Quantum control

Quantum control chooses externally adjustable interactions to steer a quantum state or implement a desired unitary operator. Its achievable transformations depend on the Hamiltonian or Lindblad equation, the available control inputs, and physical constraints such as bandwidth and amplitude. Quantum optimal control optimizes performance within those constraints.

Individual quantum control of spins by exploiting different resonance frequencies. A sufficiently narrow-band radio frequency pulse can predominantly affect one resonance when the separation is large compared with the excitation bandwidth and coupling effects are accounted for. Equal resonance frequencies remove this frequency-based distinction. Whether individual gates remain possible then depends on other available interactions, spatial resolution, or differences in control coupling; frequency equality alone is not a universal impossibility theorem.

Let $S$ be the SWAP gate. If the drift Hamiltonian operator and every allowed control Hamiltonian operator commute with $S$, every reachable time-evolution operator also commutes with $S$. Indeed, $SUS$ satisfies the same initial-value Schrödinger equation as $U$ and starts at $I$, so uniqueness gives $SUS=U$. Yet $S(Z\otimes I)S=I\otimes Z\ne Z\otimes I$. An individual Pauli Z gate is consequently unavailable within that symmetric control model. Equal nuclear resonances, isotropic exchange interaction, and identical coupling to a common homogeneous radio frequency field provide an example. An added interaction that breaks this symmetry can remove the obstruction.

For $H=J(XX+YY+ZZ)/2$, $U=e^{-itH}$, and an individually applied Pauli Z gate $P=Z\otimes I$, conjugation gives $PHP=J(-XX-YY+ZZ)/2$. The paired Pauli operators $XX,YY,ZZ$ commute with one another, soTwo equal free-evolution intervals, each of duration $t$, therefore cancel exchange while retaining the Ising coupling. This spin echo is exact for instantaneous ideal gates; a finite pulse requires accounting for the interaction during the pulse. A simultaneous Pauli Z gate on each spin commutes with $H$ and cannot perform the same cancellation. The physical availability of the individual gate is a separate quantum control requirement.

On a connected transition chain, independently selected resonant pulses realize embedded two-level rotations with tunable angle and phase. Eliminate target amplitudes from one end of the chain using complex Givens rotations, and invert those rotations to prepare the target from a basis state. The pulse angles come from amplitude ratios and the phases from relative complex arguments. Spectator components are unchanged by each embedded rotation. Work in the interaction picture and compensate the known drift phases at the chosen final time when the target is specified in the laboratory frame.

An Ising coupling generates $e^{-i\varphi Z\otimes Z}$ with $\varphi =\int J(t)dt$ in units $\hslash =1$. It is entangling precisely when $\varphi$ is not an integer multiple of $\pi /2$. A $\pi$ rotation on one qubit reverses its sign, permitting an ideal echo to cancel an always-on coupling.

A $\pi$ rotation about the x-axis on one qubit conjugates $Z\otimes Z$ to its negative. With ideal instantaneous pulses, two equal free-evolution intervals separated by this pulse and its inverse cancel an Ising coupling exactly. Finite-duration pulses require accounting for the coupling during the pulses; the error is small only when $|J|$ times the pulse duration is small, or when the control is explicitly designed to compensate it.

In coherent quantum feedback, a quantum controller interacts with the controlled system without first converting the control signal into a classical measurement record. Optical feedback through a beam splitter is one example. The input-output interconnection must preserve the physical quantum dynamics, including noise and commutation relations; a classical state-feedback formula does not automatically define a valid quantum controller.

The rotating-wave approximation to a weak quadrature coupling of a cavity mode to a broadband field retains photon-exchange terms and removes counter-rotating pair creation and annihilation. Flat coupling around resonance produces the displayed interaction in units $\hslash =1$. Its short-time propagator is generated by $\sqrt{\gamma }(ad{B}^{†}-a^{†}dB)$; the Gaussian quantum noise Ito table supplies the quadratic drift correction. The resulting passive cavity has the passive cavity input-output transfer function.

For a broadband bosonic input with $[b(t),b^{†}(s)]=\delta (t-s)$, disjoint field increments commute and $[d{B}_t,d{B}_t^{†}]=dt$. An adapted system operator commutes with the next incoming increment. Noise increments scale as $\sqrt{dt}$, so quadratic products contribute to first-order stochastic evolution.

For a zero-mean Gaussian bosonic input, $d{B}^{†}d{B}^{†}=M^{\ast }dt$ completes the displayed table, with $N\ge 0$ and $|M{|}^2\le N(N+1)$. The vacuum has $N=M=0$; an unsqueezed thermal input has $M=0$. Products with $dt$ and higher differentials vanish in the first-order Ito equation. The difference between the first two products preserves the bosonic commutator.

For $\dot{a}=-\gamma a/2-\sqrt{\gamma }{b}_0$ and $b_1=\sqrt{\gamma }a+b_0$, zero-initial-data Laplace transforms give the displayed transfer function. For $\gamma >0$ its pole is in the open left half-plane, while $|G(i\omega )|=1$ and its zero lies in the right half-plane. Passing through $-1$ in this plant's frequency plot is not evidence of an unstable open cavity: a Nyquist stability criterion needs an explicitly specified feedback return ratio.

With $b_0=\beta b_{\mathrm{in}}-\alpha b_1$ and $b_2=\alpha b_{\mathrm{in}}+\beta b_1$, real ${\alpha }^2+{\beta }^2=1$ and $|\alpha |<1$, eliminate the internal fields to obtain $\dot{a}=-{\gamma }_{\mathrm{eff}}a/2-c{b}_{\mathrm{in}}$, where $c=\beta \sqrt{\gamma }/(1+\alpha )$ and $c^2={\gamma }_{\mathrm{eff}}$. The internal output transfer is $b_1/b_{\mathrm{in}}=\beta G/(1+\alpha G)$; the external output transfer is $b_2/b_{\mathrm{in}}=(\alpha +G)/(1+\alpha G)=(s-{\gamma }_{\mathrm{eff}}/2)/(s+{\gamma }_{\mathrm{eff}}/2)$. The loop is asymptotically stable in this nondegenerate range. At $\alpha =1$ the cavity mode is undamped and decoupled; at $\alpha =-1$ the instantaneous algebraic feedback connection is singular.

Design Lindblad operators and a Quantum Hamiltonian so that desired density operators are stationary and unwanted perturbations decay. A dissipative interaction can change the spectrum and purity of a density operator, reaching targets unavailable to purely coherent Hamiltonian engineering. The engineered stationary state must be physical, and uniqueness alone must be distinguished from an explicit proof of asymptotic convergence.

Hamiltonian engineering designs accessible interactions or their effective time average to implement desired quantum states or unitary operators. The available control inputs, Quantum Hamiltonian generators and laboratory constraints determine which transformations can actually be achieved. Model-based open-loop control computes a waveform before running it; adaptive experiments instead evaluate its performance on the physical system.

Such a quantum state remains an eigenstate of $H_0+f(t)H_1$ for every control input. Its projector commutes with all available generators, so its population is conserved. This obstructs full-space pure-state controllability even when the orthogonal bright subspace can be controlled. An instantaneous dark state of a slowly varying Quantum Hamiltonian instead supports adiabatic quantum control if a spectral gap and slowness condition are maintained.

Fast prescribed pulses change the frame in which the system evolves during successive intervals. To leading order over a short control cycle, the effective Quantum Hamiltonian is the duration-weighted mean of these conjugated generators. The Baker--Campbell--Hausdorff formula supplies commutator corrections. Symmetrizing the cycle can cancel unwanted interactions or retain a desired one. The approximation requires control cycles short compared with the relevant dynamical timescales and a suitable account of finite pulse duration.

Design a slowly varying Quantum Hamiltonian path whose relevant initial eigenstate connects to a desired final eigenstate. Under the gap and slowness hypotheses of the quantum adiabatic theorem, the state follows that instantaneous eigenstate up to phase. The path, spectral gap and duration all matter; a level crossing or excessive speed can spoil the intended state transfer.

For a resonant three-level chain with couplings ${\Omega }_1,{\Omega }_2$, the zero-energy dark state of a driven Hamiltonian is $|D⟩=({\Omega }_2|1⟩-{\Omega }_1|3⟩)/\sqrt{{\Omega }_1^2+{\Omega }_2^2}$. Turning on the target-to-intermediate coupling first and sweeping ${\Omega }_1/{\Omega }_2$ from zero to infinity rotates this state from $|1⟩$ to $-|3⟩$. The other eigenvalues are $\pm \sqrt{{\Omega }_1^2+{\Omega }_2^2}$, so $\hslash |\dot{\theta }|\ll \sqrt{{\Omega }_1^2+{\Omega }_2^2}$ controls nonadiabatic leakage. Ideally the intermediate state remains unoccupied, suppressing spontaneous-emission loss. The gap must not close while the dark state is rotating.

A near-resonant drive selects a transition and, in an appropriate rotating frame and rotating-wave approximation, controls a two-level effective Quantum Hamiltonian. If $H_{\mathrm{eff}}=\Omega (t){\sigma }_x/2$ with fixed axis, then $U(T)=\exp [-i{\sigma }_x{\int }_0^T\Omega (t)dt/2]$. Pulse area fixes the rotation angle; drive phase chooses a transverse axis. Selectivity and neglect of rapidly oscillating terms require the appropriate detuning and drive-strength regime.

For a fixed dimensionless Hermitian operator $A$ driven by $H(t)=\Omega (t)A$, all instantaneous generators commute. The exact unitary operator is $U(T)=\exp (-i\theta A)$, so the area, rather than the detailed envelope, fixes the rotation. For $A^2=I$, it is $\cos \theta I-i\sin \theta A$. This convention uses the Quantum Hamiltonian coefficient directly; a convention with $\hslash {\Omega }_{R}A/2$ instead has $\theta =\frac{1}{2}\int {\Omega }_{R}dt$.

For a Hermitian operator satisfying $A^3=A$, its squared operator $P=A^2$ is the projector onto the active eigenvalue-$\pm 1$ subspace. All remaining states have eigenvalue zero. Separate even and odd powers in the matrix exponential to obtain the displayed rotation on the active subspace and the identity on its spectator subspace. Thus an embedded two-level coupling need not square to the full Hilbert-space identity.

For $\dot{\rho }=-i[H_0+f{H}_1,\rho ]$ and ${\dot{\rho }}_d=-i[H_0,{\rho }_d]$, the Hilbert-Schmidt distance gives $V=\Vert \rho -{\rho }_d{\Vert }^2/2$. The choice $f=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])$ yields $\dot{V}=-f^2$. Decrease alone does not prove convergence: orthogonal pure eigenstates of a diagonal $H_0$ give zero input for a transverse Pauli control even though their distance remains nonzero. Such a law can be simulated to produce an open-loop control.

For $V=\Vert \rho -{\rho }_d{\Vert }_{\mathrm{HS}}^2$, with both states evolving under the same drift and only $\rho$ receiving $f{H}_1$, cyclicity of the trace cancels the drift terms and gives the displayed identity. The scalar $C$ is real because $-i[H_1,\rho ]$ is Hermitian. Choosing $f=kC$, $k>0$, makes $\dot{V}=-2k{C}^2\le 0$. This is nonincrease, not by itself convergence to the target: additional invariant-set and reachability conditions are needed.

Take $H_0=0$, $H_1={\sigma }_y/2$, ${\rho }_d=(I+{\sigma }_z)/2$ and $\rho =(I+{\sigma }_x)/2$. The overlap is $f=1/2$, but $C=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])=-1/2$, so the Hilbert-Schmidt Lyapunov feedback identity gives $\dot{V}=1/2>0$. A target overlap and the derivative of that overlap along the control direction are different quantities.