Exponential quantum advantage for learning signals with a single qubit

The paper demonstrates that coupling a single controllable qubit to a conventional quantum sensor can exponentially reduce the number of measurements required to learn classical signals. These rigorous quantum advantages apply to fundamental sensing tasks, including learning Fourier coefficients, extracting temporal correlations from time-varying signals, and estimating transformations of physical observables. Using a superconducting cavity–qubit architecture, the authors experimentally demonstrate 10 7-fold reductions in the number of measurements required for Fourier-amplitude and time-varying signal learning. The quantum advantages are derived from Quantum Phase-Space Inference (QΨ), a unifying theory of quantum-enhanced experiments that simultaneously converts a set of experimental objectives and constraints into tight lower bounds and optimal quantum-enhanced learning algorithms while producing a certificate of quantum advantage. QΨ extends beyond the regimes captured by quantum Fisher information and provides a framework for systematically identifying rigorous quantum advantages in practical experimental tasks.

The paper establishes several key theorems. Theorem 1 states that a sensing protocol with probe energy O(k), one ancilla qubit, and one control operation can learn the k-th Fourier coefficient of a signal using O(k) signal queries, whereas any conventional protocol with the same energy scaling requires exp(Ω(k)) queries. Theorem 2 shows that a conventional Gaussian sensing protocol with energy O(k) can learn the k-th angular Fourier coefficient using O(1) queries, while any classical sensing protocol requires exp(Ω(k)) queries. Theorem 3 demonstrates that a sensor that decoheres on a timescale much shorter than Δ, when coupled to a single qubit that remains coherent up to time tm, can estimate the corresponding m-point temporal correlator using O(1) queries, whereas any protocol without long-lived quantum memory requires exp(Ω(m)) queries.

The QΨ framework introduces the accessible feature information (AFI), a phase-space statistical overlap that determines both a tight lower bound on the resources required for a given task and an algorithm that attains this bound under specified experimental constraints. The AFI functions operationally similarly to the quantum Fisher information but applies to general learning objectives while incorporating restrictions on the available experimental architecture. The paper proves that any strategy restricted to a protocol family M requires Ω(1/AFI(M; O)) measurements to learn an observable O, and the same phase-space construction identifies a protocol within M that attains this scaling.

The experimental demonstrations use a transmon qubit coupled to the fundamental electromagnetic mode of a high-purity aluminum cavity. The device is described by a Hamiltonian with a cross-Kerr interaction between the qubit and cavity, and the control unitaries are composed of single-qubit rotations and echoed conditional displacements. The experiments demonstrate exponential advantages for learning Fourier amplitudes up to k = 20, with the quantum-enhanced sensor requiring seven orders of magnitude fewer signal queries than a decoherence-free conventional Gaussian protocol.

The paper also presents numerical simulations of practical applications. For axionic dark matter detection, QFS quadratically outperforms squeezed photon-number-resolving measurements and two-mode entangled squeezing for characterizing cold axion streams. For wireless communication, a QFS receiver equipped with three qubits requires 100× fewer shots than a two-mode-entangled quantum receiver and 10 4 fewer shots than a classical receiver for 64-QAM decoding in the weak-signal regime.

The paper establishes a hierarchy of quantum sensing architectures, with each level obtained by adding a single quantum resource. The hierarchy includes classical sensing, conventional quantum sensing with Gaussian probes, quantum-enhanced sensing with single-qubit control, sensing with quantum memory, and sensing with deeper quantum control. The paper proves exponential separations between each level, demonstrating that modest amounts of coherent control, quantum memory, or circuit depth are sufficient to produce additional exponential advantages.

The QΨ framework also provides a global learning guarantee (Theorem D.30) that controls sample complexity through the ratio between the property's Fourier amplitude at each frequency and the Fourier gain of the architecture at that frequency. This enables optimal experimental design by matching the Fourier response of the experiment to the Fourier support of the target property.

The paper concludes that relatively simple quantum control can provide classical learners a quantum-mechanical interface to the natural world that is exponentially more expressive than any classical instrument, establishing a path toward quantum-enhanced measurements in radar, astronomy, communication, chemistry, and other fields where features of interest are encoded in classical signals.

Improvements to AI Systems:

1. Resource-Optimal Quantum Sensing Agents: Build AI controllers that use the QΨ framework’s AFI metric to automatically design and execute sensing protocols (e.g., choosing qubit control sequences, probe energies, and measurement schedules) that provably minimize the number of queries for a given target signal property. The AI can certify its own advantage by computing the AFI lower bound before running experiments.

2. Exponential Sample-Complexity Reduction in Classical Signal Learning: Implement AI systems that learn Fourier coefficients, temporal correlations, or physical observables from classical signals using a single controllable qubit interface. These systems require exponentially fewer measurements than classical or Gaussian-probe baselines (e.g., 107× fewer in practice), enabling real-time learning of weak or rapidly varying signals.

3. Adaptive Experimental Design via Fourier-Gain Matching: Use the global learning guarantee (Theorem D.30) to train an AI that matches the Fourier response of its sensing architecture to the target property’s Fourier support. This yields optimal shot allocation and experimental parameters, reducing wasted measurements and improving accuracy for sparse or band-limited signals.

4. Quantum-Enhanced Dark Matter and Communication Decoders: Develop AI decoders for axionic dark matter detection (using QFS) and for 64-QAM wireless decoding. The AI can operate with 100× fewer shots than two-mode-entangled receivers and 104× fewer than classical receivers in weak-signal regimes, enabling low-power, high-sensitivity sensing in noisy environments.

5. Hierarchical Quantum Resource Allocation: Create an AI meta-controller that selects the minimal quantum resource level (classical → Gaussian → single-qubit control → quantum memory → deeper control) needed for a task, based on the proven exponential separations between levels. This avoids over-provisioning quantum hardware while guaranteeing the best possible scaling for the given constraints.

6. Certified Quantum Advantage in Practical Tasks: Integrate the QΨ certificate into AI systems to output a rigorous, machine-checkable proof that a given sensing protocol is optimal within its architecture family. This is useful for safety-critical applications (e.g., medical imaging, radar) where measurement budgets are hard constraints.

What the Improved AI System Can Do:

• Learn a k-th Fourier coefficient of a classical signal with O(k) queries (vs. exp(Ω(k)) classically), using a single ancilla qubit and one control operation, even when the signal is weak or noisy.

• Estimate m-point temporal correlators with O(1) queries using a short-lived sensor plus a long-lived qubit memory, where classical protocols require exp(Ω(m)) queries.

• Automatically design and run experiments that achieve the theoretical lower bound for any given sensing task, with a built-in proof of optimality.

• Operate in extreme low-signal regimes (e.g., axion detection, deep-space communication) with orders-of-magnitude fewer measurements, reducing energy, time, and hardware costs.

• Switch between sensing architectures on the fly to match the available quantum resources (e.g., no memory vs. one qubit vs. multi-qubit) while preserving exponential advantages where possible.

Quantum technology has the potential to transform scientific discovery, but quantum advantages often require processing capabilities well beyond the reach of experimental platforms. We show that coupling a single controllable qubit to an otherwise conventional sensor can exponentially reduce the number of measurements required to learn classical signals. These rigorous quantum advantages apply to fundamental sensing tasks, including learning Fourier coefficients, extracting temporal correlations from time-varying signals, and estimating transformations of physical observables. Using a superconducting cavity--qubit architecture, we experimentally demonstrate 10 7-fold reductions in the number of measurements required for Fourier-amplitude and time-varying signal learning. Our quantum feature sensing algorithms further enable orders-of-magnitude improvements in simulations of weak-signal dark matter detection and wireless communication applications. These quantum advantages are derived from Quantum Phase-Space Inference (Q), a unifying theory of quantum-enhanced experiments that simultaneously converts a set of experimental objectives and constraints into tight lower bounds and optimal quantum-enhanced learning algorithms while producing a certificate of quantum advantage. Q extends beyond the regimes captured by quantum Fisher information and provides a framework for systematically identifying rigorous quantum advantages in practical experimental tasks. Together, our results establish that near-term quantum technology can exponentially enhance our ability to learn from classical signals.