Demonstrating an unconditional separation between quantum and classical information resources (2509.07255v1)

Abstract: A longstanding goal in quantum information science is to demonstrate quantum computations that cannot be feasibly reproduced on a classical computer. Such demonstrations mark major milestones: they showcase fine control over quantum systems and are prerequisites for useful quantum computation. To date, quantum advantage has been demonstrated, for example, through violations of Bell inequalities and sampling-based quantum supremacy experiments. However, both forms of advantage come with important caveats: Bell tests are not computationally difficult tasks, and the classical hardness of sampling experiments relies on unproven complexity-theoretic assumptions. Here we demonstrate an unconditional quantum advantage in information resources required for a computational task, realized on Quantinuum's H1-1 trapped-ion quantum computer operating at a median two-qubit partial-entangler fidelity of 99.941(7)%. We construct a task for which the most space-efficient classical algorithm provably requires between 62 and 382 bits of memory, and solve it using only 12 qubits. Our result provides the most direct evidence yet that currently existing quantum processors can generate and manipulate entangled states of sufficient complexity to access the exponentiality of Hilbert space. This form of quantum advantage -- which we call quantum information supremacy -- represents a new benchmark in quantum computing, one that does not rely on unproven conjectures.

• The paper demonstrates quantum information supremacy by showing an exponential separation using one-way communication complexity on a 12-qubit trapped-ion quantum computer.
• It introduces a distributed linear cross-entropy benchmarking task that forces classical protocols to require exponentially more memory than quantum implementations.
• The experiment, validated against noise with high-fidelity state preparation and Clifford measurements, challenges classical simulation limits and paves the way for scalable quantum protocols.

Unconditional Separation Between Quantum and Classical Information Resources

Introduction

This paper establishes an unconditional, experimentally realized separation between quantum and classical information resources for a well-defined computational task. The authors leverage one-way communication complexity to construct a task that provably requires exponentially more classical memory than quantum memory, and demonstrate its solution on a 12-qubit trapped-ion quantum computer. This result, termed "quantum information supremacy," is not contingent on unproven complexity-theoretic assumptions and directly probes the exponentiality of Hilbert space as a physically accessible resource.

Theoretical Framework: Communication Complexity and Quantum Information Supremacy

The central theoretical tool is one-way communication complexity, where two parties (Alice and Bob) receive inputs $x$ and %%%%1%%%% and must compute a function with minimal communication. Quantum protocols can achieve exponential reductions in communication for certain tasks compared to classical protocols. The authors recast this spatial separation as a temporal separation on a single device, transforming the communication bottleneck into a storage bottleneck. This enables experimental realization without distributed quantum systems.

Figure 1: One-way communication model and its reinterpretation as a time-separated process on a single quantum device.

The task constructed is a distributed version of linear cross-entropy benchmarking (XEB), widely used in quantum supremacy experiments. Alice receives a classical description of an $n$-qubit quantum state $\ket{\psi_x}$, and Bob receives a description of an $n$-qubit measurement (a circuit $U_y$). The goal is to produce bit strings $z$ such that the average XEB fidelity

$$F = 2^n \mathbb{E}_{x,y}[\langle z|U_y|\psi_x\rangle^2] - 1$$

is maximized. The quantum protocol transmits $\ket{\psi_x}$, applies $U_y$, and measures in the computational basis. The main theoretical result is a lower bound: any classical protocol achieving the same $F$ requires at least $$\Omega(\min\{\varepsilon^2 2^n, 2^{n-O(\sqrt{n})}\})$$ bits of communication, where $\varepsilon$ is the achieved fidelity.

Figure 2: Comparison of classical one-way communication lower bounds for the presented problem and Hidden Matching, showing improved separation for the new task.

Experimental Realization

The experiment is performed on Quantinuum's H1-1 trapped-ion quantum computer, utilizing 12 qubits. The device achieves a median two-qubit partial-entangler fidelity of $99.941(7)\%$. Alice's Haar-random state is approximated via variationally trained parameterized quantum circuits, using a brickwork ansatz with 86 $ZZ(\theta)$ layers. Bob's measurement is implemented as a random Clifford circuit, efficiently realized with $O(n^2/\log n)$ gates.

Figure 3: 4-qubit example parameterized circuit for quantum state preparation, scaled to 12 qubits and 86 $ZZ(\theta)$ layers in the experiment.

Figure 4: Classical communication bounds as a function of XEB fidelity $F$ for the $n=12$ task, with experimental results marked.

The experiment uses true hardware randomness for input generation, ensuring the validity of the classical lower bound. Over 10,000 trials, the sample mean XEB fidelity achieved is $F = 0.427(13)$. Applying the theoretical lower bound, any classical protocol matching this performance requires at least 78 bits of memory; conservatively, 62 bits are required at five standard errors below the mean. Classical upper bounds show that 330–382 bits suffice to achieve the observed fidelity, tightly bracketing the quantum resource usage.

Circuit Synthesis and Hardware Benchmarking

The state preparation circuit is variationally optimized to maximize fidelity under realistic noise models, accounting for two-qubit gate errors and memory errors. Benchmarking is performed via randomized benchmarking protocols, yielding a two-qubit gate error of $5.9(7)\times 10^{-4}$ at the typical gate angle and a memory error of $(8\pm 2)\times 10^{-5}$ per qubit per layer.

Figure 5: Average infidelity as a function of $ZZ$ angle $\theta$ in 2Q parameterized gate randomized benchmarking.

Figure 6: Distribution of magnitudes of variationally optimized $ZZ(\theta)$ angles for 2Q gates in state preparation.

The fidelity of state preparation is strongly correlated with the achieved XEB fidelity for circuit depths exceeding $d\sim 60$, validating the use of XEB as a proxy for overall circuit fidelity.

Figure 7: Comparison of fidelity estimators for $N=12$ brickwork circuits as a function of circuit depth in noisy simulation.

Clifford Measurement Implementation

Random Clifford measurements are implemented using a fixed circuit template with classical controls, leveraging the structure of stabilizer states. The measurement circuit consists of layers of Hadamard, controlled-$Z$, inverse Phase ($S^\dagger$), Hadamard, and Pauli-$X$ gates, with variation introduced solely through classical control bits.

Implications and Future Directions

This work provides direct, unconditional evidence that current quantum hardware can access the exponentiality of Hilbert space, preparing and manipulating entangled states that cannot be simulated or compressed by any polynomial number of classical bits. The separation is robust to device noise and does not rely on complexity-theoretic conjectures. The result is permanent: no future classical algorithm can close the gap.

The experiment sets a new benchmark for quantum advantage, termed quantum information supremacy. Scaling to larger $n$ would further widen the separation, with $n=26$ requiring over a million classical bits to match quantum performance. The main bottleneck is two-qubit gate fidelity, but improvements in variational state preparation and tighter classical lower bounds could enable larger separations on existing hardware.

Potential loopholes, such as the lack of strict temporal separation between Alice and Bob's inputs, are analogous to setting-independence loopholes in Bell tests and could be addressed in future experiments with online randomness sources. Further, demonstrating separations at larger $n$ would address skepticism regarding the physical realization of high-dimensional Hilbert spaces.

Conclusion

The paper demonstrates an unconditional, experimentally realized separation between quantum and classical information resources for a computational task, using 12 qubits to solve a problem that provably requires at least 62 classical bits. This establishes quantum information supremacy as a physically accessible resource, independent of computational assumptions. The result has significant implications for the foundations of quantum mechanics and the future of quantum computing, providing a direct challenge to the view that quantum systems are reducible to low-dimensional classical descriptions. Future work will focus on scaling the separation and closing remaining loopholes, further solidifying the exponential power of quantum information.