Direct Quantum Optimal Control

• Direct quantum optimal control is a method that transcribes continuous quantum dynamics into finite-dimensional nonlinear programs using discretization techniques like piecewise-constant, collocation, and spectral transcription.
• It allows the direct integration of experimental constraints and objectives—such as field fluence, state-dependent penalties, and time-optimal criteria—into the optimization framework.
• The approach leverages both derivative-free and gradient-based algorithms, facilitating efficient pulse optimization for applications ranging from gate synthesis to wavepacket control.

Direct quantum optimal control commonly denotes a “discretize-then-optimize” approach to designing time-dependent control fields $u_k(t)$ that steer a quantum system from a given initial condition to a desired target with high fidelity while respecting experimental constraints (Mahesh et al., 2022). In this formulation, the continuous control problem is replaced by a finite-dimensional nonlinear program whose decision variables may include pulse amplitudes, state variables at grid points, basis coefficients, or even segment durations. In laser-driven quantum dynamics, direct optimal control has been proposed as a “robust and flexible alternative” to indirect formulations that require iterative forward and backward propagation of quantum wavepackets (Ramos et al., 2020). The broader literature shows that direct quantum optimal control encompasses piecewise-constant transcription, collocation, spectral and spline-based parameterizations, direct search in reduced pulse bases, and large-scale sparse nonlinear programming, together with newer direct optimization variants based on automatic differentiation, polynomial optimization, tensor-network sampling, and hybrid quantum-classical circuits.

1. Conceptual scope and terminology

In the closed-system case, a standard controlled dynamics model is

$$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$

while in the open-system setting one may instead use a Lindblad-form master equation for $\rho(t)$ (Mahesh et al., 2022). Canonical objectives include gate synthesis with

$F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$

and state-to-state transfer with

$F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$

Typical costs add a quadratic penalty on control power, so that the optimization balances fidelity and field fluence (Mahesh et al., 2022).

A separate but related line of QOCT literature uses “direct formulation” in a variational sense. In that setting one extremizes a functional

$J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$

with $J_3$ enforcing the Schrödinger equation by a Lagrange multiplier $\chi(t)$, and the resulting stationarity conditions lead to forward and backward Schrödinger equations together with a pointwise field update law (0707.1883). By contrast, the direct methods emphasized in discretize-then-optimize work replace the continuous-time dynamics by algebraic constraints and then hand the entire problem to an optimizer (Ramos et al., 2020).

This suggests that the label “direct” is not uniform across subliteratures. In contemporary quantum-control practice, however, the most specific usage is the one adopted in the review literature: a direct quantum optimal control problem is transcribed first and optimized second, rather than solved indirectly through a two-point boundary value problem (Mahesh et al., 2022).

2. Direct transcription of quantum dynamics

The simplest transcription replaces each control waveform by piecewise-constant values on a uniform grid. If $[0,T]$ is partitioned into $N$ subintervals of length $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$0, one sets $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$1 on each interval and collects all control parameters into a finite-dimensional vector $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$2. The interval propagator is then

$$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$3

and the total propagator becomes $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$4 (Mahesh et al., 2022). This turns a continuous-time QOC problem into a finite-dimensional nonlinear program.

More elaborate direct methods also discretize the state trajectory. In direct collocation for quantum optimal control, one introduces states $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$5 or $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$6 at knot points, control samples $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$7, and optionally step lengths $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$8, then imposes discretized dynamics as equality constraints. In the Pade-Integrator Collocation method, the implicit dynamics constraint on interval $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$9 is

$\rho(t)$0

with $\rho(t)$1 and $\rho(t)$2 defined from the order-four diagonal Pade approximation (Trowbridge et al., 2023).

A reduced-order variant appears in laser-driven particle dynamics, where the full wavefunction is not propagated directly. Instead, the wavepacket is parameterized by a single Gaussian,

$\rho(t)$3

and the four real parameters $\rho(t)$4 obey Hamilton-like ODEs that are enforced by Hermite-Simpson collocation (Ramos et al., 2020).

Spectral transcription provides a different direct route. For one-dimensional Schrödinger control on an infinite line, artificial boundary conditions can reduce the problem to a bounded interior domain, after which one expands the interior state in a Fourier basis,

$\rho(t)$5

and then stacks state coefficients and control coefficients at all collocation times into a sparse algebraic optimization problem (Wodecki et al., 2024).

Formulation	Decision variables	Representative source
Piecewise-constant transcription	$\rho(t)$6 on a time grid	(Mahesh et al., 2022)
Direct collocation	$\rho(t)$7 or $\rho(t)$8, $\rho(t)$9, optionally $F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$0	(Trowbridge et al., 2023)
Reduced-state ansatz	$F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$1	(Ramos et al., 2020)
Spectral transcription	Fourier coefficients and control coefficients	(Wodecki et al., 2024)

Taken together, these formulations show that direct quantum optimal control is not tied to a single discretization. What is common is the replacement of continuous quantum dynamics by a finite collection of optimization variables plus equality and inequality constraints.

3. Objective functionals and constraint architecture

The standard direct objective combines a terminal target with running penalties. For gate synthesis, one commonly minimizes an infidelity such as

$F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$2

plus a regularization term $F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$3 (Nguyen et al., 2020). For state transfer, one analogously minimizes final-time infidelity plus control cost (Mahesh et al., 2022).

A defining feature of direct methods is that additional constraints are inserted directly into the nonlinear program. In the single-Gaussian laser-control problem, the decision variables are $F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$4, the objective includes the terminal overlap with the target state plus the running cost

$F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$5

and the constraints include Hermite-Simpson collocation, event constraints fixing the initial state, and a zero-net-force condition

$F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$6

to remove dc-bias (Ramos et al., 2020).

The same plug-and-play character is explicit in the Fermi-resonance study. There the running cost can include field fluence and state-population penalties,

$F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$7

so that population of an unwanted overtone state such as $F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$8 can be penalized directly in the integrand. The paper emphasizes that this is a methodological advantage because there is “no need to compute functional derivatives and coupling terms as in the case of indirect optimal control” (Ramos et al., 2021).

Direct collocation also makes free-time and minimum-time problems natural. In PICO, each $F_G = \frac{1}{d^2}\bigl\lvert\Tr\bigl[\,U(T)^\dagger\,U_{\rm target}\bigr]\bigr\rvert^2$9 may be treated as a decision variable, bounded by $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$0, and a minimum-time problem is formed by appending $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$1 to the cost while enforcing a final-fidelity constraint $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$2 (Trowbridge et al., 2023). Noise-aware CRAB further extends this idea by optimizing both pulse coefficients and the total evolution time $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$3 under commuting Markovian noise, with the paper reporting that direct joint optimization avoids the local traps that plague a sequential “optimize $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$4 with fixed $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$5, then adjust $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$6” strategy (Jeon et al., 31 Mar 2025).

These examples illustrate a central point: direct quantum optimal control is often chosen not only for pulse optimization itself, but because it absorbs amplitude bounds, bandwidth or smoothness limits, state-dependent penalties, and time-optimal objectives into a single finite-dimensional constrained program.

4. Algorithmic families and solver ecosystems

Direct quantum optimal control spans both derivative-free and gradient-based implementations. In the review literature, derivative-free direct search is represented by Nelder-Mead simplex, CMA-ES, Simulated Annealing, and CRAB, the latter reducing dimension by expanding controls in a “small random low-dimensional Fourier basis” (Mahesh et al., 2022). These methods are especially useful when analytic gradients are unavailable or unreliable.

Large-scale constrained direct methods instead exploit sparse nonlinear programming. In the laser-driven Gaussian-wavepacket problem, the nonlinear program is formulated in PSOPT, Hermite-Simpson collocation yields a sparse set of nonlinear equality constraints, and the resulting large but sparse NLP is solved with IPOPT. Typical discretizations use $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$7 nodes for a control window $F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$8 a.u., balancing accuracy and CPU time (Ramos et al., 2020). The Fermi-resonance application likewise uses PSOPT with IPOPT and trapezoidal collocation (Ramos et al., 2021). PICO similarly targets sparse-structure solvers such as IPOPT and emphasizes infeasible-start capability, banded Jacobians and Hessians, and support for general nonlinear constraints on states and controls (Trowbridge et al., 2023).

A different direct route is to derive gradients for the discretized problem itself. In the discrete-adjoint approach based on a Störmer-Verlet discretization, the main theoretical contribution is a compatible time-discretization of the adjoint state equation such that the gradient of the discrete objective function can be calculated exactly, at a computational cost of solving two Schrödinger systems, independently of the number of parameters in the control functions (Petersson et al., 2020). A Newton-Raphson gate-synthesis algorithm goes further by reformulating the problem as matrix-logarithm root finding and obtaining local error scaling

$F_S = \Tr\bigl[\rho_F^\dagger\,\rho(T)\bigr].$9

which yields doubly exponential convergence in the iteration index (Fouquieres, 2012).

Global optimization has also entered the direct-control toolbox. Quantum Control via Polynomial Optimization reformulates suitable control problems as polynomial programs and then uses moment-SOS relaxations or homotopy continuation to find globally optimal solutions. In the examples reported, QCPOp via TSSOS obtains infidelities $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$0 around $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$1 for a piecewise-constant ansatz with $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$2 slices, while local GRAPE and CRAB with $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$3 slices linger around $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$4–$J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$5 (Bondar et al., 2022).

Modern direct optimization frameworks also expand the class of admissible objectives. Semi-automatic differentiation combines GRAPE-style propagation with automatic differentiation of low-dimensional terminal functionals, allowing direct optimization of non-analytic figures of merit such as gate concurrence while keeping memory usage linear in $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$6 and $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$7 rather than $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$8 (Goerz et al., 2022). Differentiable programming embeds the ODE solver itself in an end-to-end computational graph, so that a neural-network control agent can be trained by backpropagation through both the network and the system dynamics (Schäfer et al., 2020).

Algorithmic family	Core idea	Reported property
Direct search	Optimize transcribed pulse parameters without analytic gradients	Nelder-Mead, CMA-ES, Simulated Annealing, CRAB (Mahesh et al., 2022)
Sparse collocation NLP	Treat states and controls as decision variables with equality constraints	Supports general nonlinear constraints, free-time and minimum-time problems (Trowbridge et al., 2023)
Discrete adjoints	Differentiate the discretized dynamics exactly	Gradient computed at cost of two Schrödinger systems (Petersson et al., 2020)
Newton-type gate synthesis	Root finding on matrix-logarithm error	Doubly exponential convergence near a simple root (Fouquieres, 2012)
Polynomial global optimization	Convert suitable QOC tasks to polynomial programs	Directly finds globally optimal solutions (Bondar et al., 2022)

Software infrastructure has developed in parallel with algorithms. XACC extends a quantum-classical compilation stack with a PulseTransform middle-end plugin that lowers a digital circuit or gate instruction into an optimized pulse schedule via GRAPE, GOAT, or Krotov, then emits a pulse-level intermediate representation executable on pulse-capable back ends such as QuaC or OpenPulse (Nguyen et al., 2020). This indicates that direct QOC is not only a numerical-analysis paradigm but also a compilation layer for pulse-level quantum computing.

5. Representative applications and empirical behavior

Direct quantum optimal control has been applied across chemical dynamics, few-body and many-body quantum systems, and hardware pulse design. In laser-driven wavepacket dynamics in a bistable potential, the direct-OCT formulation with a single-Gaussian ansatz produced simple, smoothly shaped fields that drive the Gaussian wavepacket from the left to the right well for $J[ψ,ε] = J_1[ψ] + J_2[ε] + J_3[ψ,χ,ε],$9 and $J_3$0 with $J_3$1 a.u. A systematic scan over $J_3$2 and $J_3$3 a.u. showed that nearly perfect transfer, with error $J_3$4, is achieved for a wide band of $J_3$5, and the optimized fields still yielded reasonable control yields when inserted into full quantum simulations using MCTDH (Ramos et al., 2020).

The same direct philosophy was extended to exact wavepacket propagation in a generic Fermi-resonance model. Using direct discretization with state-dependent running costs, the study reported that with $J_3$6 the final stretch-state population reached $J_3$7 but large transient overtone population arose, while increasing $J_3$8 to $J_3$9 suppressed $\chi(t)$0 by an order of magnitude and still achieved $\chi(t)$1 in $\chi(t)$2 (Ramos et al., 2021). This application highlights the role of direct methods in shaping intermediate populations rather than only optimizing the endpoint.

For gate synthesis and time-optimal control, PICO reports several high-fidelity examples. In a single-qubit Y-gate minimum-time problem, PICO found the analytic bang-bang pulse of duration $\chi(t)$3 and infidelity $\chi(t)$4. For a two-qubit CNOT, it converged to a $\chi(t)$5 pulse with rollout infidelity $\chi(t)$6, and for a three-qubit SWAP it achieved solver infidelity $\chi(t)$7 and rollout infidelity $\chi(t)$8 in under $\chi(t)$9 ns. In a 3D circuit-cavity QED hardware demonstration of $[0,T]$0, photon-number-resolved qubit spectroscopy showed final state $[0,T]$1 with fidelity $[0,T]$2 (Trowbridge et al., 2023).

Direct-gradient and differentiable approaches broaden the application range further. In a superconducting two-transmon example, semi-automatic differentiation enabled the first direct optimization of the non-analytic gate concurrence, with performance benchmarks reporting $[0,T]$3 ms per gradient evaluation and peak RAM $[0,T]$4 MB for semi-AD Chebychev GRAPE, compared with $[0,T]$5 ms and $[0,T]$6 MB for a full-AD ODE-based implementation (Goerz et al., 2022). Differentiable programming for quantum control achieved mean fidelity $[0,T]$7 for single-qubit eigenstate preparation in $[0,T]$8 epochs, average target fidelity $[0,T]$9 for GHZ-state preparation in chains up to $N$0 qubits on noisy initial inputs, and average fidelity $N$1 for a cat state in a quantum parametric oscillator (Schäfer et al., 2020).

Other direct formulations emphasize continuous families and global searches. Neural-network optimization of families of quantum gates learned smooth control maps $N$2, reaching average infidelity $N$3 for a single-qubit Euler-rotation family and $N$4 for selected two- and three-qubit families, with reported gate times $N$5 faster for a two-qubit case and $N$6 faster for a three-qubit case than standard compiled decompositions (Sauvage et al., 2021). Noise-aware TCRAB found globally optimal times and fidelities in several commuting-noise examples, including $N$7 and infidelity $N$8 for entanglement generation in two Josephson charge qubits, and $N$9 with infidelity $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$00 for a noisy CZ gate with $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$01 control (Jeon et al., 31 Mar 2025).

6. Limitations, misconceptions, and research directions

A common misconception is that direct quantum optimal control is synonymous with derivative-free pulse search. The literature does not support that restriction. Direct search methods are important, but direct collocation, discrete adjoints, Newton methods, semi-automatic differentiation, and sparse NLP formulations are equally central parts of the direct-control landscape (Mahesh et al., 2022). Taken together, these works show that “direct” refers most fundamentally to how the optimization problem is transcribed, not to whether gradients are used.

Another misconception is that direct transcription alone resolves model inadequacy. The Gaussian-wavepacket study makes the opposite point explicit: the single-Gaussian ansatz cannot describe splitting, strong interference, or far non-Gaussian dynamics, and its accuracy depends on keeping the wavepacket localized, with better agreement for heavier masses or fields that impart large crossing momentum (Ramos et al., 2020). In other words, direct optimization can be numerically robust while still being physically limited by the reduced model.

Global optimality remains a central controversy because quantum-control landscapes are non-convex and often populated by many local extrema. QCPOp addresses this directly by polynomial reformulation and global optimization, but its scope is tied to polynomially representable approximations and moderate problem sizes (Bondar et al., 2022). By contrast, local direct methods such as GRAPE-style direct optimization, collocation NLPs, and direct search heuristics remain more broadly deployable but do not generally certify the global optimum.

Current research directions expand both modeling reach and computational strategy. Proposed extensions include more flexible ansätze such as superpositions of Gaussians or other basis sets, direct-OCT coupled to MCTDH or on-the-fly ab initio dynamics for polyatomic systems, and inclusion of robustness constraints such as field-noise tolerance and parameter uncertainty (Ramos et al., 2020). Spectral methods with artificial boundary conditions aim at rigorous treatment of infinite-domain Schrödinger control with sparse direct transcription (Wodecki et al., 2024). Adaptive tensor-network sampling introduces an MPS/TT score over discrete control strings and reports stable convergence behavior with competitive empirical performance relative to established gradient-free baselines (Zeybek et al., 27 Apr 2026). Hybrid quantum-classical approaches encode control parameters into variational circuits and assess performance by a control-optimality metric based on a 1-Wasserstein distance, while efficient quantum algorithms for QOC seek $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$02-approximate first-order stationary points with complexity polynomial in $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$03, $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$04, and $$\dot U(t)= -\,i\bigl[\,H_0 + \sum_{k}u_k(t)\,H_k\bigr]\,U(t),\quad U(0)=I,$$05, at the cost of requiring fault-tolerant quantum computers (Huang et al., 29 May 2025, Li et al., 2023).

The overall trajectory suggests an increasingly heterogeneous field. Direct quantum optimal control now includes sparse constrained NLP, exact discrete adjoints, direct optimization of non-analytic objectives, pulse-level compiler integration, continuous-family control, noise-aware time optimization, tensor-network heuristics, and quantum-assisted optimization. What unifies these directions is the replacement of the original infinite-dimensional control problem by a directly optimized finite representation in which constraints, objectives, and hardware structure are encoded explicitly.