Holonomies are of great interest to quantum computation and simulation. The geometrical nature of holonomies offers increased stability to quantum gates. Furthermore, symmetries of particle physics are naturally reflected in holonomies, making them ideally suited for quantum simulation of quantum chromodynamics and grand unified theories. Yet, practically designing quantum holonomies with the required properties and scale is challenging. Here, we construct a new class of holonomies by increasing the particle number. We show that multiparticle holonomies can even exist in systems devoid of any single-particle holonomies. We present a comprehensive framework for multiparticle quantum holonomies and experimentally realize various two-particle holonomies in integrated photonics. Our results enable the number of particles to be harnessed as a design parameter, offering increased freedom in constructing holonomic systems.
Non Abelian holonomy, a geometric aspect of quantum mechanics, underpins the geometric formulation of fundamental interactions and plays a vital role in holonomic quantum computation. Photonics is a fertile platform for exploring non-Abelian holonomies. However, universal unitary holonomies are still out of reach. The lack of control over scale phase degrees of freedom hinders the direct quantum simulation of unitary-governed phenomena, such as electron spin, weak isospin, and quantum chromodynamics,as well as the realization of generic geometric quantum gates for holonomic quantum computing. Here we employ artificial gauge fields (AGFs) to introduce complex-valued couplings, synthesizing arbitrary unitary holonomies and enabling generic geometric quantum gates in photonics. Higher dimensional U(m) transformations are experimentally synthesized based on Givens rotations. Our results pave the way for all-geometric-phase approach for both quantum and classical computing in photonics.
Abstract When a quantum system undergoes slow changes, the evolution of its state depends only on the corresponding trajectory in Hilbert space. This phenomenon, known as quantum holonomy, brings to light the geometric aspects of quantum theory. Depending on the number of degrees of freedom involved, these purely geometric entities can be scalar or belong to a matrix-valued symmetry group. In their various forms, holonomies are vital elements in the description of the fundamental forces in particle physics as well as theories beyond the standard model such as loop quantum gravity or topological quantum field theory. Yet, implementing matrix-valued holonomies thus far has proven challenging, being further complicated by the difficulties involved in identifying suitable dark states for their construction in bosonic systems. Here we develop a representation of holonomic theory founded on the Heisenberg picture and leverage these insights for the experimental realization of a three-dimensional quantum holonomy. Its non-Abelian geometric phase is implemented via the judicious manipulation of bosonic modes constructed from indistinguishable photons and obeys the U(3) symmetry relevant to the strong interaction. Our findings could enable the experimental study of higher-dimensional non-Abelian gauge symmetries and the exploration of exotic physics on a photonic chip.
Quantum computation faces a major challenge: the need for stable quantum gates. Holonomies offer a way to increase the stability of quantum gates on a fundamental level, as their functionality directly arises from the geometry of the underlying Hilbert space. We present the quantum optical realization of holonomies as single-qubit quantum gates. Specifically, we implement them in a non-adiabatic scheme, which paves the way for unprecedented miniaturization. To demonstrate their versatility, we realize the Hadamard and Pauli-X gates, experimentally show their non-Abelian nature, and combine them into a single-qubit quantum algorithm, the PQ penny flipover. The planar geometry of our designs enables them to substitute directional couplers currently in widespread use in photonic quantum architectures across all platforms.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert space
Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspace
Hgeoof the Hilbert-space that is spanned by states
{∣Φk⟩}kthat fulfill
(Φk∣H^∣Φj⟩=0, where
H^is the system's Hamiltonian. Restricting the propagation to
Hgeoensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expanding
Hgeoand enabling the synthesis of holonomies from higher symmetry groups
U(N)as larger and more capable computational units [6], [7].
This preprint develops a state-level framework in which holonomy is treated as a property of cyclic subspaces rather than single-particle eigenstates or gauge fields. Within number-conserving quadratic Hamiltonian dynamics on bosonic or fermionic modes, a precise notion of single-particle irreducibility is introduced, meaning that no compatible single-particle cyclic subspace of the same dimension can satisfy a holonomic (purely geometric) condition. The main result is an explicit existence theorem showing that nontrivial, purely geometric holonomies can arise in two-particle cyclic subspaces even when the associated single-particle problem admits no holonomic cyclic subspace of the same dimension. A fully worked minimal example—two bosons in two modes—is provided, yielding a nontrivial SU(2) holonomy on an encoded two-dimensional subspace while the single-particle sector is provably non-holonomic. The paper is self-contained and includes a conservative, device-agnostic certification scheme based on loop composition and conjugacy-invariant readouts, enabling experimental verification without model-specific gauge-field assumptions.
Geometric and holonomic quantum computation utilizes intrinsic geometric properties of quantum-mechanical state spaces to realize quantum logic gates. Since both geometric phases and quantum holonomies are global quantities depending only on the evolution paths of quantum systems, quantum gates based on them possess built-in resilience to certain kinds of errors. This review provides an introduction to the topic as well as gives an overview of the theoretical and experimental progress for constructing geometric and holonomic quantum gates and how to combine them with other error-resistant techniques.