# Application of Systems and Control Theory to Quantum Engineering

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_100158-1

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## Abstract

This entry is devoted to show that some quantum engineering problems, state transfer, state protection, non-demolition measurement, and back-action-evading measurement, can be formulated and solved within the framework of linear systems and control theory.

## Keywords

Quantum information Linear systems Controllability and observability## Introduction

*x*denotes the vector of variables (called “observables” in quantum mechanics) such as the position

*q*and the momentum

*p*of a particle: note that these variables are operators, not scalar-valued quantities, which thus do not commute with each other, e.g.,

*qp*−

*pq*=

*i*. Also, if one aims to control the system or extract information from it, typically an electromagnetic field called the probe field is injected into the system;

*u*denotes the vector of probe variables such as amplitude and phase of the filed.

*y*denotes the variables of (reflected or transmitted) output field or the signal obtained as a result of measurement on the output field. The matrices (

*A*,

*B*,

*C*,

*D*) are determined from the system. The goal of this entry is to show that some quantum engineering problems can be well formulated and solved within the framework of systems and control theory. See Nurdin and Yamamoto (2017) for other applications to quantum engineering.

## State Transfer

To realize quantum information technology such as quantum computation and quantum communication, it is very important to devise a scalable hybrid system composed of optical and solid-state systems, to compensate for their advantages and disadvantages; the main advantage of solid-state systems such as superconducting devices is that it is relatively easy to generate a highly nonlinear coupling and produce a genuine quantum state, but the disadvantage is that sending those states to other places is hard; on the other hand, optical systems such as an optical fiber network is suited to efficiently sending a quantum state to several sites, but in general, the quantumness of optical states is weak, due to the weak nonlinearity of optical systems. A key technique to connect these different systems is state transfer from optics to solid state and vice versa. In fact there have been a number of research studies on this topic, including the analysis of particular systems achieving perfect state transfer.

*A*= −

*i*Ω −

*C*

^{†}

*C*∕2 with Ω a Hermitian matrix and

*C*a complex row vector describing the system-probe coupling. The idea is simple; the perfect input

*u*(

*t*) (more precisely

*u*(

*t*) is the pulse shape of the input single-photon state) should be the one such that

*y*(

*t*) = 0 always holds for the duration of the transfer, meaning that the input field is completely absorbed into the solid-state system with variable

*x*(

*t*), and accordingly, the output field must be a vacuum. Then

*y*(

*t*) = 0, (

*t*≤

*t*

_{1}) readily leads to \(u^{\mathrm {opt}}(t)=-x(t_1)^\top e^{-A^*(t-t_1)}C^\top \) (•

^{†}, •

^{⊤}, and •

^{∗}represent the Hermitian conjugate, transpose, and element-wise complex conjugate, respectively). This special type of system, whose output is always zero, is said to have the

*zero dynamics*in the systems and control theory, which are completely characterized by the zeros of the transfer function in the linear case; Yamamoto et al. (2016) used this fact to extend the above result to the multi-input case. Moreover, Nakao and Yamamoto (2017) considered an optimal control problem where an additional control signal is introduced through the

*A*matrix (more precisely through the system Hamiltonian) for reducing the complexity of the desired pulse shape

*u*

^{opt}(

*t*).

## State Protection

Once a desirable quantum state is generated in a solid-state system by, e.g., the state transfer method described above, the next task is to manipulate and store that state. This process must be carried out in a subsystem isolated from the probe field as well as the surrounding environment. This subsystem is called a *decoherence-free subsystem (DFS)*.

*u*-uncontrollable and

*y*-unobservable subsystem for the system (1). Based on this fact, Yamamoto (2014) gave a simple necessary and sufficient condition for the general linear quantum system to have a DFS. An example is an atomic ensemble described by the following linear dynamical equation:

*x*

_{1},

*x*

_{2},

*x*

_{3}) are the system variables (spin operators) and (

*κ*,

*g*,

*δ*,

*ω*) are the system parameters. Clearly,

*x*

_{3}is decoupled from both the input and output fields when

*ω*= 0; thus, it functions as a DFS. This system can be used as a quantum memory as follows. By setting

*ω*≠ 0 and using the technique described in the previous subsection, one can transfer an input photon state to

*x*

_{3}; then by setting

*ω*= 0, that state is preserved. If one wants to use the stored state later, then the DFS again couples with the probe field and releases the state. Also, Yamamoto (2014) considered a linear system having a two-dimensional DFS and showed a simple manipulation of the DFS state.

An interesting related problem is how to engineer a system that contains a DFS. *Coherent feedback (CF)* control, the method connecting some quantum subsystems through electromagnetic fields in a feedback way without introducing any measurement process, gives a solution for this purpose. In fact it was demonstrated in Yamamoto (2014) that, actually, the CF method is applied to engineer a linear system having a DFS.

## Quantum Non-demolition Measurement

There is a need to precisely determine and control a particular element of *x*, say *x*_{s}, such as the motional variable of a mechanical oscillator and the number of photons in an optical cavity. For this purpose, rich information on *x*_{s} should be available via some measurement on the output probe field, while *x*_{s} should not be perturbed by the corresponding input probe field; if there exists such a variable *x*_{s}, then it is called a *quantum non-demolition (QND)* variable, and the corresponding measurement schematic is called the QND measurement. The system theoretic formulation of a QND variable is that *x*_{s} is *u*-uncontrollable and *y*-observable. Thanks to this general characterization, it is possible to take several new approaches for analyzing and synthesizing QND variables. For instance, using a special type of CF control, one can create a QND variable which does not exist in the original uncontrolled system (Yamamoto 2014).

## Back-Action-Evading Measurement

To detect a very small (i.e., quantum level) signal such as a gravitational wave and a tiny magnetic field, it is important to devise a special type of detector that fully takes into account quantum mechanical property; the *back-action-evading (BAE)* measurement is one such established method. Here we describe the problem for the system (1). This system functions as a sensor for a small signal, in such a way that the signal drives some *y*-observable elements of *x*. In the case where the probe field is single mode, the input is represented as *u* = (*Q*, *P*), where *Q* and *P* are the amplitude and phase variables of the input field; these are the so-called conjugate variables satisfying the Heisenberg uncertainty relation 〈 Δ*Q*^{2}〉〈 Δ*P*^{2}〉≥ 1. Also now the corresponding measurement output is given by *y* = *Cx* + *Q*. Hence it seems that S/N increases by reducing the magnitude of the noise *Q*, but this induces an increase of *P* from the abovementioned Heisenberg uncertainty relation; for this reason, *P* is called the back-action noise. Consequently, in the usual setup of the detector, the noise level of *y* is lower bounded by the so-called *standard quantum limit (SQL)*.

There have been a number of proposals for the detector configuration that can beat the SQL and achieve better sensitivity of the signal, that is, the aim is to devise a BAE detector such that *y* is not affected by *P*. In the language of systems and control theory, this condition is that the transfer function from *P* to *y*, Ξ(*s*), is zero for all *s*. This is equivalent to the geometric condition that the *y*-observable subspace is contained in the *P*-uncontrollable subspace. Using this characterization, (Yamamoto 2014; Yokotera and Yamamoto 2016) gave a new BAE force detector, based on the CF control. Moreover, in Yokotera and Yamamoto (2016), the BAE problem is formulated as the optimization problem ∥ Ξ(*s*)∥→ min. for synthesizing a detector that beats the SQL.

## References

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