Ryan Babbush
Ryan is the director of the Quantum Algorithm & Applications Team at Google. The mandate of this research team is to develop new and more efficient quantum algorithms, discover and analyze new applications of quantum computers, build and open source tools for accelerating quantum algorithms research and compilation, and to design algorithmic experiments to execute on existing and future fault-tolerant quantum devices.
Authored Publications
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Exponential quantum advantage in processing massive classical data
Haimeng Zhao
Alexander Zlokapa
John Preskill
Hsin-Yuan (Robert) Huang
arXiv:2604.07639 (2026)
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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 validate 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.
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Securing Elliptic Curve Cryptocurrencies against Quantum Vulnerabilities: Resource Estimates and Mitigations
Michael Broughton
Thiago Bergamaschi
Justin Drake
Dan Boneh
arXiv:2603.28846 (2026)
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This whitepaper seeks to elucidate implications that the capabilities of developing quantum architectures have on blockchain vulnerabilities and mitigation strategies. First, we provide new resource estimates for breaking the 256-bit Elliptic Curve Discrete Logarithm Problem, the core of modern blockchain cryptography. We demonstrate that Shor's algorithm for this problem can execute with either <1200 logical qubits and <90 million Toffoli gates or <1450 logical qubits and <70 million Toffoli gates. In the interest of responsible disclosure, we use a zero-knowledge proof to validate these results without disclosing attack vectors. On superconducting architectures with 1e-3 physical error rates and planar connectivity, those circuits can execute in minutes using fewer than half a million physical qubits. We introduce a critical distinction between fast-clock (such as superconducting and photonic) and slow-clock (such as neutral atom and ion trap) architectures. Our analysis reveals that the first fast-clock CRQCs would enable on-spend attacks on public mempool transactions of some cryptocurrencies. We survey major cryptocurrency vulnerabilities through this lens, identifying systemic risks associated with advanced features in some blockchains such as smart contracts, Proof-of-Stake consensus, and Data Availability Sampling, as well as the enduring concern of abandoned assets. We argue that technical solutions would benefit from accompanying public policy and discuss various frameworks of digital salvage to regulate the recovery or destruction of dormant assets while preventing adversarial seizure. We also discuss implications for other digital assets and tokenization as well as challenges and successful examples of the ongoing transition to Post-Quantum Cryptography (PQC). Finally, we urge all vulnerable cryptocurrency communities to join the ongoing migration to PQC without delay.
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We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in control theory and physics. Rather than encoding the solution in a quantum state in a fashion analogous to prior quantum linear algebra solvers, our approach constructs the solution matrix X in a block-encoding, rescaled by some factor. This allows us to obtain certain properties of the entries of X exponentially faster than would be possible from preparing X as a quantum state. The query and gate complexities of the quantum circuit that implements this block-encoding are almost linear in a condition number that depends on A and B, and depend logarithmically in the dimension and inverse error. We show how our quantum circuits can solve BQP-complete problems efficiently, discuss potential applications and extensions of our approach, its connection to Riccati equation, and comment on open problems.
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The solution of linear systems of equations is the basis of many other quantum algorithms, and recent results provided an algorithm with optimal scaling in both the condition number κ and the allowable error ϵ [PRX Quantum 3, 0403003 (2022)]. That work was based on the discrete adiabatic theorem, and worked out an explicit constant factor for an upper bound on the complexity. Here we show via numerical testing on random matrices that the constant factor is in practice about 1,200 times smaller than the upper bound found numerically in the previous results. That means that this approach is far more efficient than might naively be expected from the upper bound. In particular, it is over an order of magnitude more efficient than using a randomized approach from [arXiv:2305.11352] that claimed to be more efficient.
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Optimization by Decoded Quantum Interfereometry
Stephen Jordan
Mary Wootters
Alexander Schmidhuber
Robbie King
Sergei Isakov
Nature, 646 (2025), pp. 831-836
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Achieving superpolynomial speed-ups for optimization has long been a central goal for quantum algorithms. Here we introduce decoded quantum interferometry (DQI), a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems. When approximating optimal polynomial fits over finite fields, DQI achieves a superpolynomial speed-up over known classical algorithms. The speed-up arises because the algebraic structure of the problem is reflected in the decoding problem, which can be solved efficiently. We then investigate whether this approach can achieve a speed-up for optimization problems that lack an algebraic structure but have sparse clauses. These problems reduce to decoding low-density parity-check codes, for which powerful decoders are known. To test this, we construct a max-XORSAT instance for which DQI finds an approximate optimum substantially faster than general-purpose classical heuristics, such as simulated annealing. Although a tailored classical solver can outperform DQI on this instance, our results establish that combining quantum Fourier transforms with powerful decoding primitives provides a promising new path towards quantum speed-ups for hard optimization problems.
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Faster electronic structure quantum simulation by spectrum amplification
Guang Hao Low
Robbie King
Alec White
Rolando Somma
Dominic Berry
Qiushi Han
Albert Eugene DePrince III
arXiv (2025) (to appear)
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We discover that many interesting electronic structure Hamiltonians have a compact and close-to-frustration-free sum-of-squares representation with a small energy gap. We show that this gap enables spectrum amplification in estimating ground state energies, which improves the cost scaling of previous approaches from the block-encoding normalization factor $\lambda$ to just $\sqrt{\lambda E_{\text{gap}}}$. For any constant-degree polynomial basis of fermionic operators, a sum-of-squares representation with optimal gap can be efficiently computed using semi-definite programming. Although the gap can be made arbitrarily small with an exponential-size basis, we find that the degree-$2$ spin-free basis in combination with approximating two-body interactions by a new Double-Factorized (DF) generalization of Tensor-Hyper-Contraction (THC) gives an excellent balance of gap, $\lambda$, and block-encoding costs. For classically-hard FeMoco complexes -- candidate applications for first useful quantum advantage -- this combination improves the Toffoli gates cost of the first estimates with DF [Phys. Rev. Research 3, 033055] or THC [PRX Quantum 2, 030305] by over two orders of magnitude.
https://drive.google.com/file/d/1hw4zFv_X0GeMpE4et6SS9gAUM9My98iJ/view?usp=sharing
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Fast electronic structure quantum simulation by spectrum amplification
Guang Hao Low
Robbie King
Dominic Berry
Qiushi Han
Albert Eugene DePrince III
Alec White
Rolando Somma
arXiv:2502.15882 (2025)
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The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block-encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectrum amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectrum amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor $\Lambda$ to just $\sqrt{2\Lambda E_{\text{gap}}}$, where $E_{\text{gap}} \ll \Lambda$ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes-- namely, double factorization and tensor hypercontraction-- that when combined with spectrum amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO$_{2}$-fixation catalyst.
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We introduce sum-of-squares spectral amplification (SOSSA), a framework for improving quantum simulation algorithms relevant to low-energy problems. SOSSA first represents the Hamiltonian as a sum-of-squares and then applies spectral amplification to amplify the low-energy spectrum. The sum-of-squares representation can be obtained using semidefinite programming. We show that SOSSA can improve the efficiency of traditional methods in several simulation tasks involving low-energy states. Specifically, we provide fast quantum algorithms for energy and phase estimation that improve over the state-of-the-art in both query and gate complexities, complementing recent results on fast time evolution of low-energy states. To further illustrate the power of SOSSA, we apply it to the Sachdev-Ye-Kitaev model, a representative strongly correlated system, where we demonstrate asymptotic speedups by a factor of the square root of the system size. Notably, SOSSA was recently used in [G.H. Low \textit{et al.}, arXiv:2502.15882 (2025)] to achieve state-of-art costs for phase estimation of real-world quantum chemistry systems.
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A Simplified Version of the Quantum OTOC2 Problem
Robbie King
Kostyantyn Kechedzhi
Tom O'Brien
Vadim Smelyanskiy
arXiv:2510.19751 (2025)
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This note presents a simplified version of the OTOC2 problem that was recently experimentally implemented by Google Quantum AI and collaborators. We present a formulation of the problem for growing input size and hope this spurs further theoretical work on the problem.
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Shadow Hamiltonian Simulation
Rolando Somma
Robbie King
Tom O'Brien
Nature Communications, 16 (2025), pp. 2690
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Simulating quantum dynamics is one of the most important applications of quantum computers. Traditional approaches for quantum simulation involve preparing the full evolved state of the system and then measuring some physical quantity. Here, we present a different and novel approach to quantum simulation that uses a compressed quantum state that we call the "shadow state". The amplitudes of this shadow state are proportional to the time-dependent expectations of a specific set of operators of interest, and it evolves according to its own Schrödinger equation. This evolution can be simulated on a quantum computer efficiently under broad conditions. Applications of this approach to quantum simulation problems include simulating the dynamics of exponentially large systems of free fermions or free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. These simulations are hard for classical methods and also for traditional quantum approaches, as preparing the full states would require exponential resources. Shadow Hamiltonian simulation can also be extended to simulate expectations of more complex operators such as two-time correlators or Green's functions, and to study the evolution of operators themselves in the Heisenberg picture.
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