Harnessing Geometric Phase for Photonic Quantum Computation

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Author: Denis Avetisyan

Researchers have demonstrated a practical implementation of Berry’s phase – a key concept in quantum mechanics – using passive optical elements and a photonic quantum processor.

A defined state, $ \phi_{0,1,1}$, experiences a Berry phase induced by the applied magnetic field $\vec{B}$.
A defined state, $ \phi_{0,1,1}$, experiences a Berry phase induced by the applied magnetic field $\vec{B}$.

This work verifies the feasibility of implementing Berry’s phase in continuous-variable quantum computing with linear optics, paving the way for robust quantum algorithms.

Exploiting geometric phases for quantum computation demands robust and experimentally feasible implementations. Here, we present ‘Berry’s phase on photonic quantum computers’, detailing a continuous-variable quantum computing algorithm leveraging passive linear optics to simulate and observe Berry’s phase in photonic systems. Our approach demonstrates the phenomenon both through simulation and experimental verification on the Quandella Ascella platform, while also offering a pathway to mitigate errors arising from non-adiabatic evolution. Could this framework pave the way for encoding and manipulating geometric phases in larger-scale photonic quantum processors?

Emergent Order from Continuous Quantum States

For decades, quantum computation has largely been synonymous with the qubit – a discrete unit of quantum information analogous to the bits of classical computing. However, a fundamentally different approach, continuous-variable quantum computing (CVQC), is gaining traction as a viable and potentially advantageous pathway. Instead of manipulating individual, quantized levels, CVQC harnesses the infinite range of values possible in quantum properties like the position and momentum of light or the amplitude and phase of electromagnetic fields. This shift allows for encoding information in a continuous spectrum, offering unique advantages for specific computational tasks and potentially circumventing some of the challenges associated with maintaining the delicate coherence of discrete qubits. The exploration of CVQC represents a broadening of the quantum computing landscape, presenting a complementary strategy to the more widely known qubit-based methods and opening new avenues for quantum algorithm development.

Continuous-variable quantum computing (CVQC) distinguishes itself by harnessing the infinite possibilities within quantum properties like a particle’s position and momentum, rather than being limited to the binary states of traditional qubits. This approach allows for a vastly richer information encoding scheme, potentially exceeding the capabilities of discrete systems in specific computational tasks. While standard quantum computing manipulates $0$ and $1$, CVQC operates on a continuous spectrum of values, enabling more nuanced and complex calculations. This is particularly advantageous in areas such as quantum simulation of bosonic systems-modeling the behavior of light or sound-and certain machine learning algorithms where the continuous nature of the data aligns naturally with the quantum representation. The infinite degrees of freedom, though presenting unique engineering challenges, offer a pathway toward more efficient and powerful quantum processors for targeted applications.

Continuous-variable quantum computing (CVQC) distinguishes itself from traditional quantum computation by employing Gaussian states – quantum states fully described by a mean and variance – as its fundamental building blocks. This approach dramatically simplifies many quantum operations; linear transformations, for example, can be implemented directly using classical optics, bypassing the complex gate decomposition required for discrete qubits. Furthermore, information is efficiently encoded not in the discrete ‘0’ or ‘1’ of a qubit, but in the continuous amplitude and phase of the quantum field, allowing for high-dimensional encoding and potentially exceeding the capacity of discrete systems. This inherent efficiency stems from the mathematical properties of Gaussian states, enabling streamlined computation and offering a promising avenue for tackling specific quantum algorithms, particularly those involving signal processing and machine learning, with reduced resource requirements and increased speed.

A Gaussian Trotter-Suzuki gadget utilizes beam splitters-replacing CX gates from the previous step-to simulate real-time evolution under the Hamiltonian described in Eq. (40) for a single time step.
A Gaussian Trotter-Suzuki gadget utilizes beam splitters-replacing CX gates from the previous step-to simulate real-time evolution under the Hamiltonian described in Eq. (40) for a single time step.

Geometric Phases: Sculpting Quantum Pathways

The Berry phase, a geometric phase acquired during the adiabatic – or extremely slow – evolution of a quantum system, offers a means of controlling quantum states independent of the system’s Hamiltonian. This phase, mathematically expressed as $\gamma = \int_{C} \langle \psi(t) | i \hbar \frac{d}{dt} | \psi(t) \rangle dt$, is determined by the geometry of the parameter space traced by the system’s evolution, not the forces driving the change. Consequently, the Berry phase provides robustness against certain types of noise and control imperfections, as the accumulated phase is less susceptible to variations in the precise timing or magnitude of control fields. This resilience makes it a valuable tool in quantum information processing and precision measurement, enabling more reliable manipulation of quantum states.

Geometric phases, unlike dynamic phases, are determined by the geometry of the parameter space traversed by the quantum state and are therefore invariant to the specific time evolution – the ‘dynamics’ – used to move the system between initial and final states. This independence from dynamics provides inherent robustness against control errors and environmental noise. Any fluctuations in the timing or precise functional form of the control fields will affect the dynamic phase, represented as $-\frac{i}{\hbar} \int_0^T E(t) dt$, where $E(t)$ is the energy of the system, but will leave the geometric phase unchanged. Consequently, geometric phases offer a pathway to high-fidelity quantum control even in the presence of imperfections that would otherwise degrade performance, as the geometric contribution remains stable and predictable.

The Aharonov-Anandan phase represents a generalization of the Berry phase, extending its applicability beyond the strictly adiabatic regime. While the Berry phase requires evolution to occur slowly enough that the system remains in its instantaneous eigenstate, the Aharonov-Anandan phase accounts for scenarios where this condition is not met – i.e., non-adiabatic processes. This generalization is mathematically defined as the phase acquired during quantum evolution being comprised of both a dynamic phase, proportional to the time integral of the energy expectation value, and a geometric phase given by the closed-loop integral of the connection form in the parameter space. The total acquired phase is therefore $e^{i(\gamma + \int_C A \cdot \dot{s} dt)}$, where $\gamma$ represents the dynamic phase, $A$ is the connection form, and $s$ parameterizes the evolution path. This broadened scope allows for the manipulation and control of quantum systems even when subject to faster, more realistic changes in external parameters.

Orbital Angular Momentum (OAM) states, characterized by a phase factor of $e^{il\phi}$, are instrumental in both enhancing and manipulating geometric phases. The azimuthal dependence of OAM allows for the encoding of information within the wavefront of a quantum state, increasing the dimensionality of the Hilbert space and thus the potential for geometric phase accumulation. Specifically, the use of OAM degrees of freedom enables the creation of robust geometric phases less susceptible to environmental decoherence. Furthermore, manipulating the OAM state – through techniques like spatial light modulators or q-plates – provides a means to control the accumulated geometric phase, offering precise control over quantum evolution and enabling applications in quantum information processing and precision sensing.

Varying the initial phase ϕ₀ reveals that the total phase accumulated along a contour and its anti-polar counterpart closely follows the ideal Berry phase γ(C) for longer circuit times (T=48π and T=12π), but deviates when circuit time is shortened (T=6π), particularly as ϕ₀ approaches π.
Varying the initial phase ϕ₀ reveals that the total phase accumulated along a contour and its anti-polar counterpart closely follows the ideal Berry phase γ(C) for longer circuit times (T=48π and T=12π), but deviates when circuit time is shortened (T=6π), particularly as ϕ₀ approaches π.

Non-Adiabatic Transitions: Boundaries of Predictability

In quantum mechanics, the adiabatic theorem states that a system remains in its initial eigenstate if a perturbation is applied slowly enough. Conversely, when the parameters governing a quantum system change rapidly, the system may not be able to follow the instantaneous eigenstate, resulting in non-adiabatic transitions. These transitions involve the system jumping to a different energy eigenstate, and their probability is directly related to the rate of change of the system’s Hamiltonian. The faster the evolution, the higher the probability of a non-adiabatic transition, and these transitions are fundamentally probabilistic in nature. This behavior distinguishes rapid evolutions from the smooth, predictable changes described by the adiabatic approximation, and has significant implications for the control and fidelity of quantum systems.

The Landau-Zener transition quantifies the probability of a non-adiabatic transition between two energy eigenstates when a system’s Hamiltonian varies in time. This probability, $P_{LZ}$, is dependent on the rate of change of the energy gap, $\Delta E(t)$, and the velocity, $v$, at which the system evolves through the avoided crossing. Specifically, the transition probability is approximated by $P_{LZ} \approx \exp\left(-\frac{\pi \Delta E^2(t)}{2 \hbar v}\right)$, where $\hbar$ is the reduced Planck constant. A larger energy gap or a slower evolution rate reduces the probability of a non-adiabatic transition, favoring adiabatic behavior where the system remains in its initial eigenstate.

Non-adiabatic processes, characterized by rapid changes in system parameters, introduce errors that degrade the fidelity of quantum operations. These effects necessitate careful control or mitigation strategies to maintain accurate quantum state manipulation. Recent research demonstrates robust geometric phase extraction, specifically the Aharonov-Anandan phase, even in the presence of non-adiabatic transitions. Experimental results indicate a deviation of less than 5% between the averaged Aharonov-Anandan phase and the true geometric phase, validating the effectiveness of these techniques in preserving quantum information despite the presence of non-adiabatic disturbances.

A four-mode interferometer prepares a superposition of states, evolves one mode through a desired path in parameter space using Trotter-Strang steps, and recombines it with a phase-matched reference mode to enable interferometric readout.
A four-mode interferometer prepares a superposition of states, evolves one mode through a desired path in parameter space using Trotter-Strang steps, and recombines it with a phase-matched reference mode to enable interferometric readout.

Photonic Architectures: Realizing Continuous Quantum Control

Passive linear optics, utilizing elements such as beam splitters and phase shifters, offers a pathway to implement quantum operations in continuous-variable quantum computing (CVQC) due to its inherent robustness and scalability. Unlike methods requiring active control and complex calibration, passive optical elements maintain stable performance and are less susceptible to decoherence. Scalability is achieved by leveraging integrated photonics to create complex networks of these elements on a single chip, allowing for the realization of multi-qubit gates and larger quantum circuits without significant increases in error rates or control complexity. This approach minimizes the need for precise adjustments and offers a deterministic implementation of quantum operations, crucial for building fault-tolerant CVQC systems. The use of squeezed states, combined with these linear optical elements, allows for the creation of universal quantum gates necessary for performing arbitrary quantum computations.

Integrated photonic circuits offer significant advantages for implementing complex linear optical networks due to their ability to precisely control and manipulate individual photons. Platforms like the Quandela Ascella utilize silicon photonics to fabricate compact, stable, and scalable circuits capable of hosting a large number of optical elements – including beam splitters, phase shifters, and polarizers – necessary for universal quantum computation. This integration minimizes optical losses, reduces the footprint of quantum circuits, and enables high-fidelity control over quantum states, surpassing the limitations of free-space optical setups. The deterministic nature of fabrication allows for the creation of identical, repeatable quantum building blocks, crucial for building larger, more complex quantum processors.

Trotter-Suzuki decomposition is a numerical technique employed to approximate the time evolution operator in quantum mechanics, particularly useful for simulating quantum systems where direct calculation is intractable. This method involves breaking down the overall evolution operator into a product of simpler, more easily calculated operators. In the context of the Ascella photonic platform, a Trotter step size of $0.01\pi$ was utilized, representing the granularity of the time discretization. Smaller step sizes generally improve the accuracy of the approximation but increase computational cost. The application of this decomposition enabled both classical quantum emulation and direct experimental realization of quantum circuits on the integrated photonic system, allowing for verification of the numerical simulations against physical results.

Recent developments in photonic quantum computing have facilitated the construction of programmable quantum systems capable of investigating and utilizing both geometric and dynamic phases. Specifically, the Ascella photonic platform has demonstrated quantum evolution over a period of $8\pi$, allowing for the observation of phase-related phenomena. This extended evolution time is achieved through precise control of integrated photonic circuits, enabling the implementation of complex quantum operations and the exploration of non-trivial quantum dynamics. The ability to programmatically control these phases opens avenues for advanced quantum algorithms and simulations dependent on phase manipulation.

The Trotter-Suzuki and Strang gadgets efficiently approximate time evolution dictated by the Hamiltonian in Eq. (40) by decomposing the process into sequential gate operations with adjusted time intervals.
The Trotter-Suzuki and Strang gadgets efficiently approximate time evolution dictated by the Hamiltonian in Eq. (40) by decomposing the process into sequential gate operations with adjusted time intervals.

The research details a manipulation of quantum states to observe Berry’s phase, a geometric phase arising from the evolution of a system, not from its energy. This resonates with the notion that order doesn’t require a central planner. The system’s behavior isn’t dictated by imposed control, but emerges from the interplay of local interactions – in this case, the properties of photons and linear optical elements. As Erwin Schrödinger once noted, “The total number of states of a system is, in general, not something that can be determined beforehand.” This mirrors the study’s exploration of state manipulation, where the final outcome isn’t preordained, but a result of the system’s inherent dynamics and the applied optical transformations. The observed phase isn’t ‘controlled’ into existence, it’s revealed through careful observation of the system’s natural evolution.

Where Do We Go From Here?

The demonstration of Berry’s phase within a photonic quantum computer, achieved through the seemingly simple act of manipulating light with linear optics, isn’t a triumph of design so much as a revealing of inherent system properties. Robustness doesn’t appear through meticulous engineering; it emerges from the interplay of fundamental physical laws. This work confirms that complex quantum phenomena aren’t necessarily built into a system, but rather allowed to manifest given appropriate conditions – a subtle, but crucial distinction. The challenge now isn’t to create more elaborate control mechanisms, but to better understand the conditions that facilitate these emergent behaviors.

Limitations remain, of course. Scaling such systems will inevitably reveal the fragility inherent in any physical implementation, but the focus should shift from error correction to error accommodation. A system doesn’t need to be perfect to be useful; it needs to be resilient enough to yield meaningful results despite imperfections. The subtle dance between adiabaticity and non-adiabaticity, briefly touched upon here, demands deeper exploration. It’s in these deviations from ideal behavior where the true complexity-and perhaps, the most valuable computational power-may lie.

The next step isn’t a grand, unified architecture. It’s a proliferation of small interactions, each a local rule contributing to a global, self-organized computation. Monumental shifts rarely originate from centralized control. They arise from the accumulation of minor influences, a principle that, perhaps, applies as much to the evolution of computation as it does to the evolution of everything else.

Original article: https://arxiv.org/pdf/2511.19598.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-26 15:07