Application of Pontryagin's minimum principle to Grover's quantum search problem

Abstract

Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schrödinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.PACS numbers: I. INTRODUCTIONQuantum computation deliberately uses the quantum-mechanical phenomena, such as superposition and entanglement, to reduce the computation time or the number of queries to accomplish certain tasks [1,2]. Well known quantum algorithms include Shor's algorithm for factoring [3] and Grover's algorithm for searching an unstructured database or an unordered list [4,5]. While the standard paradigm for quantum computation involves a discrete sequence of unitary logic gates [3-9] there exists another paradigm, pioneered by Farhi and Gutmann [10], where the quantum register evolves under some designed Hamiltonian which can vary continuously in time [10-13]. The concept of "continuous-time" quantum computation explicitly allows established physics principles such as the adiabatic theorem [14-16] and the Trotter-Suzuki decomposition [17,18] to guide how quantum algorithms can be designed. The adiabatic theorem is, for example, the foundation of the quantum annealing technique [11,[19][20][21][22] and the fast, non-adiabatic evolution is found to be helpful for other problems [23,24]. More recently, there are quantum algorithms based on the variational principle, notably the Variational Quantum Eigensolver (VQE) [25][26][27] and the Quantum Approximate Optimization Algorithm (QAOA) [28][29][30]. They are more fault-tolerant than quantum algorithms of the standard paradigm and are promising for Noisy Intermediate-Scale Quantum (NISQ) technology [31].Generally, when applying quantum algorithms to solve a classical NP (non-deterministic polynomial-time) problem, we are given a quantum "problem Hamiltonian" (oracle) whose ground state is the solution of the original classical problem [21,[32][33][34]. Designing a quantum algorithm is equivalent to find a "driving Hamiltonian" and an initial state, both easily implemented, that can steer the initial state to the target state (e.g., ground state of the problem Hamiltonian) within the shortest time. From this point of view, time-optimal control [35-37] appears to be fundamentally connected to quantum computation as both a...

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Optimal Control for Quantum Metrology via Pontryagin's principle

“…Pontryagin's Maximum Principle (PMP) [49][50][51][52] is a powerful tool in classical control theory, and it has been applied to quantum state preparation [7,53] and non-adiabatic quantum computation [54,55]. In essence, PMP adopts the variational principle to derive a set of necessary conditions for the optimal control.…”

“…In particular, it provides an efficient way to compute the gradient of the cost function with respect to the control field as well as the evolution time by introducing an auxiliary system (described by costate variables) that follows the dynamics similar to the original problem. When the system degrees of freedom are small (such as a single qubit), these necessary conditions are very restrictive and analytical solutions can sometimes be constructed [7,53,55]. For systems of higher dimensions, these necessary conditions become less informative but the efficient procedure of computing gradient is still useful for numerical solutions.…”