Hsin-Yuan Huang, Michael Broughton, Jordan S. Cotler, Sitan Chen, Jerry Zheng Li, Masoud Mohseni, Hartmut Neven, Ryan Babbush, Richard Kueng, John P. Preskill, Jarrod R. McClean
Quantum technology promises to revolutionize how we learn about the physical world. An experiment that processes quantum data with a quantum computer could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. We proved that quantum machines could learn from exponentially fewer experiments than the number required by conventional experiments. This exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis, and learning about physical dynamics. Furthermore, the quantum resources needed for achieving an exponential advantage are quite modest in some cases. Conducting experiments with 40 superconducting qubits and 1300 quantum gates, we demonstrated that a substantial quantum advantage is possible with today's quantum processors.
Quantum technology promises to revolutionize how we learn about the physical world. An experiment that processes quantum data with a quantum computer could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. We proved that quantum machines could learn from exponentially fewer experiments than the number required by conventional experiments. This exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis, and learning about physical dynamics. Furthermore, the quantum resources needed for achieving an exponential advantage are quite modest in some cases. Conducting experiments with 40 superconducting qubits and 1300 quantum gates, we demonstrated that a substantial quantum advantage is possible with today's quantum processors.
We study the performance of classical and quantum machine learning (ML) models in predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve execution of a (possibly unknown) quantum process E. Our figure of merit is the number of runs of E required to achieve a desired prediction performance. We consider classical ML models that perform a measurement and record the classical outcome after each run of E, and quantum ML models that can access E coherently to acquire quantum data; the classical or quantum data are then used to predict the outcomes of future experiments. We prove that for any input distribution D(x), a classical ML model can provide accurate predictions on average by accessing E a number of times comparable to the optimal quantum ML model. In contrast, for achieving an accurate prediction on all inputs, we prove that the exponential quantum advantage is possible. For example, to predict the expectations of all Pauli observables in an n-qubit system ρ, classical ML models require 2^{Ω(n)} copies of ρ, but we present a quantum ML model using only O(n) copies. Our results clarify where the quantum advantage is possible and highlight the potential for classical ML models to address challenging quantum problems in physics and chemistry.
Broadly applicable quantum advantage, particularly in classical data processing and machine learning, has been a fundamental open problem. In this work, we prove that a small quantum computer of polylogarithmic size can perform large-scale classification and dimension reduction on massive classical data by processing samples on the fly, whereas any classical machine achieving the same prediction performance requires exponentially larger size. Furthermore, classical machines that are exponentially larger yet below the required size need superpolynomially more samples and time. We provide evidence for these quantum advantages in real-world applications, including single-cell RNA sequencing and movie review sentiment analysis, demonstrating four to six orders of magnitude reduction in size with fewer than 60 logical qubits. These quantum advantages are enabled by quantum oracle sketching, an algorithm for accessing the classical world in quantum superposition using only random classical data samples. Combined with classical shadows, our algorithm circumvents the data loading and readout bottleneck to construct succinct classical models from massive classical data, a task provably impossible for any classical machine that is not exponentially larger than the quantum machine. These quantum advantages persist even when classical machines are granted unlimited time or if BPP = BQP, and rely only on the correctness of quantum mechanics. Together, our results establish machine learning on classical data as a broad and natural domain of quantum advantage and a fundamental test of quantum mechanics at the complexity frontier.
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 107-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 (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. Together, our results establish that near-term quantum technology can exponentially enhance our ability to learn from classical signals.
Abstract Quantum computers are believed to bring computational advantages in simulating quantum many-body systems. However, recent works have shown that classical machine learning algorithms are able to predict numerous properties of quantum systems with classical data. Despite examples of learning tasks with provable quantum advantages being proposed, they all involve cryptographic functions and do not represent any physical scenarios encountered in laboratory settings. In this paper, we prove quantum advantages for the physically relevant task of learning quantum observables from classical (measured-out) data. We consider two types of observables: first, we prove a learning advantage for linear combinations of Pauli strings, then we extend our results to a broader case of unitarily parametrized observables. For each case, we delineate sharp boundaries separating physically relevant tasks that admit efficient classical learning from those for which a quantum computer remains necessary for data analysis. Unlike previous works, our classical hardness results rely only on the weaker assumption that BQP B Q P hard processes cannot be simulated by polynomial-size classical circuits, and we also provide a nontrivial quantum learning algorithm. Our results clarify when quantum resources are useful for learning problems in quantum many-body physics, and suggest practical directions in which quantum learning improvements may emerge.
Learning about physical systems from quantum-enhanced experiments can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for state learning, quantum process learning is less understood. We establish an exponential quantum advantage for learning an unknown n -qubit quantum process N . We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix (PTM) of an arbitrary N , (b) predicting expectation values of Pauli-sparse observables measured on the output of an arbitrary N upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary observables measured on the output of an unknown N with sparse PTM upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in- n many copies of the Choi state of N . In contrast, any learner without quantum memory requires exponentially-in- n many queries, even when using adaptively designed experiments. In proving this separation, we extend existing shadow tomography bounds from states to channels. Moreover, we combine PTM learning with polynomial interpolation to learn arbitrary Hamiltonians from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.
Quantum computing promises the ability to compute properties of quantum systems exponentially faster than classical computers. Quantum advantage is achieved when a practical problem is solved more efficiently on a quantum computer than on a classical computer. Demonstrating quantum advantage requires a powerful quantum computer with low error rates and an efficient quantum algorithm that has a useful application. Despite rapid progress in hardware development, we still lack useful applications that are feasible for the next generation of quantum computers. Here we argue that an exponential quantum advantage exists in producing numerical resource estimates of larger quantum algorithms by accurately measuring simulation errors. We provide a quantum algorithm for measuring simulation errors of Trotter-based algorithms. Our results indicate that this method will reduce runtimes of quantum algorithms by approximately three orders of magnitude for one-hundred qubit systems. We also predict that these reductions will increase with system size. The methods we propose require relatively few qubits and operations, meaning the next generation of quantum computers could compute simulation errors for classically intractable systems. Since the underlying computations that lead to reduced resource estimates are infeasible for classical computers, this task is a candidate for demonstrating practical quantum advantage.