Physically Realizable Quantum Controller

• Physically realizable quantum controllers are models that adhere to quantum commutation relations and hardware constraints, ensuring they can be experimentally instantiated.
• Synthesis methods leverage QSDEs, Hamiltonian parameterizations, and additional noise channels to meet strict algebraic and stochastic criteria.
• Applications span coherent-feedback synthesis, pulse-level control in many-body systems, and reinforcement learning, integrating control theory with hardware feasibility.

A physically realizable quantum controller is a controller specification that can be instantiated as an actual quantum dynamical system or as an experimentally executable control law without violating the structural constraints of quantum mechanics. In quantum systems theory, this means that a stochastic or state-space model arises from a bona fide open quantum model and preserves canonical or Pauli commutation relations under unitary or Markovian evolution (Espinosa et al., 2012, Vuglar et al., 2013, Vuglar et al., 2015). In finite-dimensional and many-body control, the same phrase denotes controllers whose Hamiltonians, pulse policies, feedback interconnections, and readout-actuation interfaces obey hard physical constraints such as bounded amplitudes, finite bandwidth and slew rate, discrete sampling, dissipation, calibration offsets, locality of access, and platform-specific hardware limits (Maruyama et al., 2017, Ernst et al., 24 Jan 2025, Chen et al., 2022).

1. Formal meaning and scope

The term spans several adjacent control-theoretic regimes. For open quantum systems driven by Bosonic fields, the controller is often modeled by a quantum stochastic differential equation (QSDE) in the Heisenberg picture, with an underlying Hudson–Parthasarathy evolution generated by a scattering operator $S$, a coupling operator $L$, and a Hamiltonian $H$. For finite-dimensional qubit or spin systems, the controller is typically a Hamiltonian law of the form

$H(t)=H_0+\sum_j u_j(t)H_j,$

with $u_j(t)$ restricted by actuator physics. In learning-based formulations, the controller is a policy $u(t)=\pi(o(t),\theta)$ whose output must already satisfy amplitude, bandwidth, quantization, and noise constraints rather than being post-processed afterward (Espinosa et al., 2012, Ernst et al., 24 Jan 2025).

Across these settings, physical realizability is not merely implementability in a loose engineering sense. It is a structural compatibility condition between the controller model and the kinematics of the controlled quantum system. For linear and bilinear quantum stochastic models, this compatibility is encoded algebraically in CCR- or SU(2)-preserving identities. For coherent feedback, it requires that the controller itself be an open quantum system, not an arbitrary matrix realization. For pulse-level control, it requires that the synthesized waveform or switching schedule be admissible for the actual control stack, including DAC resolution, AWG timing, cryogenic switching, or field-delivery bandwidth (Vuglar et al., 2013, Vladimirov et al., 2010, Chen et al., 2022).

A further broadening appears in many-body control architectures. Gateway control on spin networks and global-pulse control on analog simulators treat limited accessibility itself as part of the realizability condition: only a small subset of spins, or only spatially uniform fields, may be actuated, and controllability must then be certified under those access restrictions rather than under arbitrary local control assumptions (Maruyama et al., 2017, Hu et al., 26 Aug 2025).

2. Algebraic and stochastic criteria

For linear quantum stochastic systems in real quadratures, the standard model is

$$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$

with canonical commutation matrix $\Theta$. Physical realizability requires the state evolution to preserve the CCR and the input-output map to correspond to an open quantum harmonic oscillator. In quadrature form this yields the familiar constraints

$$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$

together with equivalent annihilation/creation-form relations such as $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$ and $L$0 (Vuglar et al., 2015, Zhang et al., 8 Apr 2026).

A central consequence is that an arbitrary synthesized LTI controller is generally not quantum-mechanically admissible as written. If additional vacuum inputs are allowed, however, any given linear model can be made physically realizable. The exact minimal number of additional quantum noise quadratures is determined by the rank of the residual PR-violation matrix. In the formulation using signal input $L$1, this matrix is

$L$2

and the minimum added noise dimension is $L$3, so the total number of introduced vacuum quadratures is $L$4 (Vuglar et al., 2015). In the closely related formulation of linear time-invariant quantum implementation, the same minimality statement appears as the exact necessity-and-sufficiency result $L$5 with $L$6 (Vuglar et al., 2013).

For open two-level systems, the situation is intrinsically bilinear rather than linear because the system variables are Pauli operators. With $L$7, Hamiltonian $L$8, coupling $L$9, and the SU(2) structure map $H$0, the bilinear QSDE

$H$1

is physically realizable if and only if there exist $H$2 and $H$3 such that

$H$4

and

$H$5

These conditions are equivalent to the existence of a genuine Markovian qubit model and imply preservation of the Pauli commutation relations throughout the stochastic evolution (Espinosa et al., 2012).

The system-theoretic significance of these results is twofold. First, physical realizability turns controller synthesis into a constrained algebraic problem rather than a purely numerical optimization. Second, the constraints are constructive: once the realizability identities hold, one can recover the controller Hamiltonian and coupling operators explicitly, or determine the minimal extra noise ports needed to embed an otherwise nonphysical linear controller into a valid quantum implementation (Espinosa et al., 2012, Vuglar et al., 2013).

3. Coherent-feedback synthesis and performance limits

In coherent feedback, the controller is itself a quantum system interconnected with the plant without intermediate measurement. For coherent quantum LQG control, the controller QSDE has the form

$H$6

and physical realizability imposes

$H$7

A Hamiltonian parametrization then writes

$H$8

with $H$9 symmetric. This parametrization is the basis of the quasi-separation principle and Newton-like iterative scheme developed for coherent quantum LQG synthesis: the optimization is carried out over $H(t)=H_0+\sum_j u_j(t)H_j,$0 rather than over arbitrary state-space matrices, so physical realizability is enforced by construction (Vladimirov et al., 2010).

For coherent $H(t)=H_0+\sum_j u_j(t)H_j,$1 control of quantum linear systems, the same structural constraint persists, but the recent synthesis methodology emphasizes computational simplification. Instead of solving two coupled algebraic Riccati equations, a physically realizable coherent-feedback $H(t)=H_0+\sum_j u_j(t)H_j,$2 controller can be obtained by solving at most four Lyapunov equations in the general case. In the passive case, a necessary-and-sufficient condition is given in terms of two uncoupled pairs of Lyapunov equations. This reduction relies on the quantum structural symmetry $H(t)=H_0+\sum_j u_j(t)H_j,$3 and yields stabilizing controllers with guaranteed disturbance attenuation and PR-compliant auxiliary ports (Zhang et al., 8 Apr 2026).

Performance limits for quantum controllers can also be stated kinematically. When a plant $H(t)=H_0+\sum_j u_j(t)H_j,$4 is coupled coherently to an auxiliary quantum controller $H(t)=H_0+\sum_j u_j(t)H_j,$5, the observable yield

$H(t)=H_0+\sum_j u_j(t)H_j,$6

lies between the classical kinematic bound (CKB), attainable with classical control only, and the quantum kinematic bound (QKB), attainable with a quantum controller. Surpassing the upper CKB is possible if and only if the controller spectral bandwidth satisfies

$H(t)=H_0+\sum_j u_j(t)H_j,$7

and exact attainment of the QKB requires the rank–degeneracy condition

$H(t)=H_0+\sum_j u_j(t)H_j,$8

These conditions quantify how controller entropy capacity, controller dimension, and objective degeneracy limit or enable coherent quantum feedback performance (Wu et al., 2014).

Taken together, these results establish a distinctive systems-theoretic notion of a physically realizable controller: it is not enough for a controller to stabilize or optimize an abstract model; its state-space realization must be compatible with an oscillator or spin implementation, with the necessary ancillary vacuum channels, Hamiltonian parametrization, and field commutation structure built into the synthesis procedure itself (Vladimirov et al., 2010, Zhang et al., 8 Apr 2026).

4. Access-limited controllability in many-body systems

In large spin networks, physical realizability is often constrained by limited access rather than by stochastic state-space structure. Gateway schemes formalize this setting by restricting actuation and measurement to a small accessible subset $H(t)=H_0+\sum_j u_j(t)H_j,$9 of spins while using the intrinsic drift Hamiltonian to propagate control through the network. For Heisenberg/XYZ couplings on a graph $u_j(t)$0, the key controllability theorem states that if the coupling is algebraically propagating and the gateway set $u_j(t)$1 infects $u_j(t)$2 under the zero-forcing rule, then

$u_j(t)$3

Heisenberg-like couplings satisfy the algebraic propagation condition, so any Heisenberg/XYZ network with an infecting gateway is controllable; in particular, chains with arbitrary couplings are controllable from one end spin. For XX chains, control by a single local field $u_j(t)$4 on spin 1 yields transport unitaries in the Jordan–Wigner picture, with numerically observed swap time scaling $u_j(t)$5 and gate fidelities greater than $u_j(t)$6 up to $u_j(t)$7 (Maruyama et al., 2017).

A related but more restrictive architecture appears in globally controlled analog simulators. For a qubit chain with only global fields

$u_j(t)$8

the dynamical Lie algebra equals $u_j(t)$9 if and only if at least one additional global control field breaks lattice reflection symmetry. This gives a necessary-and-sufficient universality criterion for global-pulse control: reflection symmetry is the obstruction, and any globally addressable symmetry-breaking field removes it. The same work extends the criterion to fermionic and bosonic superlattices, where alternating global hoppings and chemical potentials generate all nearest-neighbor hopping and onsite terms, and combined with a Hubbard interaction yield universal dynamics on fixed-particle-number sectors (Hu et al., 26 Aug 2025).

The practical importance of these controllability theorems is that they bridge abstract Lie closure and hardware-constrained control synthesis. In the gateway setting, control acts only on a physically accessible subset and must respect amplitude, bandwidth, and slew-rate bounds. In the global-control setting, the actuation is spatially uniform by design, and the controller is realized as a short, smooth pulse pair optimized directly under experimental bounds. On a QuEra Rydberg array outside blockade, direct quantum optimal control with the constraints $u(t)=\pi(o(t),\theta)$0 MHz, $u(t)=\pi(o(t),\theta)$1 MHz, $u(t)=\pi(o(t),\theta)$2 MHz/$u(t)=\pi(o(t),\theta)$3s, $u(t)=\pi(o(t),\theta)$4 MHz/$u(t)=\pi(o(t),\theta)$5s, and $u(t)=\pi(o(t),\theta)$6s synthesized effective ZXZ dynamics. Two pulse families were reported: a short pulse with $u(t)=\pi(o(t),\theta)$7s and unitary fidelity $u(t)=\pi(o(t),\theta)$8 on 3 atoms, and a longer pulse with $u(t)=\pi(o(t),\theta)$9s and unitary fidelity $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$0, with the shorter pulse giving better experimental signatures because decoherence accumulated less strongly (Hu et al., 26 Aug 2025).

Physical realizability in many-body control therefore has a topological aspect as well as a hardware aspect. Infection, symmetry breaking, degeneracy lifting, and localization length determine whether control can propagate through the many-body Hilbert space at all, while actuator limits determine whether the resulting controllable trajectories are executable on the platform (Maruyama et al., 2017, Hu et al., 26 Aug 2025).

5. Hardware-constrained pulse synthesis and physical-layer control

A hardware-constrained controller is physically realizable only if its action space is already matched to laboratory electronics. In reinforcement-learning-based quantum control, this requirement is made explicit by constraining the policy output to bounded amplitudes, finite bandwidth and slew rate, discrete sampling and DAC quantization, realistic noise and dissipation, and calibration imperfections. The policy therefore searches only over experimentally admissible signals. In the reported implementation, actions are filtered by Gaussian convolution as a hardware bandwidth proxy, endpoints of amplitude pulses are forced to zero, and a hard cap on the maximum number of ODE solver steps $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$1 biases the search toward adiabatic-like trajectories while reducing solver stiffness. Using PPO in PureJAXRL, with Qiskit-Dynamics and Diffrax and up to 1024 policies in parallel on a single GPU, the method reached fidelities exceeding $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$2 across a multilevel $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$3 system, a neutral-atom Rydberg two-qubit gate, and a superconducting transmon reset, with specific results including $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$4 for the $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$5 system, $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$6 for a two-photon Rydberg C–Z gate at $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$7s, and $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$8 for a transmon reset at $$dx(t)=Ax(t)\,dt+B\,dW(t),\qquad dy(t)=Cx(t)\,dt+D\,dW(t),$$9s (Ernst et al., 24 Jan 2025).

A complementary route eliminates arbitrary waveform generators altogether. In the on/off or few-level switching protocol, each control channel is restricted to a finite alphabet such as $\Theta$0, and optimization is performed over switching instants rather than over arbitrary amplitudes. Because the Hamiltonian at any moment belongs to a finite set, each allowed Hamiltonian can be pre-diagonalized once, reducing the cost of repeated matrix exponentials. The method preserves controllability under the usual Lie-rank conditions and produced superconducting-circuit examples with a 10 ns NOT gate at fidelity $\Theta$1, a 20 ns CNOT at fidelity $\Theta$2, and a 30 ns CCZ at fidelity $\Theta$3. In the same examples, naively converted staircase AWG waveforms performed worse, with reported fidelities $\Theta$4, $\Theta$5, and $\Theta$6 respectively (Chen et al., 2022).

At the physical-layer controller interface, Walsh-function hardware offers a deterministic low-latency classical companion to quantum control. An FPGA-based Walsh controller generated all first $\Theta$7 Walsh functions concurrently, with total trigger-to-output latency of 4.5 clock cycles at a 100 MHz clock, i.e. 45 ns. Full programming required only 345 bits, LUT utilization for the first 8 Paley orders was under 5% on a Xilinx Zynq-7010, and Walsh-sequence calculation was reported as about $\Theta$8 faster than on a microcontroller. The same architecture synthesized Walsh timing signals and Walsh-based AM, $\Theta$9M, and QAM waveforms for dynamic error suppression and signal reconstruction, providing a physically scalable classical-quantum interface (Ball et al., 2016).

These approaches share a common design principle: experimental feasibility must be embedded into controller synthesis, not imposed afterward as a heuristic correction. In that sense, bounded action parameterizations, few-level switching alphabets, deterministic FPGA timing, solver-step caps, and filter-function-compatible waveforms are all realizability constraints of the controller itself (Ernst et al., 24 Jan 2025, Chen et al., 2022, Ball et al., 2016).

6. Specialized controller architectures and realizability limits

Several physically realizable controller paradigms are neither purely linear-system-theoretic nor conventional pulse optimizers. One example is finite-time dissipative state preparation by a resettable ancilla qubit. With complete unitary control on an $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$0-qubit target register, a single resettable ancilla qubit, and controlled-NOT gates between ancilla and target, one can build a sequence of $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$1 control blocks based on splitting subspaces $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$2 whose intersection is the desired pure state. Each block implements the CPTP map

$$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$3

so population in $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$4 is deterministically pumped into $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$5. The composition of $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$6 such blocks prepares an arbitrary $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$7-qubit pure state in exactly $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$8 steps, providing a dead-beat coherent-feedback controller rather than an asymptotic dissipative one (Baggio et al., 2012).

Another example is holonomic control in continuous-variable cat-code manifolds stabilized by engineered dissipation. For a single mode with annihilation operator $$A\Theta+\Theta A^T + BJB^T = 0,\qquad \Theta C^T = -BJD^T\Theta,\qquad DJD^T=J,$$9, the Lindblad jump operator

$A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$0

stabilizes a decoherence-free subspace spanned by coherent states $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$1. Adiabatically moving one $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$2 around a closed loop yields a Berry phase

$A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$3

where $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$4 is the enclosed phase-space area. Together with the collision gate for merging coherent states and the two-mode “infinity” gate based on the jump operators $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$5 and $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$6, these holonomies form a universal set on cat-code encodings. The scheme was proposed as realizable by reservoir engineering in systems with tunable nonlinearities, notably trapped ions and circuit QED (Albert et al., 2015).

A third architecture stores the control program in a quantum state. For a spin-$A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$7 program system coupled coherently to a target qubit, the optimal program is a spin-coherent state $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$8 encoding an unknown rotation axis $A^\cal + A^{\cal\flat} + B^\cal B^{\cal\flat}=0$9, and the controller action is generated by an isotropic Heisenberg interaction

$L$00

The resulting channel is physically realizable by a unitary on program and system alone. Its coherent performance strictly exceeds the best measure-and-operate benchmark for all $L$01 and $L$02; at $L$03 and $L$04, the classical benchmark is at most $L$05, whereas the coherent controller reaches approximately $L$06 (Mo et al., 2017).

Realizability can also become a spectral or arithmetic constraint. In the quantum Zermelo problem with a fixed Hilbert–Schmidt resource bound $L$07, the optimal controller co-moves with the drift,

$L$08

and saturates the geometric energy–time relation

$L$09

For low-dimensional examples, however, physically realizable controls exist only for quantized energy resources. In the single-qubit oscillator example,

$L$10

and in the Heisenberg spin dimer,

$L$11

These quantization conditions arise because the optimal abstract control must also match the operator structure available in hardware, such as a dipole drive or a Zeeman term (Bofill et al., 2020).

The aggregate picture is that physically realizable quantum controllers are bounded not only by controllability and optimization but also by representability: the controller must fit an admissible open-system model, an available actuator algebra, a realizable dissipation channel, or a programmable interaction architecture. This suggests a persistent research frontier at the intersection of algebraic realizability, controllability under restricted access, and hardware-level co-design, especially for large Hilbert spaces, strong disorder, model uncertainty, and platforms where verification remains limited by available measurements (Baggio et al., 2012, Albert et al., 2015, Mo et al., 2017, Bofill et al., 2020).