Keywords

Footnote 1

Introduction

Quantum control is the control of systems whose dynamics are described by the laws of quantum physics rather than classical physics. The dynamics of quantum systems must be described using quantum mechanics which allows for uniquely quantum behavior such as entanglement and coherence. There are two main approaches to quantum mechanics which are referred to as the Schrödinger picture and the Heisenberg picture. In the Schrödinger picture, quantum systems are modeled using the Schrödinger equation or a master equation which describe the evolution of the system state or density operator. In the Heisenberg picture, quantum systems are modeled using quantum stochastic differential equations which describe the evolution of system observables. These different approaches to quantum mechanics lead to different approaches to quantum control. Important areas in which quantum control problems arise include physical chemistry, atomic and molecular physics, and optics. Detailed overviews of the field o quantum control can be found in the survey papers Dong and Petersen (2010) and Brif et al. (2010) and the monographs Wiseman and Milburn (2010) and D’Alessandro (2007).

A fundamental problem in a number of approaches to quantum control is the controllability problem. Quantum controllability problems are concerned with finite dimensional quantum systems modeled using the Schrödinger picture of quantum mechanics and involves the structure of corresponding Lie groups or Lie algebras; e.g., see D’Alessandro (2007). These problems are typically concerned with closed quantum systems which are quantum systems isolated from their environment. For a controllable quantum system, an open loop control strategy can be constructed in order to manipulate the quantum state of the system in a general way. Such open loop control strategies are referred to as coherent control strategies. Time optimal control is one method of constructing these control strategies which has been applied in applications including physical chemistry and in nuclear magnetic resonance systems; e.g., see Khaneja et al. (2001). An alternative approach to open loop quantum control is the Lyapunov approach; e.g., see Wang and Schirmer (2010). This approach extends the classical Lyapunov control approach in which a control Lyapunov function is used to construct a stabilizing state feedback control law. However in quantum control, state feedback control is not allowed since classical measurements change the quantum state of a system and the Heisenberg uncertainty principle forbids the simultaneous exact classical measurement of noncommuting quantum variables. Also, in many quantum control applications, the timescales are such that real time classical measurements are not technically feasible. Thus, in order to obtain an open loop control strategy, the deterministic closed loop system is simulated as if the state feedback control were available and this enables an open loop control strategy to be constructed. As an alternative to coherent open loop control strategies, some classical measurements may be introduced leading to incoherent control strategies; e.g., see Dong et al. (2009).

In addition to open loop quantum control approaches, a number of approaches to quantum control involve the use of feedback; e.g., see Wiseman and Milburn (2010). This quantum feedback may either involve the use of classical measurements, in which case the controller is a classical (nonquantum) system or it may involve the case where no classical measurements are used since the controller itself is a quantum system. The case in which the controller itself is a quantum system is referred to as coherent quantum feedback control; e.g., see Lloyd (2000) and James et al. (2008). Quantum feedback control may be considered using the Schrödinger picture, in which case the quantum systems under consideration are modeled using stochastic master equations. Alternatively using the Heisenberg picture, the quantum systems under consideration are modeled using quantum stochastic differential equations. Applications in which quantum feedback control can be applied include quantum optics and atomic physics. In addition, quantum control can potentially be applied to problems in quantum information (e.g., see Nielsen and Chuang 2000) such as quantum error correction (e.g., see Kerckhoff et al. 2010) or the preparation of quantum states. Quantum information and quantum computing in turn have great potential in solving intractable computing problems such as factoring large integers using Shor’s algorithm; see Shor (1994).

Schrödinger Picture Models of Quantum Systems

The state of a closed quantum system can be represented by a unit vector \(\vert \psi \rangle\) in a complex Hilbert space \(\mathcal{H}\). Such a quantum state is also referred to as a wavefunction. In the Schrödinger picture, the time evolution of the quantum state is defined by the Schrödinger equation which is in general a partial differential equation. An important class of quantum systems are finite-level systems in which the Hilbert space is finite dimensional. In this case, the Schrödinger equation is a linear ordinary differential equation of the form

$$\displaystyle\begin{array}{rcl} i\hslash \frac{\partial } {\partial t}\vert \psi (t)\rangle & =& H_{0}\vert \psi (t)\rangle {}\\ \end{array}$$

where H0 is the free Hamiltonian of the system, which is a self-adjoint operator on \(\mathcal{H}\); e.g., see Merzbacher (1970). Also, \(\hslash\) is the reduced Planck’s constant, which can be assumed to be one with a suitable choice of units. In the case of a controlled closed quantum system, this differential equation is extended to a bilinear ordinary differential equation of the form

$$\displaystyle\begin{array}{rcl} i \frac{\partial } {\partial t}\vert \psi (t)\rangle & =& \left [H_{0} +\sum _{ k=1}^{m}u_{ k}(t)H_{k}\right ]\vert \psi (t)\rangle {}\end{array}$$
(1)

where the functions u k (t) are the control variables and the H k are corresponding control Hamiltonians, which are also assumed to be self-adjoint operators on the underlying Hilbert space. These models are used in the open loop control of closed quantum systems.

To represent open quantum systems, it is necessary to extend the notion of quantum state to density operators ρ which are positive operators with trace one on the underlying Hilbert space \(\mathcal{H}\). In this case, the Schrödinger picture model of a quantum system is given in terms of a master equation which describes the time evolution of the density operator. In the case of an open quantum system with Markovian dynamics defined on a finite dimensional Hilbert space of dimension N, the master equation is a matrix differential equation of the form

$$\displaystyle\begin{array}{rcl} \dot{\rho }(t)& =& -i\left [\left (H_{0} +\sum _{ k=1}^{m}u_{ k}(t)H_{k}\right ),\rho (t)\right ] \\ & & +\frac{1} {2}\sum _{j,k=0}^{N^{2}-1 }\alpha _{j,k}\left (\left [F_{j}\rho (t),F_{k}^{\dag }\right ]\right . \\ & & +\left .\left [F_{j},\rho (t)F_{k}^{\dag }\right ]\right ); {}\end{array}$$
(2)

e.g., see Breuer and Petruccione (2002). Here the notation \([X,\rho ] = X\rho -\rho X\) refers to the commutation operator and the notation† denotes the adjoint of an operator. Also, \(\left \{F_{j}\right \}_{j=0}^{N^{2}-1 }\) is a basis set for the space of bounded linear operators on \(\mathcal{H}\) with F0 = I. Also, the matrix \(A = \left (\alpha _{j,k}\right )\) is assumed to be positive definite. These models, which include the Lindblad master equation for dissipative quantum systems as a special case (e.g., see Wiseman and Milburn 2010), are used in the open loop control of finite-level Markovian open quantum systems.

In quantum mechanics, classical measurements are described in terms of self-adjoint operators on the underlying Hilbert space referred to as observables; e.g., see Breuer and Petruccione (2002). An important case of measurements are projective measurements in which an observable M is decomposed as \(M =\sum _{ k=1}^{m}kP_{k}\) where the P k are orthogonal projection operators on \(\mathcal{H}\); e.g., see Nielsen and Chuang (2000). Then, for a closed quantum system with quantum state \(\vert \psi \rangle\), the probability of an outcome k from the measurement is given by \(\langle \psi \vert P_{k}\vert \psi \rangle\) which denotes the inner product between the vector \(\vert \psi \rangle\) and the vector \(P_{k}\vert \psi \rangle\). This notation is referred to as Dirac notation and is commonly used in quantum mechanics. If the outcome of the quantum measurement is k, the state of the quantum system collapses to the new value of \(\frac{P_{k}\vert \psi \rangle } {\sqrt{\langle \psi \vert P_{k } \vert \psi \rangle }}\). This change in the quantum state as a result of a measurement is an important characteristic of quantum mechanics. For an open quantum system which is in a quantum state defined by a density operator ρ, the probability of a measurement outcome k is given by tr(P k ρ). In this case, the quantum state collapses to \(\frac{P_{k}\rho P_{k}} {\mathrm{tr}(P_{k}\rho )}.\)

In the case of an open quantum system with continuous measurements of an observable X, we can consider a stochastic master equation as follows:

$$\displaystyle\begin{array}{rcl} \mathrm{d}\rho (t)& =& -i\left [\left (H_{0} +\sum _{ k=1}^{m}u_{ k}(t)H_{k}\right ),\rho (t)\right ]\mathrm{d}t \\ & & -\kappa \left [X,\left [X,\rho (t)\right ]\right ]\mathrm{d}t \\ & & +\sqrt{2\kappa }\left (X\rho (t) +\rho (t)X\right . \\ & & -2\mathrm{tr}\left .\left (X\rho (t)\right )\rho (t)\right )\mathrm{d}W {}\end{array}$$
(3)

where κ is a constant parameter related to the measurement strength and dW is a standard Wiener increment which is related to the continuous measurement outcome y(t) by

$$\displaystyle{ \mathrm{d}W =\mathrm{ d}y - 2\sqrt{\kappa }\mathrm{tr}\left (X\rho (t)\right )\mathrm{d}t; }$$
(4)

e.g., see Wiseman and Milburn (2010). These models are used in the measurement feedback control of Markovian open quantum systems. Also, the Eqs. (3) and (4) can be regarded as a quantum filter in which ρ(t) is the conditional density of the quantum system obtained by filtering the measurement signal y(t); e.g., see Bouten et al. (2007) and Gough et al. (2012).

Heisenberg Picture Models of Quantum Systems

In the Heisenberg picture of quantum mechanics, the observables of a system evolve with time and the quantum state remains fixed. This picture may also be extended slightly by considering the time evolution of general operators on the underlying Hilbert space rather than just observables which are required to be self-adjoint operators. An important class of open quantum systems which are considered in the Heisenberg picture arise when the underlying Hilbert space is infinite dimensional and the system represents a collection of independent quantum harmonic oscillators interacting with a number of external quantum fields. Such linear quantum systems are described in the Heisenberg picture by linear quantum stochastic differential equations (QSDEs) of the form

$$\displaystyle\begin{array}{rcl} \mathrm{d}x(t)& =& Ax(t)\mathrm{d}t + B\mathrm{d}w(t); \\ \mathrm{d}y(t)& =& Cx(t)\mathrm{d}t + D\mathrm{d}w(t){}\end{array}$$
(5)

where A, B, C, D are real or complex matrices, x(t) is a vector of possibly noncommuting operators on the underlying Hilbert space \(\mathcal{H}\); e.g., see James et al. (2008). Also, the quantity dw(t) is decomposed as

$$\displaystyle{\mathrm{d}w(t) =\beta _{w}(t)\mathrm{d}t +\mathrm{ d}\tilde{w}(t)}$$

where β w (t) is an adapted process and \(\tilde{w}(t)\) is a quantum Wiener process with Itô table:

$$\displaystyle{\mathrm{d}\tilde{w}(t)\mathrm{d}\tilde{w}(t)^{\dag } = F_{\tilde{ w}}\mathrm{d}t.}$$

Here, \(F_{\tilde{w}} \geq 0\) is a real or complex matrix. The quantity w(t) represents the components of the input quantum fields acting on the system. Also, the quantity y(t) represents the components of interest of the corresponding output fields that result from the interaction of the harmonic oscillators with the incoming fields.

In order to represent physical quantum systems, the components of vector x(t) are required to satisfy certain commutation relations of the form

$$\displaystyle{\left [x_{j}(t),x_{k}(t)\right ] = 2i\Theta _{jk},\ j,k = 1,2,\ldots ,n,\ \forall t}$$

where the matrix \(\Theta = \left (\Theta _{jk}\right )\) is skew symmetric. The requirement to represent a physical quantum system places restrictions on the matrices A, B, C, D, which are referred to as physical realizability conditions; e.g., see James et al. (2008) and Shaiju and Petersen (2012). QSDE models of the form (5) arise frequently in the area of quantum optics. They can also be generalized to allow for nonlinear quantum systems such as arise in the areas of nonlinear quantum optics and superconducting quantum circuits; e.g., see Bertet et al. (2012). These models are used in the feedback control of quantum systems in both the case of classical measurement feedback and in the case of coherent feedback in which the quantum controller is also a quantum system and is represented by such a QSDE model.

(S, L, H) Quantum System Models

An alternative method of modeling an open quantum system as opposed to the stochastic master equation (SME) approach or the quantum stochastic differential equation (QSDE) approach, which were considered above, is to simply model the quantum system in terms of the physical quantities which underlie the SME and QSDE models. For a general open quantum system, these quantities are the scattering matrix S which is a matrix of operators on the underlying Hilbert space, the coupling operator L which is a vector of operators on the underlying Hilbert space, and the system Hamiltonian which is a self-adjoint operator on the underlying Hilbert space; e.g., see Gough and James (2009). For a given (S, L, H) model, the corresponding SME model or QSDE model can be calculated using standard formulas; e.g., see Bouten et al. (2007) and James et al. (2008). Also, in certain circumstances, an (S, L, H) model can be calculated from an SME model or a QSDE model. For example, if the linear QSDE model (5) is physically realizable, then a corresponding (S, L, H) model can be found. In fact, this amounts to the definition of physical realizability.

Open Loop Control of Quantum Systems

A fundamental question in the open loop control of quantum systems is the question of controllability. For the case of a closed quantum system of the form (1), the question of controllability can be defined as follows (e.g., see Albertini and D’Alessandro 2003):

Definition 1 (Pure State Controllability)

The quantum system (1) is said to be pure state controllable if for every pair of initial and final states \(\vert \psi _{0}\rangle\) and \(\vert \psi _{f}\rangle\), there exist control functions {u k (t)} and a time T > 0 such that the corresponding solution of (1) with initial condition \(\vert \psi _{0}\rangle\) satisfies \(\vert \psi (T)\rangle = \vert \psi _{f}\rangle\).

Alternative definitions have also been considered for the controllability of the quantum system (1); e.g., see Albertini and D’Alessandro (2003) and Grigoriu et al. (2013) in the case of open quantum systems. The following theorem provides a necessary and sufficient condition for pure state controllability in terms of the Lie algebra \(\mathcal{L}_{0}\) generated by the matrices \(\left \{-iH_{0},-iH_{1},\ldots ,-iH_{m}\right \}\), u(N) the Lie algebra corresponding to the unitary group of dimension N, su(N) the Lie algebra corresponding to the special unitary group of dimension N, \(\mathrm{sp}(\frac{N} {2} )\) the \(\frac{N} {2}\) dimensional symplectic group, and \(\tilde{\mathcal{L}}\) the Lie algebra conjugate to \(\mathrm{sp}(\frac{N} {2} )\).

Theorem 1 (See D’Alessandro 2007)

The quantum system (1) is pure state controllable if and only if the Lie algebra\(\mathcal{L}_{0}\)satisfies one of the following conditions:

  1. (1)

    \(\mathcal{L}_{0} =\mathrm{ su}(N)\) ;

  2. (2)

    \(\mathcal{L}_{0}\) is conjugate to \(\mathrm{sp}(\frac{N} {2} )\) ;

  3. (3)

    \(\mathcal{L}_{0} =\mathrm{ u}(N)\) ;

  4. (4)

    \(\mathcal{L}_{0} =\mathrm{ span}\left \{iI_{N\times N}\right \} \oplus \tilde{\mathcal{L}}\) .

Similar conditions have been obtained when alternative definitions of controllability are used.

Once it has been determined that a quantum system is controllable, the next task in open loop quantum control is to determine the control functions {u k (t)} which drive a given initial state to a given final state. An important approach to this problem is the optimal control approach in which a time optimal control problem is solved using Pontryagin’s maximum principle to construct the control functions {u k (t)} which drives the given initial state to the given final state in minimum time; e.g., see Khaneja et al. (2001). This approach works well for low dimensional quantum systems but is computationally intractable for high dimensional quantum systems.

An alternative approach for high dimensional quantum systems is the Lyapunov control approach. In this approach, a Lyapunov function is selected which provides a measure of the distance between the current quantum state and the desired terminal quantum state. An example of such a Lyapunov function is

$$\displaystyle{V =\langle \psi (t) -\psi _{f}\vert \psi (t) -\psi _{f}\rangle \geq 0;}$$

e.g., see Mirrahimi et al. (2005). A state feedback control law is then chosen to ensure that the time derivative of this Lyapunov function is negative. This state feedback control law is then simulated with the quantum system dynamics (1) to give the required open loop control functions {u k (t)}.

Classical Measurement Based Quantum Feedback Control

A Schrödinger Picture Approach to Classical Measurement Based Quantum Feedback Control

In the Schrödinger picture approach to classical measurement based quantum feedback control with weak continuous measurements, we begin the stochastic master equations (3) and (4) which are considered as both a model for the system being controlled and as a filter which will form part of the final controller. These filter equations are then combined with a control law of the form

$$\displaystyle{u(t) = f(\rho (t))}$$

where the function \(f(\cdot )\) is designed to achieve a particular objective such as stabilization of the quantum system. Here u(t) represents the vector of control inputs u k (t). An example of such a quantum control scheme is given in the paper Mirrahimi and van Handel (2007) in which a Lyapunov method is used to design the control law \(f(\cdot )\) so that a quantum system consisting of an atomic ensemble interacting with an electromagnetic field is stabilized about a specified state \(\rho _{f} = \vert \psi _{m}\rangle \langle \psi _{m}\vert\).

A Heisenberg Picture Approach to Classical Measurement Based Quantum Feedback Control

In this Heisenberg picture approach to classical measurement based quantum feedback control, we begin with a quantum system which is described by linear quantum stochastic equations of the form (5). In these equations, it is assumed that the components of the output vector all commute with each other and so can be regarded as classical quantities. This can be achieved if each of the components are obtained via a process of homodyne detection from the corresponding electromagnetic field; e.g., see Bachor and Ralph (2004). Also, it is assumed that the input electromagnetic field w(t) can be decomposed as

$$\displaystyle{ \mathrm{d}w(t) = \left [\begin{array}{c} \beta _{u}(t)\mathrm{d}t +\mathrm{ d}\tilde{w}_{1}(t) \\ \mathrm{d}w_{2}(t) \end{array} \right ] }$$
(6)

where β u (t) represents the classical control input signal and \(\tilde{w}_{1}(t),\ w_{2}(t)\) are quantum Wiener processes. The control signal displaces components of the incoming electromagnetic field acting on the system via the use of an electro-optic modulator; e.g., see Bachor and Ralph (2004).

The classical measurement feedback based controllers to be considered are classical systems described by stochastic differential equations of the form

$$\displaystyle\begin{array}{rcl} \mathrm{d}x_{K}(t)& =& A_{K}x_{k}(t)\mathrm{d}t + B_{K}\mathrm{d}y(t) \\ \beta _{u}(t)\mathrm{d}t& =& C_{K}x_{k}(t)\mathrm{d}t. {}\end{array}$$
(7)

For a given quantum system model (5), the matrices in the controller (7) can be designed using standard classical control theory techniques such as LQG control (see Doherty and Jacobs 1999) or H∞ control (see James et al. 2008).

Coherent Quantum Feedback Control

Coherent feedback control of a quantum system corresponds to the case in which the controller itself is a quantum system which is coupled in a feedback interconnection to the quantum system being controlled; e.g., see Lloyd (2000). This type of control by interconnection is closely related to the behavioral interpretation of feedback control; e.g., see Polderman and Willems (1998).

An important approach to coherent quantum feedback control occurs in the case when the quantum system to be controlled is a linear quantum system described by the QSDEs (5). Also, it is assumed that the input field is decomposed as in (6). However in this case, the quantity β u (t) represents a vector of noncommuting operators on the Hilbert space underlying the controller system. These operators are described by the following linear QSDEs, which represent the quantum controller:

$$\displaystyle\begin{array}{rcl} \mathrm{d}x_{K}(t)& =& A_{K}x_{k}(t)\mathrm{d}t + B_{K}\mathrm{d}y(t) +\bar{ B}_{K}\mathrm{d}\bar{w}_{K}(t) \\ \mathrm{d}y_{K}(t)& =& C_{K}x_{k}(t)\mathrm{d}t +\bar{ D}_{K}\mathrm{d}\bar{w}_{K}(t). {}\end{array}$$
(8)

Then, the input β u (t) is identified as

$$\displaystyle{\beta _{u}(t) = C_{K}x_{k}(t).}$$

Here the quantity

$$\displaystyle{ \mathrm{d}w_{K}(t) = \left [\begin{array}{c} \mathrm{d}y(t)\\ \mathrm{d}\bar{w}_{ K}(t) \end{array} \right ] }$$
(9)

represents the quantum fields acting on the controller quantum system and where w K (t) corresponds to a quantum Wiener process with a given Itô table. Also, y(t) represents the output quantum fields from the quantum system being controlled. Note that in the case of coherent quantum feedback control, there is no requirement that the components of y(t) commute with each other and this in fact represents one of the main advantages of coherent quantum feedback control as opposed to classical measurement based quantum feedback control.

An important requirement in coherent feedback control is that the QSDEs (8) should satisfy the conditions for physical realizability; e.g., see James et al. (2008). Subject to these constraints, the controller (8) can then be designed according to an H∞ or LQG criterion; e.g., see James et al. (2008) and Nurdin et al. (2009). In the case of coherent quantum H∞ control, it is shown in James et al. (2008) that for any controller matrices (A K , B K , C K ), the matrices \((\bar{B}_{K},\bar{D}_{K})\) can be chosen so that the controller QSDEs (8) are physically realizable. Furthermore, the choice of the matrices \((\bar{B}_{K},\bar{D}_{K})\) does not affect the H∞ performance criterion considered in James et al. (2008). This means that the coherent controller can be designed using the same approach as designing a classical H∞ controller.

In the case of coherent LQG control such as considered in Nurdin et al. (2009), the choice of the matrices \((\bar{B}_{K},\bar{D}_{K})\) significantly affects the closed loop LQG performance of the quantum control system. This means that the approach used in solving the coherent quantum H∞ problem given in James et al. (2008) cannot be applied to the coherent quantum LQG problem. To date there exist only some nonconvex optimization methods which have been applied to the coherent quantum LQG problem (e.g., see Nurdin et al. 2009), and the general solution to the coherent quantum LQG control problem remains an open question.

Cross-References