circuits require a few classical computations. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator *U* = *e* ^{−iHt } for a given atomic and molecular Hamiltonian *H* in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. Since the presented circuit designs are independent from the matrix decomposition techniques and the global optimization processes used to find quantum circuits for a given operator, high accuracy simulations can be done for the unitary propagators of molecular Hamiltonians on quantum computers. As an example, we show how to build the circuit design for the hydrogen molecule.

© 2012 American Institute of Physics

Received 13 August 2012
Accepted 30 November 2012
Published online 21 December 2012

Acknowledgments:
This work is supported by the National Science Foundation (NSF) Centers for Chemical Innovation: Quantum Information for Quantum Chemistry, CHE-1037992.

Article outline:

I. INTRODUCTION
II. THE GENERAL SIMULATION IDEA
III. GENERATION OF PROGRAMMABLE CIRCUITS
A. The first circuit design
1. Formation step
2. Combination step
3. Input modification step
B. The second circuit design
1. Formation step
2. Combination step
3. Input modification
IV. COMPLEXITY ANALYSIS OF THE CIRCUITS
A. The complexity of the first circuit design
1. The classical complexity
2. The quantum complexity
B. The complexity of the second circuit
1. Classical complexity
2. The quantum complexity
C. Comparison with the non-programmable circuit designs
V. DISCUSSION AND CONCLUSION
A. Programmable quantum chips
B. Finding angles
C. Complex cases
D. Simulation of molecular Hamiltonians

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2012-12-21

2016-02-11

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