--- title: Propagation-Dependent Unitary Transformations url: https://www.emergentmind.com/topics/propagation-dependent-unitary-transformations type: topic --- # Propagation-Dependent Unitary Transformations Propagation-dependent unitary transformations, as used across several recent research programs, can be understood as unitary maps whose form, implementability, or operational effect depends on a propagation mechanism: Hamiltonian evolution in time, free-space diffraction between phase planes, transport through a spin chain, sequential or indefinite-causal-order use of an unknown gate, or auxiliary-Hilbert-space transport between fibre functors. In that broad sense, the topic spans operator-intrinsic quantum speed limits for implementing a target unitary, dynamically reconfigurable optical transformations on beam arrays, unitary basis changes in mixed quantum-classical propagation, exact mappings between inequivalent dynamical generators, and higher-order transformations of unitary operations themselves [2406.03964] [2407.06981] [2404.15614] [2207.09515] [1909.01366] [2109.08202]. ## 1. Conceptual scope and recurring structures A recurrent feature of the literature is the shift from state-to-state evolution to operator-level or process-level transformation. In the quantum-speed-limit setting, the object of interest is not the transformation of a specific pair of states but the minimum time required to implement an \(N\)-dimensional unitary \(U=e^{-iHT}\), with bounds depending only on \(\operatorname{tr}(U)\) and spectral characteristics of \(H\) [2406.03964]. In optical implementations, the target is an arbitrary spatial unitary on a beam-array basis, synthesized as a concatenation of phase modulation and diffraction [2407.06981]. In mixed quantum-classical dynamics, the central move is to rewrite classical phase-space variables in a complex form that admits arbitrary unitary basis transformations analogous to those used for quantum coordinates [2404.15614]. A second recurring structure is that propagation dependence is not always time dependence in the elementary sense. In the free-to-harmonic mapping, the relevant unitary is explicitly time dependent and is accompanied by a nonlinear time reparameterization \(T=\arctan(\omega t)/\omega\) [2207.09515]. In spin-chain communication, the effective transformation at the receiver depends on the prior propagation operator \(V(t)=e^{-iHt}\) and on a final local unitary \(U^{(ER)}\) acting on an extended receiver, so that the operational map is a composite object rather than a single autonomous gate [1812.01408]. In higher-order circuit transformations, the decisive “propagation” variable is causal architecture—parallel, adaptive, or indefinite causal order—which changes what transformations of an unknown input unitary are possible and with what success probability or fidelity [1909.01366] [2109.08202]. Taken together, these works indicate that “propagation dependence” is best regarded as a structural constraint on unitary action: the same formal target unitary may have different realizability, cost, or operational meaning depending on whether propagation is generated by a Hamiltonian, diffraction, transport, or higher-order circuit wiring. This suggests that the subject is less a single formalism than a family of closely related operator-theoretic questions. ## 2. Operator-intrinsic speed limits for implementing unitaries The 2024 work on quantum speed limits for implementation of unitary transformations formulates lower bounds on the minimal time \(T\) needed to realize an \(N\)-dimensional gate \(U=e^{-iHT}\) without reference to any initial or final state [2406.03964]. The principal bound is \[ T \geq \max \left\{ \frac{\pi}{2E}\left[1-\frac{|\operatorname{tr}(U)|}{N}\sqrt{1+\frac{4}{\pi^2}}\right], \; \frac{1}{\Delta E}\sqrt{1-\frac{|\operatorname{tr}(U)|^2}{N^2}} \right\}, \] where \[ E:=\frac{1}{N}\sum_k E_k - E_0, \qquad (\Delta E)^2:=\frac{1}{N}\sum_{k=0}^{N-1}(E_k-\overline E)^2. \] The first term is presented as a generalization of the Margolus-Levitin bound, and the second as a generalization of the Mandelstam-Tamm bound. A corollary using the spectral width \(\delta E=E_{\max}-E_{\min}\) yields \[ T \geq \max \left\{ \frac{\pi}{\delta E}\left[1-\frac{|\operatorname{tr}(U)|}{N}\sqrt{1+\frac{4}{\pi^2}}\right], \; \frac{2}{\delta E}\sqrt{1-\frac{|\operatorname{tr}(U)|^2}{N^2}} \right\}. \] The dependence on \( |\operatorname{tr}(U)| \) is the distinctive operator-level feature. The bounds are invariant under global phase shifts \(U\to e^{i\phi}U\) and under basis changes \(U\to VUV^\dagger\). The identity transformation corresponds to \( |\operatorname{tr}(U)|\approx N \), for which the lower bound shortens, while “far” unitaries with \( |\operatorname{tr}(U)|\to 0 \) incur larger minimal times. For mutually unbiased basis transformations the paper states \( |\operatorname{tr}(U)|\le \sqrt N \), giving an immediate specialization of the bound. For permutation operators fixing \(m\) basis states, \(\operatorname{tr}(U)=m\), again reducing the time bound to a trace-based expression. Low-dimensional cases make the interpretation especially sharp. For qubit gates, the paper gives \[ T=\frac{\cos^{-1}(|\operatorname{tr}(U)|/2)}{E}, \] and reports that for low dimensions, including qubits and qutrits, the bound aligns closely or even saturates with the true minimal implementation time [2406.03964]. In this formulation, propagation dependence enters through the generator \(H\), but the bound itself is intrinsic to the target operator and to gross spectral data rather than to any chosen state. ## 3. Optical beam arrays and diffraction-mediated reconfigurable unitaries In free-space photonics, propagation-dependent unitary transformations are realized physically by alternating phase modulation and diffraction. The multi-plane light converter (MPLC) architecture is modeled as \[ \bm{U} \approx \mathcal{D}_P \mathcal{P}_P \dotsb \mathcal{D}_1 \mathcal{P}_1, \] where \(\mathcal{P}_p\) denotes phase modulation at plane \(p\) and \(\mathcal{D}_p\) the intervening diffraction [2407.06981]. The paper studies superpositions of parallel free-space beams arranged in an array, with basis states \(\ket m\). For two beams, the target family is parameterized as \[ \bm{U}(\theta,\phi)= \begin{pmatrix} \cos\theta & i e^{-i\phi}\sin\theta \\ i e^{i\phi}\sin\theta & \cos\theta \end{pmatrix}, \] with \(\theta\) controlling coupling and \(\phi\) the relative phase. The reported implementation uses a single SLM placed parallel to a flat mirror, with light entering at an angle so that multiple reflections realize five phase-mask planes. Phase profiles are computed with a wavefront matching algorithm, and an additional phase-only SLM after the primary MPLC is used as a correcting mask for residual phase errors. Output amplitudes and phases are reconstructed by interfering the beam array with a reference beam and measuring the resulting off-axis holograms. Performance is quantified by the gate fidelity \[ F_G(\bm{U}_t,\bm{U}_d)=\frac{1}{M}\sum_{m=1}^M \left|\langle \psi_{t_m} \mid \psi_{d_m}\rangle\right|^2. \] For the full two-dimensional unitary group sampled over \((\theta,\phi)\), the paper reports an average experimental gate fidelity \(\bar F_G=0.67\pm 0.17\) without a correcting phase mask and \(\bar F_G=0.85\pm 0.03\) with correction; the latter is stated to closely match the theoretical design fidelity \(\sim 0.92\) [2407.06981]. The work therefore treats propagation not as an unwanted constraint but as the synthesis resource itself: diffraction between programmable phase planes is the mechanism that generates the unitary. The same work is explicit about limitations. The minimum number of phase planes required for arbitrary \(N\)-mode unitaries is said to grow at least as \(N\), practical “\(N\)-plane \(N\)-dimensional unitary” recipes are lacking, and performance is sensitive to misalignment, SLM surface curvature, cumulative phase-plane errors, and loss. A common misconception is therefore corrected by the experiment itself: the theoretical universality of MPLC does not automatically imply experimentally general high-dimensional performance. ## 4. Dynamical basis changes, exact mappings, and tensor-state propagation A distinct line of work embeds unitary transformations into the equations of motion themselves. In mixed quantum-classical dynamics, Miyazaki, Krotz, and Tempelaar define a quantum basis change \[ |\xi\rangle=\sum_n u_{\xi n}|n\rangle \] and introduce complex classical coordinates \[ z_n=\sqrt{\frac{m_n h_n}{2}\left(q_n+i\frac{p_n}{m_n h_n}\right)}, \qquad z_\xi=\sum_n U_{\xi n} z_n, \] so that classical and quantum coordinates can be transformed by analogous unitaries [2404.15614]. The transformed classical equations of motion are written as \[ \dot z_\xi=-i\frac{\partial H}{\partial z_\xi^*}, \] with the Hamiltonian rewritten in the transformed basis through coefficients \[ h_{\xi\xi'}=\sum_n U_{\xi n}^* h_n U_{\xi' n}, \qquad \tilde h_{\xi\xi'}=\sum_n U_{\xi n} h_n U_{\xi' n}. \] The paper focuses on choosing the basis at the outset of propagation and notes that a time-dependent variationally optimized \(U(t)\) would require extending the framework to include the time dependence of the transformation itself. The practical motivation is basis truncation. Surface-hopping calculations for an electronic carrier scattering onto a single impurity in the presence of phonons show that appropriate basis transformations, capturing impurity localization together with delocalized higher-energy excitations, can reproduce the dynamics within a fraction of the classical and quantum basis sets [2404.15614]. The contrast between physical, reciprocal, and eigenbasis truncations is central: truncating a non-optimal basis can fail badly even when the transformed equations are analytically equivalent to the original ones. A more explicit time-dependent construction appears in the free-to-harmonic mapping. There the free-particle Schrödinger dynamics are related to harmonic-oscillator dynamics by the coordinate change \[ t(T)=\tan(\omega T), \qquad x(\xi,T)=\frac{\xi}{\omega\cos(\omega T)}, \] combined with a squeezing unitary \[ U(t)=\exp\left(\frac{i}{2}\ln c(t)\,(\hat x\hat p+\hat p\hat x)\right), \] a gauge unitary \[ R(t)=\exp\left(\frac{i}{2}c(t)\dot c(t)\hat x^2\right), \] and the scaling function \[ c(t)=\sqrt{1+\omega^2 t^2}, \qquad dT=\frac{dt}{c^2(t)}. \] The same structure is extended to Koopman-von Neumann dynamics, and the mapping imparts the identical scaling \( \gamma\to c^4(T)\gamma \) and \( D\to c^4(T)D \) to quantum decoherence and classical diffusion parameters, respectively [2207.09515]. Here propagation dependence is inseparable from time reparameterization. In tensor-state simulation, mode folding provides an alternative to Trotter-Suzuki expansion for locally interacting bosonic systems [1408.2251]. Single-particle propagation is encoded in mode operators \[ \hat\alpha_q^\dagger(t)=e^{-it\hat H}\hat a_q^\dagger e^{it\hat H} =\sum_{l=1}^N c_{l,q}(t)\hat a_l^\dagger, \] and the coefficients are manipulated through diagonal phase removals \[ \hat r_k^{[l]}(t)=e^{i\phi_{l,k}(t)\hat a_l^\dagger \hat a_l} \] and pairwise rotations \[ \hat R_k^{[j+1,j]}(t)=e^{i\theta_{j,k}(t)\hat J_y^{[j+1,j]}}, \qquad \hat J_y^{[j+1,j]}=\frac{1}{2i}\left(\hat a_{j+1}^\dagger \hat a_j-\hat a_j^\dagger \hat a_{j+1}\right). \] For imaginary-time evolution, non-unitary transformations \[ \hat Q^{[j+1,j]}=e^{\epsilon_j \hat J_{xz}^{[j+1,j]}} \] are introduced. The reported benchmarks indicate that mode folding and Trotter-Suzuki have similar errors when matched for bond dimension \(\chi\) and time step \(t_s\), with split-operator error dominating in both cases [1408.2251]. The broader significance is that propagation can be represented as a structured sequence of local mode transformations rather than solely as an exponential of local Hamiltonian fragments. ## 5. Transport, causal structure, and transformations of unknown unitaries In spin-chain state transfer, propagation dependence appears as a post-transport controllability problem. The global evolution is \[ W=\left(E^{(S,TL)}\otimes U^{(ER)}\right)V(t), \qquad V(t)=e^{-iHt}, \] and the receiver state is \[ \rho^{(R)}(t)=\operatorname{Tr}_{S,TL,A}\!\left[W\,\rho(0)\,W^\dagger\right]. \] Its matrix elements are linear combinations of the sender’s density-matrix elements through a \(T\)-operator determined jointly by chain propagation and the unitary on the extended receiver [1812.01408]. Within a 42-node spin-\(1/2\) chain with two-qubit sender and receiver, the paper numerically demonstrates four operations: turning some transferred matrix elements to zero, rearranging matrix elements, constructing linear combinations with required coefficients, and solving a system of linear algebraic equations with a system of two equations as example. It also notes a limitation: not every single element can be turned to zero individually; certain groups of elements can be simultaneously nulled instead. A more abstract but closely related problem is the transformation of unknown unitary operations by higher-order circuits. The probabilistic exact framework distinguishes parallel, adaptive, and indefinite-causal-order protocols for turning \(k\) uses of an arbitrary \(d\)-dimensional unitary \(U_d\) into \(U_d^T\), \(U_d^*\), or \(U_d^{-1}\) [1909.01366]. The paper proves that for unitary complex conjugation and unitary inversion, if \(k0\) and \(K\) compact, on suitable \(M\)-good arcs [1502.02301]. Consequences include absence of singular continuous spectrum on those arcs, persistence of absolutely continuous spectrum, and eigenvalues that are isolated with finite multiplicity and can accumulate only at threshold points. In one dimension, symmetric quantum walks are said to be universal, as are CMV matrices, so propagation properties of diverse unitary network models can be analyzed through a common spectral framework. At the opposite end of abstraction, unitary pseudonatural transformations (UPTs) between fibre functors \(F,F':\operatorname{Rep}(G)\to \mathrm{Hilb}\) provide a categorical formalization of auxiliary-space-dependent unitary transport [2004.12761]. A UPT consists of a Hilbert space \(H\) and unitary maps \[ \alpha_X:F(X)\otimes H\to H\otimes F'(X) \] satisfying naturality and monoidality conditions. The central classification theorem identifies the category of UPTs \(F_1\to F_2\) and modifications with the category of finite-dimensional \(*\)-representations of the corresponding bi-Hopf-Galois object \(Z\). Accessible fibre functors are classified, up to the equivalences stated in the paper, by simple Frobenius monoids in \(\operatorname{Rep}(A_G)\), and finite-dimensional quantum isomorphisms of quantum graphs become UPTs between fibre functors on \(\operatorname{Rep}(G_X)\). A plausible implication is that propagation dependence can be expressed not only by differential generators or causal wiring, but also by functorial transport through an auxiliary Hilbert space. Several misconceptions are therefore ruled out by the combined literature. Propagation-dependent unitary transformations are not synonymous with ordinary time evolution; they include diffraction-mediated spatial synthesis, receiver-side corrections after transport, circuit-order-dependent supermaps, and categorical pseudonatural transformations. They are also not automatically arbitrary in practice: MPLC realizations degrade under experimental imperfections [2407.06981], mixed quantum-classical basis optimization is developed primarily for a basis fixed at the outset rather than for fully time-dependent \(U(t)\) [2404.15614], and exact universal conjugation or inversion of unknown \(d\)-dimensional unitaries is impossible when \(k