Use of the time evolution operator

Takeyuki Hida

Satoru Shoji, Kosuke Ito, Yukihiro Shimizu +1 more

Implementing time-evolution operators in shallow quantum circuits is important for quantum simulations. The standard method of Trotterization requires a large number of gates to achieve practical accuracy. Variational Quantum Simulation (VQS) is an algorithm that calculates the time evolution of a quantum state and can be executed with shallower circuits than Trotterization. However, the operator obtained by VQS evolves only a fixed initial state and is not the time evolution operator itself. In this paper, we propose Variational Quantum Operator Simulation (VQOS), a method to realize time evolution operators in shallow quantum circuits. This method is based on the variational principle for operators and does not require the implementation of the desired Trotter decomposition of the time evolution operator. We performed numerical simulations of the VQOS algorithm and successfully implemented the time evolution operator for closed systems in a quantum circuit that is up to 5 times shallower than the Trotterization. By providing a more practical way to implement time evolution operators, VQOS increases the applicability of near-term quantum computers.

P C García Quijas, L M Arévalo Aguilar

There is a widespread belief in the quantum physical community, and textbooks used to teach quantum mechanics, that it is a difficult task to apply the time evolution operator on an initial wavefunction. Because the Hamiltonian operator is, generally, the sum of two operators, then it is not possible to apply the time evolution operator on an initial wavefunction ψ( x , 0), for it implies using terms like . A possible solution is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wavefunction. However, the exponential operator does not directly factorize, i.e. . In this study we present a useful procedure for factorizing the time evolution operator when the argument of the exponential is a sum of two operators, which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like a particle subject to a constant force and a harmonic oscillator. Also, we discuss an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.

S. Balasubramanian

Donald H. Kobe, Kuo‐Ho Yang

The time-evolution operator is in general gauge dependent. Its gauge transformation property follows from the gauge transformation of the wave function and ensures gauge-invariant matrix elements. The same transformation property is shown here to follow from the formal solution of the Schr\"odinger equation for the time-evolution operator, which is a time-ordered exponential of the time integral of the Hamiltonian. The gauge transformation property of the time-evolution operator in the interaction picture is also obtained. The perturbation expansion of the time-evolution operator in one gauge can be transformed to give the perturbation expansion for the time-evolution operator in another gauge. The A\ensuremath{\cdot}p versus E\ensuremath{\cdot}r controversy in the electric dipole approximation is resolved by specifying the correct initial and final states.