Links to Quantum Control

Quantum control turns the forward problem of driven dynamics into a constrained inverse problem. Instead of asking only what a prescribed Hamiltonian does, one asks which allowed waveform, pulse sequence, or drive cycle produces a stated target with acceptable error and robustness.

This page is the bridge from closed-system dynamics to that design problem. It translates propagators, rotating frames, and Floquet maps into quantities used to design pulses. Pulse protocols, optimization, decoherence, feedback, and experimental limits are developed in Quantum Control and Feedback.

Forward Dynamics and Control Design

For a prescribed Hamiltonian $H(t)$, the forward problem is

$$i\hbar\dot U(t) = H(t)U(t), \qquad U(0)=I.$$

Given $H(t)$, one computes $U(t_f)$. A control problem reverses the emphasis:

$$\text{choose allowed controls} \quad\longrightarrow\quad \text{make }U(t_f)\text{ or }\rho(t_f) \text{ meet a target}.$$

The inverse is rarely unique. Many waveforms may implement the same target, and the most useful one depends on constraints such as

• maximum amplitude and slew rate;
• available carrier frequencies and phases;
• pulse duration and timing resolution;
• leakage outside a computational or spectroscopic subspace;
• uncertainty in detuning and coupling strength;
• decoherence, heating, and calibration drift;
• the distinction between state preparation, gate synthesis, and channel engineering.

A mathematically reachable target need not be experimentally accurate, fast, or robust. That distinction runs through all of quantum control.

Control Hamiltonians

A common closed-system model is control affine:

$$H(t) = H_0 + \sum_{a=1}^{m} u_a(t)H_a.$$

Here $H_0$ is the drift Hamiltonian, the $H_a$ are fixed control operators, and the real functions $u_a(t)$ are adjustable waveforms. Examples include electric-dipole, magnetic-dipole, gate-voltage, flux, tunneling, and trap-modulation controls.

The separation between operator and waveform is physical. A laboratory may permit two quadratures of one microwave field but not an arbitrary Hermitian matrix. The available $H_a$, together with amplitude and bandwidth constraints on $u_a$, define the control resources.

For finite-dimensional closed systems, ideal controllability is related to the Lie algebra generated by

$$-\frac{i}{\hbar}H_0, \qquad -\frac{i}{\hbar}H_1, \ldots, -\frac{i}{\hbar}H_m.$$

For piecewise-constant controls allowed to vary independently through an interval containing zero, with free final time in the ideal finite-dimensional model, generating $\mathfrak{su}(d)$ gives access to arbitrary special unitaries on a $d$-dimensional Hilbert space, hence arbitrary unitaries up to global phase. Generating a smaller algebra restricts the reachable set.

This algebraic statement is an ideal reachability result. It does not supply a pulse, a minimum time, or a robustness guarantee, and it can fail to describe leakage, uncertain parameters, one-sided controls, or open-system evolution.

Qubit example

Consider

$$H_0 = \frac{\hbar\omega_0}{2}Z, \qquad H_x = \frac{\hbar}{2}X.$$

For $\omega_0\neq0$, the available skew-Hermitian directions include $iZ$ and $iX$. Their commutator supplies the missing axis:

$$[iZ,iX] = -2iY.$$

Thus $iX$, $iY$, and $iZ$ span $\mathfrak{su}(2)$. The ideal model is fully controllable up to global phase. A real qubit may still be limited by finite drive strength, detuning uncertainty, leakage, and decoherence.

Pulses and Gates

Suppose controls are piecewise constant over intervals of duration $\Delta t_j$. On interval $j$,

$$H_j = H_0 + \sum_a u_{a,j}H_a,$$

and

$$U_j = \exp\left( -\frac{i}{\hbar}H_j\Delta t_j \right).$$

If interval $1$ acts first, the full sequence is

$$U(t_f) = U_NU_{N-1}\cdots U_2U_1.$$

The rightmost factor acts first. Changing pulse order generally changes the result because the interval Hamiltonians need not commute.

Fixed-axis pulse-area rule

In a rotating frame, an ideal resonant qubit drive with fixed phase $\phi$ can take the form

$$H_R(t) = \frac{\hbar\Omega(t)}{2} \left( \cos\phi\,X + \sin\phi\,Y \right).$$

The operator axis is fixed, so

$$[H_R(t_1),H_R(t_2)]=0.$$

Time ordering is then unnecessary. Define the pulse area

$$\Theta = \int_0^{t_f}\Omega(t)\,dt.$$

The resulting unitary is

$$U_R(t_f) = \exp\left[ -\frac{i\Theta}{2} \left( \cos\phi\,X + \sin\phi\,Y \right) \right].$$

This is a rotation by angle $\Theta$ about an equatorial Bloch-sphere axis. The area rule ceases to be exact when detuning, time-dependent phase, leakage, noncommuting drift terms, or strong-drive corrections matter.

The named protocols, phase conventions, and finite-pulse cautions are explained in Pulse Sequences and Rabi and Ramsey Control.

State Targets and Gate Targets

A state-transfer objective for a pure initial state may use

$$F_{\rm state} = \left\lvert \langle\psi_{\rm tar} \vert U(t_f) \vert\psi_0\rangle \right\rvert^2.$$

A pulse can make this quantity equal to one while acting incorrectly on other inputs, particularly superpositions. One successful state transfer therefore does not establish a target gate.

For two unitaries on a $d$-dimensional target space, a phase-insensitive process overlap is

$$F_{\rm pro} = \frac{1}{d^2} \left\lvert \operatorname{Tr} \left( U_{\rm tar}^\dagger U(t_f) \right) \right\rvert^2.$$

For ideal unitary evolution without leakage, the corresponding average gate fidelity is

$$F_{\rm avg} = \frac{dF_{\rm pro}+1}{d+1}.$$

With leakage or open-system noise, one must specify the retained subspace and compare quantum channels rather than insert a nonunitary matrix into these formulas without justification.

Rotating Frames as a Design Tool

Rotating frames remove known rapid motion and expose slowly varying control parameters. With the convention

$$\lvert\psi_R(t)\rangle = R(t)^\dagger \lvert\psi(t)\rangle,$$

the exact transformed Hamiltonian is

$$H_R(t) = R^\dagger(t)H(t)R(t) - i\hbar R^\dagger(t)\dot R(t).$$

This transformation changes the description, not the physical operation. It can turn a near-resonant carrier into an approximately static detuning plus two controlled quadratures:

$$H_R(t) \approx \frac{\hbar}{2} \left[ \Delta Z + \Omega_x(t)X + \Omega_y(t)Y \right].$$

The approximation sign usually reflects a rotating-wave or truncation step made after the exact frame change. Frame choice does not erase counter-rotating terms, bandwidth limits, or laboratory-frame leakage.

For control work, every reported pulse should identify

• the laboratory and rotating frames;
• the carrier and detuning convention;
• which phase defines the $X$ axis;
• whether amplitudes are angular frequencies or ordinary frequencies;
• how the final frame is converted back before measurement.

The exact transformation and sign conventions are derived in Rotating Frames.

Average Hamiltonian Theory Preview

Repeated control cycles are naturally analyzed in a toggling frame. If $U_c(t)$ is the control propagator and $H_d$ is the drift or error to be reshaped, then

$$\widetilde H_d(t) = U_c^\dagger(t)H_dU_c(t).$$

When the control closes after a period $T_c$, the logarithm of the one-cycle propagator defines a stroboscopic Hamiltonian. Its leading term is

$$\overline H^{(0)} = \frac{1}{T_c} \int_0^{T_c} dt\, \widetilde H_d(t).$$

Average Hamiltonian Theory develops higher commutator terms and finite-pulse corrections. Magnus Expansion derives the exponential series, while Dynamical Decoupling applies control cycles to unwanted couplings.

Optimal Control Preview

Analytic pulses use symmetry, resonance, and solvable limits. Optimal control instead searches an allowed function space for controls that minimize a stated cost. A schematic objective is

$$J[u] = 1-F[u] + \lambda \int_0^{t_f}dt\, \sum_a u_a(t)^2 + J_{\rm bandwidth} + J_{\rm leakage} + J_{\rm robustness}.$$

The fidelity term encodes the target. The remaining terms encode physical cost and constraints. Their weights are part of the problem definition, not harmless numerical details.

The sensitivity of the final propagator follows from first-order variation. If $U(t)=U(t,0)$ and the Hamiltonian changes by $\delta H(t)$, then

$$\delta U(t_f) = -\frac{i}{\hbar} U(t_f) \int_0^{t_f}dt \times U^\dagger(t) \delta H(t) U(t).$$

This forward–backward structure underlies efficient gradient calculations. GRAPE-style methods exploit piecewise propagation; Krotov methods use a related variational update; automatic differentiation can differentiate the numerical program. None of these choices removes the need for a correct model, a physical objective, convergence tests, and independent validation.

Optimal Control develops these gradients and robustness objectives.

From Closed Control to Open Systems

Closed-system pulse synthesis assumes the retained state evolves unitarily. A controlled open-system model may instead have

$$\dot\rho = -\frac{i}{\hbar} \left[ H_0+\sum_a u_a(t)H_a, \rho \right] + \sum_\mu \mathcal D[L_\mu(t)]\rho.$$

Here $\mathcal D[L]\rho=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}$, with rates included in $L$. Such a Lindblad equation requires a justified Markovian description. The control can alter both coherent motion and exposure to noise. The target may be a state, a channel, a steady state, or a feedback-stabilized trajectory. A pulse optimized under unitary dynamics can fail when relaxation, dephasing, colored noise, or leakage is restored.

Use Driven Open Systems for controlled master equations and Control Limits from Noise for bandwidth, speed, leakage, and robustness limits. Controls that depend on a measurement record belong to measurement-based feedback rather than predetermined pulse synthesis.

Common Mistakes

• Treating ideal controllability as a pulse-synthesis or fidelity result.
• Omitting the drift Hamiltonian when computing pulse rotations.
• Reading a product of pulse unitaries in the order written rather than right to left.
• Applying the pulse-area rule when the rotation axis changes in time.
• Forgetting the inertial term in a rotating-frame Hamiltonian.
• Calling the time average exact when Hamiltonians at different times do not commute.
• Comparing state-transfer fidelity with gate fidelity as though they were interchangeable.
• Optimizing without amplitude, bandwidth, leakage, or robustness constraints.
• Validating only at the same parameter values used to design the pulse.
• Applying a closed-system optimum to an open system without repropagating the noisy model.

References

• C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present and future,” New Journal of Physics 12, 075008, 2010, doi:10.1088/1367-2630/12/7/075008.
• D. D’Alessandro, Introduction to Quantum Control and Dynamics, Chapman and Hall/CRC, 2007.
• S. J. Glaser et al., “Training Schrödinger’s cat: quantum optimal control,” European Physical Journal D 69, 279, 2015, doi:10.1140/epjd/e2015-60464-1.
• U. Haeberlen, High Resolution NMR in Solids: Selective Averaging, Academic Press, 1976.
• N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms,” Journal of Magnetic Resonance 172, 296–305, 2005, doi:10.1016/j.jmr.2004.11.004.
• W. Magnus, “On the exponential solution of differential equations for a linear operator,” Communications on Pure and Applied Mathematics 7, 649–673, 1954, doi:10.1002/cpa.3160070404.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes, Wiley, 2003.

Exercises

1. Derive the fixed-axis pulse-area rule and state the assumption that makes it exact.

Solution

Write

$$H_R(t) = \frac{\hbar\Omega(t)}{2}A_\phi,$$

where

$$A_\phi = \cos\phi\,X + \sin\phi\,Y$$

is time independent. Then

$$[H_R(t_1),H_R(t_2)]=0$$

for all times. The time-ordered exponential reduces to

$$\begin{aligned} U_R(t_f) &= \exp\left[ -\frac{i}{\hbar} \int_0^{t_f}H_R(t)\,dt \right] \\ &= \exp\left( -\frac{i\Theta}{2}A_\phi \right), \end{aligned}$$

with $\Theta=\int_0^{t_f}\Omega(t)dt$. The rule is exact because every instantaneous Hamiltonian is proportional to the same operator.

2. Show to second order in $\Delta t$ why two noncommuting pulse intervals depend on their order.

Solution

Expand

$$U_BU_A = e^{-iH_B\Delta t/\hbar} e^{-iH_A\Delta t/\hbar}$$

and the reversed product. Their common terms through first order are

$$I - \frac{i\Delta t}{\hbar} (H_A+H_B).$$

At second order, the cross terms are respectively

$$-\frac{\Delta t^2}{\hbar^2}H_BH_A$$

and

$$-\frac{\Delta t^2}{\hbar^2}H_AH_B.$$

Therefore

$$U_BU_A-U_AU_B = \frac{\Delta t^2}{\hbar^2} [H_A,H_B] + O(\Delta t^3).$$

The two orders agree to this accuracy only if the commutator vanishes.

3. Verify that drift along $Z$ and control along $X$ generate all qubit rotations in the ideal Lie-algebra test.

Solution

Ignoring nonzero scalar factors, the available skew-Hermitian generators are $iZ$ and $iX$. Their commutator is

$$[iZ,iX] = -2iY.$$

The real span of $iX$, $iY$, and $iZ$ is $\mathfrak{su}(2)$. Under the assumptions of the finite-dimensional closed-system controllability theorem, the reachable unitaries therefore include all of $SU(2)$. This does not determine the minimum-time or most robust pulse.

4. Give an example showing that perfect state transfer does not certify a target gate.

Solution

Take the target gate to be the identity and the tested initial state to be $\lvert0\rangle$. Both

$$U_{\rm tar}=I$$

and

$$U=Z$$

send $\lvert0\rangle$ to the same ray, so the state-transfer fidelity is one. On the input $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$, however, $Z$ gives the orthogonal ray $\lvert-\rangle$, while the identity leaves $\lvert+\rangle$ unchanged. This is a measurable gate error; comparing only the phase of $\lvert1\rangle$ would miss it.

5. Derive the first-order variation formula for the final propagator.

Solution

Let the perturbed Hamiltonian be $H+\delta H$. To first order, the Dyson variation is

$$\delta U(t_f) = -\frac{i}{\hbar} \int_0^{t_f}dt\, U(t_f,t) \delta H(t) U(t,0).$$

Using

$$U(t_f,t) = U(t_f,0)U(t,0)^\dagger$$

and writing $U(t)=U(t,0)$ gives

$$\delta U(t_f) = -\frac{i}{\hbar} U(t_f) \int_0^{t_f}dt \times U^\dagger(t) \delta H(t) U(t).$$

The integral propagates a local Hamiltonian variation into the interaction frame, while the left factor returns it to the final time.