Encoding is one of the first and most consequential decisions in designing a quantum program. Just as classical computers must decide how to represent integers or characters as bits, quantum algorithms must map problem variables—whether they denote colors, routes, or cluster assignments—onto the quantum states of a register. Over the past decade, researchers have adapted classical schemes such as one-hot, binary, and Gray codes to qubits, and have invented quantum-native alternatives like angle encoding, amplitude encoding, and Schmidt decomposition. Each approach embodies a trade-off between the number of qubits consumed and the complexity of the circuits needed to operate on them. One-hot encoding, in which one bit is reserved for every possible value a variable can take and exactly one of those bits is set to 1, is particularly popular because it is simple to encode, decode, and check for validity—even though it is wasteful in qubit count. A variable with four possible values, for instance, needs four qubits under one-hot rules, yielding the four allowed states 0001, 0010, 0100, and 1000.

The insight behind Multi-Unary is disarmingly simple. Instead of requiring that exactly one bit be set to 1, the scheme requires only that at least one bit be set to 1, with the single forbidden state being the all-zero string. Under this relaxation, the four-qubit register above can represent not just four values but fifteen—the full set of non-zero bit strings. Because more than one value can live in the same group of qubits, the Spanish team describes Multi-Unary as a “dense” encoding scheme, and, to their knowledge, it is the first encoding in the quantum computing literature with this property. The consequence is profound for algorithms like Grover’s: a single measured bit string can now encode several valid assignments at once, so the quantum state effectively contains and delivers multiple solutions in a single shot.

Grover’s algorithm, first proposed in 1996, searches an unstructured space of size N containing M solutions with a complexity of O(√(N/M)), a quadratic speedup over any classical method. The algorithm begins by placing the input qubits into an equal superposition of all possible states. It then applies an oracle—a quantum subroutine that recognizes valid solutions and flips their phase by π radians—followed by a diffusion operation that inverts the amplitudes about their mean, amplifying the marked states. Each oracle-plus-diffusion cycle is one Grover iteration, and the probability of measuring a solution after i iterations is given by P = sin²((2i+1)·θg), where θg = arcsin(√(M/N)). The optimal number of iterations scales as R ≤ ⌈(π/4)√(N/M)⌉. Crucially, both the success probability and the required iteration count depend on the ratio M/N: the more states qualify as solutions, the better.

This is exactly where Multi-Unary shines. By enlarging the set of bit strings that count as legitimate solutions, the scheme increases M, which the team proves mathematically raises the success probability P and lowers the required number of Grover iterations R. They derive the partial derivatives of both quantities with respect to M, showing rigorously that P grows monotonically and R shrinks monotonically over the entire operating range. The practical payoff comes in three forms: a higher chance that any given measurement yields a solution, fewer oracle calls needed to maximize that chance, and measured outputs that can bundle multiple solutions into one value, reducing the total number of circuit executions—or “shots”—needed to recover a desired fraction of all solutions.

To demonstrate the idea concretely, the researchers applied Multi-Unary to the graph coloring problem, a canonical combinatorial challenge that asks for the assignment of k colors to the nodes of a graph such that no two adjacent nodes share a color. Graph coloring matters well beyond theory—it underpins scheduling, register allocation, and resource assignment tasks—and it has become a standard benchmark for quantum algorithms, tackled previously with quantum annealing, variational approaches, and Grover-based oracles using both binary and one-hot encodings. In the Spanish team’s demonstration, a six-node, six-edge graph is colored with three colors, the minimum possible.

Their oracle design rests on quantum counters, built from multi-controlled X gates, which tally constraint violations. The logic proceeds in three steps. First, for each node, the counter is incremented if all of that node’s color qubits are zero—penalizing strings that violate the Multi-Unary rule that every variable must carry at least one value. Second, for every edge and every color, the counter is incremented if both endpoints display that same color simultaneously, penalizing genuine coloring conflicts. Third, only strings for which the counter reads zero—strings satisfying every constraint—have their phases rotated by a multi-controlled Z gate, marking them for amplification. A single ancilla qubit acts as a trigger that fires each counter increment and is uncomputed immediately afterward, while the remaining ancillas store the running count. The design also incorporates a subtle optimization: because early counter increments only touch the least significant bits of the count, the number of controls on successive increment gates can be grown gradually, trimming expensive multi-controlled gates.

The resource analysis for the example circuit is instructive. The complete oracle comprises 367 gates, of which 284 are controlled X gates—and 188 of those, fully 66.2 percent, implement the counter alone, underscoring how central counting is to this style of oracle construction. The circuit depth is 287 layers, dominated by the trigger qubit through which every increment must pass. Simulating the full circuit with 36 Grover iterations—the number that maximizes success probability—yielded a 99.88 percent chance of measuring a solution, and a histogram of 1,000 shots in Qiskit’s Aer simulator recovered 126 distinct solution values, with only two spurious non-solutions appearing once each, a rate consistent with statistical expectations.

The headline result is the sheer expansion of the solution space. Under a conventional one-hot oracle, the example graph has exactly 48 valid colorings. With Multi-Unary, the number of solution values jumps to 126—78 additional bit strings that are also valid, thanks to nodes that can carry more than one compatible color. This flexibility arises naturally in real graphs: leaf nodes and nodes with low connectivity often admit several admissible colors, and Multi-Unary lets the quantum state express that freedom directly. Decoding is straightforward: any measured variable whose bits contain more than one 1 is expanded into as many one-hot candidates as there are set bits, each assigning a single value while preserving the rest. In one worked example, the output string 011 010 001 100 010 101—where two nodes each hold two colors—unpacks into four distinct, fully valid colorings of the graph. A single measurement delivered four answers.

The authors are careful to position their contribution honestly. Binary encoding still wins on raw qubit count—a 50-node graph with 10 colors needs 200 binary-encoded input qubits versus 500 for one-hot or Multi-Unary—but binary circuits must then explicitly reject the invalid states corresponding to out-of-range values, adding gates and ancillas that Multi-Unary avoids by construction. Indeed, validity checking is where Multi-Unary beats its closest cousin outright: the only forbidden state under Multi-Unary is the all-zero string, detectable with a simple NOR gate, whereas one-hot verification requires a counter to confirm that exactly one bit is set. Because encoding and decoding work identically in both schemes, the team argues that Multi-Unary could outright replace one-hot encoding, with migration requiring changes only to the oracle’s validity-check portion—and a simpler oracle as a reward. The advantage does taper off for densely connected graphs, where fewer nodes admit multiple colors and the scheme converges back toward plain one-hot behavior.

The implications reach beyond graph coloring. The researchers point to shortest-path, graph coverage, and clustering problems as natural targets, and note that their encoding could slot into emerging frameworks like the discrete quantum intermediate representation proposed for encoding-independent optimization. For today’s noisy, resource-starved quantum hardware, where every ancilla qubit and every additional Grover iteration carries a real cost, a scheme that squeezes multiple solutions into the same qubits—and trims the iteration count as a bonus—offers a rare kind of free lunch. If quantum computing’s near-term future depends on getting the most out of every qubit, Multi-Unary suggests that sometimes the smartest move is not a new algorithm, but a new way of writing the problem down.

At the heart of the work is the qudit, a quantum system that can occupy more than the two states available to a conventional qubit. A qubit may be represented by a photon’s horizontal or vertical polarization, for example, while a qudit can use several distinguishable levels encoded in paths, time bins, orbital angular momentum, frequency, or other optical properties. Increasing the number of available levels can allow a single quantum carrier to store more information and may improve the efficiency of quantum communication, sensing, and computation.

The researchers focus on two distinct patterns of entanglement. In a GHZ state, several quantum systems are linked so strongly that their measured properties show collective correlations across the entire state. A simplified three-qubit GHZ state can be written as a superposition of all systems being in one state and all systems being in another. W states have a different structure: the excitation is distributed among several possibilities, creating a state that remains entangled even if one component is lost. These contrasting forms make W and GHZ states valuable for testing the limits of quantum networks and for developing protocols that rely on different types of nonclassical correlations.

What makes the new demonstration especially striking is that the chip uses two photons while producing states that behave as though several high-dimensional quantum subsystems are participating. In photonic experiments, a single photon can carry multiple degrees of freedom, and carefully engineered optical circuits can arrange these degrees of freedom into effective quantum modes. By controlling how the photons interfere and how their properties are measured, the device can create complex entangled states without requiring a separate physical photon for every logical subsystem.

The chip’s operation relies on the quantum interference of indistinguishable photons. When photons enter an integrated circuit through carefully selected pathways, their probability amplitudes combine rather than behaving like independent classical particles. Waveguides, beam splitters, phase shifters, and other on-chip elements manipulate these amplitudes with high precision. The resulting output is not a single predetermined configuration but a coherent superposition of many possibilities. Measurements then reveal correlations that cannot be explained by assigning fixed classical states to the photons before detection.

Moving this process onto a chip is important because conventional optical experiments often depend on large collections of mirrors, lenses, interferometers, and alignment systems. Even tiny mechanical shifts can change the phase relationships required for quantum interference. Integrated photonics replaces many of these free-space components with structures fabricated directly into a solid substrate. The components become more stable, compact, and potentially easier to reproduce, creating a route toward quantum devices that can leave the laboratory and operate as practical hardware.

The ability to generate high-dimensional entanglement with a small number of photons may also address one of the major bottlenecks in photonic quantum technology: the difficulty of producing and controlling many identical photons. Single-photon sources remain technically demanding, and losses increase rapidly as optical systems grow. If different degrees of freedom can be exploited efficiently, researchers may be able to encode more quantum information without simply multiplying the number of particles. This approach does not eliminate the challenges of photon loss, imperfect detectors, or environmental noise, but it offers a different strategy for increasing the information capacity of quantum circuits.

The reported platform could have implications beyond a single state-generation experiment. High-dimensional entangled states are candidates for more efficient quantum key distribution, in which additional levels can increase the information carried by each detected photon and may provide stronger resistance to certain noise processes. They are also relevant to quantum teleportation, distributed quantum computing, measurement-based computation, and precision sensing. W states are particularly interesting for networks in which the loss of one link must not destroy all useful correlations, while GHZ states are central to coordinated measurements and nonlocality tests.

The work also highlights a broader shift in quantum engineering: the field is moving from demonstrating isolated quantum effects toward designing compact architectures that can generate several classes of states on demand. Producing W and GHZ states on the same two-photon chip suggests that programmable photonic platforms may eventually support a wider library of quantum resources. The next steps will include improving source brightness, reducing fabrication imperfections, increasing detection efficiency, and proving that the generated states can be integrated into complete communication or computation protocols. If those challenges can be overcome, a small optical chip may become a powerful gateway to quantum systems far more complex than its physical size suggests.

Subject of Research: High-dimensional photonic entanglement, qudit W states, GHZ states, and integrated two-photon quantum chips

Article Title: Qudit W and Greenberger–Horne–Zeilinger states on a two-photon chip

Article References: Chi, Y., Ding, H., Wang, F. et al. Qudit W and Greenberger–Horne–Zeilinger states on a two-photon chip. Light Sci Appl 15, 340 (2026). https://doi.org/10.1038/s41377-026-02285-7

Image Credits: AI Generated

DOI: 10.1038/s41377-026-02285-7

Keywords: quantum photonics, qudits, W states, GHZ states, quantum entanglement, integrated photonic chips, two-photon quantum technology, high-dimensional quantum information, quantum communication, quantum computing

At present, our global communication infrastructure relies on a hybrid model: electronic circuits are used to store and process information, while optical fibers transmit that information as light across vast distances. Quantum networks are expected to benefit from a similar division of labor. In such networks, information would be encoded not in classical bits, which can only represent a 0 or 1, but in qubits, the quantum counterparts of ordinary bits. Unlike classical bits, qubits can exist in superpositions of states, enabling powerful new modes of computation and communication. However, realizing such a system requires that qubits designed for storage and processing (such as those based on electrons) interact seamlessly with qubits designed for communication and transport (such as those based on photons).

In conventional telecommunications, this challenge is addressed by electro-optic modulators, devices that use electronic signals to modify the properties of light. In the quantum world, however, scientists must find new mechanisms to allow delicate quantum states to influence one another without being destroyed in the process. The recent breakthrough, achieved by the research group of Edo Waks—Fellow at the Joint Quantum Institute (JQI) and Associate Professor of Electrical and Computer Engineering at the University of Maryland—represents a promising interface between individual photons and electrons.

By confining a photon and an electron in the same extremely small cavity, the team has created a system in which the electron can rapidly change the quantum properties of the photon, and conversely, the photon can directly alter the state of the electron. Their research, reported online in Nature Nanotechnology on February 8, 2016, demonstrates an approach that could eventually enable the “quantum wiring” needed for distributed quantum systems.

“Our platform has two major advantages over previous work,” explains Shuo Sun, graduate student at JQI and lead author of the paper. “First, the electronic qubit is integrated directly on a chip, making it highly scalable. Second, the interaction between light and matter is extremely fast, occurring in just a trillionth of a second—about 1,000 times quicker than earlier studies.”

Constructing a Quantum Interface

At the heart of this breakthrough is a carefully engineered photonic crystal. Photonic crystals are microscopic structures built from semiconductor layers patterned with a repeating grid of nanometer-sized holes. These periodic arrangements allow researchers to precisely manipulate the way light propagates through the material. By tailoring the size, shape, and distribution of the holes, scientists can create pathways for light, bend it around corners, or even trap it in tiny cavities where it bounces back and forth.

“These photonic crystals can focus light into an incredibly small volume, down to the fundamental quantum limit where the presence of a single photon is enough to drastically affect the system,” Waks explains. This ability to control light at the quantum level is essential for creating devices that operate reliably with individual quanta of energy rather than large pulses of light.

The experiment builds on another line of research involving quantum dots—engineered nanocrystals that behave like artificial atoms. Quantum dots can confine electrons within a very small region and exhibit discrete energy levels, much like natural atoms. In prior work, JQI researchers demonstrated that quantum dots could strongly influence beams of light, redirecting them or altering their properties.

In their new study, the team combined both approaches: the light-trapping power of photonic crystals with the electron-trapping ability of quantum dots. They fabricated a photonic crystal punctuated by holes only 72 nanometers wide. By intentionally leaving three adjacent holes undrilled, they introduced a controlled “defect” into the lattice. This defect formed a resonant cavity that selectively admitted and confined photons with very specific energies.

Inside this cavity, embedded in layers of semiconductor material, they placed a quantum dot capable of holding a single electron. The quantum property of that electron, known as its spin, then dictated how photons entering the cavity behaved. If the spin pointed upward, photons passed through unchanged. But if the spin pointed downward, every photon that entered emerged with its polarization flipped—its electric field oscillation rotated to the opposite orientation.

Crucially, the process also worked in reverse: a single photon prepared with the appropriate polarization could flip the electron’s spin. This bidirectional coupling between electron spin states and photon polarization demonstrates a fundamental type of quantum switch—a building block that could form the basis for scalable quantum circuits.

Toward Quantum Networking

The successful demonstration of this photon-electron interface has far-reaching implications. A robust quantum network will likely combine the storage and processing strengths of electrons with the long-distance transport abilities of photons. For example, electrons confined in quantum dots or other solid-state systems could hold information locally and perform computations, while photons transmitted through optical fibers could carry that information securely to distant locations.

Such a network would also make possible the distribution of entanglement, the uniquely quantum correlation that links particles across arbitrary distances. Entanglement is the foundation of many proposed quantum technologies, including distributed quantum computation, quantum teleportation of information, and secure communication protocols based on unbreakable quantum keys.

Before these applications become reality, however, more work is required. Sun and his colleagues emphasize that the next challenge is to demonstrate entanglement between the electron and photon qubits in their system—a step that requires even more precise measurements. Only once entanglement is verified and controlled can the platform serve as a reliable node in a future quantum network.

“The ultimate goal is to integrate photon generation, routing, and switching all onto a single chip,” Sun explains. “If we can accomplish that, we will be able to construct increasingly sophisticated quantum devices and circuits, paving the way toward practical quantum computers and secure quantum communication systems.”

A Glimpse Into the Future

This work underscores how progress in nanofabrication and materials engineering is enabling scientists to control light and matter at unprecedented scales. By merging photonic crystals with quantum dots, the JQI researchers have shown a pathway toward functional interfaces that can mediate interactions between photons and electrons, two of the most promising candidates for quantum information carriers.

Although still at an early stage, this research demonstrates that the basic ingredients for building quantum networks are beginning to fall into place. With continued development, such technologies could transform how information is stored, transmitted, and secured—ushering in a new era of communication where the principles of quantum mechanics are harnessed on a global scale.

Journal Reference:

Shuo Sun, Hyochul Kim, Glenn S. Solomon, Edo Waks. A quantum phase switch between a single solid-state spin and a photon. Nature Nanotechnology, 2016; DOI: 10.1038/nnano.2015.334