Uncomputability and complexity of quantum control

TL;DR

An example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.

Abstract

In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent or dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These findings do not preclude an algorithmic solvability for a particular class of quantum control problems. Moreover, any quantum control problem can be made algorithmically solvable if the set of accessible interactions (i.e., controls) is rich enough. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.

Figures (1)

• Figure 1: A physical system for simulating Diophantine equations with $n$ variables. The system is either $n$ trapped ions or an $n$--mode coherent field. The controls $\hat{D}_1^{\dagger}$, $\ldots$, $\hat{D}_n^{\dagger}$ independently address each subsystem. For ions, the controls excite transitions between nearest levels, and transfer population of the highest excited state down to the ground state. For coherent states, the control for the $i$-th mode is the displacement $\hat{D}_i$ by the magnitude one. The Diophantine polynomial is embedded in the observable $\hat{O}$ whose expectation value has to be optimized as the control goal. A highly non-trivial example corresponds already to the simple case of a two-mode coherent field ($n=2$) with $\hat{O} = -(\alpha \hat{a}_1^{\dagger}\hat{a}_1^{\dagger}+ \beta \hat{a}_2^{\dagger} - \gamma ) (\alpha \hat{a}_1\hat{a}_1+ \beta \hat{a}_2 - \gamma )$, where $\alpha,\beta$, and $\gamma$ are positive integers. The observable is non-linear but physical; its leading term is of the Kerr-type nonlinearity. Maximizing the expectation of this observable is NP-hard, i.e., it is at least as hard as the famous Traveling Salesman Problem. Note that an $n=9$ system is sufficient to solve any Diophantine equation.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof