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arXiv:2102.07018

Factorizing time evolution into elementary steps

David Edward Bruschi

Published Feb 13, 2021• 0 citations
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Abstract

We propose an approach to factorize the time-evolution operator of a quantum system through a (finite) sequence of elementary operations that are time-ordered. Our proposal borrows from previous approaches based on Lie algebra techniques and other factorization procedures, and requires a set of optimization operations that provide the final result. Concretely, the algorithm produces at each step three optimal quantities, namely the optimal duration of the desired unitary operation, the optimal functional dependence of the driving function on the optimal time, and the optimal elementary Hermitian operation that induces the additional unitary operation to be implemented. The resulting sequence of unitary operations that is obtained this way is sequential with time. We compare our proposal with existing approaches, and highlight which key assumptions can be relaxed for practical implementations.

DOI
10.48550/arxiv.2102.07018

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We propose an approach to factorize the time-evolution operator of a quantum system through a (finite) sequence of elementary operations that are time-ordered. Our proposal borrows from previous approaches based on Lie algebra techniques and other factorization procedures, and requires a set of optimization operations that provide the final result. Concretely, the algorithm produces at each step three optimal quantities, namely the optimal duration of the desired unitary operation, the optimal functional dependence of the driving function on the optimal time, and the optimal elementary Hermitian operation that induces the additional unitary operation to be implemented. The resulting sequence of unitary operations that is obtained this way is sequential with time. We compare our proposal with existing approaches, and highlight which key assumptions can be relaxed for practical implementations.
The evolution operator of a quantum system in a time-dependent potential is factorized in unitary exponential operators at order 6. This expression is derived with the time-ordering method. It is compared with lower-order factorizations on several simple one-dimensional examples. Better accuracies are reached at sixth order for a given time step than at lower orders. Due to a significant increase of computation duration per time step, the sixth-order approximation is mainly useful when high accuracies are required.
Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the Lie–Trotter formula. The strategy is then extended first to simulation of any Hamiltonian that is a linear combination of two projection operators, and then to any local efficiently computable Hamiltonian. The key feature is to construct an evolution in terms of the largest possible steps instead of taking small time steps. Reflection operations and Chebyshev expansions are used to efficiently control the total error on the overall evolution, without worrying about discretization errors for individual steps. We also use a digital implementation of quantum states that makes linear algebra operations rather simple to perform.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
There is increasing interest in quantum algorithms (QAs) that are based on the imaginary time evolution (ITE), a successful classical numerical approach to obtain ground states. However, most of the proposals so far require heavy postprocessing computational steps on a classical computer, such as solving linear equations. Here we provide an alternative approach to implement ITE. A key feature in our approach is the use of an orthogonal basis set: the propagated state is efficiently expressed in terms of orthogonal basis states at every step of the evolution. We argue that the number of basis states needed at those steps to achieve an accurate solution can be kept on the order of nnn, the number of qubits, by controlling the precision (number of significant digits) and the imaginary time increment. The number of quantum gates per imaginary time step is estimated to be polynomial in nnn. Additionally, while in many QAs the locality of the Hamiltonian is a key assumption, in our algorithm this restriction is not required. This characteristic of our algorithm renders it useful for studying highly nonlocal systems, such as the occupation-representation nuclear shell model. We illustrate our algorithm through numerical implementation on an IBM quantum simulator.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator-splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speed-up over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
Quantum computation and quantum control operate by building unitary transformations out of sequences of elementary quantum logic operations or applications of control fields. This paper puts upper bounds on the minimum time required to implement a desired unitary transformation on a d-dimensional Hilbert space when applying sequences of Hamiltonian transformations. We show that strategies of building up a desired unitary out of non-infinitesimal and infinitesimal unitaries, or equivalently, using power and band limited controls, can yield the best possible scaling in time O(d2)O(d^2)O(d2).