OPEN Quantum control using quantum memory

Quantum control, the ability to precisely manipulate quantum systems, is crucial for future quantum computation and simulation. Existing methods often rely on dynamically adjusting external potentials throughout the process. This paper introduces a novel approach where control is encoded solely in the initial state of the system. The system studied is a discrete-time quantum walk (QW), a versatile computational model applicable to various phenomena. QWs typically involve a position register and an internal state (coin state), with the walker's propagation determined by the coin state. While QWs are powerful, controlling their evolution is challenging. This paper addresses this by introducing quantum memory to each site of the grid, allowing the walker's interactions to be conditioned by the memory states. This innovative approach enables complete control of the walker's dynamics, such as variance and mean trajectory, using only the initial state. The authors contend that this not only provides a distributed quantum computational model for single-qubit control but also hints at potential applications in modeling curved propagation. The simplicity and efficiency of this method offer significant advantages for simulating a wider range of dynamic physical models, potentially including those involving curved spacetimes.

The authors review existing quantum control methods, highlighting the typical reliance on dynamically adjusted external potentials. They discuss the universality of quantum walks (QWs) as computational models, mentioning their applications in search algorithms, graph isomorphism, and the simulation of both quantum and classical dynamics in various dimensions and topologies. Existing work on quantum walks with memory is also reviewed, focusing on models incorporating additional coins to record the walker's history. This paper differentiates itself by proposing a scheme where quantum memory is used to encode the control parameters directly into the initial state, enabling full control without requiring modifications to the evolution operator at each time step.

The paper begins by defining the model of a quantum walk in one spatial dimension, both with and without quantum memory. Without memory, the walker's state is defined in a Hilbert space composed of position and velocity (coin) components. The evolution operator (W) consists of a coin operator (C) and a shift operator (S). With quantum memory, an additional qubit is added to each site of the grid, resulting in a larger Hilbert space that incorporates the memory's state. The coin operator is replaced by a new operator (Q) that acts on the joint velocity-memory space, involving the memory qubits adjacent to the walker's position. The shift operator remains unchanged. The authors analyze a localized set of initial conditions where all memory qubits to the right of the walker with a non-zero internal state are set to |0⟩. Recurrence relations for the walker's amplitudes are derived, leading to Theorem 1 which provides analytical expressions for the left- and right-moving amplitudes. Using this theorem, the probability density of the walker's position, mean trajectory (E(t)), and variance (Var(t)) are calculated. The authors then provide examples demonstrating how to control the mean trajectory (linear and parabolic) and variance by manipulating the initial parameters {Ax, Bx}. Finally, Theorem 2 shows that with an appropriately chosen set of initial parameters (Bk), an arbitrary probability density Px(t) and consequently an arbitrary trajectory can be generated. The mathematical derivations for the amplitudes, probability density, mean trajectory, and variance are detailed in the appendix and are supported by numerical simulations.

The paper's key finding is that by incorporating quantum memory into a quantum walk, the walker's dynamics can be completely controlled by manipulating the initial state alone. The authors prove this analytically, showing how the initial conditions can be chosen to generate specific probability densities, mean trajectories, and variances. Specifically: * **Theorem 1:** Provides analytical expressions for the left- and right-moving amplitudes of the quantum walker, revealing how they depend solely on the initial parameters. This forms the basis for calculating the walker's probability density. * **Linear Mean Trajectory:** The authors demonstrate how to obtain a linear mean trajectory by appropriately setting the initial parameters, showing that the mean velocity is adjustable. * **Parabolic Mean Trajectory:** A non-linear variance, corresponding to a parabolic mean trajectory, is obtained by selecting different initial parameters, showcasing the versatility of the control scheme. * **Theorem 2:** This theorem proves that a completely arbitrary probability density, and thus an arbitrary trajectory, can be generated by carefully choosing the initial parameters (Bk). This means any desired walker behaviour can be pre-programmed in the initial state, eliminating the need for dynamic control during the walk. Numerical simulations are presented to corroborate the theoretical results, showing excellent agreement between theoretical predictions and simulation outcomes for both linear and parabolic mean trajectories.

The results demonstrate a significant advance in quantum control by eliminating the need for continuous adjustment of the evolution operator. The method achieves complete control solely through the initial state, reducing computational resources. The ability to generate arbitrary probability densities opens the door to simulating a wider range of physical phenomena, including those on curved manifolds. This aligns with the authors' suggestion of applications in modeling quantum particle propagation on curved spacetime. The simplicity and efficiency of the proposed scheme are noteworthy contributions, particularly considering the potential for future extensions to higher dimensions and applications in developing more efficient quantum algorithms such as spatial search algorithms.

This paper presents a novel approach to quantum control using quantum walks and memory qubits. By encoding the desired dynamics in the initial state, the authors have shown that complete control of a quantum walker's trajectory can be achieved without the need for continuous adjustments to the system's evolution. This simplifies simulations and opens up new possibilities for modeling complex physical phenomena on quantum devices, particularly those involving curved spacetimes. Future research directions include extensions to higher dimensions and exploration of applications in developing novel quantum algorithms.

The paper primarily focuses on a one-dimensional quantum walk. While the authors mention potential extensions to higher dimensions, this remains future work. The analytical calculations rely on specific initial conditions, and the generalizability to more complex initial states or system configurations remains to be explored. The impact of noise and decoherence on the control scheme is not explicitly addressed, which could be relevant for practical implementations.