We propose a fundamentally new paradigm for quantum computation based on the intrinsic geometry of quantum state space. Rather than specifying computation as a sequence of discrete gates applied over external time, Geometric Quantum Computing (GQC) defines computation as geodesic evolution through Hilbert space, where the computational 'time' emerges from the Fubini-Study metric itself. This framework inverts the standard relationship between quantum dynamics and time: instead of viewing decoherence as degradation of quantum information over time, we recognize that decoherence is time—the environment's information processing. We derive the theoretical foundations of GQC, demonstrate that it naturally circumvents barren plateaus by operating directly in state space rather than parameter space, and propose experimental protocols for implementation. The framework suggests that optimal quantum computation is achieved not by fighting decoherence but by engineering the effective geometry of state space to make desired computations correspond to natural geodesic flow.
We propose a fundamentally new paradigm for quantum computation based on the intrinsic geometry of quantum state space. Rather than specifying computation as a sequence of discrete gates applied over external time, Geometric Quantum Computing (GQC) defines computation as geodesic evolution through Hilbert space, where the computational 'time' emerges from the Fubini-Study metric itself. This framework inverts the standard relationship between quantum dynamics and time: instead of viewing decoherence as degradation of quantum information over time, we recognize that decoherence is time—the environment's information processing. We derive the theoretical foundations of GQC, demonstrate that it naturally circumvents barren plateaus by operating directly in state space rather than parameter space, and propose experimental protocols for implementation. The framework suggests that optimal quantum computation is achieved not by fighting decoherence but by engineering the effective geometry of state space to make desired computations correspond to natural geodesic flow.
The geometric phase stands as a foundational concept in quantum physics, revealing deep connections between geometric structures and quantum dynamical evolution. Unlike dynamical phases, geometric phases exhibit intrinsic resilience to certain types of perturbation, making them particularly valuable for quantum information processing, where maintaining coherent quantum operations is essential. This article provides a review of geometric phases in the context of universal quantum gate construction, i.e., the geometric quantum computation (GQC), with special attention to recent progress in nonadiabatic implementations that enhance gate fidelity and/or operational robustness. We first review a unified theoretical framework that can encompass all existing nonadiabatic GQC approaches, then systematically examine the design principles of nonadiabatic geometric gates with a particular focus on how optimal control techniques can be leveraged to improve the accuracy and noise resistance. In addition, we conducted detailed numerical comparisons of various nonadiabatic GQC protocols, offering a quantitative assessment of their respective performance characteristics and practical limitations. Through this focused investigation, our aim is to provide researchers with both fundamental insights and practical guidance for advancing geometric approaches in quantum computing.
geodesic in (2 n+1 -1)-dimensional sphere.Further, the reduced search sequence is given rise to the complex projective space CP 2 n -1 through a geometric reduction, which is also shown to be on a geodesic in CP 2 n -1 .Roughly speaking, the reduction in [15] is made through the phase-factor elimination from quantum states, so that CP 2 n -1 is thought of as the space of rays.Note that the geodesics above are associated with the standard metric on (2 n+1 -1)-dimensional sphere and with the Fubini-Study metric on CP 2 n -1 , respectively.The Fubini-Study metric on CP 2 n -1 is utilized also in [15] to measure the minimum distance from each state involved in the search sequence to the submanifold consisting of non-entangled states, which characterizes the entanglement of the states along the search.As expected benefits of geometric and dynamical views on quantum algorithmic studies like [15], the following would be worth listed;1 By revealing underlying geometry of quantum algorithms (not necessarily universal), numbers of results in geometry are expected to be applied to make advances in quantum computation and information.2 On looked upon the iterations made in algorithms as (discrete) time-evolutions of states, numbers of results in dynamical systems are expected to be applied to make advances in quantum computation and information.3 In view of a close connection between geometry and dynamical systems, geometric and dynamical-systems studies on quantum algorithms may provide interesting examples of dynamical systems.It would be worth noting here that there exists another approach to quantum searches using adiabatic evolution [16][17][18].That approach, however, is outside the scope of this chapter since the search dealt with in this chapter is organized on the so-called amplitude magnification technique [8] which differs from the adiabatic evolution.
Geometric-Structured Quantum Computing represents a paradigm shift in the design of quantum algorithms and hardware, leveraging the inherent geometric properties of quantum systems to fundamentally alter the process of quantum computation. This paper explores the core concept of utilizing geometric structures as the foundation for quantum computing, proposing a mechanism to transform quantum qubit and gate structures into geometric representations, enabling novel optimization and computation strategies. The paper argues that this approach offers significant advantages over traditional methods, particularly in the context of algorithm design, error mitigation, and overall hardware architecture. We detail the core mechanisms behind this paradigm, providing a foundational understanding of the potential for this new approach.
Geometry and Quantum Computing represent a burgeoning field with significant potential for transformative advancements across diverse disciplines. This paper explores the integration of geometric information theory and quantum computation, proposing a novel quantum computation algorithm predicated on the inherent properties of geometric information. We outline the core mechanism by leveraging geometric information to enhance quantum computation's efficiency and robustness. The research aims to contribute to the development of more powerful and reliable quantum systems by integrating these two fundamental principles. This integration promises to unlock new possibilities in areas such as quantum simulation, optimization, and cryptography.
This paper establishes the foundational differential-geometric architecture for the coordinate-independent tracking of pure quantum states within closed physical systems, validated via explicit computational modeling of a driven two-level system. While standard quantum mechanics relies on the linear algebraic structure of complex Hilbert spaces, the unique identification of physical states requires factoring out non-zero complex scaling factors, positioning the true arena of quantum dynamics as the projective Hilbert space P(H). We analyze this space as a compact complex manifold possessing an intrinsic Kähler structure that fuses a Riemannian metric tensor, a symplectic form, and a complex structure into a unified geometric framework. Through this manifold lens, we systematically derive the Fubini–Study metric tensor gF S and its corresponding connection geometries. By evaluating the evolution of state vectors under a time-dependent driving Hamiltonian, we show that the natural propagation of a closed quantum system traces out a geodesic path on the projective manifold. This geometric tracking defines a localized vector gauge field—the Berry connection 1-form—which gives rise to an antisymmetric curvature tensor that modifies the effective classical mechanics of the parameter space. Furthermore, we state and prove three core theorems establishing the geometric constraints of pure state transport: the Holonomy Theorem for cyclic evolutions, the Quantum Cramér–Rao Bound, and the Mandelstam–Tamm Quantum Speed Limit. To bridge theory with physical architectures, we implement an analytical, parameter-shift algorithmic pipeline to extract these metrics natively. Numerical grid analysis confirms the exact structural scaling relation gQF IM = 4 · gF S down to machine precision (1.24 × 10−15). Finally, by tracking the metric volume form det(gij ) alongside the inverse metric components, we visually map out and isolate the topological caustic singularities (θ → 0, π) where the embedding map drops rank. This computational diagnostic provides a robust framework to identify and structurally bypass non-immersive control bottlenecks during the initial Hamiltonian design phase.
Abstract We propose physical distinguishability as the fundamental primitive underlying quantum mechanics, general relativity, and thermodynamics. A state is defined operationally: two states are physically distinct if a physical interaction or measurement can reveal a difference between them. From this single principle, the distinguishability divergence D generates a smooth state space endowed with a Riemannian metric , a geometric velocity, an accessibility scale Λ, an accessibility ratio , and a geometric time . The framework provides a unified geometric description of the three major domains of physics. Quantum mechanics emerges through the Fubini–Study geometry of projective Hilbert space, thermodynamics through Fisher–Rao and Ruppeiner information geometry, and general relativity through the geometry of geodesic deviation. The threshold condition A=1acquires a universal physical meaning and appears in all three realizations as the boundary between physically accessible and inaccessible evolution. Variational formulations of both the Einstein and Schrödinger equations are constructed within the same geometric framework. The Lorentzian signature of spacetime emerges from a single sign reversal associated with the geometric-time direction, while the coupling constant of the distinguishability action is connected to the observed vacuum energy density. The framework further predicts a characteristic vacuum distinguishability scale ≈8.8×10^(-5) m, corresponding to an energy scale≈2.25" " meV. The theory is tested experimentally on IBM Quantum superconducting processors. Across one-qubit, two-qubit, 127-qubit, and 156-qubit experiments performed on three independent Heron-generation backends, excitation probabilities collapse onto universal curves when parameterized by endpoint Fubini–Study distinguishability. Protocols with identical laboratory control parameters but different endpoint distinguishabilities produce excitation probabilities differing by up to a factor of 3.8, while protocols with identical distinguishability produce identical outcomes across independent processors. Cross-backend transfer errors remain below 1%, providing direct experimental support for endpoint distinguishability as a universal physical variable. Taken together, the theoretical construction, variational formulations, vacuum-sector predictions, and large-scale quantum-processor experiments support distinguishability geometry as a unified and experimentally testable foundation for physical dynamics.