Superposed Quantum Paths

• Superposed quantum paths are coherent superpositions of different evolution alternatives, mathematically described by Feynman's path integrals and realized in cat states and quantum switches.
• They empower advanced quantum information processing by enabling superpositions of causal orders and nonclassical interference key to quantum computing and metrology.
• Experimental implementations in optical qubits, weak measurements, and quantum batteries demonstrate their potential to overcome classical limitations and drive technological innovation.

Superposed quantum paths refer to coherent quantum superpositions of distinct propagation alternatives—histories, trajectories, gate orders, or even spacetimes—playing a central role across quantum mechanics, quantum information processing, and foundational studies. This concept generalizes the canonical path superposition in the Feynman sum-over-histories formalism, is experimentally manifested in cat states and "which-path" interference, and is now foundational for advanced quantum protocols such as quantum switches, spatial channel superpositions, and superposed parameterised circuits. The sections below detail the mathematical structure, experimental realization, computational significance, and foundational implications of superposed quantum paths.

1. Mathematical Structure and Physical Realizations

Superposed quantum paths arise whenever a quantum system simultaneously explores multiple evolution alternatives, and their amplitudes are coherently combined. In the canonical language of Feynman's path integral, the total propagator is an integral (or sum) over all possible paths, with each path $x(t)$ assigned a complex amplitude $e^{iS[x]/\hbar}$, where $S[x]$ is the classical action. For a double slit experiment, this reduces to

$$K(x_f, t_f | x_i, t_i) = K_1(x_f, t_f | x_i, t_i) + K_2(x_f, t_f | x_i, t_i),$$

where $K_{1,2}$ correspond to the contributions from each slit (Duprey et al., 2021).

In quantum optics, superposed coherent states (cat states) take the form $$|\psi\rangle = c_1 |+\alpha\rangle + c_2 |-\alpha\rangle$$. These are used as logical qubits, with gate protocols realized by displacement operations, single-photon subtraction, and projective measurements, as shown in the phase gate protocol:

$$|\psi_{\text{in}}\rangle = x|\alpha\rangle + y|-\alpha\rangle \xrightarrow{D(\gamma),\,a,\,D(-\gamma)} x(\alpha+\gamma)|\alpha\rangle + y(-\alpha+\gamma)|-\alpha\rangle.$$

Choosing $(\gamma-\alpha)/(\gamma+\alpha)=e^{i\varphi}$ enables an arbitrary phase gate (Marek et al., 2010).

In advanced scenarios, superposed paths can refer to the order of operations itself: for instance, the quantum switch employs a control qubit to encode the superposition of two causal orders, yielding Kraus operators

$$S_{ij} = E_i D_j \otimes |0\rangle\langle 0| + D_j E_i \otimes |1\rangle\langle 1|.$$

This encodes, at the process level, a quantum superposition over alternative gate orderings (Pellitteri et al., 25 Aug 2025, Simonov et al., 2023). Similarly, generalized transposition $T[W]$ transforms the time axis of quantum operations, enabling processes with the time axis in a genuine superposed direction (Lie et al., 2023).

2. Nonclassical Path Contributions and Path Entanglement

Superposed quantum paths extend far beyond the addition of "classical" or extremal trajectories. Full path integral treatments in interference experiments demonstrate measurable contributions from non-classical (looped or multi-slit crossing) paths. In the three-slit scenario,

$\psi_{ABC} = \psi_A + \psi_B + \psi_C + \psi_L,$

with measurable deviations from naive superposition detected via the parameter $\kappa = \varepsilon/\delta$, where

$$\varepsilon = p_{ABC} - (p_{AB} + p_{AC} + p_{BC}) + (p_A + p_B + p_C).$$

Nonzero $\kappa$ values signify the necessity of nonclassical path inclusion, directly measurable in triple-slit experiments (Sawant et al., 2013).

Furthermore, temporal analogues of entanglement—quantum path entanglement (QPE)—emerge in operator-theoretic models of multi-plane diffraction, where the quantum history state resides in a tensor product Hilbert space over times:

$$\mathcal{H} = \mathcal{H}_N \odot \mathcal{H}_{N-1} \odot ... \odot \mathcal{H}_0,$$

with measurement and projection operators acting sequentially. Temporal correlations (QPE) and interference (QPI) are intrinsic resources, enabling violation of the Leggett–Garg inequality even with classical light (Gulbahar, 2018).

3. Experimental Manifestations and Control

Superposed quantum paths are experimentally manifested in several regimes:

• Coherent-state optical qubits: Protocols using photon subtraction and displacement with single-shot heralding achieve universal gates, requiring only standard on-off detectors (Marek et al., 2010).
• Quantum switches: Realized experimentally in photonic, nuclear magnetic resonance, and other platforms, the quantum switch superposes the order in which two channels act, enabling transformations impossible in any definite order (Simonov et al., 2023).
• Weak measurement protocols: Arrays of weakly-coupled probes can sample the "sum over paths," providing experimental evidence for Feynman trajectory interference in double-slit setups (Duprey et al., 2021).
• On-chip photon superposition: Integrated silicon photonic chips can produce multi-photon states where the origin of creation (which probabilistic source produced a given photon set) is placed in genuine superposition, producing quantum interference between different possible generation pathways (Feng et al., 2021).
• Quantum batteries: Superposed trajectories, controlled by an ancillary qudit, allow a battery qubit to be charged by several cavities (or cavity positions) simultaneously, with quantum interference enhancing the ergotropy (maximum extractable work) and leading to immediate charging—far surpassing classical protocols (Lai et al., 2023).
• Quantum machine learning: Superposed parameterized quantum circuits (SPQCs) use an address register in superposition to embed exponentially many variational quantum sub-models. Sequential amplitude transformation circuits realize polynomial nonlinearities akin to classical neural activations, achieving substantial accuracy improvements and reduction in run-to-run variance (Patapovich et al., 10 Jun 2025).

4. Computational and Metrological Applications

Superposed quantum paths enable computational primitives and information-processing advantages unattainable in classically ordered frameworks:

• Universal quantum computing: Superposing the causal order of single-qubit gates under the quantum switch enables deterministic realization of any two-qubit controlled gate including the Barenco gate, which is universal for quantum computation. This methodology removes the need for fixed gate sequencing and extends deterministic controlled operations to platforms where two-qubit gates are otherwise probabilistic (Simonov et al., 2023).
• Genuine quantum networking: Quantum control registers allow networks to execute tasks (e.g., sending, merging, or measuring entangled states) in superposition, enabling robust, flexible protocols such as superposed routing, remote protocol switching, and enhanced robustness to loss (Miguel-Ramiro et al., 2020).
• Quantum parameter estimation: A probe-control qubit pair passing through two noisy unitaries in coherent superposition allows phase parameter estimation via measurement on the control alone. Crucially, the control qubit remains sensitive to the parameter even under maximal noise, outperforming both conventional and causally switched estimation methods in high-noise regimes (Chapeau-Blondeau, 2021).

5. Foundations: Relativity, Indefinite Causality, and Quantum Gravity

Superposed quantum paths underpin several foundational domains:

• Relativistic origins: The phase structure of quantum amplitudes and the superposition principle itself can be reinterpreted as relics of relativistic time dilation, where “Lorentz filtering” retains only paths with equivalent clock parity, directly producing quantum interference patterns (Ord, 2017).
• Indefinite temporal and spatial structure: Recent advances generalize the direction of the time axis using operator-valued (generalized) transpositions $T[W]$, enabling a continuous superposition of forward and backward time evolution, and revealing that compatibility with multiple time axes restricts information flow to enforce causality (Lie et al., 2023).
• Quantum field theory in curved spacetime: Superposed spacetime geometries, such as observers in a coherent superposition of distinct causal diamonds, serve as a resource for entanglement generation and the partial mitigation of thermal-induced decoherence; analytical results show enhanced logarithmic negativity and mutual information compared to classical settings (Liu et al., 31 Dec 2024).
• No-go theorems for superposed actions: While superpositions of physical states are unitary, superposing nonorthogonal “actions” or agent-dependent choices (e.g., in extended Wigner’s friend setups) is incompatible with unitary quantum theory, leading to fundamental constraints on how agent “freedom” enters quantum evolution (Łukaszyk, 2018).

6. Quantum Path Engineering in Materials and Matter

Superposed quantum paths are intrinsic to mesoscopic transport and advanced material systems:

• Quantum interference in superposed lattices: In materials with incommensurate lattices (such as Cr with a coexisting spin-density-wave), superposed reciprocal lattices yield networks of quantum orbits whose interference manifests in phase-shifted Shubnikov–de Haas oscillations. The observed $\pi$ phase shift between conductivity channels directly traces to the reconnection and quantum interference between open and closed magnetic field–driven orbits (Feng et al., 2023).
• Wave mixing and photon state visualization: On-chip quantum wave mixing in superconducting artificial atoms reveals the mapping of discrete superposed and coherent photon states into the elastically scattered spectrum, with peak counts directly visualizing the underlying Fock-state statistics. This enables on-chip quantum metrology and state discrimination in the microwave regime (Dmitriev et al., 2017).

7. Implications and Outlook

Superposed quantum paths form a unifying concept across quantum foundations, quantum information, and quantum technologies. Their mathematical structure enables scalable encoding, robust information-processing protocols (including universal computing and noiseless communication), and practical quantum devices exhibiting distinctly nonclassical capabilities. Experimentally, photon subtraction, beam splitters, weak measurements, spatial path multiplexing, and control-based supermaps implement diverse forms of path superposition with demonstrable advantage. Foundationally, superposed paths elucidate the connections between quantum coherence, relativity, causality, and the emergent structure of spacetime, while fundamentally constraining the applicability of unitary evolution when extended to choices or actions.

Key formulas and architectural elements underlying these developments are summarized in the following table:

Context	Superposed Path Structure	Key Mathematical Expression / Protocol
Feynman propagation, slits	All possible (classical and non-classical)	$K = K_c + K_{nc}$; $\kappa = (\varepsilon)/\delta$
Cat-state gate protocols	$|\alpha\rangle$, $|-\alpha\rangle$	$(\gamma-\alpha)/(\gamma+\alpha) = e^{i\varphi}$
Quantum switch, superposed orders	Process-level causal order superposition	$$S_{ij} = E_i D_j \otimes |0\rangle + D_j E_i \otimes |1\rangle$$
Path entanglement (histories)	Tensor product over times: QPE/QPI	$\mathcal{H} = \bigotimes_{j} \mathcal{H}_j$
Parameterised quantum circuits (SPQC)	Exponential superposition over models	$$|\Phi\rangle = \frac{1}{\sqrt{L}} \sum_j |j\rangle U(\theta^{(j)}) S(x)|0\rangle$$
Generalized time axis	Superposed temporal direction	$$M^{T(\theta)} = e^{-i\theta/2}[ \cos(\theta/2) M + i\sin(\theta/2) M^T ]$$
Quantum battery charging	Superposed spatial interactions	$$|\psi_D\rangle = (1/\sqrt{N}) \sum_{j=1}^N |j_D\rangle$$

In all contexts, engineering, measuring, and harnessing superposed quantum paths drive both foundational understanding and quantum technological innovation.