OPEN Quantum control using quantum memory

Quantum control refers to the ability to steer a dynamical quantum system from an initial to a desired target or outcome, with a desired accuracy¹. Several theoretical and experimental approaches to model controlled wave packets and their application are very useful to pave the way for future simulation or quantum calculation schemes². In many of these, the physical system to be controlled is driven by an external potential, which needs to be controlled all along the experience, until the target is achieved. Although in this work we do not claim to offer a general theory of quantum control, we provide a new approach in which the control scheme is encoded once and for all into its initial state. The main protagonist here is not a generic quantum system, but a quantum walks (QW) in discrete time³.⁴. What may seem like a particular choice, in reality offers great potential, given the recognised versatility of this simple system. In fact, QW are a universal computational model, that spans a large spectrum of physical and biological phenomena, relevant both for fundamental science and for applications. Applications include search algorithms⁹⁻¹² and graph isomorphism algorithms¹³ to modeling and simulating quantum¹⁴⁻¹⁸ and classical dynamics¹⁹,²⁰. These models have sparked various theoretical investigations covering areas in mathematics, computer science, quantum information and statistical mechanics and have been defined in any physical dimensions²¹,²² and over several topologies²³⁻²⁵. QW appear in multiple variants and can be defined on arbitrary graphs. Essentially, these simple systems have two registers: one for its position on the graph and the other is its internal state, often called coin state. It propagates on the graph, conditioned by its internal state, similarly to the classical case, where at each step we flip a coin to determine the direction of the walker. The essential difference is that in the quantum case, the walker propagates in superposition on the graph in various directions starting from a node. This feature allows the quantum walker to explore the graph quadratically faster a classical one, property that make it very useful to design, e.g., efficient search algorithms. However, we do not know many way to control the quantum walker evolution. For instance we can choose the initial condition and the evolution operator to tune the walker's variance σ²(t) = af(t), where a is a real prefactor and f(t) is typically a linear function of t. However, once these are fixed at the initial time, both f and a remain the same all along the evolution, unless we do not allow the evolution operator to change in an in-homogeneous way at each time-step, as in²⁶,²⁷, which may be very costly. How can we control the walker's dynamics at our will without having to change the evolution operator? Would it be possible to control, having only the initial condition, the variance or its average trajectory? In this manuscript we argue that, at the price of introducing a quantum memory, the answer is affirmative. Quantum walks with memory have already been studied and come in several variants²⁸,²⁹. As an example, these modified quantum walks may have extra coins to record the walker's latest path, as in³⁰,³¹. Here, the idea is to define an additional qubit for each site in the grid, with which the walker interacts throughout the evolution. Surprisingly, we will prove that the initial condition of the whole system, memory + walker, is sufficient to control, e.g., the variance and the mean position of the walker for all times. The interest is double: from one hand we provide a simple distributed quantum computational model to control a single qubit along its dynamics, which will not require us to control and adjust the local update rule at each time step; from a totally different perspective, this simple system may suggest an operational way to model and to unitary discretise curved propagation, as argued in³². The manuscript is organised as follows: in "The model" we will provide the definition of the model with and without memory, in one spatial dimension; then, in "Control the walker's dynamics", we will prove analytically and numerically how to control the variance and the mean trajectory of a quantum walker, solely via the initial condition of the whole system. Finally, in "Discussion" we discuss and conclude.

The authors study a discrete-time quantum walk (QW) with and without site-dependent quantum memory. The standard QW is defined on H = X ⊗ V, where X is the position Hilbert space with basis |x⟩, x ∈ Z^N, and V = C^2 is the coin (velocity) space with basis {|v^+⟩, |v^-⟩}. The walker’s state at time t is Ψ_t = ∑_x ψ_x^+(t)|x⟩|v^+⟩ + ψ_x^-(t)|x⟩|v^-⟩. Each time step applies a coin operator C ∈ U(2) on V (example C with angle θ given) followed by a conditional shift S that moves the |v^+⟩ component to x+1 and |v^-⟩ to x−1, yielding W = S (Id_X ⊗ C). Quantum memory extension: At each lattice site x, a memory qubit |m_x⟩ ∈ M_x = C^2 is introduced. The global Hilbert space becomes X ⊗ V ⊗ ∏_x M_x. The coin operator is replaced by a local update Q that acts on the walker’s velocity and the two neighboring memory qubits at positions x−1 and x+1. Q is specified in the text as a sum over projectors on memory states coupled to velocity flips/keeps, designed to yield a history-dependent walk. The shift S acts as usual on X ⊗ V and trivially on memory. The overall step operator is G = S Q. Initial condition for control: They choose localized initial conditions where the walker starts at x=0 with some internal state |ν̄⟩, and for all sites to the right of any site with nonzero internal state, memory qubits are initialized to |0⟩. More explicitly, Ψ(0) = |0⟩|ν̄⟩ ⊗ (∑_{x=-∞}^{N−1} (A_x|0⟩ + B_x|1⟩)|x=0⟩), with the memory configuration depicted in their Fig. 2. Here {A_x, B_x} are parameters set at t=0 that encode the control. Dynamics in the single-walker subspace yields recurrence relations (derived in the Appendix): - Ψ_x^−(t+1) = A_{−x} Ψ_{x+1}^−(t) - Ψ_x^+(t+1) = Ψ_{x−1}^+(t) + B_{−x+2} Ψ_{x−1}^−(t) Theorem 1 provides closed-form solutions for Ψ_x^±(t) under the given initial conditions, leading to explicit expressions of the probability distribution P_x(t), mean E(t), and variance Var(t) entirely in terms of products over {A_i} and factors {B_k} chosen in the initial memory state. The methodology proceeds to construct specific choices of {A_k, B_k} to realize desired mean trajectories (linear, parabolic) and non-linear variances. Finally, Theorem 2 prescribes how to choose B_k in terms of an arbitrary nonnegative sequence {f_i} (satisfying normalization constraints) so that the full time-dependent spatial probability distribution P_x(t) follows a pre-specified profile across positions at each time t, all determined solely by the initial memory parameters.

- Exact solvability with memory-based control: With a per-site memory qubit and a fixed local unitary Q coupling the walker’s coin to neighboring memories, the walker’s amplitudes obey simple recurrences that admit closed-form solutions (Theorem 1). The probability distribution for |x| ≤ t is: P_x(t) = { (∏_{i=1}^t A_i^2) if x = −t; 0 if x−t is odd; B_{t+1}^2 ∏_{j=1}^t A_j^2 otherwise }. - Mean and variance controlled by initial parameters {A_i, B_i}: E(t) = −∏_{i=1}^t A_i^2 + ∑_{k=0}^{t−1} (−2k) B_{t+1} ∏_{j=1}^t A_j^2. Var(t) = ∏_{i=1}^t A_i^2 − (∏_{i=1}^t A_i^2 − ∑_{k=0}^{t−1} (−2k) B_{t+1} ∏_{j=1}^t A_j^2)^2 + ∑_{i=1}^t (2i−1)^2 B_{2i+1}^2 ∏_{j=1}^t A_j^2. Both moments depend solely on the initial memory parameters. - Linear mean trajectory example: Setting A_1 = 1 and B_k = 0 for k>1 (with A_1^2 = 1 − B_1^2) gives P_x(t) concentrated at x=−t and x=t with weights ((1−B_1^2)^{2t}, 1), and mean E(t) = −t(1 − 2B_1^2). The variance scales ballistically ∝ t^2. - Parabolic mean trajectory and non-linear variance: Choosing B_k = sqrt((z(k−1) − 2)/(z+2)) with z>0 yields P_x(t) taking fixed values at x=−t and elsewhere, mean E(t) = t(zt − 2)/(z+2), and a time-dependent standard deviation σ(t) with non-linear dependence on t and parameter z, explicitly given in the text. - Arbitrary target probability profiles (Theorem 2): For any sequence {f_i} with 0 ≤ f_i ≤ 1 and ∑_{i=0}^{t−1} f_i ≤ 1, choosing B_k^2 = f_{k−1}/(1 − ∑_{i=1}^{k−1} f_i) leads to P_x(t) = { 1 − ∑_{i=0}^{t−1} f_i if x = −t; 0 if x−t is odd or |x|>t; f_x otherwise }. Thus, the entire space-time evolution of P_x(t), and hence E(t) and Var(t), can be pre-programmed via the initial memory state without time-dependent control of the evolution operator.

The work demonstrates that adding a local memory qubit at each lattice site and coupling it to the walker's internal state enables full control of the walker’s dynamics—probability distribution, mean trajectory, and variance—through parameters fixed once in the initial state. This removes the need for inhomogeneous, time-dependent control of the local unitary during evolution. Such capability can emulate effective motion on curved manifolds by encoding curvature effects into initial memory configurations, suggesting an operational route to unitary discretizations of curved propagation. Computationally, embedding the desired trajectory into the initial state offers a resource advantage for simulating diverse dynamical physical models on near-term quantum devices. The approach could also inspire extensions to quantum algorithms, such as spatial search schemes based on controlled quantum walks, and motivates generalizations to higher-dimensional lattices.

The paper introduces a history-dependent discrete-time quantum walk where a per-site quantum memory interacts with the walker via a fixed local unitary, enabling exact, analytic control of the walker's probability distribution, mean trajectory, and variance solely through the initial memory state. The authors provide closed-form solutions (Theorem 1) and a constructive prescription (Theorem 2) to realize arbitrary target distributions under mild constraints, with illustrative linear and parabolic trajectory examples. This framework facilitates simulating dynamics akin to propagation on curved manifolds and reduces the need for time-varying controls in implementations. Future research directions include: extending the model to higher spatial dimensions; exploring efficient quantum algorithms (e.g., spatial search) leveraging memory-based control; investigating robustness under noise and imperfect memory initialization; and experimental realizations on quantum platforms supporting local memory-register interactions.